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The Ways We Use Math in the Real World: 8 Examples

Created: December 21, 2023

Last updated: January 9, 2024

You may not realize it, but many children ask themselves: What relevance do abstract equations have in my life? Why practice math problems like ‘John bought 45 watermelons’ equations when nobody ever buys 45 watermelons in real life?

Yes, it happens, and sometimes schools fail to address these doubts correctly or even recognize them. Getting kids to understand math formulas and processes has become second nature to teachers. However, getting them to learn math using real world math problems is another story altogether.

In What Ways Does Math Exist in The Real World?

To a lot of individuals, math serves little practical purpose. We do not use all those equations after leaving school. However, the simple truth is that we unknowingly use mathematical concepts daily. Numerical calculations are part of going shopping, cooking, and working, amongst other activities.

Fluency in math equals success in countless real-world activities that involve logical thinking and spatial skills. Moreover, it also enables different brain parts to utilize them well. Nevertheless, if you view math at its full scale, you will see that the civilization that exists at present was brought by ancient Greek mathematics. Everything, from architecture, machines, and medicines to iPads, came out of calculation.

Math for Kids

Math in the Real World

Often, teachers take situations from everyday life to teach math in a meaningful and real-world sense. Check these real life math problems below:

Which Popcorn Container Is Better?

While in the cinema, you look forward to getting a big popcorn box in your mouth, which should keep it occupied till the end of the film. However, upon getting closer to the popcorn stand, you notice two bags of the same product: cone-shaped and cylindrical. They are the same size, but the only ones selling anything for $5.

Which one should you choose? You can hardly tell them apart at a glance. However, mathematicians are brilliant and understand that you’ll be forced to pay much more for a coned bag. Therefore, the cone has a constant ratio of one-third of cylinders with the same height. Therefore, this cylindrical bag will hold the acorn and allow you to pick up various snack foods.

What Makes Bees Build Hexagons Inside Their Hives?

Bees are among Earth’s most complex working creatures, yet their buzzing critters are economical with space for all their work. Their hives always take the form of a hexagon, not any other shape.

The question is: Why not round honeycombs? They will not fit since they would leave large cavities with round cells. Why should triangle or square cells, then? They won’t be efficient either. Many of such operations with numbers take place in those hexagon cells. A hexagon has the lowest circumference and maximum area.

As one of the math in the real world examples, we can also consider if the hexagon has a side length of five centimeters. The formula of a perimeter is the sum of all sides: 5 x 6 = 30. Thus, the hexagon of the 5 cm side becomes a 30 cm long line. The hexagon’s surface area formula is The quotient of (3 √3 x S2)/2, with S as the hexagon’s side length. For our hexagon, we have the following formula: The area of the body is equal to (3 × √3 × 25) ÷ 2 = 64.95 sq cms.

As such, let us apply the same to an equilateral triangle with a perimeter measuring 30 centimeters. It will have sides measuring 10 centimeters each and an area of 43.3 square centimeters. To construct a triangle with a surface area of 64.95cm^{2}, its perimeter measures 37.75cm, about 25% larger than a hexagon’s side.

Therefore, had the bees been triangulators, they would have needed 25% more wax to cover a hexagonal cell’s surface area. Additionally, a hexagon provides six wall stiffeners that share loads more evenly than triangles and squares.

Folding Paper Problem

Have you heard about paper-folding and its magical theory? You have probably learned that folding a regular letter-size paper more than eight times is impossible.

A paper sheet 1:219 KM long has been used for the current world record of noon. It is also believed that if you fold a piece of paper forty-two times, it will be as thick as the distance between you and the moon. Firstly, it increases in thickness as it folds exponentially. Consider a typical sheet with a thickness of 0.1 mm (similar to an A4 paper). Here is how thick it will be with each fold:

• No fold – 0.1 mm thickness
• One fold is equal to 2 times the thickness. The paper has two layers. It has a depth of not more than 0.2mm.
• Two folds = 2². Upon folding, you will have two layers of material, after which the process is repeated.
• Three folds = 2³. You will double four layers. There will be eight layers.
• Four folds = 2⁴. This is doubling eight layers for a total of 16.
• Five folds = 2⁵. Doubling 16 results in 32.
• Six folds = 2⁶. Doubling 32 equals 64.
• Seven folds = 2⁷. Doubling 64 equals 128.

To illustrate, a paper sheet of 7 folds consists of 128 layers. It will be 12.8 mm thick (128 x 0.1 equals 12.8mm). The paper is now too thick, and insufficient surface spaces support further folding.

Folding Paper to the Moon

What about 42 folds? These products will have 4,398,046,511,104 or 4.39 trillion levels each year. Theoretically, the paper thickness should be 439,800 millimeters / 439.8 billion meters. Consequently, 439.8 million meters is similar to 439804 km, which is precisely equal to the Earth and moon’s distance of 384,400km.

To answer the question of how is math used in the real world, take this example. When you fold a flat piece of paper enough times, it will exceed the distance from here to the moon. However, while being folded, the visible area of the paper diminishes by a factor ½ (1/2), but the actual area does not. The first fold reduces the paper to half its original size, making it 50% of its original size. It then will be one-fourth of the first fold.

After 42 folds, the visible surface area becomes less than an atom’s. Essentially, you cannot have it shrunk to that tiny size. Hence, kids can understand why saying you cannot fold a piece of paper more than seven times is an oversimplification. This is one of the direct math in real life examples.

Saving Money

Help kids start a simple savings plan to introduce them to financial literacy. Motivate them to have manageable targets, such as putting aside money for a toy or an excursion. Teach them to track savings and decisions related to spending as they earn or receive Money. Such an approach is based on practice, helping to reinforce addition and subtraction while teaching children how to save or budget.

Investment Using Geometric Progression

There are many jokes related to math when speaking of school money matters. For examples of math in the real world, when individuals walk into a store to purchase 49 watermelons, in the case of math. You could, for instance, talk about Money in terms of stock markets and interest rates as an elementary instance of simple interest — or just fortune and wealth.

Considering that you invested $1,000 in banks and stocks, what would be your returns in a few years? Over time, the prices of various stocks go up, and $1,000 could rise to $10,000 or even $100,000 as a geometrical proportion. However, this is just a one-time investment. How about using $150 each month in stocks and banks?

One of the real world math examples is the crypto market, which has become faster several times. However, there are some occasions whereby fortunate investors amass millions within seconds due to rapid variations in the value of varied digital currencies.

Such instances demonstrate that math is an integral part of daily life, which will help improve the children’s knowledge of finance. This practice gives them answers to some of the most relatable questions: What’s the way to break the rat race? How to become independent?

Measurements

Jobs requiring measurements are numerous, such as in construction and post, meteorology, etc. Measurements count for everyone; hence, you should prove them to your child if they need to mail a package or find out whether their PlayStation fits the table.

Now, give them a meter rule and ask them to measure things instead. For real life math problems examples, get them to assist you in posting a parcel to your grandparents. Allow them to record the dimensions along with the weight of the packing box, or keep a note about the temperature record on a thermometer.

Searching For Discounts While Practicing Percentages

Take them through Amazon and eBay if your child is uncomfortable with percentages. One can learn percentages best while online shopping because of the off-percentage discounts. Their favorite shoes cost $100. How many can be saved with a twenty percent sign-in discount? This integration of a math problem into something they love will serve as an encouragement to young brains to master percentages.

In some cases, it is just as easy to calculate the percentage of a number by multiplication. For example, to determine what 20% of 5 is, multiply it by 0.2.

Grocery Shopping

They are giving kids a chance to understand addition and subtraction while grocery shopping, which is very beneficial. Have them tally items in the cart as they walk past each item. Ask them to take discount calculations and explain why prices decrease when a promotion occurs. This kind of hands-on experience will revisit the basics of real life situation math problems and show how math is needed daily in real-life situations.

Time Management

For most students, the notion of time is difficult to comprehend. It is an inevitable skill that the child must be familiar with from the beginning of their childhood. Children will also learn to use the clock to tell the time and thus be able to assign and finish tasks.

Organizational skills include being able to tell time and manage schedules, among others, which are very important for teaching children—using a clock to schedule activities/setting timers for tasks to help them understand time in terms of hours, minutes, and seconds. This approach further enhances math skills, which also helps instill discipline and self-responsibility in children. However, time management is an invaluable skill that transcends math and constitutes one of the essential attributes in individual development in general.

Building and Construction

Building projects like a birdhouse or a small shelf can be an engaging activity, showing kids how math real life problems are applied in real life. They also learn about measurements, geometry, and spatial relationships using materials. Building something with their hands gives them a tangible understanding of length, width, and height. This increases achievement experience, giving kids a chance to appreciate mathematics truly.

Sports Scores and Statistics

Venture into scores, statistics, averages, and other numbers that give meaning to the thrill of sports. Mathematical ideas such as averages, percentages, and basic arithmetic can be presented when discussing player statistics or team performance. The relationship between sport and math is established by kids who can determine a batting average, points per game, etc., relating them to their cherished activity.

Map Reading and Navigation

Map reading and navigation are other examples of how math is used in the real world to introduce basic geometry and measurement skills. Encourage your kids while planning trips and exploring neighborhoods by allowing them to understand distances, directions, and scale markings on maps.

Besides developing their spatial awareness, they learn that units are used in measuring for real world math problems examples. Length, weight, area, or volume, as well as geometric relations, help them solve real world problems math, which also become a part of how they explore and understand the world around them.

Budgeting for Allowance

Giving kids an allowance that will teach them about budgeting is a great way to empower them with financial skills, as they will learn how best they can save, spend, and give out, as well as how to budget for things. Motivating them to save a portion of their income and spending the rest while giving to others would not be wrong. It shows them how Money is essential while they are still young and makes it possible for them to learn the essential qualities of responsibility for finances at an early age. These skills in budgeting pave the way for further financial literacy.

Math in real world for children demonstrates why they learn such mathematics and engages them to understand such mathematics better. Children will develop a strong understanding of the relevance and application of mathematical concepts in their daily lives using math in real life scenarios to create the necessary base for further advancement.

Temperature and Weather

Speak about changes in temperature — those happening during the day and in different seasons. Assist them in recording temperatures, highs, and lows and their observation patterns. This activity also strengthens mathematics education by teaching students to gather, evaluate, and analyze information, leading to another skill—understanding statistics. Kids can also make simple graphs and charts that will help them understand the use of weather data as a mathematical representation.

Planting and Gardening

Evaluate plant spacing, developmental rates, and observational data on plant growth during time. Talk about counting the plant’s height, seeds sprouting time, and land size for various plants. Through gardening, one can relate math to the natural world as maths in the real world is applied in simple tasks such as planting and watering plants, where you are expected to achieve specific results at certain periods.

When cooking, little kids wonder about how much math is involved. So, allow your child to get closer with kitchen math: three pounds of tomatoes, four ounces of ice cream, one teaspoon of ground coffee, or 1/2 ounce of cinnamon. How much is the 1–1 proportion?

Moreover, when it comes to cooking, most recipes require you to have math skills, so this is an excellent opportunity to connect mathematics in the real world. Ratios, proportions, and basic math operations in cooking make sense.

A good presentation of math in fundamental topics using relatable examples helps children understand that this field is essential for their future. Compared to other approaches, this provides more chances of attracting children’s attention and inspiring interest in mathematical studies.

You may also register your child with Brighterly’s online math classes for further math real world problems practice. This recommendation is for kids to learn more effectively when they are in a relaxed manner. Your children will benefit from being made to use everyday language, math in the real world word problem sheets and worksheets, and engaging activities.

Jessica is a a seasoned math tutor with over a decade of experience in the field. With a BSc and Master’s degree in Mathematics, she enjoys nurturing math geniuses, regardless of their age, grade, and skills. Apart from tutoring, Jessica blogs at Brighterly. She also has experience in child psychology, homeschooling and curriculum consultation for schools and EdTech websites.

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26 Snappy Answers to the Question “When Are We Ever Going to Use This Math in Real Life?”

Next time they ask, you’ll be ready.

As a math teacher, how many times have you heard frustrated students ask, “When are we ever going to use this math in real life!?” We know, it’s maddening! Especially for those of us who love math so much we’ve devoted our lives to sharing it with others.

It may very well be true that students won’t use some of the more abstract mathematical concepts they learn in school unless they choose to work in specific fields. But the underlying skills they develop in math class—like taking risks, thinking logically and solving problems—will last a lifetime and help them solve work-related and real-world problems.

Here are 26 images and accompanying comebacks to share with your students to get them thinking about all the different and unexpected ways they might use math in their futures!

1. If you go bungee jumping, you might want to know a thing or two about trajectories.

https://giphy.com/gifs/funny-fail-5OuUiP0we57b2

Source: GIPHY

2. When you invest your money, you’ll do better if you understand concepts such as interest rates, risk vs. reward, and probability.

3. once you’re a driver, you’ll need to be able to calculate things like reaction time and stopping distance., 4. in case of a zombie apocalypse, you’re going to want to explore geometric progressions, interpret data and make predictions in order to stay human..

Trigger an outbreak of learning and infectious fun in your classroom with this Zombie Apocalypse activity from TI’s STEM Behind Hollywood series.

5. Before you tackle that home wallpaper project, you’ll need to calculate just how much wall paper glue you need per square foot.

6. when you buy your first house and apply for a 30-year mortgage, you may be shocked by the reality of what interest compounded over 30 years looks like., 7. to be a responsible pet owner, you’ll need to calculate how much hamster food to have on hand., 8. even if you’re just an armchair athlete, you can’t believe the math involved in kicking field goals.

Check out this Field Goal for the Win activity that encourages students to model, explore and explain the dynamics of kicking a football through the uprights.

9. When you double a recipe, you’re going to need to understand ratios so your dinner guests don’t look like this.

10. before you take that family road trip , you’re going to want to calculate time and distance., 11. before you go candy shopping, you’re going to have to figure out x trick or treaters times x pieces of candy equals…, 12. if  you grow up to be an ice cream scientist, you’re going to have to understand the effect of temperature and pressure at the molecular level..

https://giphy.com/gifs/ice-lick-cream-3Z1kRYmLRQm5y

Explore states of matter and the processes that change cow milk into a cone of delicious decadence with this Ice Cream, Cool Science activity .

13. Once you have little ones, you’ll need to know how many diapers to buy for the month.

14. because what if it’s your turn to organize the annual ping pong tournament, and there are 7 players at a club with 4 tables, where each player plays against each other player, 15. when dressing for the day, you might want to consider the percent likelihood of rain., 16. if you go into medical research, you’re going to have to know how to solve equations..

Learn more about inspiring careers that improve lives with STEM Behind Health , a series of free activities from TI.

17. Understanding percentages will help you get the best deal at the mall. For example, how much will something cost with 40% off? What about once the 8% tax is added? What if it’s advertised as half-off?

https://giphy.com/gifs/blue-kawaii-pink-5aplc3D2G0IrC

18. Budgeting for vacation will require figuring out how many hours at your pay rate you’ll have to work to afford the trip you want.

19. when you volunteer to host the company holiday party, you’ll need to figure out how much food to get., 20. if you grow up to be a super villain, you’re going to need to use math to determine the most effective way to slow down the superhero and keep him from saving the day..

Put your students in the role of an arch-villain’s minions with Science Friction, a STEM Behind Hollywood activity .

21. You’ll definitely want to understand how to budget your money so you don’t look like this at the grocery checkout.

22. if you don’t work the numbers out in advance, you might at some point regret choosing that expensive out-of-state college., 23. before taking on a building project, remember the old saying—measure twice, cut once., 24. if have aspirations of being a fashion designer, you’ll have to understand geometry in order to make the perfect twirling skirt.

https://giphy.com/gifs/loop-bunny-ballet-yarFJggnH24da

Geometry and fashion design intersect in this STEM Behind Cool Careers activity .

25. Everyone loves a good bargain! Figuring out the best deal is not only fun, it’s smart!

26. if you can’t manage calculations, running the numbers at the car dealership might leave you feeling like this:, you might also like.

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120 Math Word Problems To Challenge Students Grades 1 to 8

Written by Marcus Guido

Hey teachers! 👋

Use Prodigy to spark a love for math in your students – including when solving word problems!

• Teaching Tools
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• Multiplication
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You sit at your desk, ready to put a math quiz, test or activity together. The questions flow onto the document until you hit a section for word problems.

A jolt of creativity would help. But it doesn’t come.

Whether you’re a 3rd grade teacher or an 8th grade teacher preparing students for high school, translating math concepts into real world examples can certainly be a challenge.

This resource is your jolt of creativity. It provides examples and templates of math word problems for 1st to 8th grade classes.

There are 120 examples in total.

The list of examples is supplemented by tips to create engaging and challenging math word problems.

120 Math word problems, categorized by skill

Addition word problems.

Best for: 1st grade, 2nd grade

1. Adding to 10: Ariel was playing basketball. 1 of her shots went in the hoop. 2 of her shots did not go in the hoop. How many shots were there in total?

2. Adding to 20: Adrianna has 10 pieces of gum to share with her friends. There wasn’t enough gum for all her friends, so she went to the store to get 3 more pieces of gum. How many pieces of gum does Adrianna have now?

3. Adding to 100: Adrianna has 10 pieces of gum to share with her friends. There wasn’t enough gum for all her friends, so she went to the store and got 70 pieces of strawberry gum and 10 pieces of bubble gum. How many pieces of gum does Adrianna have now?

4. Adding Slightly over 100: The restaurant has 175 normal chairs and 20 chairs for babies. How many chairs does the restaurant have in total?

5. Adding to 1,000: How many cookies did you sell if you sold 320 chocolate cookies and 270 vanilla cookies?

6. Adding to and over 10,000: The hobby store normally sells 10,576 trading cards per month. In June, the hobby store sold 15,498 more trading cards than normal. In total, how many trading cards did the hobby store sell in June?

7. Adding 3 Numbers: Billy had 2 books at home. He went to the library to take out 2 more books. He then bought 1 book. How many books does Billy have now?

8. Adding 3 Numbers to and over 100: Ashley bought a big bag of candy. The bag had 102 blue candies, 100 red candies and 94 green candies. How many candies were there in total?

Subtraction word problems

Best for: 1st grade, second grade

9. Subtracting to 10: There were 3 pizzas in total at the pizza shop. A customer bought 1 pizza. How many pizzas are left?

10. Subtracting to 20: Your friend said she had 11 stickers. When you helped her clean her desk, she only had a total of 10 stickers. How many stickers are missing?

11. Subtracting to 100: Adrianna has 100 pieces of gum to share with her friends. When she went to the park, she shared 10 pieces of strawberry gum. When she left the park, Adrianna shared another 10 pieces of bubble gum. How many pieces of gum does Adrianna have now?

Practice math word problems with Prodigy Math

Join millions of teachers using Prodigy to make learning fun and differentiate instruction as they answer in-game questions, including math word problems from 1st to 8th grade!

12. Subtracting Slightly over 100: Your team scored a total of 123 points. 67 points were scored in the first half. How many were scored in the second half?

13. Subtracting to 1,000: Nathan has a big ant farm. He decided to sell some of his ants. He started with 965 ants. He sold 213. How many ants does he have now?

14. Subtracting to and over 10,000: The hobby store normally sells 10,576 trading cards per month. In July, the hobby store sold a total of 20,777 trading cards. How many more trading cards did the hobby store sell in July compared with a normal month?

15. Subtracting 3 Numbers: Charlene had a pack of 35 pencil crayons. She gave 6 to her friend Theresa. She gave 3 to her friend Mandy. How many pencil crayons does Charlene have left?

16. Subtracting 3 Numbers to and over 100: Ashley bought a big bag of candy to share with her friends. In total, there were 296 candies. She gave 105 candies to Marissa. She also gave 86 candies to Kayla. How many candies were left?

Multiplication word problems

Best for: 2nd grade, 3rd grade

17. Multiplying 1-Digit Integers: Adrianna needs to cut a pan of brownies into pieces. She cuts 6 even columns and 3 even rows into the pan. How many brownies does she have?

18. Multiplying 2-Digit Integers: A movie theatre has 25 rows of seats with 20 seats in each row. How many seats are there in total?

19. Multiplying Integers Ending with 0: A clothing company has 4 different kinds of sweatshirts. Each year, the company makes 60,000 of each kind of sweatshirt. How many sweatshirts does the company make each year?

20. Multiplying 3 Integers: A bricklayer stacks bricks in 2 rows, with 10 bricks in each row. On top of each row, there is a stack of 6 bricks. How many bricks are there in total?

21. Multiplying 4 Integers: Cayley earns $5 an hour by delivering newspapers. She delivers newspapers 3 days each week, for 4 hours at a time. After delivering newspapers for 8 weeks, how much money will Cayley earn?

Division word problems

Best for: 3rd grade, 4th grade, 5th grade

22. Dividing 1-Digit Integers: If you have 4 pieces of candy split evenly into 2 bags, how many pieces of candy are in each bag?

23. Dividing 2-Digit Integers: If you have 80 tickets for the fair and each ride costs 5 tickets, how many rides can you go on?

24. Dividing Numbers Ending with 0: The school has $20,000 to buy new computer equipment. If each piece of equipment costs $50, how many pieces can the school buy in total?

25. Dividing 3 Integers: Melissa buys 2 packs of tennis balls for $12 in total. All together, there are 6 tennis balls. How much does 1 pack of tennis balls cost? How much does 1 tennis ball cost?

26. Interpreting Remainders: An Italian restaurant receives a shipment of 86 veal cutlets. If it takes 3 cutlets to make a dish, how many cutlets will the restaurant have left over after making as many dishes as possible?

Mixed operations word problems

27. Mixing Addition and Subtraction: There are 235 books in a library. On Monday, 123 books are taken out. On Tuesday, 56 books are brought back. How many books are there now?

28. Mixing Multiplication and Division: There is a group of 10 people who are ordering pizza. If each person gets 2 slices and each pizza has 4 slices, how many pizzas should they order?

29. Mixing Multiplication, Addition and Subtraction: Lana has 2 bags with 2 marbles in each bag. Markus has 2 bags with 3 marbles in each bag. How many more marbles does Markus have?

30. Mixing Division, Addition and Subtraction: Lana has 3 bags with the same amount of marbles in them, totaling 12 marbles. Markus has 3 bags with the same amount of marbles in them, totaling 18 marbles. How many more marbles does Markus have in each bag?

Ordering and number sense word problems

31. Counting to Preview Multiplication: There are 2 chalkboards in your classroom. If each chalkboard needs 2 pieces of chalk, how many pieces do you need in total?

32. Counting to Preview Division: There are 3 chalkboards in your classroom. Each chalkboard has 2 pieces of chalk. This means there are 6 pieces of chalk in total. If you take 1 piece of chalk away from each chalkboard, how many will there be in total?

33. Composing Numbers: What number is 6 tens and 10 ones?

34. Guessing Numbers: I have a 7 in the tens place. I have an even number in the ones place. I am lower than 74. What number am I?

35. Finding the Order: In the hockey game, Mitchell scored more points than William but fewer points than Auston. Who scored the most points? Who scored the fewest points?

Fractions word problems

Best for: 3rd grade, 4th grade, 5th grade, 6th grade

36. Finding Fractions of a Group: Julia went to 10 houses on her street for Halloween. 5 of the houses gave her a chocolate bar. What fraction of houses on Julia’s street gave her a chocolate bar?

37. Finding Unit Fractions: Heather is painting a portrait of her best friend, Lisa. To make it easier, she divides the portrait into 6 equal parts. What fraction represents each part of the portrait?

38. Adding Fractions with Like Denominators: Noah walks ⅓ of a kilometre to school each day. He also walks ⅓ of a kilometre to get home after school. How many kilometres does he walk in total?

39. Subtracting Fractions with Like Denominators: Last week, Whitney counted the number of juice boxes she had for school lunches. She had ⅗ of a case. This week, it’s down to ⅕ of a case. How much of the case did Whitney drink?

40. Adding Whole Numbers and Fractions with Like Denominators: At lunchtime, an ice cream parlor served 6 ¼ scoops of chocolate ice cream, 5 ¾ scoops of vanilla and 2 ¾ scoops of strawberry. How many scoops of ice cream did the parlor serve in total?

41. Subtracting Whole Numbers and Fractions with Like Denominators: For a party, Jaime had 5 ⅓ bottles of cola for her friends to drink. She drank ⅓ of a bottle herself. Her friends drank 3 ⅓. How many bottles of cola does Jaime have left?

42. Adding Fractions with Unlike Denominators: Kevin completed ½ of an assignment at school. When he was home that evening, he completed ⅚ of another assignment. How many assignments did Kevin complete?

43. Subtracting Fractions with Unlike Denominators: Packing school lunches for her kids, Patty used ⅞ of a package of ham. She also used ½ of a package of turkey. How much more ham than turkey did Patty use?

44. Multiplying Fractions: During gym class on Wednesday, the students ran for ¼ of a kilometre. On Thursday, they ran ½ as many kilometres as on Wednesday. How many kilometres did the students run on Thursday? Write your answer as a fraction.

45. Dividing Fractions: A clothing manufacturer uses ⅕ of a bottle of colour dye to make one pair of pants. The manufacturer used ⅘ of a bottle yesterday. How many pairs of pants did the manufacturer make?

46. Multiplying Fractions with Whole Numbers: Mark drank ⅚ of a carton of milk this week. Frank drank 7 times more milk than Mark. How many cartons of milk did Frank drink? Write your answer as a fraction, or as a whole or mixed number.

Decimals word problems

Best for: 4th grade, 5th grade

47. Adding Decimals: You have 2.6 grams of yogurt in your bowl and you add another spoonful of 1.3 grams. How much yogurt do you have in total?

48. Subtracting Decimals: Gemma had 25.75 grams of frosting to make a cake. She decided to use only 15.5 grams of the frosting. How much frosting does Gemma have left?

49. Multiplying Decimals with Whole Numbers: Marshall walks a total of 0.9 kilometres to and from school each day. After 4 days, how many kilometres will he have walked?

50. Dividing Decimals by Whole Numbers: To make the Leaning Tower of Pisa from spaghetti, Mrs. Robinson bought 2.5 kilograms of spaghetti. Her students were able to make 10 leaning towers in total. How many kilograms of spaghetti does it take to make 1 leaning tower?

51. Mixing Addition and Subtraction of Decimals: Rocco has 1.5 litres of orange soda and 2.25 litres of grape soda in his fridge. Antonio has 1.15 litres of orange soda and 0.62 litres of grape soda. How much more soda does Rocco have than Angelo?

52. Mixing Multiplication and Division of Decimals: 4 days a week, Laura practices martial arts for 1.5 hours. Considering a week is 7 days, what is her average practice time per day each week?

Comparing and sequencing word problems

Best for: Kindergarten, 1st grade, 2nd grade

53. Comparing 1-Digit Integers: You have 3 apples and your friend has 5 apples. Who has more?

54. Comparing 2-Digit Integers: You have 50 candies and your friend has 75 candies. Who has more?

55. Comparing Different Variables: There are 5 basketballs on the playground. There are 7 footballs on the playground. Are there more basketballs or footballs?

56. Sequencing 1-Digit Integers: Erik has 0 stickers. Every day he gets 1 more sticker. How many days until he gets 3 stickers?

57. Skip-Counting by Odd Numbers: Natalie began at 5. She skip-counted by fives. Could she have said the number 20?

58. Skip-Counting by Even Numbers: Natasha began at 0. She skip-counted by eights. Could she have said the number 36?

59. Sequencing 2-Digit Numbers: Each month, Jeremy adds the same number of cards to his baseball card collection. In January, he had 36. 48 in February. 60 in March. How many baseball cards will Jeremy have in April?

Time word problems

66. Converting Hours into Minutes: Jeremy helped his mom for 1 hour. For how many minutes was he helping her?

69. Adding Time: If you wake up at 7:00 a.m. and it takes you 1 hour and 30 minutes to get ready and walk to school, at what time will you get to school?

70. Subtracting Time: If a train departs at 2:00 p.m. and arrives at 4:00 p.m., how long were passengers on the train for?

71. Finding Start and End Times: Rebecca left her dad’s store to go home at twenty to seven in the evening. Forty minutes later, she was home. What time was it when she arrived home?

Money word problems

Best for: 1st grade, 2nd grade, 3rd grade, 4th grade, 5th grade

60. Adding Money: Thomas and Matthew are saving up money to buy a video game together. Thomas has saved $30. Matthew has saved $35. How much money have they saved up together in total?

61. Subtracting Money: Thomas has $80 saved up. He uses his money to buy a video game. The video game costs $67. How much money does he have left?

62. Multiplying Money: Tim gets $5 for delivering the paper. How much money will he have after delivering the paper 3 times?

63. Dividing Money: Robert spent $184.59 to buy 3 hockey sticks. If each hockey stick was the same price, how much did 1 cost?

64. Adding Money with Decimals: You went to the store and bought gum for $1.25 and a sucker for $0.50. How much was your total?

65. Subtracting Money with Decimals: You went to the store with $5.50. You bought gum for $1.25, a chocolate bar for $1.15 and a sucker for $0.50. How much money do you have left?

67. Applying Proportional Relationships to Money: Jakob wants to invite 20 friends to his birthday, which will cost his parents $250. If he decides to invite 15 friends instead, how much money will it cost his parents? Assume the relationship is directly proportional.

68. Applying Percentages to Money: Retta put $100.00 in a bank account that gains 20% interest annually. How much interest will be accumulated in 1 year? And if she makes no withdrawals, how much money will be in the account after 1 year?

Physical measurement word problems

Best for: 1st grade, 2nd grade, 3rd grade, 4th grade

72. Comparing Measurements: Cassandra’s ruler is 22 centimetres long. April’s ruler is 30 centimetres long. How many centimetres longer is April’s ruler?

73. Contextualizing Measurements: Picture a school bus. Which unit of measurement would best describe the length of the bus? Centimetres, metres or kilometres?

74. Adding Measurements: Micha’s dad wants to try to save money on gas, so he has been tracking how much he uses. Last year, Micha’s dad used 100 litres of gas. This year, her dad used 90 litres of gas. How much gas did he use in total for the two years?

75. Subtracting Measurements: Micha’s dad wants to try to save money on gas, so he has been tracking how much he uses. Over the past two years, Micha’s dad used 200 litres of gas. This year, he used 100 litres of gas. How much gas did he use last year?

76. Multiplying Volume and Mass: Kiera wants to make sure she has strong bones, so she drinks 2 litres of milk every week. After 3 weeks, how many litres of milk will Kiera drink?

77. Dividing Volume and Mass: Lillian is doing some gardening, so she bought 1 kilogram of soil. She wants to spread the soil evenly between her 2 plants. How much will each plant get?

78. Converting Mass: Inger goes to the grocery store and buys 3 squashes that each weigh 500 grams. How many kilograms of squash did Inger buy?

79. Converting Volume: Shad has a lemonade stand and sold 20 cups of lemonade. Each cup was 500 millilitres. How many litres did Shad sell in total?

80. Converting Length: Stacy and Milda are comparing their heights. Stacy is 1.5 meters tall. Milda is 10 centimetres taller than Stacy. What is Milda’s height in centimetres?

81. Understanding Distance and Direction: A bus leaves the school to take students on a field trip. The bus travels 10 kilometres south, 10 kilometres west, another 5 kilometres south and 15 kilometres north. To return to the school, in which direction does the bus have to travel? How many kilometres must it travel in that direction?

Ratios and percentages word problems

Best for: 4th grade, 5th grade, 6th grade

82. Finding a Missing Number: The ratio of Jenny’s trophies to Meredith’s trophies is 7:4. Jenny has 28 trophies. How many does Meredith have?

83. Finding Missing Numbers: The ratio of Jenny’s trophies to Meredith’s trophies is 7:4. The difference between the numbers is 12. What are the numbers?

84. Comparing Ratios: The school’s junior band has 10 saxophone players and 20 trumpet players. The school’s senior band has 18 saxophone players and 29 trumpet players. Which band has the higher ratio of trumpet to saxophone players?

85. Determining Percentages: Mary surveyed students in her school to find out what their favourite sports were. Out of 1,200 students, 455 said hockey was their favourite sport. What percentage of students said hockey was their favourite sport?

86. Determining Percent of Change: A decade ago, Oakville’s population was 67,624 people. Now, it is 190% larger. What is Oakville’s current population?

87. Determining Percents of Numbers: At the ice skate rental stand, 60% of 120 skates are for boys. If the rest of the skates are for girls, how many are there?

88. Calculating Averages: For 4 weeks, William volunteered as a helper for swimming classes. The first week, he volunteered for 8 hours. He volunteered for 12 hours in the second week, and another 12 hours in the third week. The fourth week, he volunteered for 9 hours. For how many hours did he volunteer per week, on average?

Probability and data relationships word problems

Best for: 4th grade, 5th grade, 6th grade, 7th grade

89. Understanding the Premise of Probability: John wants to know his class’s favourite TV show, so he surveys all of the boys. Will the sample be representative or biased?

90. Understanding Tangible Probability: The faces on a fair number die are labelled 1, 2, 3, 4, 5 and 6. You roll the die 12 times. How many times should you expect to roll a 1?

91. Exploring Complementary Events: The numbers 1 to 50 are in a hat. If the probability of drawing an even number is 25/50, what is the probability of NOT drawing an even number? Express this probability as a fraction.

92. Exploring Experimental Probability: A pizza shop has recently sold 15 pizzas. 5 of those pizzas were pepperoni. Answering with a fraction, what is the experimental probability that he next pizza will be pepperoni?

93. Introducing Data Relationships: Maurita and Felice each take 4 tests. Here are the results of Maurita’s 4 tests: 4, 4, 4, 4. Here are the results for 3 of Felice’s 4 tests: 3, 3, 3. If Maurita’s mean for the 4 tests is 1 point higher than Felice’s, what’s the score of Felice’s 4th test?

94. Introducing Proportional Relationships: Store A is selling 7 pounds of bananas for $7.00. Store B is selling 3 pounds of bananas for $6.00. Which store has the better deal?

95. Writing Equations for Proportional Relationships: Lionel loves soccer, but has trouble motivating himself to practice. So, he incentivizes himself through video games. There is a proportional relationship between the amount of drills Lionel completes, in x , and for how many hours he plays video games, in y . When Lionel completes 10 drills, he plays video games for 30 minutes. Write the equation for the relationship between x and y .

Geometry word problems

Best for: 4th grade, 5th grade, 6th grade, 7th grade, 8th grade

96. Introducing Perimeter:  The theatre has 4 chairs in a row. There are 5 rows. Using rows as your unit of measurement, what is the perimeter?

97. Introducing Area: The theatre has 4 chairs in a row. There are 5 rows. How many chairs are there in total?

98. Introducing Volume: Aaron wants to know how much candy his container can hold. The container is 20 centimetres tall, 10 centimetres long and 10 centimetres wide. What is the container’s volume?

99. Understanding 2D Shapes: Kevin draws a shape with 4 equal sides. What shape did he draw?

100. Finding the Perimeter of 2D Shapes: Mitchell wrote his homework questions on a piece of square paper. Each side of the paper is 8 centimetres. What is the perimeter?

101. Determining the Area of 2D Shapes: A single trading card is 9 centimetres long by 6 centimetres wide. What is its area?

102. Understanding 3D Shapes: Martha draws a shape that has 6 square faces. What shape did she draw?

103. Determining the Surface Area of 3D Shapes: What is the surface area of a cube that has a width of 2cm, height of 2 cm and length of 2 cm?

104. Determining the Volume of 3D Shapes: Aaron’s candy container is 20 centimetres tall, 10 centimetres long and 10 centimetres wide. Bruce’s container is 25 centimetres tall, 9 centimetres long and 9 centimetres wide. Find the volume of each container. Based on volume, whose container can hold more candy?

105. Identifying Right-Angled Triangles: A triangle has the following side lengths: 3 cm, 4 cm and 5 cm. Is this triangle a right-angled triangle?

106. Identifying Equilateral Triangles: A triangle has the following side lengths: 4 cm, 4 cm and 4 cm. What kind of triangle is it?

107. Identifying Isosceles Triangles: A triangle has the following side lengths: 4 cm, 5 cm and 5 cm. What kind of triangle is it?

108. Identifying Scalene Triangles: A triangle has the following side lengths: 4 cm, 5 cm and 6 cm. What kind of triangle is it?

109. Finding the Perimeter of Triangles: Luigi built a tent in the shape of an equilateral triangle. The perimeter is 21 metres. What is the length of each of the tent’s sides?

110. Determining the Area of Triangles: What is the area of a triangle with a base of 2 units and a height of 3 units?

111. Applying Pythagorean Theorem: A right triangle has one non-hypotenuse side length of 3 inches and the hypotenuse measures 5 inches. What is the length of the other non-hypotenuse side?

112. Finding a Circle’s Diameter: Jasmin bought a new round backpack. Its area is 370 square centimetres. What is the round backpack’s diameter?

113. Finding a Circle's Area: Captain America’s circular shield has a diameter of 76.2 centimetres. What is the area of his shield?

114. Finding a Circle’s Radius: Skylar lives on a farm, where his dad keeps a circular corn maze. The corn maze has a diameter of 2 kilometres. What is the maze’s radius?

Variables word problems

Best for: 6th grade, 7th grade, 8th grade

115. Identifying Independent and Dependent Variables: Victoria is baking muffins for her class. The number of muffins she makes is based on how many classmates she has. For this equation, m is the number of muffins and c is the number of classmates. Which variable is independent and which variable is dependent?

116. Writing Variable Expressions for Addition: Last soccer season, Trish scored g goals. Alexa scored 4 more goals than Trish. Write an expression that shows how many goals Alexa scored.

117. Writing Variable Expressions for Subtraction: Elizabeth eats a healthy, balanced breakfast b times a week. Madison sometimes skips breakfast. In total, Madison eats 3 fewer breakfasts a week than Elizabeth. Write an expression that shows how many times a week Madison eats breakfast.

118. Writing Variable Expressions for Multiplication: Last hockey season, Jack scored g goals. Patrik scored twice as many goals than Jack. Write an expression that shows how many goals Patrik scored.

119. Writing Variable Expressions for Division: Amanda has c chocolate bars. She wants to distribute the chocolate bars evenly among 3 friends. Write an expression that shows how many chocolate bars 1 of her friends will receive.

120. Solving Two-Variable Equations: This equation shows how the amount Lucas earns from his after-school job depends on how many hours he works: e = 12h . The variable h represents how many hours he works. The variable e represents how much money he earns. How much money will Lucas earn after working for 6 hours?

How to easily make your own math word problems & word problems worksheets

Armed with 120 examples to spark ideas, making your own math word problems can engage your students and ensure alignment with lessons. Do:

• Link to Student Interests:  By framing your word problems with student interests, you’ll likely grab attention. For example, if most of your class loves American football, a measurement problem could involve the throwing distance of a famous quarterback.
• Make Questions Topical:  Writing a word problem that reflects current events or issues can engage students by giving them a clear, tangible way to apply their knowledge.
• Include Student Names:  Naming a question’s characters after your students is an easy way make subject matter relatable, helping them work through the problem.
• Be Explicit:  Repeating keywords distills the question, helping students focus on the core problem.
• Test Reading Comprehension:  Flowery word choice and long sentences can hide a question’s key elements. Instead, use concise phrasing and grade-level vocabulary.
• Focus on Similar Interests:  Framing too many questions with related interests -- such as football and basketball -- can alienate or disengage some students.
• Feature Red Herrings:  Including unnecessary information introduces another problem-solving element, overwhelming many elementary students.

A key to differentiated instruction , word problems that students can relate to and contextualize will capture interest more than generic and abstract ones.

Final thoughts about math word problems

You’ll likely get the most out of this resource by using the problems as templates, slightly modifying them by applying the above tips. In doing so, they’ll be more relevant to -- and engaging for -- your students.

Regardless, having 120 curriculum-aligned math word problems at your fingertips should help you deliver skill-building challenges and thought-provoking assessments.

The result?

A greater understanding of how your students process content and demonstrate understanding, informing your ongoing teaching approach.

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April 5, 2023 k-2-math-practices , k-2-measurement-and-data , 3-5-measurement-and-data , 3-5-math-practices , other-seasonal

Math in real life– strategies for planning lessons involving real-world contexts, by: allie johnston.

When are we going to use this? Why are we learning this? Why can’t I just use a calculator? If you’ve ever been asked these questions by your math students, then this article is for you! As teachers, our job is much more than teaching students math content; it is about helping students to problem solve in the real world in which they live today. The best way to do this is to allow students to learn math through the real-world situations that they encounter and that matter to them! In this article, you will learn about the relevance and usefulness of real-world applications in the math classroom and ways to implement these types of lessons in your classroom. The article includes strategies for planning lessons involving real-world contexts as well as a downloadable example that you can use in your elementary math classroom.

Benefits of Relating Math to Real Life  

The list of reasons to use real-world math applications and their benefits for students is long; they include increasing student engagement, improving how well students can remember and recall key concepts, decreasing behavior issues, and helping teachers with management. Both teachers and students enjoy the learning process more when the vehicle is a real-world problem.

Student interest and engagement are most directly impacted by the type of activity presented in class. By using math problems that mimic the real world and are relatable, students are more interested and therefore willing to work and engage with the tasks. As a result, students are more focused, and teachers worry less about classroom management. Additionally, students are better able to understand what they are learning and why they are learning it when the material is presented in a context that they can envision being in.

By using math problems that mimic the real world and are relatable, students are more interested and therefore willing to work and engage with the tasks.

Through these real-life math problems, students can develop the key thinking skills outlined by the National Council of Teachers of Mathematics in the 8 Standards of Mathematical Practice. Real-world applications often require students to decontextualize and recontextualize the task during the solving process, model with mathematics to represent the real-world situations, and push students to ask questions and persevere in problem-solving. These skills are increasingly important as technology becomes more and more accessible.

Lastly, real-world math situations enable students to transfer their learning out of the classroom and into their lives. Real-world contexts enable students to draw on existing funds of knowledge, transferring their background knowledge into the math classroom. They also enable students to apply what they have learned in new contexts, transferring their knowledge from the classroom to their lives.

Using “Windows” and “Mirrors” to Reflect Real-World Math Problems

Bringing real-life applications into the classroom requires a careful balance of offering students access to problems that they will connect with and problems that students are unfamiliar with that will help them to explore the world around them. Presenting both types of problems can help students draw connections to their own life, “mirrors,” and help students understand the lives of others through “windows.”  

“Mirror” problems can be created by relying on students' daily life experiences. Common community events and shared spaces are great places to look for possible real-world math problems. Shared developmentally appropriate interests and characteristics are another way to connect math to the students’ lives. For example, learners in early grades often enjoy playing kitchen, doctor, and teacher. Using these contexts can help them draw connections between their play and the math they are learning. For older students, using the context of video games or sports is often an effective way to increase interest in learning math concepts.

Interdisciplinary settings allow students to draw parallels and make new connections acting as both a “window” and a “mirror” to using real-world math. Oftentimes, students struggle to make connections between the learning that they are doing in each content area class. Using an application related to history or science can assist students in integrating their knowledge between settings. History is often a great connection for early math learners to begin understanding time, number lines, and basic addition and subtraction problems to discover the amount of time that has passed between different historic events.

Bringing real-life applications into the classroom requires a careful balance of offering students access to problems that they will connect with and problems that students are unfamiliar with that will help them to explore the world around them.

Equally valuable to making math problems relatable is the opportunity to expose students to the vast world around them. Sharing real-world math problems that demonstrate how mathematics is useful can open students' eyes to careers other than the common interests of most kids such as being a firefighter, teacher, or doctor. For example, math is necessary for carpentry, architecture, business analysts, and many other careers that may pique the interest of students. Math applications can also show a glimpse of how math was discovered and is continued to be used around the world. For example, looking at how numbers were initially written in ancient times may help students appreciate the number system in use today.

STEAM connections, or interdisciplinary connections among science, technology, engineering, the arts, and mathematics, abound in real-world applications and reveal to students that math connects to everything.

In any of the above types of real-world math applications, it is necessary for the teacher and the students to be able to see the context as realistic and useful. As you being to think about using and designing real-world applications in your own classroom, consider these strategies for incorporating realistic real-world math problems:

• Include realistic word problems that your students might encounter.
• Offer analogies as simple connections between a problem and the real world such as subtraction and the temperature dropping. Analogies can also be used to analyze a more complex concept such as comparing 0 on a number line to a mirror.
• Set the stage through the wording of the problem or the topic and data being presented.
• Give context and offer an entire problem that is an example of a real-world situation that students need to use math to solve.
• Model real situations using geometric shapes and figures, equations, or technology.

The best way to get started is to give real-world application problems a try!

Math in Real Life Problems Examples

Each lesson of the Sadlier Math program opens with a real-world application and offers a STEAM connection lesson offering students both “windows” and “mirrors” with which to view problems. On April 21 we celebrate Earth Day, which makes this Protecting Our Planet STEAM Lesson both timely and relevant for students. This activity invites students to use units of measure for length to study precipitation records. After preparing models of a particular region’s record rainfall and snowfall, they explore concepts of climate change and its causes and effects. This activity connects to the United Nations Sustainable Development Goal (SDG) 13: Climate Action, which promotes awareness and education about climate change.

Your students will benefit from the opportunity to enjoy math learning as they develop the skills that they need in their lives, now and in the future through real-world applications and rich interdisciplinary connections.

References:

Lee, J. E. Prospective elementary teachers’ perceptions of real-life connections reflected in posing and evaluating story problems. J Math Teacher Educ 15, 429–452 (2012). https://doi.org/10.1007/s10857-012-9220-5

Premadasa, Kirthi and Bhatia, Kavita (2013) "Real Life Applications in Mathematics: What Do Students Prefer?," International Journal for the Scholarship of Teaching and Learning: Vol. 7: No. 2, Article 20.

5 Real Life Algebra Problems That You Solve Everyday

Algebra has a reputation for not being very useful in daily life. In fact, in my experience as a high school math teacher, the complaint that I get the most often is that we don’t spend enough time solving real life algebra problems.

You might be surprised to hear that I understand the frustration that my students experience. Unless we are solving real life algebra problems related to money in some way, algebra can feel very “artificial” or disconnected from real life.

My goal here is to walk you through 5 real life algebra problems that will give you a whole new appreciation for the application of algebra to the real-world. I am excited to help you see how many algebraic equations and algebraic concepts are applicable beyond just algebra word problems in your math class!

What is an Example of Algebra in Real Life?

While it is often seen as an abstract branch of mathematics, there are many real-life applications of algebra in everyday life. Now, it is unlikely that you will be solving quadratic equations while walking your dog, or solving real-world problems with linear equations while you play video games. But you can see examples of real life algebra problems all around you!

A simple example is when you want to quickly determine the total cost of a product including taxes, or the total cost after a discount from the original price. Knowing the total amount of money something will cost is a real-life scenario that everyone can relate to!

Depending on your chosen career path, you may see the use of algebra more often than others (I know I see it a lot in my daily life as a math teacher…!).

For example, if you are a business owner, you may use algebra to determine the number of labor hours to spread amongst your staff, or the lowest price you can sell your product for to break even.

For more uses of algebra, check out my list of 20  examples of algebra in real life !

What is an Example of an Algebra Problem in Real Life?

An algebra problem is a mathematical problem that requires the use of algebraic concepts and strategies to determine unknown values or unknown variables. Much like how the order of operations are required to evaluate numerical expressions, algebra problems require the problem solver to apply a set of rules in order to arrive at a solution.

Real world problems that require the use of algebra usually involve modelling real-life situations with  algebraic formulas . A formula is a specific equation that can be applied to solve a problem. Formulas make it possible to make predictions about a given real-life scenario.

For example, consider the following problem:

You are saving up for a new smartphone and currently have $200 in your savings account. Your plan is to save a certain amount of money each week from your allowance. If the smartphone costs $600, and you want to have enough money to buy it in 8 weeks, how much money should you save each week?

To solve this problem, we first need to use the information provided in the problem to create an equation that models the real-life scenario. Thinking about the problem in terms of variables, we can define T as the total of the savings, and variable x as the amount saved each week.

Since we know that we have a fixed value of 200, we can use the following equation to model this real world problem:

$$T=200+8x$$

This equation says “the total saved is equal to the original $200 plus whatever amount is saved per week, for 8 weeks”.

Substituting the total of the smartphone allows us to begin solving for the unknown variable x. Remember, when solving algebraic equations, you must apply the same operation to both sides of the equation.

$$ \begin{split} T&=200+8x  \\ \\ 600&=200+8x  \\ \\ 600-200 &= 8x \\ \\ 400 &= 8x \\ \\ \frac{400}{8} &= \frac{8x}{8} \\\\ 50 &= x \end{split} $$

Therefore, since x = 50, you should save $50 each week in order to save enough money for the smartphone. For more practice with the algebra used in this solution, check out this free collection of  solving two step equations worksheets !

5 Real Life Algebra Problems with Step-By-Step Solutions

There are so many real-life examples of algebra problems, but I want to focus on 5 here that I believe will convince you of just how applicable algebra is to the real-world! So let’s dig into these 5 real-world algebraic word problems!

Example #1: Comparing Cell Phone Plans

Link is considering two different cell phone plans. Plan A charges a monthly fee of $30 and an additional $0.10 per minute of talk time. Plan B charges a monthly fee of $45 regardless of how much time is used talking. How many minutes of total time talking will make the plans equal in cost?

The best way to start this problem is by writing two equations to represent each scenario. If C represents total cost, and x represents minutes of talk time used, the equations can be written as follows:

• Plan A: \(C=30+0.1x\)
• Plan B: \(C=45\)

Setting the first equation equal to the second equation will allow us to employ algebra to solve for the number of minutes that makes the two plans equal.

$$ \begin{split}  30+0.1x&=45 \\ \\ 30-30+0.1x&=45-30 \\ \\ 0.1x&=15 \\ \\ \frac{0.1x}{0.1}&=\frac{15}{0.1} \\ \\ x&=150 \end{split} $$

Therefore, the two cell phone plans are equal when 150 minutes of total time talking are used.

Example #2: Calculating Gallons of Gas

Zelda is driving from Hyrule to the Mushroom Kingdom, which are 180 miles apart. Her car can travel 30 miles per gallon of gas. Write an equation to represent the number of gallons of gas, G, that Zelda needs for the trip in terms of the distance, d, she needs to travel. Then calculate how many gallons of gas she needs for this trip.

The number of gallons of gas (G) Zelda needs for any trip can be represented by the equation \(G = \frac{d}{30}\). Since the distance between Hyrule and the Mushroom Kingdom is 180 miles, we can substitute 180 into the equation for  d  to determine the number of gallons of gas needed:

$$G=\frac{180}{30}=6$$

Therefore, Zelda needs 6 gallons of gas for her trip.

Example #3: Basketball Players in Action!

A basketball player shoots a basketball from a height of 6 feet above the ground. Unfortunately he completely misses the net and the ball bounces off court. A sports analyst models the path of the basketball using the equation \(h(t) = -16t^2 + 16t + 6\), where h(t) represents the height of the basketball above the ground in feet at time t seconds after the shot. Determine the time it takes for the basketball to hit the ground.

Since we are asked for when the ball hits the ground and  h(t)  is given as the height above the ground, we know that we are looking for the x-intercepts of this quadratic function. We therefore set the equation equal to zero and solve for x. 

Note that we cannot use  trinomial factoring  here since the quadratic is not factorable! Thankfully quadratic equations are solvable using the quadratic formula!

$$ \begin{split}  x&=\frac{-b \pm \sqrt{b^2-4ac}}{2a} \\\\ &=\frac{-16 \pm \sqrt{16^2-4(-16)(6)}}{2(-16)}\\\\ &=\frac{-16 \pm \sqrt{640}}{-32}\\\\ x&=-0.291 \\\\ x&=1.291 \end{split}$$

Therefore, the ball hits the ground after approximately 1.3 seconds. Remember that time cannot be negative, so the first answer is inadmissible and rejected!

Example #4: Saving for a Computer Game

You are saving to buy a new computer game that costs $90. You decide to save up for the computer game by depositing some money into a savings account that earns an annual interest rate of 5% (compounded monthly). You start with an initial deposit of $30 and plan to save for 22 months. Will you have enough to purchase the computer game?

This is an example of a math problem that connects to financial problems people encounter everyday! Since the account you chose earns  interest , we can apply a compound interest formula to help us out here:

$$A=P(1+i)^n$$

In this formula:

• A(t)  is the total amount of money.
• P  is the initial deposit (which is $30 in this case).
• i  is the monthly interest rate (5% annual interest, compounded monthly means that  i  is approximately 0.004167).
• n  is the time that has elapsed (since we are working with months, we multiply by 12)

We can set up our equation and see if our total amount of money is greater than $90:

$$\begin{split}  A(22)&=30(1.004167)^{22 \times 12} \\\\ &=$89.93 \end{split} $$

Remember to always include a dollar sign in your answer and to round to two decimal places when working with money!

Since our answer is approximately equal to $90, we can say that you will have enough money after 22 months! It’s time to get saving!

Example #5: How Many Tickets Did the Movie Theater Sell?

A movie theater charges $10 per ticket for adults and $6 per ticket for children. On a particular day, the theater sold a total of 150 tickets, and the total revenue for the day was $1350. Write a system of equations to represent this real-life scenario and then solve for the number of adult and child tickets sold.

Let’s assume that variable  x  represents the number of adult tickets sold and variable  y  represents the number of child tickets sold. We can set up two linear equations as follows:

• First Equation (the total number of tickets sold): \(x+y=150\) 
• Second Equation (the total revenue from ticket sales is 1350): \(10x+6y=1350\) 

We can use substitution to solve this linear system by rearranging the first equation and substituting it into the second equation. You can catch a quick overview of the substitution process by checking out  this substitution video  on my YouTube channel!

Rearranging the first equation into a different form to solve for  y  results in \(y=-x+150\). Substituting this expression for  y  into the second equation results in: 

$$ \begin{split}  10x+6(-x+150)&=1350 \\ \\ 10x-6x+900&=1350 \\ \\ 4x&=450\\ \\ x&=112.5\\ \\ \end{split}  $$

We then substitute this value for  x  into our expression for  y: 

$$ \begin{split}  y&=-x+150 \\ \\ &=-112.50+150 \\ \\ &=37.5\\ \\ \end{split}  $$

Since we can’t have fractional ticket sales, we can say that approximately 112 adult tickets were sold and 38 child tickets were sold.

Appreciating Real Life Algebra Problems

While algebra is often seen as an abstract topic, I am hopeful that I have shown you just how applicable it can be to real-life situations! Some of these examples you may have even encountered in your own life!

Even if you aren’t drawing up complex equations and solving them while you are playing basketball, combining basic math and problem solving is one of the most important skills people can have in both their work and their lives. 

I hope that I have helped you further your understanding of algebra, while growing an appreciation for the different ways it can be used in your own life!

Did you find this guide to real life algebra problems helpful? Share this post and subscribe to Math By The Pixel on YouTube for more helpful mathematics content!

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Algebra in Real Life

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Have you ever wondered how Algebra may be applied to solve real-life problems?

We regularly see people using Algebra in many parts of everyday life; for instance, it is utilized in our morning schedule each day to measure the time you will spend in the shower, making breakfast, or driving to work.

The absence of "X" or "Y" doesn't imply that algebra is not around us; algebra’s actual occurrences are uncountable. This exact and compact numerical language works wonderfully with practically all different subjects and everyday life. 

In this article, we will grasp instances in real life where applications of algebra are needed and examples of applications of algebra in real life.  

• How to factorise a polynomial?

What is Algebra?

Algebra is a part of mathematics that deals with symbols and the standards for controlling those symbols. The more basic parts of algebra are called elementary algebra, and the more abstract types are called Abstract Algebra or modern algebra. 

Algebraic Expression

Let us consider the pattern below. It has been created using marbles. Here we see that the first column has 2 marbles, the second column has 3 marbles, the third column has 4 marbles and so on.

Thus we observe that every new column increases by 1 marble. We can write the representation as 

The number of marbles used in a column =

position of the column + 1

or as 

the number of marbles used in a column = n + 1.

Here n represents the position of the column. So ‘n’ is an example for a variable that can take any value 1,2,3… so on. Thus n + 1 is the algebraic expression formed with n as variable and constant 1.

A variable is a number that does not have a fixed value. The picture and the list below show some real-life examples, where the value of a variable changes with the change in place and time. 

• The temperature in different places also change.
• The height of a growing child changes with time.        
• The speed of a car changes with time.
• The age of people keeps on increasing year by year.

Constant 

The value remains fixed for specific numbers that represent quantities or ideas that will not change. For example, the date of birth of a particular person, the normal human body temperature and capacity of a given container.

Framing algebraic expressions with given conditions

Now we will see how to frame an algebraic expression. The rules are that variables are to be represented with alphabetic letters, say lower case a-z and constants in numeric form.

1. Amanda has 10 storybooks more than Alex. Express the number of storybooks Amanda has in terms of the number of storybooks Alex has. 

Let the number of storybooks Alex has = y

Therefore the number of storybooks Amanda has = y + 10

2. Sweets from a big box are equally distributed in 10 small boxes. Express the number of sweets in one small box in terms of the total number of sweets.

Let x be the total number of sweets. 

Number of sweet boxes = 10

Therefore, the number of sweets in one box is = x/10

Solving Equations

Let us see how practical applications of algebra can be used to solve equations. You will often see equations like 3x + 4 = 5, where you want to find x.

Consider a situation from our daily life.

The cost of a book is £5 more than the cost of a pen. Let us take the cost of the pen as £x. Then the cost of the book is £ (x + 5) . If the cost of the book is £20, what is the cost of the pen?

We know that the book’s cost is x + 5 and it is given that x + 5 = 20. This is an equation in the variable x.

A table is prepared as shown below for various values of x:

It is clear from the table that x + 5 = 20 only for x = 15. So, the cost of the pen is £15. 

In general we say that x = 15 is the solution of the equation x + 5 = 20. This is the trial and error method where we substitute different values for the variable that satisfies the given equation.

An equation has two parts which are connected by an equal to sign. The two parts or sides of an equation are denoted as LHS (Left Hand Side) and RHS (Right Hand Side). If LHS = RHS we get an equation. 2x = 6 is an

algebraic equation, whereas 3x > 10 or 4x < 12 are not equations.

Solving an equation using the Principle of Balances

Consider the balance given in the figure.

Four circles balance one square and a circle on the other side. The idea is we have to find out how many circles will balance a square. If we remove the circle from the left pan, we have only the square there. Since we removed a circle from the left pan, we have to remove the circle from the right pan also. Then there will be three circles in the right pan.

Now the balance looks like the one shown on the right. This is called the principle of balances. Using balancing equations, we can solve equations in a systematic way.

Solve using the principle of balances:

Benjamin's mother is three times as old as Benjamin. If Benjamin's mother is 39 years old, find Benjamin's age. 

Let Benjamin's age be x. 

Benjamin’s mother's age 3x = 39

3x/3 = 39/3 {Dividing by 3 on both the sides }

So, Benjamin’s age = 13.

The same quantity can be added or subtracted to both sides of the equation. If the same amount is multiplied or divided on both the sides of an equation, it remains the same.

Forming an equation to find the unknown

Translating verbal descriptions into algebraic expressions is an essential initial step in solving word problems. So let’s see another real-life example in the form of a puzzle.

Detailed Solution:

Our first supposition is that Uma buys at least one ball of each kind. Now let’s say she buys x footballs, y cricket balls, and z table-tennis balls.

The question requires x + y + z = 100  [ 1 ]

It also requires 15 x + 1y + z/4 = 100  [ 2 ]Since we have 3 variables but only 2 equations we’ll have to use the trial and error method to get at the solution.

Let’s vary x the number of footballs and see what we get:

Suppose x = 1, then 

y + z =99 and y + z/ 4 = 83z/4 = 14 3z = 56 z = 56/3 which is not a whole number.

Trying for x = 2 also fails and now

If x = 3, then  z = 97 and y + z / 4 = 55 3z/4 =42 3z= 168 z = 168/3 = 56  which is a whole number! 

And if z = 56  then y = 97 - z = 97 - 56 = 41 

So the set of balls Uma buys is { 3 footballs, 41 cricket balls and 56 table tennis balls }

Algebra in Geometry

In Algebraic Geometry we study geometric objects and their assortment that are characterized by polynomial equations. 

Examples of algebraic varieties’ most studied classes are plane algebraic curves, including lines, circles, parabolas, ellipses, hyperbolas. There are also cubic curves like elliptic curves and quartic curves like lemniscates and Cassini ovals.

In real life, algebraic geometry can be used to study the dynamics properties of robotics mechanisms.

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A robot can move in continuous space with an infinite set of possible actions and states. When the robot has arms and legs that must also be controlled and the search space becomes many-dimensional. Robot’s kinematics can be formulated as a polynomial equation system that can be solved using algebraic geometry tools.

Algebraic geometry is also widely used in statistics, control theory, and geometric modelling. There are also connections to string theory, game theory, graph matchings and integer programming.

Algebra in Computer Programming

The mathematical languages unite fields such as science, technology, and engineering into itself. That is why an individual intrigued by the field of computer programming and coding should figure out how to comprehend and control mathematical logic.

Strong comprehension of algebra incorporates characterizing the connections between objects, critical thinking with restricted factors, and analytical skill development to help execute decision making. 

One such use of Algebra can be seen in Inference procedures used in Knowledge engineering. Variables and constant symbols are used as terms representing objects in real life. 

The knowledge engineer adds a set of facts and specifies what is true, and the inference procedure figures out how to turn the facts into a solution to the problem. 

Besides, because a fact is true regardless of what task one is trying to solve, knowledge bases can be reused for various tasks without modification. 

Example for the  task of inference

Take a sentence,

Everyone likes ice cream.

It is represented in First-order logic as 

 x Likes ( x, ice cream ) 

where x is the variable and is the universal quantifier that generalizes to all persons liking icecream. 

If another sentence found in the knowledge base is as follows:

    John likes ice cream

It is represented as Likes( John, ice cream)

The inference procedure will reason out from x Likes ( x, ice cream ) with the substitution {x/John} and infers Likes(John, ice cream) and concludes that John likes icecreams.

                                                                                                                                                      Biostatistics University of Florida

Other uses of algebra in programming are  Ontology, error correction algorithms, Natural language processing, Neural networks,  designing artificial intelligence programming languages such as LISP and PROLOG and theorem provers such as OTTER.

In real life there are a plethora of instances where Algebra is being used. It’s utility is being universally quantified in all walks of our lives. For instance, take a shopping domain where we need to be budgeted with the cart items and some algebraic formulation is applied.

The economy of every country is analysed with the help of economists taking the help of algebra to solve the problems related to debts or loans.

Tom Evans's ANALOGY program (1968) solved geometric analogy problems that appear in IQ tests such as the one shown below.

                                                                                                                                                 Source: Artificial Intelligence by Stuart Russell

The use of algebra is multipurpose, and it goes handy in every sphere of our lives.  It isn't just mathematicians, however, even most academicians, educationists, researchers, and experts from all different backgrounds collectively

agree with the adaptability of algebra. 

Real-World Applications of Linear Algebra

What is linear algebra.

Linear algebra is the branch of mathematics concerning linear equations such as linear maps and their representations in vector spaces and matrices.

The concept of classification can be simulated with the help of neural network structures that use a linear regression model. Here the training set is compared with the test data so that the learning algorithms generate outcomes to predict data related to decision making, medical diagnosis, statistical inferences, etc.

Applications

The most generally utilized use of linear algebra is certainly optimization, and the most broadly utilized sort of advancement is linear algebra. You can upgrade spending plans, your eating regimen, and your course to work 

utilizing linear algebra, and this uniqueness starts to expose a lot of applications. 

Other real-world applications of linear algebra include ranking in search engines, decision tree induction, testing software code in software engineering, graphics, facial recognition, prediction and so on.

In real life, algebra can be compared to a universally handy device or a sorcery wand that can help manage regular issues of life. Whenever life throws a maths problem at you, for example when you have to solve an equation or work out a geometrical problem, algebra is usually the best way to attack it. 

Written by Jesy Margaret, Cuemath teacher

About Cuemath

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Using Mathematical Modeling to Get Real With Students

Unlike canned word problems, mathematical modeling plunges students into the messy complexities of real-world problem solving.  

How do you bring math to life for kids? Illustrating the boundless possibilities of mathematics can be difficult if students are only asked to examine hypothetical situations like divvying up a dessert equally or determining how many apples are left after sharing with friends, writes third- and fourth- grade teacher Matthew Kandel for Mathematics Teacher: Learning and Teaching PK-12 .

In the early years of instruction, it’s not uncommon for students to think they’re learning math for the sole purpose of being able to solve word problems or help fictional characters troubleshoot issues in their imaginary lives, Kandel says. “A word problem is a one-dimensional world,” he writes. “Everything is distilled down to the quantities of interest. To solve a word problem, students can pick out the numbers and decide on an operation.” 

But through the use of mathematical modeling, students are plucked out of the hypothetical realm and plunged into the complexities of reality—presented with opportunities to help solve real-world problems with many variables by generating questions, making assumptions, learning and applying new skills, and ultimately arriving at an answer.

In Kandel’s classroom, this work begins with breaking students into small groups, providing them with an unsharpened pencil and a simple, guiding question: “How many times can a pencil be sharpened before it is too small to use?”

Setting the Stage for Inquiry 

The process of tackling the pencil question is not unlike the scientific method. After defining a question to investigate, students begin to wonder and hypothesize—what information do we need to know?—in order to identify a course of action. This step is unique to mathematical modeling: Whereas a word problem is formulaic, leading students down a pre-existing path toward a solution, a modeling task is “free-range,” empowering students to use their individual perspectives to guide them as they progress through their investigation, Kandel says. 

Modeling problems also have a number of variables, and students themselves have the agency to determine what to ignore and what to focus their attention on. 

After inter-group discussions, students in Kandel’s classroom came to the conclusion that they’d need answers to a host of other questions to proceed with answering their initial inquiry: 

• How much does the pencil sharpener remove? 
• What is the length of a brand new, unsharpened pencil? 
• Does the pencil sharpener remove the same amount of pencil each time it is used?

Introducing New Skills in Context

Once students have determined the first mathematical question they’d like to tackle (does the pencil sharpener remove the same amount of pencil each time it is used?), they are met with a roadblock. How were they to measure the pencil if the length did not fall conveniently on an inch or half inch? Kandel took the opportunity to introduce a new target skill which the class could begin using immediately: measuring to the nearest quarter inch. 

“One group of students was not satisfied with the precision of measuring to the nearest quarter inch and asked to learn how to measure to the nearest eighth of an inch,” Kandel explains. “The attention and motivation exhibited by students is unrivaled by the traditional class in which the skill comes first, the problem second.” 

Students reached a consensus and settled on taking six measurements total: the initial length of the new, unsharpened pencil as well as the lengths of the pencil after each of five sharpenings. To ensure all students can practice their newly acquired skill, Kandel tells the class that “all group members must share responsibility, taking turns measuring and checking the measurements of others.” 

Next, each group created a simple chart to record their measurements, then plotted their data as a line graph—though exploring other data visualization techniques or engaging students in alternative followup activities would work as well.

“We paused for a quick lesson on the number line and the introduction of a new term—mixed numbers,” Kandel explains. “Armed with this new information, students had no trouble marking their y-axis in half- or quarter-inch increments.” 

Sparking Mathematical Discussions

Mathematical modeling presents a multitude of opportunities for class-wide or small-group discussions, some which evolve into debates in which students state their hypotheses, then subsequently continue working to confirm or refute them. 

Kandel’s students, for example, had a wide range of opinions when it came to answering the question of how small of a pencil would be deemed unusable. Eventually, the class agreed that once a pencil reached 1 ¼ inch, it could no longer be sharpened—though some students said they would be able to still write with it. 

“This discussion helped us better understand what it means to make an assumption and how our assumptions affected our mathematical outcomes,” Kandel writes. Students then indicated the minimum size with a horizontal line across their respective graphs. 

Many students independently recognized the final step of extending their line while looking at their graphs. With each of the six points representing their measurements, the points descended downward toward the newly added horizontal “line of inoperability.” 

With mathematical modeling, Kandel says, there are no right answers, only models that are “more or less closely aligned with real-world observations.” Each group of students may come to a different conclusion, which can lead to a larger class discussion about accuracy. To prove their group had the most accurate conclusion, students needed to compare and contrast their methods as well as defend their final result. 

Developing Your Own Mathematical Models

The pencil problem is a great starting point for introducing mathematical modeling and free-range problem solving to your students, but you can customize based on what you have available and the particular needs of each group of students.

Depending on the type of pencil sharpener you have, for example, students can determine what constitutes a “fair test” and set the terms of their own inquiry. 

Additionally, Kandel suggests putting scaffolds in place to allow students who are struggling with certain elements to participate: Simplified rulers can be provided for students who need accommodations; charts can be provided for students who struggle with data collection; graphs with prelabeled x- and y-axes can be prepared in advance.

.css-1sk4066:hover{background:#d1ecfa;} 7 Real-World Math Strategies

Students can also explore completely different free-range problem solving and real world applications for math . At North Agincourt Jr. Public School in Scarborough, Canada, kids in grades 1-6 learn to conduct water audits. By adding, subtracting, finding averages, and measuring liquids—like the flow rate of all the water foundations, toilets, and urinals—students measure the amount of water used in their school or home in a single day. 

Or you can ask older students to bring in common household items—anything from a measuring cup to a recipe card—and identify three ways the item relates to math. At Woodrow Petty Elementary School in Taft, Texas, fifth-grade students display their chosen objects on the class’s “real-world math wall.” Even acting out restaurant scenarios can provide students with an opportunity to reinforce critical mathematical skills like addition and subtraction, while bolstering an understanding of decimals and percentages. At Suzhou Singapore International School in China, third- to fifth- graders role play with menus, ordering fictional meals and learning how to split the check when the bill arrives. 

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Real-Life Math Problems with Solutions

How can you save money? How should a building be designed? How do you train for a competition? In the real world, math problems involve many disciplines, are complicated, and don’t have an answer key. In a math classroom, the problems usually need to be smaller scale and more focused. These problems still mimic and provide insight into the real world and can serve as a “diving board” into the sorts of math problems students encounter outside of math class.

Below are three open-ended math problems for Grades K, 2, and 4, which encourage students to explore math in real-life contexts. They get students thinking about how math can be used to solve problems in gardening, biology, and music.

Real-World Math Problems with Answers

These problems, which can all be found in HMH Into Math , do have clear solutions. Even though in all three cases many answers are possible, you can assess students’ understanding based on how they respond.

Kindergarten: 5-and-More Garden

• Key Standard: Represent addition within 10 with drawings and objects.
• Key Standard: Model with mathematics.

“ 5-and-More Garden ” asks students to create a visual representation of a garden bed by spinning a wheel and then “planting” different quantities of lettuce and carrots.

In this activity, students will make groups to show 5 and the number on the spinner. Answers can be represented in a variety of ways: drawn on the provided garden teacher resource, created using paper and counters, or possibly built out of clay and other materials. If students use the spinner provided, there should be between 5–10 carrots and 5–10 lettuce plants. If there are only 5 of both, make sure the student did in fact spin 0 on the spinner both times! The image on the first page of the activity shows a possible model if students build a garden using clay.

Sample answers for the Challenge questions:

• Carrots: 5 + 1 = 6
• Lettuce: 5 + 3 = 8
• If I pick 3 carrots, then 3 carrots will be left.

Grade 2: By the Sea

• Key Standard: Understand that the three digits of a three-digit number represent the amounts of hundreds, tens, and ones.
• Key Standard: Reason quantitatively.

“ By the Sea ” challenges students to imagine various quantities of plants and animals that they might observe by the sea and demonstrates the efficiency of using place value to denote three-digit numbers. They make a math storybook about the wildlife, choosing a number for each organism, writing the number two ways, and drawing to show the number using hundreds, tens, and ones.

In this activity, students should choose a number between 100 and 999 for snails, pieces of seaweed, starfish, and clams. They should record each value accurately, matching the models they make with manipulatives. Drawings should match the values chosen. For example, if they chose 583 starfish, then for starfish, they should note there are 5 hundreds, 8 tens, and 3 ones and draw 5 large starfish, 8 medium starfish, and 3 small starfish.

• Each page is made up of two types of plants and animals. On page 2, there are 583 starfish and 215 clams, so there are 798 animals total.
• There are four types of plants or animals in the whole journal. On page 1, there are 296 snails and 317 pieces of seaweed. On page 2, there are 583 starfish and 215 clams. So, there are 1,411 plants and animals total. That is 14 hundreds, 1 ten, and 1 one.

Grade 4: Concert Calculations

• Key Standard: Fluently add and subtract numbers through 1,000,000.
• Key Standard: Make sense of problems and persevere in solving them.

“ Concert Calculations ” gives students a budget of $300,000 to spend on a tour for their band; students must use their critical thinking and decision-making skills to weigh cities with the appropriate stadium capacity against tour costs. Their goal is to reach a million attendees.

In this activity, students select different cities around the U.S. They will need to keep track of costs (stadium + hotel + food for each city) and capacities (add the capacities for each city). If they find a successful answer for the activity, check the totals. The total cost should be under $300,000, and the total capacities should be over 1 million.

Sample answers for the Reflection Questions :

• The numbers being added included 2-, 3-, and 5-digit numbers. I had to ensure I was adding ones, tens, hundreds, thousands, and ten thousands, paying attention to which digit was contained in which place value.
• I decided to visit Atlanta because the stadiums had a high capacity (138,258) but low cost ($15,678). Evidence that this was a good decision is that I succeeded in having a total capacity of over 1,000,000 but did not go over $300,000.
• Authors do not need stadiums like popular musicians do. I would plan a book tour similar to the concert tour, but I would look at the capacity for different bookstores in each city and go to as many as possible to make the total amount of people who can meet the author as high as possible.

HMH Into Math is a core mathematics curriculum for Grades K­­–8 that inspires students to see the value and purpose of math in their daily lives through rewarding, real-life activities and lessons.

To learn more about the indicators of a powerful math task and strategies to promote math talk in the classroom, watch our webinar The Power of a Great Math Task.

Get our FREE guide "Optimizing the Math Classroom: 6 Best Practices."

• Activities & Lessons

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When am I ever going to use math?

Each activity-lesson in the book contains several questions about the situation, starting with basics and going into more in-depth evaluations, and should be adequate for one-two complete class periods.

• The problems are written by an experienced math teacher (Frank Wilson)
• The problems are matched to the learning objectives of the National Council of Teachers of Mathematics (NCTM) and in the Common Core Standards. This means that the concepts and skills required to complete the problems ARE found in typical middle and high school mathematics curriculum. You can simply replace some of the problems in your textbook with these real-life scenarios.
• These activities are excellent to be used in a small-group setting.
• Typically, the activities contain challenging parts and therefore allow students to practice real problem solving - not just apply knowledge from textbook examples to other almost identical problems.
• Gifted students can enjoy the challenge of solving all the questions on their own.

The video below explains the basics of the Make It Real Learning activity workbooks:

List of available workbooks:

Arithmetic I - for grades 3-6

Arithmetic II - for grades 4-7

Fractions, Percents, and Decimals I - for grades 4-8. $4.99

Fractions, Percents, and Decimals II - for grades 6-11. $4.99

Fractions, Percents, and Decimals III - for grades 4-7. $4.99

Fractions, Percents, and Decimals IV - for grades 4-7. $4.99

Geometry I - for grades 6-7. $4.99

Geometry II - for grades 4,7, and high school. $4.99

Sets, Probability, and Statistics I - for grades 6-10. $4.99

Sets, Probability, and Statistics II - for grades 6-12. $4.99

Linear Functions I - for algebra 1 and algebra 2. $4.99

Linear Functions II - for algebra 1 and algebra 2. $4.99

Linear Functions III - for algebra 1 and algebra 2. $4.99

Graphing and Other Algebra Skills I - for grades 7, 8, and high school. $4.99

Graphing and Other Algebra Skills II - for grades 6-8, and high school. $4.99

Quadratic Functions I - for algebra 1 and algebra 2/precalculus. $4.99

Quadratic Functions II - for algebra 1 and algebra 2/precalculus. $4.99

Exponential and Logarithmic Functions I - for algebra 2/precalculus. $4.99

Periodic and Piecewise Functions I - for algebra 2/precalculus. $4.99

Periodic and Piecewise Functions II - for algebra 2/precalculus. $4.99

Polynomial, Power, Logistic, and Rational Functions I - for algebra 2/precalculus. $4.99

Calculus I - for grade 12. $4.99

Make It Real Learning Activity Libraries

The activity libraries allow you to get 11 workbooks from the series for a discounted price of $39.99 !

Arithmetic I Fractions, Percents, Decimals I Fractions, Percents, Decimals II Sets, Probability, and Statistics I Linear Functions I Linear Functions II Quadratic Functions I Exponential and Logarithmic Functions I Periodic and Piecewise Functions I Polynomial, Power, Logistic, and Rational Functions I Calculus I

Arithmetic II Fractions, Percents, Decimals III Fractions, Percents, Decimals IV Geometry I Geometry II Sets, Probability, and Statistics II Linear Functions III Graphing and Other Algebra Skills I Graphing and Other Algebra Skills II Quadratic Functions II Periodic and Piecewise Functions II

States by the Numbers series

Free Sample (PDF) : North Dakota

Besides the workbooks mentioned above, Make It Real Learning also has a series of smaller workbooks called States by the Numbers . This series contains 50 workbooks — one for each state in the United States. The data in each workbook is taken directly from the Census Bureau's 2008 Statistical Abstract of the United States.

Additionally, the “What's the big idea?” pages give learners the opportunity to reflect on the things they've learned.

There are multiple ways to use the activities in a teaching environment. Since the activities teach both mathematics and social studies, many teachers and families enjoy using the workbooks to reinforce mathematics across the curriculum. Although the activities may be effectively used in a formal teaching setting, they are designed specifically for the independent learner.

Read more about States by the Numbers series workbooks here .

Make It Real Learning Superbundle

• Activity Library I (11 workbooks)
• Activity Library II (11 workbooks)
• States by the Numbers bundle (50 workbooks)

You can get all of the above for $79.99:

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Fun teaching resources & tips to help you teach math with confidence

Show Kids Math in Real Life

One of the best ways to engage kids and get them interested in math is to show them how we see and use math in our everyday life . Seeing math in real life helps kids to recognize it’s relevance to them, shows them why it matters and can deepen their understanding by providing a meaningful context .

Why is it Helpful to Teach Math in Real Life?

There are so many reasons to help kids see math in real life and how it is relevant to them.

First of all, a lot of kids assume that math is NOT relevant to their life. By showing them ways to use and apply math that are actually relevant and meaningful to them we can debunk this myth .

But more than that, applying math skills to real life will help them to better understand and visualize the math they’re learning:

• To understand math deeply, our brain needs to form connections . Seeing math in real life helps students form connections between abstract math concepts and real, everyday life, thus strengthening their understanding.
• When math is relevant for students, they are more interested, engaging and willing to participate . Have kids who refuse to do their work? Try making it relate to their hobbies, favorite things or everyday life!
• Exploring math in a real life context will help kids to develop their problem solving skills . This is the reason we teach math–to train thinkers and problem solvers who can take what they know and apply it to an actual situation.

Where Do We See Math in Nature?

Although I think it is important for kids to see how math is used and relevant to their everyday lives, I also love showing them how we see math in the world around us.

This helps kids to see math in a creative, beautiful and open way. It also allows kids the opportunity to explore patterns such as Fibonacci sequence or fractals, which are not always covered in a typical math classroom.

Want to learn more about how and where we see math in the world around us?

Read: 5 Stunning Ways We See Math in the World

Want to grab a FREE set of posters so your kids can see and think about math in nature? Simply enter your name and email below to get your poster set!

Find Additional Math in Real Life Lessons & Problems Below:

Looking for more ideas for teaching and exploring math in real life ? I’ve got you covered! From the kitchen to lunch menus to budgets, there’s a variety of resources for kids of all ages.

This is an ever growing resource list, which includes a variety of articles and printable resources to help kids see, think about and explore math in real life.

Find More Resources To Help Make Math Engaging!

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Math Time Doesn't Have to End in Tears

Join 165,000+ parents and teachers who learn new tips and strategies, as well as receive engaging resources to make math fun. Plus, receive my guide, "5 Games You Can Play Today to Make Math Fun," as my free gift to get you started!

Using Math in the Real World

There’s a popular math meme that says, “Math: The only place where people buy 60 watermelons and no one wonders why.”

After a chuckle, you might realize there’s some truth to this joke. Sometimes, it may seem like math has nothing to do with the real world. In fact, it’s likely your students have asked you, “Why do I need to know this? When will I ever use this in real life?”

Of course, as a teacher, you know that math matters. Making math relevant to your students can be challenging, however.

You can talk about the importance of learning addition, subtraction, statistics, geometry, algebra, and more , but your students might not see the real-world applications. The truth is, we all need math. Math teaches us how to solve problems , a skill that’s useful in all career fields and just for navigating everyday life.

So, how can you make math relevant to your students? Try these examples of math in the real world.

Cooking and baking are great ways to show your students how math applies to life outside of the classroom . Lead your students in reading recipes, discuss fractions, and talk about how to double a recipe or cut one in half. Then, reward their hard work with a hands-on lesson. Try this classic recipe that will let them get sticky while they learn!

• Checks and Balances

Create a checkbook system in your classroom, where students can learn how to balance a bank account—a real-world skill we all need! Make your own “classroom dollars” and give them opportunities to spend and save throughout the year, from renting their desks monthly to earning cash for good behavior. They’ll practice basic math in the process, and, if you end the unit with the opportunity to purchase items with their hard-earned money, you’ll teach them the value of saving and budgeting.

• Buying Power

After your class has earned classroom dollars throughout the year, hold a yard sale or class auction. Have students donate items and let them spend their money however they’d like. Add to the challenge by announcing discounts—“everything’s 30 percent off for the next 10 minutes”—and let them do the real-world math to see what the new price would be.

• Measure for Measure

Many jobs, from construction workers to architects, require accurate measuring skills. Measuring and math go hand-in-hand, and you can up the fun factor in math by setting your students free throughout the classroom—or even the school. How tall is the principal? How long is the hallway? What’s the square footage of the gym? The possibilities are endless.

• Map a Course

With GPS as a regular part of our daily lives, many students have lost the art of reading a map . Map reading uses math skills, too, requiring an understanding of scale, coordinates, distances, fractions, and more. Chances are, your students have never really had to use a map to get anywhere. Go on a virtual field trip by following a map to your favorite destination, and practice math in the process.

Shop ’til You Drop

Make problem solving meaningful for students. Lessons feature kid-friendly topics such as movie marathons and summer fun to reinforce problem solving skills and understanding of math in daily life.

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[FREE] Fun Math Games & Activities Packs

Always on the lookout for fun math games and activities in the classroom? Try our ready-to-go printable packs for students to complete independently or with a partner!

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Ratio problem solving

Here you will learn about ratio problem solving, including how to set up and solve problems. You will also look at real life ratio word problems.

Students will first learn about ratio problem solving as part of ratio and proportion in 6 th grade and 7 th grade.

What is ratio problem solving?

Ratio problem solving is a collection of ratio and proportion word problems that link together aspects of ratio and proportion into more real life questions. This requires you to be able to take key information from a question and use your knowledge of ratios (and other areas of the curriculum) to solve the problem.

A ratio is a relationship between two or more quantities. They are usually written in the form a : b where a and b are two quantities. When problem solving with a ratio, the key facts that you need to know are:

• What is the ratio involved?
• What order are the quantities in the ratio?
• What is the total amount / what is the part of the total amount known?
• What are you trying to calculate ?

As with all problem solving, there is not one unique method to solve a problem. However, this does not mean that there aren’t similarities between different problems that you can use to help you find an answer.

The key to any problem solving is being able to draw from prior knowledge and use the correct piece of information to allow you to get to the next step and then the solution.

Let’s look at a couple of methods you can use when given certain pieces of information.

When solving ratio word problems, it is very important that you are able to use ratios. This includes being able to use ratio notation.

For example, Charlie and David share some sweets in the ratio of 3 : 5. This means that for every 3 sweets Charlie gets, David receives 5 sweets.

Charlie and David share 40 sweets, how many sweets do they each get?

You use the ratio to divide 40 sweets into 8 equal parts.

40 \div 8=5

Then you multiply each part of the ratio by 5.

3\times 5:5\times 5=15 : 25

This means that Charlie will get 15 sweets and David will get 25 sweets.

There can be ratio word problems involving different operations and types of numbers.

Here are some examples of different types of ratio word problems:

Common Core State Standards

How does this relate to 6 th grade math?

• Grade 6 – Ratios and Proportional Relationships (6.RP.A.3) Use ratio and rate reasoning to solve real-world and mathematical problems, for example, by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
• Grade 7 – Ratio and Proportional Relationships (7.RP.A.2) Recognize and represent proportional relationships between quantities.

How to do ratio problem solving

In order to solve problems including ratios:

Identify key information within the question.

Know what you are trying to calculate.

Use prior knowledge to structure a solution.

[FREE] Ratio Check for Understanding Quiz (Grade 6 and 7)

Use this quiz to check your 6th and 7th grade students’ understanding of ratios. 10+ questions with answers covering a range of 6th and 7th grade ratio topics to identify areas of strength and support!

Ratio problem solving examples

Example 1: part:part ratio.

Within a school, the total number of students who have school lunches to packed lunches is 5 : 7. If 465 students have a school lunch, how many students have a packed lunch?

Within a school, the number of students who have school lunches to packed lunches is \textbf{5 : 7} . If \textbf{465} students have a school lunch, how many students have a packed lunch?

Here you can see that the ratio is 5 : 7, where the first part of the ratio represents school lunches (S) and the second part of the ratio represents packed lunches (P).

You could write this as:

Where the letter above each part of the ratio links to the question.

You know that 465 students have school lunch.

2 Know what you are trying to calculate.

From the question, you need to calculate the number of students that have a packed lunch, so you can now write a ratio below the ratio 5 : 7 that shows that you have 465 students who have school lunches, and p students who have a packed lunch.

You need to find the value of p.

3 Use prior knowledge to structure a solution.

You are looking for an equivalent ratio to 5 : 7. So you need to calculate the multiplier.

You do this by dividing the known values on the same side of the ratio by each other.

465\div 5 = 93

This means to create an equivalent ratio, you can multiply both sides by 93.

So the value of p is equal to 7 \times 93=651.

There are 651 students that have a packed lunch.

Example 2: unit conversions

The table below shows the currency conversions on one day.

Use the table above to convert £520 \; (GBP) to Euros € \; (EUR).

Use the table above to convert \bf{￡520} \textbf{ (GBP)} to Euros \textbf{€ } \textbf{(EUR)}.

The two values in the table that are important are \text{GBP} and EUR. Writing this as a ratio, you can state,

You know that you have £520.

You need to convert GBP to EUR and so you are looking for an equivalent ratio with GBP=£520 and EUR=E.

To get from 1 to 520, you multiply by 520 and so to calculate the number of Euros for £520, you need to multiply 1.17 by 520.

1.17 \times 520=608.4

So £520=€608.40.

Example 3: writing a ratio 1 : n

Liquid plant food is sold in concentrated bottles. The instructions on the bottle state that the 500 \, ml of concentrated plant food must be diluted into 2 \, l of water. Express the ratio of plant food to water, respectively, in the ratio 1 : n.

Liquid plant food is sold in concentrated bottles. The instructions on the bottle state that the \bf{500 \, ml} of concentrated plant food must be diluted into \bf{2 \, l} of water. Express the ratio of plant food to water respectively as a ratio in the form 1 : n.

Using the information in the question, you can now state the ratio of plant food to water as 500 \, ml : 2 \, l. As you can convert liters into milliliters, you could convert 2 \, l into milliliters by multiplying it by 1000.

2 \, l=2000 \, ml

So you can also express the ratio as 500 : 2000 which will help you in later steps.

You want to simplify the ratio 500 : 2000 into the form 1:n.

You need to find an equivalent ratio where the first part of the ratio is equal to 1. You can only do this by dividing both parts of the ratio by 500 (as 500 \div 500=1 ).

So the ratio of plant food to water in the form 1 : n is 1 : 4.

Example 4: forming and solving an equation

Three siblings, Josh, Kieran and Luke, receive an allowance each week proportional to their age. Kieran is 3 years older than Josh. Luke is twice Josh’s age. If Josh receives \$ 8 allowance, how much money do the three siblings receive in total?

Three siblings, Josh, Kieran and Luke, receive an allowance each week proportional to their ages. Kieran is \bf{3} years older than Josh. Luke is twice Josh’s age. If Luke receives \bf{\$ 8} allowance, how much money do the three siblings receive in total?

You can represent the ages of the three siblings as a ratio. Taking Josh as x years old, Kieran would therefore be x+3 years old, and Luke would be 2x years old. As a ratio, you have:

You also know that Luke receives \$ 8.

You want to calculate the total amount of allowance for the three siblings.

You need to find the value of x first. As Luke receives \$ 8, you can state the equation 2x=8 and so x=4.

Now you know the value of x, you can substitute this value into the other parts of the ratio to obtain how much money the siblings each receive.

The total amount of allowance is therefore 4+7+8=\$ 19.

Example 5: simplifying ratios

Below is a bar chart showing the results for the colors of counters in a bag.

Express this data as a ratio in its simplest form.

From the bar chart, you can read the frequencies to create the ratio.

You need to simplify this ratio.

To simplify a ratio, you need to find the highest common factor of all the parts of the ratio. By listing the factors of each number, you can quickly see that the highest common factor is 2.

\begin{aligned} & 12 = 1, {\color{red}2}, 3, 4, 6, 12 \\\\ & 16 = 1, {\color{red}2}, 4, 8, 16 \\\\ & 10 = 1, {\color{red}2}, 5, 10 \end{aligned}

HCF(12,16,10) = 2

Dividing all the parts of the ratio by 2, you get

Our solution is 6 : 8 : 5.

Example 6: combining two ratios

Glass is made from silica, lime and soda. The ratio of silica to lime is 15 : 2. The ratio of silica to soda is 5 : 1. State the ratio of silica:lime:soda.

Glass is made from silica, lime and soda. The ratio of silica to lime is \bf{15 : 2}. The ratio of silica to soda is \bf{5 : 1}. State the ratio of silica:lime:soda.

You know the two ratios

You are trying to find the ratio of all 3 components: silica, lime and soda.

Using equivalent ratios you can say that the ratio of Silica:Soda is equivalent to 15 : 3 by multiplying the ratio by 3.

You now have the same amount of silica in both ratios and so you can now combine them to get the ratio 15 : 2 : 3.

Example 7: using bar modeling

India and Beau share some popcorn in the ratio of 5 : 2. If India has 75 \, g more popcorn than Beau, what was the original quantity?

India and Beau share some popcorn in the ratio of \bf{5 : 2} . If India has \bf{75 \, g} more popcorn than Beau, what was the original quantity?

You know that the initial ratio is 5 : 2 and that India has three more parts than Beau.

You want to find the original quantity.

Drawing a bar model of this problem, you have:

Where India has 5 equal shares, and Beau has 2 equal shares.

Each share is the same value and so if you can find out this value, you can then find the total quantity.

From the question, India’s share is 75 \, g more than Beau’s share so you can write this on the bar model.

You can find the value of one share by working out 75 \div 3=25 \, g.

You can fill in each share to be 25 \, g.

Adding up each share, you get

India=5 \times 25=125 \, g

Beau=2 \times 25=50 \, g

The total amount of popcorn was 125+50=175 \, g.

Teaching tips for ratio problem solving

• Continue to remind students that when solving ratio word problems, it’s important to identify the quantities being compared and express the ratio in its simplest form.
• Create practice problems for students using the information in your classroom. For example, ask students to find the ratio of boys to the ratio of girls using the total number of students in your classroom, then the school.
• To find more practice questions, utilize educational websites and apps instead of worksheets. Some of these may also provide tutorials for struggling students. These can also be helpful for test prep as they are more engaging for students.
• Use a variety of numbers in your ratio word problems – whole numbers, fractions, decimals, and mixed numbers – to give students a variety of practice.
• Provide students with a step-by-step process for problem solving, like the one shown above, that can be applied to every ratio word problem.

Easy mistakes to make

• Mixing units Make sure that all the units in the ratio are the same. For example, in example 6, all the units in the ratio were in milliliters. You did not mix ml and l in the ratio.
• Writing ratios in the wrong order For example, the number of dogs to cats is given as the ratio 12 : 13 but the solution is written as 13 : 12.

• Counting the number of parts in the ratio, not the total number of shares For example, the ratio 5 : 4 has 9 shares, and 2 parts. This is because the ratio contains 2 numbers but the sum of these parts (the number of shares) is 5+4=9. You need to find the value per share, so you need to use the 9 shares in your next line of working.
• Ratios of the form \bf{1 : \textbf{n}} The assumption can be incorrectly made that n must be greater than 1, but n can be any number, including a decimal.

Related ratio lessons

• Unit rate math
• Simplifying ratios
• Ratio to fraction
• How to calculate exchange rates
• Ratio to percent
• How to write a ratio

Practice ratio problem solving questions

1. An online shop sells board games and computer games. The ratio of board games to the total number of games sold in one month is 3 : 8. What is the ratio of board games to computer games?

8-3=5 computer games sold for every 3 board games.

2. The ratio of prime numbers to non-prime numbers from 1-200 is 45 : 155. Express this as a ratio in the form 1 : n.

You need to simplify the ratio so that the first number is 1. That means you need to divide each number in the ratio by 45.

45 \div 45=1

155\div{45}=3\cfrac{4}{9}

3. During one month, the weather was recorded into 3 categories: sunshine, cloud and rain. The ratio of sunshine to cloud was 2 : 3 and the ratio of cloud to rain was 9 : 11. State the ratio that compares sunshine:cloud:rain for the month.

3 \times S : C=6 : 9

4. The angles in a triangle are written as the ratio x : 2x : 3x. Calculate the size of each angle.

You should know that the 3 angles in a triangle always equal 180^{\circ}.

\begin{aligned} & x+2 x+3 x=180 \\\\ & 6 x=180 \\\\ & x=30^{\circ} \\\\ & 2 x=60^{\circ} \\\\ & 3 x=90^{\circ} \end{aligned}

5. A clothing company has a sale on tops, dresses and shoes. \cfrac{1}{3} of sales were for tops, \cfrac{1}{5} of sales were for dresses, and the rest were for shoes. Write a ratio of tops to dresses to shoes sold in its simplest form.

\cfrac{1}{3}+\cfrac{1}{5}=\cfrac{5+3}{15}=\cfrac{8}{15}

1-\cfrac{8}{15}=\cfrac{7}{15}

6. The volume of gas is directly proportional to the temperature (in degrees Kelvin). A balloon contains 2.75 \, l of gas and has a temperature of 18^{\circ}K. What is the volume of gas if the temperature increases to 45^{\circ}K?

The given ratio in the word problem is 2. 75 \mathrm{~L}: 18^{\circ} \mathrm{K}

Divide 45 by 18 to see the relationship between the two temperatures.

45 \div 18=2.5

45 is 2.5 times greater than 18. So we multiply 2.75 by 2.5 to get the amount of gas.

2.75 \times 2.5=6.875 \mathrm{~l}

Ratio problem solving FAQs

A ratio is a comparison of two or more quantities. It shows how much one quantity is related to another.

A recipe calls for 2 cups of flour and 1 cup of sugar. What is the ratio of flour to sugar? (2 : 1)

In middle school ( 7 th grade and 8 th grade), students transition from understanding basic ratios to working with more complex and real-life applications of ratios and proportions. They gain a deeper understanding of how ratios relate to different mathematical concepts, making them more prepared for higher-level math topics in high school.

The next lessons are

• Properties of equality
• Multiplication and division

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1.5: Problem Solving and Estimating

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• David Lippman
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Finally, we will bring together the mathematical tools we’ve reviewed, and use them to approach more complex problems. In many problems, it is tempting to take the given information, plug it into whatever formulas you have handy, and hope that the result is what you were supposed to find. Chances are, this approach has served you well in other math classes.

This approach does not work well with real life problems. Instead, problem solving is best approached by first starting at the end: identifying exactly what you are looking for. From there, you then work backwards, asking “what information and procedures will I need to find this?” Very few interesting questions can be answered in one mathematical step; often times you will need to chain together a solution pathway, a series of steps that will allow you to answer the question.

Problem Solving Process

• Identify the question you’re trying to answer.
• Work backwards, identifying the information you will need and the relationships you will use to answer that question.
• Continue working backwards, creating a solution pathway.
• If you are missing necessary information, look it up or estimate it. If you have unnecessary information, ignore it.
• Solve the problem, following your solution pathway.

In most problems we work, we will be approximating a solution, because we will not have perfect information. We will begin with a few examples where we will be able to approximate the solution using basic knowledge from our lives.

How many times does your heart beat in a year?

This question is asking for the rate of heart beats per year. Since a year is a long time to measure heart beats for, if we knew the rate of heart beats per minute, we could scale that quantity up to a year. So the information we need to answer this question is heart beats per minute. This is something you can easily measure by counting your pulse while watching a clock for a minute.

Suppose you count 80 beats in a minute. To convert this beats per year:

\(\frac{80 \text { beats }}{1 \text { minute }} \cdot \frac{60 \text { minutes }}{1 \text { hour }} \cdot \frac{24 \text { hours }}{1 \text { day }} \cdot \frac{365 \text { days }}{1 \text { year }}=42,048,000\) beats per year

How thick is a single sheet of paper? How much does it weigh?

While you might have a sheet of paper handy, trying to measure it would be tricky. Instead we might imagine a stack of paper, and then scale the thickness and weight to a single sheet. If you’ve ever bought paper for a printer or copier, you probably bought a ream, which contains 500 sheets. We could estimate that a ream of paper is about 2 inches thick and weighs about 5 pounds. Scaling these down,

\(\frac{2 \text { inches }}{\text { ream }} \cdot \frac{1 \text { ream }}{500 \text { pages }}=0.004\) inches per sheet

\(\frac{5 \text { pounds }}{\text { ream }} \cdot \frac{1 \text { ream }}{500 \text { pages }}=0.01\) pounds per sheet, or 0.16 ounces per sheet.

A recipe for zucchini muffins states that it yields 12 muffins, with 250 calories per muffin. You instead decide to make mini-muffins, and the recipe yields 20 muffins. If you eat 4, how many calories will you consume?

There are several possible solution pathways to answer this question. We will explore one.

To answer the question of how many calories 4 mini-muffins will contain, we would want to know the number of calories in each mini-muffin. To find the calories in each mini-muffin, we could first find the total calories for the entire recipe, then divide it by the number of mini-muffins produced. To find the total calories for the recipe, we could multiply the calories per standard muffin by the number per muffin. Notice that this produces a multi-step solution pathway. It is often easier to solve a problem in small steps, rather than trying to find a way to jump directly from the given information to the solution.

We can now execute our plan:

\(12 \text{muffins} $\cdot \frac{250 \text { calories }}{\text { muffin }}=3000$\) calories for the whole recipe

\(\frac{3000 \text { calories }}{20 \text { mini }-\text { muffins }}\) gives 150 calories per mini-muffin

\(4\text{ mini muffins } \cdot \frac{150 \text { calories }}{\text { mini - muffin }}\) totals 600 calories consumed.

You need to replace the boards on your deck. About how much will the materials cost?

There are two approaches we could take to this problem: 1) estimate the number of boards we will need and find the cost per board, or 2) estimate the area of the deck and find the approximate cost per square foot for deck boards. We will take the latter approach.

For this solution pathway, we will be able to answer the question if we know the cost per square foot for decking boards and the square footage of the deck. To find the cost per square foot for decking boards, we could compute the area of a single board, and divide it into the cost for that board. We can compute the square footage of the deck using geometric formulas. So first we need information: the dimensions of the deck, and the cost and dimensions of a single deck board.

Suppose that measuring the deck, it is rectangular, measuring 16 ft by 24 ft, for a total area of \(384 \mathrm{ft}^{2}\).

From a visit to the local home store, you find that an 8 foot by 4 inch cedar deck board costs about $7.50. The area of this board, doing the necessary conversion from inches to feet, is:

\(8 \text { feet } \cdot 4 \text { inches } \cdot \frac{1 \text { foot }}{12 \text { inches }}=2.667 \mathrm{ft}^{2}\). The cost per square foot is then

\(\frac{\$ 7.50}{2.667 \mathrm{ft}^{2}}=\$ 2.8125 \text { per } \mathrm{ft}^{2}\).

This will allow us to estimate the material cost for the whole \(384 \mathrm{ft}^{2}\) deck

\(\$ 384 \mathrm{ft}^{2} \cdot \frac{\$ 2.8125}{\mathrm{ft}^{2}}=\$ 1080\) total cost.

Of course, this cost estimate assumes that there is no waste, which is rarely the case. It is common to add at least 10% to the cost estimate to account for waste.

Is it worth buying a Hyundai Sonata hybrid instead the regular Hyundai Sonata?

To make this decision, we must first decide what our basis for comparison will be. For the purposes of this example, we’ll focus on fuel and purchase costs, but environmental impacts and maintenance costs are other factors a buyer might consider.

It might be interesting to compare the cost of gas to run both cars for a year. To determine this, we will need to know the miles per gallon both cars get, as well as the number of miles we expect to drive in a year. From that information, we can find the number of gallons required from a year. Using the price of gas per gallon, we can find the running cost.

From Hyundai’s website, the 2013 Sonata will get 24 miles per gallon (mpg) in the city, and 35 mpg on the highway. The hybrid will get 35 mpg in the city, and 40 mpg on the highway.

An average driver drives about 12,000 miles a year. Suppose that you expect to drive about 75% of that in the city, so 9,000 city miles a year, and 3,000 highway miles a year.

We can then find the number of gallons each car would require for the year.

\(9000\text{ city miles } \cdot \frac{1 \text { gallon }}{24 \text { city miles }}+3000\text{ hightway miles}. \frac{1 \text { gallon }}{35 \text { highway miles }}=460.7\text{ gallons}\)

\(9000\text{ city miles }\cdot \frac{1 \text { gallon }}{35 \text { city miles }}+3000\text{ hightway miles}. \frac{1 \text { gallon }}{40 \text { highway miles }}=332.1\text{ gallons}\)

If gas in your area averages about $3.50 per gallon, we can use that to find the running cost:

Sonata: \(460.7 \text { gallons } \cdot \frac{\$ 3.50}{\text { gallon }}=\$ 1612.45\)

Hybrid: \(\text { 332.1 gallons } \cdot \frac{\$ 3.50}{\text { gallon }}=\$ 1162.35\)

The hybrid will save $450.10 a year. The gas costs for the hybrid are about \(\frac{\$ 450.10}{\$ 1612.45} = 0.279 = 27.9\%\) lower than the costs for the standard Sonata.

While both the absolute and relative comparisons are useful here, they still make it hard to answer the original question, since “is it worth it” implies there is some tradeoff for the gas savings. Indeed, the hybrid Sonata costs about $25,850, compared to the base model for the regular Sonata, at $20,895.

To better answer the “is it worth it” question, we might explore how long it will take the gas savings to make up for the additional initial cost. The hybrid costs $4965 more. With gas savings of $451.10 a year, it will take about 11 years for the gas savings to make up for the higher initial costs.

We can conclude that if you expect to own the car 11 years, the hybrid is indeed worth it. If you plan to own the car for less than 11 years, it may still be worth it, since the resale value of the hybrid may be higher, or for other non-monetary reasons. This is a case where math can help guide your decision, but it can’t make it for you.

Try it Now 6

If traveling from Seattle, WA to Spokane WA for a three-day conference, does it make more sense to drive or fly?

There is not enough information provided to answer the question, so we will have to make some assumptions, and look up some values.

Assumptions:

a) We own a car. Suppose it gets 24 miles to the gallon. We will only consider gas cost.

b) We will not need to rent a car in Spokane, but will need to get a taxi from the airport to the conference hotel downtown and back.

c) We can get someone to drop us off at the airport, so we don’t need to consider airport parking.

d) We will not consider whether we will lose money by having to take time off work to drive.

Values looked up (your values may be different)

a) Flight cost: \(\$184\)

b) Taxi cost: \(\$25\) each way (estimate, according to hotel website)

c) Driving distance: \(280\) miles each way

d) Gas cost: \(\$3.79\) a gallon

Cost for flying: \(\$184\text{ flight cost }+ \$50\text{ in taxi fares }= \$234\).

Cost for driving: \(560\) miles round trip will require 23.3 gallons of gas, costing \(\$88.31\).

Based on these assumptions, driving is cheaper. However, our assumption that we only include gas cost may not be a good one. Tax law allows you deduct \(\$0.55\) (in 2012) for each mile driven, a value that accounts for gas as well as a portion of the car cost, insurance, maintenance, etc. Based on this number, the cost of driving would be \(\$319\).

22 Examples of Mathematics in Everyday Life

Let’s read further to know the real-life situations where maths is applied.

1. Making Routine Budgets

How much should I spend today? When I will be able to buy a new car? Should I save more? How will I be able to pay my EMIs? Such thoughts usually come into our minds. The simple answer to such type of question is maths. We prepare budgets based on simple calculations with the help of simple mathematical concepts. So, we can’t say, I am not going to study maths ever! Everything which is going around us is somehow related to maths only.

Application: 

• Basic mathematical operations (addition, subtraction, multiplication, and division)
• Calculation of percentage
• Arithmetic calculations

2. Construction Purpose

You know what, maths is the basis of any construction work. A lot of calculations, preparations of budgets, setting targets, estimating the cost, etc., are all done based on maths. If you don’t believe it, ask any contractor or construction worker, and they will explain as to how important maths is for carrying out all the construction work.

Application:

• Preparing budgets
• Taking measurements
• Estimating the cost and profit
• Calculus and Statistics
• Trigonometry

3. Exercising and Training

I should reduce some body fat! Will I be able to achieve my dream body ever? How? When? Will I be able to gain muscles? Here, the simple concept that is followed is maths. Yes! based on simple mathematical concepts, we can answer to above-mentioned questions. We set our routine according to our workout schedule, count the number of repetitions while exercising, etc., just based on maths.

• Basic Mathematical Operations (additions, subtraction, multiplication, and division)
• Logical and Analogical Reasoning

4. Interior Designing

Interior designing seems to be a fun and interesting career but, do you know the exact reality? A lot of mathematical concepts, calculations, budgets, estimations, targets, etc., are to be followed to excel in this field. Interior designers plan the interiors based on area and volume calculations to calculate and estimate the proper layout of any room or building. Such concepts form an important part of maths.

• Ratios and Percentages
• Mathematical Operations

5. Fashion Designing

Just like interior design, maths is also an essential concept of fashion design. From taking measurements, estimating the quantity and quality of clothes, choosing the color theme, and estimating the cost and profit, to producing cloth according to the needs and tastes of the customers, maths is followed at every stage.

• Basic Mathematical Operations
• Rations and Percentages

6. Shopping at Grocery Stores and Supermarkets

The most obvious place where you would see the application of basic mathematical concepts is your neighborhood grocery store and supermarket. The schemes like ‘Flat 50% off, ‘Buy one get one free, etc., are seen in most of the stores. Customers visit the stores, see such schemes, estimate the quantity to be bought, the weight, the price per unit, discount calculations, and finally the total price of the product, and buy it. The calculations are done based on basic mathematical concepts. Thus, here also, maths forms an important part of our daily routine.

• Ratio and Percentage

7. Cooking and Baking

In your kitchen also, the maths is performed. For cooking or baking anything, a series of steps are followed, telling us how much of the quantity is to be used for cooking, the proportion of different ingredients, methods of cooking, the cookware to be used, and many more. Such are based on different mathematical concepts. Indulging children in the kitchen while cooking anything, is a fun way to explain maths as well as basic cooking methods.

• Mathematical Algorithm
• Ratios and Proportions

Maths improves the cognitive and decision-making skills of a person. Such skills are very important for a sportsperson because by this he can take the right decisions for his team. If a person lacks such abilities, he won’t be able to make correct estimations. So, maths also forms an important part of the sports field.

• Probability
• Mathematical Operations and Algorithm
• Logical Reasoning
• Game Theory

9.  Management of Time

Now managing time is one of the most difficult tasks which is faced by a lot of people. An individual wants to complete several assignments in a limited time. Not only the management, but some people also are not even able to read the timings on an analog clock. Such problems can be solved only by understanding the basic concepts of maths. Maths not only helps us to understand the management of time but also to value it.

10. Driving

‘Speed, Time, and Distance’ are all these three things that are studied in mathematical subjects, which are the basics of driving irrespective of any mode of transportation. Maths helps us to answer the following question;

• How much should be the speed to cover any particular distance?
• How much time would be taken?
• Whether to turn left or right?
• When to stop the car?
• When to increase or decrease the speed?
• Logical reasoning
• Numerical Reasoning

11. Automobiles Industry

The different car manufacturing companies produce cars based on the demands of the customers. Every company has its category of cars ranging from microcars to luxury SUVs. In such companies, basic mathematical operations are being applied to gain knowledge about the different demands of the customers.

12. Computer Applications

Ever wondered how a computer works? How easily it completes every task in a proper series of actions? The simple reason for this is the application of maths. The fields of mathematics and computing intersect both in computer science. The study of computer applications is next to impossible without maths. Concepts like computation, algorithms and many more forms the base for different computer applications like PowerPoint, word, excel, etc. are impossible to run without maths.

Applications:

• Computation
• Coding Methods
• Cryptography

13. Planning a Trip

We all are bored with our monotonous life and we wish to go on long vacations. For this, we have to plan things accordingly. We need to prepare the budget for the trip, the number of days, the destinations, and hotels adjust our other work accordingly, and many more. Here comes the role of maths. Basic mathematical concepts and operations are required to be followed to plan a successful trip.

14. Hospitals

Every Hospital has to make the schedule the timings of the doctors available, the systematic methods of conducting any major surgery, keeping the records of the patients, records of the success rate of surgeries, number of ambulances required, training for the use of medicines to nurses, prescriptions, and scheduling all tasks, etc. All these are done based on Mathematical concepts.

• Body Mass Index

15. Video Games

Playing video games is one of the most favorite entertainment activities done all over the world, irrespective of the fact that whether you are a kid or an adult. Students usually skip their maths classes to play video games. But, do you know here also they are learning maths? Here, they learn about the different steps and techniques to be followed to win any game. Not only while playing, but the engineers who introduce different games for people also follow the different mathematical concepts.

16. Weather Forecasting

The weather forecasting is all done based on the probability concept of maths. Through this, we get to know about the weather conditions like whether it’s going to be a sunny day or rainfall will come So, next time you plan your outing, don’t forget to see the weather forecasting.

Application:    

17. Base of Other Subjects

Though maths is itself a unique subject. But, you would be surprised to know that it forms the base for every subject. The subjects like physics, chemistry, economics, history, accountancy, and statistics every subject is based upon maths. So, next time you say, “I’m not going to study this maths subject ever!” remember, this subject will not going to leave you ever.

• Operations Research
• Linear Programming

18. Music and Dance

Listening to music and dancing is one of the most common hobbies of children. Here also, they learn maths while singing and learning different dance steps. Coordination in any dance can be gained by simple mathematical steps.

19. Manufacturing Industry

The part of maths called ‘Operations Research is an important concept that is being followed at every manufacturing unit. This concept of maths gives the manufacturer a simple idea of performing several tasks under the manufacturing unit like,

• What quantity is to be produced?
• What methods are to be followed?
• How to increase production?
• How the cost of production can be reduced?
• Removing unnecessary tasks.
• Following methods like target costing, ABC costing, cost-profit budgeting, and many more.
• Ratios and Probability

20. Planning of Cities

Urban planning includes the concepts of budgeting, planning, setting targets, and many more which all forms part of mathematics. No activity is possible without maths.

21. Problem-solving skills

Problem-solving skills are one of the most important skills which every individual should possess to be successful in life. Such skills help the individual in taking correct decisions in life, let it be professional or personal. This is all done when the person has the correct knowledge of basic mathematical concepts.

• Mathematical Reasoning

22. Marketing

The marketing agencies make the proper plans as to how to promote any product or service. The tasks like promoting a product online, use of social media platforms, following different methods of direct and indirect marketing, door-to-door sales, sending e-mails, making calls, and providing several schemes like ‘Buy one get one free, ‘Flat 50% off, offering discounts on special occasions, etc. are all done based on simple mathematical concepts. Thus, maths is present everywhere.

• Percentages
• Mathematical Operations  

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16 comments.

very good and rare found examples are here to be learned here.

Really informative and knowledgeable

I never realized how much math we use every day.

Example of cost

It nice and great! Idea

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March 12, 2024

The Simplest Math Problem Could Be Unsolvable

The Collatz conjecture has plagued mathematicians for decades—so much so that professors warn their students away from it

By Manon Bischoff

Mathematicians have been hoping for a flash of insight to solve the Collatz conjecture.

James Brey/Getty Images

At first glance, the problem seems ridiculously simple. And yet experts have been searching for a solution in vain for decades. According to mathematician Jeffrey Lagarias, number theorist Shizuo Kakutani told him that during the cold war, “for about a month everybody at Yale [University] worked on it, with no result. A similar phenomenon happened when I mentioned it at the University of Chicago. A joke was made that this problem was part of a conspiracy to slow down mathematical research in the U.S.”

The Collatz conjecture—the vexing puzzle Kakutani described—is one of those supposedly simple problems that people tend to get lost in. For this reason, experienced professors often warn their ambitious students not to get bogged down in it and lose sight of their actual research.

The conjecture itself can be formulated so simply that even primary school students understand it. Take a natural number. If it is odd, multiply it by 3 and add 1; if it is even, divide it by 2. Proceed in the same way with the result x : if x is odd, you calculate 3 x + 1; otherwise calculate x / 2. Repeat these instructions as many times as possible, and, according to the conjecture, you will always end up with the number 1.

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For example: If you start with 5, you have to calculate 5 x 3 + 1, which results in 16. Because 16 is an even number, you have to halve it, which gives you 8. Then 8 / 2 = 4, which, when divided by 2, is 2—and 2 / 2 = 1. The process of iterative calculation brings you to the end after five steps.

Of course, you can also continue calculating with 1, which gives you 4, then 2 and then 1 again. The calculation rule leads you into an inescapable loop. Therefore 1 is seen as the end point of the procedure.

Following iterative calculations, you can begin with any of the numbers above and will ultimately reach 1.

Credit: Keenan Pepper/Public domain via Wikimedia Commons

It’s really fun to go through the iterative calculation rule for different numbers and look at the resulting sequences. If you start with 6: 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1. Or 42: 42 → 21 → 64 → 32 → 16 → 8 → 4 → 2 → 1. No matter which number you start with, you always seem to end up with 1. There are some numbers, such as 27, where it takes quite a long time (27 → 82 → 41 → 124 → 62 → 31 → 94 → 47 → 142 → 71 → 214 → 107 → 322 → 161 → 484 → 242 → 121 → 364 → 182 → 91 → 274 → ...), but so far the result has always been 1. (Admittedly, you have to be patient with the starting number 27, which requires 111 steps.)

But strangely there is still no mathematical proof that the Collatz conjecture is true. And that absence has mystified mathematicians for years.

The origin of the Collatz conjecture is uncertain, which is why this hypothesis is known by many different names. Experts speak of the Syracuse problem, the Ulam problem, the 3 n + 1 conjecture, the Hasse algorithm or the Kakutani problem.

German mathematician Lothar Collatz became interested in iterative functions during his mathematics studies and investigated them. In the early 1930s he also published specialist articles on the subject , but the explicit calculation rule for the problem named after him was not among them. In the 1950s and 1960s the Collatz conjecture finally gained notoriety when mathematicians Helmut Hasse and Shizuo Kakutani, among others, disseminated it to various universities, including Syracuse University.

Like a siren song, this seemingly simple conjecture captivated the experts. For decades they have been looking for proof that after repeating the Collatz procedure a finite number of times, you end up with 1. The reason for this persistence is not just the simplicity of the problem: the Collatz conjecture is related to other important questions in mathematics. For example, such iterative functions appear in dynamic systems, such as models that describe the orbits of planets. The conjecture is also related to the Riemann conjecture, one of the oldest problems in number theory.

Empirical Evidence for the Collatz Conjecture

In 2019 and 2020 researchers checked all numbers below 2 68 , or about 3 x 10 20 numbers in the sequence, in a collaborative computer science project . All numbers in that set fulfill the Collatz conjecture as initial values. But that doesn’t mean that there isn’t an outlier somewhere. There could be a starting value that, after repeated Collatz procedures, yields ever larger values that eventually rise to infinity. This scenario seems unlikely, however, if the problem is examined statistically.

An odd number n is increased to 3 n + 1 after the first step of the iteration, but the result is inevitably even and is therefore halved in the following step. In half of all cases, the halving produces an odd number, which must therefore be increased to 3 n + 1 again, whereupon an even result is obtained again. If the result of the second step is even again, however, you have to divide the new number by 2 twice in every fourth case. In every eighth case, you must divide it by 2 three times, and so on.

In order to evaluate the long-term behavior of this sequence of numbers , Lagarias calculated the geometric mean from these considerations in 1985 and obtained the following result: ( 3 / 2 ) 1/2 x ( 3 ⁄ 4 ) 1/4 x ( 3 ⁄ 8 ) 1/8 · ... = 3 ⁄ 4 . This shows that the sequence elements shrink by an average factor of 3 ⁄ 4 at each step of the iterative calculation rule. It is therefore extremely unlikely that there is a starting value that grows to infinity as a result of the procedure.

There could be a starting value, however, that ends in a loop that is not 4 → 2 → 1. That loop could include significantly more numbers, such that 1 would never be reached.

Such “nontrivial” loops can be found, for example, if you also allow negative integers for the Collatz conjecture: in this case, the iterative calculation rule can end not only at –2 → –1 → –2 → ... but also at –5 → –14 → –7 → –20 → –10 → –5 → ... or –17 → –50 → ... → –17 →.... If we restrict ourselves to natural numbers, no nontrivial loops are known to date—which does not mean that they do not exist. Experts have now been able to show that such a loop in the Collatz problem, however, would have to consist of at least 186 billion numbers .

The length of the Collatz sequences for all numbers from 1 to 9,999 varies greatly.

Credit: Cirne/Public domain via Wikimedia Commons

Even if that sounds unlikely, it doesn’t have to be. In mathematics there are many examples where certain laws only break down after many iterations are considered. For instance,the prime number theorem overestimates the number of primes for only about 10 316 numbers. After that point, the prime number set underestimates the actual number of primes.

Something similar could occur with the Collatz conjecture: perhaps there is a huge number hidden deep in the number line that breaks the pattern observed so far.

A Proof for Almost All Numbers

Mathematicians have been searching for a conclusive proof for decades. The greatest progress was made in 2019 by Fields Medalist Terence Tao of the University of California, Los Angeles, when he proved that almost all starting values of natural numbers eventually end up at a value close to 1.

“Almost all” has a precise mathematical meaning: if you randomly select a natural number as a starting value, it has a 100 percent probability of ending up at 1. ( A zero-probability event, however, is not necessarily an impossible one .) That’s “about as close as one can get to the Collatz conjecture without actually solving it,” Tao said in a talk he gave in 2020 . Unfortunately, Tao’s method cannot generalize to all figures because it is based on statistical considerations.

All other approaches have led to a dead end as well. Perhaps that means the Collatz conjecture is wrong. “Maybe we should be spending more energy looking for counterexamples than we’re currently spending,” said mathematician Alex Kontorovich of Rutgers University in a video on the Veritasium YouTube channel .

Perhaps the Collatz conjecture will be determined true or false in the coming years. But there is another possibility: perhaps it truly is a problem that cannot be proven with available mathematical tools. In fact, in 1987 the late mathematician John Horton Conway investigated a generalization of the Collatz conjecture and found that iterative functions have properties that are unprovable. Perhaps this also applies to the Collatz conjecture. As simple as it may seem, it could be doomed to remain unsolved forever.

This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.