problem solving involving multiplication and addition

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Multiplication Word Problem Worksheets

This page hosts a vast collection of multiplication word problems for 3rd grade, 4th grade, and 5th grade kids, based on real-life scenarios, practical applications, interesting facts, and vibrant themes. Featured here are various word problems ranging from basic single-digit multiplication to two-digit and three-digit multiplication. Another set of printable worksheets hone children's multiplication skill by multiplying large numbers. Free worksheets are included.

Single-digit Multiplication Word Problems

The printable PDF worksheets presented here involve single-digit multiplication word problems. Each worksheet carries five word problems based on day-to-day scenarios.

• Download the set

Multiplication Word Problems: Two-digit times Single-digit

The word problems featured here require a grade 3 learner to find the product by multiplying a two-digit number by a single-digit multiplier.

Multiplication Word Problems: Two-digit times Two-digit

The worksheets presented here involve multiplication of two-digit numbers. Read the word problems and find the product. Apply long multiplication (also known as column multiplication) method for easy calculation.

Theme Based Word Problems

Our engaging theme-based pdf worksheets help young minds understand the fundamentals of multiplication. Answer the word problems based on three fascinating themes - Winter Season, Ice rink and Library.

Multiplication Word Problems: Three-digit times Two-digit

Read the word problems featured in these printable worksheets for grade 4 and find the product of three-digit and two-digit numbers. Write down your answers and use the answer key below to check if they are right.

Three-digit Multiplication Word Problems

Solve these well-researched word problems that involve three-digit multiplication. Perform multiplication operation and carry over numbers carefully to find the product.

Multiplication: Three or Four-digit times Single-digit

The word problems featured here are based on practical applications and fact-based situations. Multiply a three or four-digit number by a single-digit multiplier to find the correct product.

Multi-digit Word Problems: Multiplying Large Numbers

Sharpen your skills by solving these engaging multi-digit word problems for grade 5. Apply long multiplication method to solve the problems. Use the answer key to check your answers.

Related Worksheets

» Addition Word Problems

» Subtraction Word Problems

» Division Word Problems

» Word Problems

» Multiplication

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2.3.3: Two-Step Equations with Addition and Multiplication

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Solve Equations Using Inverse Properties of Addition and Multiplication

The marching band at Floyd Middle School is undergoing some changes. They currently have 140 students, plus the drum major, and will be adding four new students this year. This means they will need to redo their formation for the big finale into eight even rows. Band member Anica announces that she can figure out the number of students in each row with an equation.

In this concept, you will solve equations involving the inverse properties of addition and multiplication.

Inverse Properties of Adding and Multiplying

An equation is a statement with an equal sign where the quantity on one side of the equals is the same as the quantity on the other side of the equals.

Here is a simple equation:

You have an equation with a variable where x is the unknown quantity. To solve this perform an inverse operation or opposite operation. You would subtract eleven from 15 to give you 4. That is the value of the variable.

Most of the time, you don’t even think about performing an inverse operation, your mind naturally solves the problem in this way.

When you have an equation with one variable, it is called a one-step equation . It only takes one operation or one inverse operation to solve it.

To solve a two-step equation , you will need to use more than one inverse operation. When you perform inverse operations to find the value of a variable, you work to get the variable alone on one side of the equal sign. This is called isolating the variable . It is one strategy for solving equations. You can use isolating the variable whether you are solving one-step or two-step equations.

Let’s look at an example.

Solve for a:

You can call each piece of the equation a term. There is a term with a variable and there is a term without a variable. Notice that there are two terms on the left side of the equation, 3a and 12.

First, use inverse operations to get the term that includes a variable, 3a, by itself on one side of the equal sign. In equations, you would always leave the term that holds the variable until the end in order to isolate the variable. So, in the equation above, 12 is added to 3a. You can use the inverse of addition, which is subtraction as your first step. We can subtract 12 from both sides of the equation.

3a+12−12=45-12

Next, use inverse operations to get the a by itself. Since 3a means 3×a , you can use the inverse of multiplication, which is division. You can divide both sides of the equation by 3 to isolate the variable.

The answer is 11.

Let’s review your steps to solving this two-step equation. Your goal is to isolate your variable so you first need to get this variable term alone on one side of the equal sign.

Example 2.3.3.1

Earlier, you were given a problem about the marching band's new configuration.

First, let’s look at the given information.

There are 144 students in the band. There is also one drum major. The students need to be organized into eight even rows.

Here is our equation.

Notice that the drum major is not included. The drum major is not included in the lines as the drum major is in the lead.

Next, you only have a one-step equation.

The answer is 18.

There will be 18 students in each row.

Example 2.3.3.2

A gardener charges $20 for each gardening job plus $15 for each hour worked. He charged $80 for a gardening job he did yesterday.

• Write an algebraic equation to represent h, the number of hours that the gardener worked on that $80 job.
• Find the number of hours that the gardener worked on that $80 job.

First, in order to solve part a, you need to write an equation. In this problem, you are looking for number of hours, so let h=the number of hours.

Use a number, an operation sign, a variable, or an equal sign to represent each part of the problem. The gardener earned $15 for each hour worked on that job, so you could multiply $15 by h, the number of hours worked, to find how much money the gardener charged for his work time.

So, the equation to use for this problem is:

Next, solve part b . You need to use your equation from part ‘a’ to find the number of hours the gardener worked on that job.

20−20+15h=80-20

Then, since 15 is multiplied by the variable, h, you can use the inverse of multiplication which is division. Divide both sides by 15.

15h/15=60/15

The answer is 4.

The gardener worked four hours.

Example 2.3.3.3

First, use inverse operations (subtraction) to get the term that includes a variable, 4x, by itself on one side of the equal sign.

4x+5−5=29-5

Next, use inverse operations (division) to get the x by itself.

The answer is x=6.

Example 2.3.3.4

First, use inverse operations (subtraction) to get the term that includes a variable, 3y, by itself on one side of the equal sign.

3y+7−7=43-7

Next, use inverse operations (division) to get the ‘y’ by itself.

The answer is y=12.

Example 2.3.3.5

First, use inverse operations (subtraction) to get the term that includes a variable, 6x, by itself on one side of the equal sign.

6x+8−8=71-8

Next, use inverse operations (division) to get the ‘x’ by itself.

The answer is x=9.

Solve the following two-step equations that have addition and multiplication in them.

Review (Answers)

To see the Review answers, open this PDF file and look for section 3.1.

Additional Resources

PLIX: Play, Learn, Interact, eXplore: T-Shirt Equation

Practice: Two-Step Equations with Addition and Multiplication

Real World Application: What Does Justice Weigh?

Grade 5 Mathematics Module: Solving Problems Involving Multiplication

This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson.

Each SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you.

Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these.

Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task.

If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator.

Hello mathletes!

In this module, you are going to gain an understanding on solving routine and non-routine problems involving multiplication. In solving word problems, you need to understand the problem very well, devise a plan to solve it, show solution with confidence, and review the correctness of your answer.

After going through this module, you are expected to solve routine and non routine problems involving multiplication without or with addition or subtraction of decimals and whole numbers including money using appropriate problem-solving strategies and tools.

Grade 5 Mathematics Quarter 2 Self-Learning Module: Solving Problems Involving Multiplication

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Multiplication Word Problem Area 2nd Grade

Welcome to our Multiplication Word Problems page for Second Grade. Here you will find our range of printable multiplication problems which will help your child apply and practice their multiplication and times tables skills to solve a range of 'real life' problems.

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• 2nd Grade Multiplication Problems Worksheets
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Multiplication Word Problems 2nd Grade Online Quiz

Multiplication word problems, 2nd grade multiplication problems.

Here you will find a range of problem solving worksheets involving multiplication.

Each sheet involves solving a range of written multiplication problems.

There are 3 levels of difficulty for each worksheet below: A,B and C.

Worksheet A is the easiest level, suitable for children at the beginning of their grade.

Worksheet B is a medium level worksheets for children who are working at the expected level in their grade.

Worksheet C is set at a harder level, suitable for children who are more able mathematicians.

The problems in each worksheet are similar in wording, but the numbers involved become trickier as the level gets harder.

To encourage careful checking and thinking skills, each sheet includes one 'trick' question which is not a multiplication problem. Children need to spot this word problem, and work out which operation they need to solve it.

Using these sheets will help your child to:

• apply their multiplication and times tables skills;
• apply their times table knowledge to work out related facts;
• recognise multiplication problems, and try to spot 'trick' problems;
• solve a range of 'real life' problems;
• solve multiplication problems with numbers up to 5x5;
• solve multiplicaiton problems for the 2, 5 and 10 times tables.

Some of the sheets have a UK version with spelling set for the UK.

2nd Grade Multiplication Problem Worksheets

Series 2 sheet 1 set.

• Series 2 Sheet 1A (easier)
• Series 2 Sheet 1B (medium)
• Series 2 Sheet 1C (hard)

• PDF version Series 2.1 (6 sheets)

Series 2 Sheet 2 Set

• Series 2 Sheet 2A (easier)
• UK version Sheet 2A
• Series 2 Sheet 2B (medium)
• UK Version Sheet 2B
• Series 2 Sheet 2C (hard)
• PDF version Series 2.2 (6 sheets)
• PDF version Series 2.2 (UK version) (6 sheets)

Series 2 Sheet 3 Set

• Series 2 Sheet 3A
• Series 2 Sheet 3B (medium)
• Series 2 Sheet 3C (hard)
• PDF version Series 2.3 (6 sheets)

Looking for some harder Multiplication Problems?

In our 3rd Grade Multiplication word problem area, you will find a range of multiplication word problems aimed at 3rd graders.

The following areas are covered:

• basic multiplication fact sheets;
• multiplication facts to 10x10;
• problems needing written multiplication methods to solve (TU x U)
• Multiplication Word Problem Worksheets 3rd Grade

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

Looking for more 2nd Grade Word Problems?

We have a range of real-life problems involving different measurements, such as the fastest insects, or the length of different dinosaurs!

There are a range of different skills covered, from rounding and place value, to using addition, subtraction and multiplication.

• 2nd Grade Math Word Problems

Multiplication Times Table Charts

Here you will find a selection of Multiplication Times Table Charts to 10x10 or 12x12 to support your child in learning their multiplication facts.

There is a wide selection of multiplication charts including both color and black and white, smaller charts, filled charts and blank charts.

Using these charts will help your child to:

• Learn their multiplication facts to 10x10 or 12x12;
• Practice their multiplication table.

All the free printable Math charts in this section are informed by the Elementary Math Benchmarks.

• Large Multiplication Chart
• Large Multiplication Charts Times Tables
• Multiplication Times Tables Chart to 10x10
• Times Table Grid to 12x12
• Blank Multiplication Charts to 10x10
• Blank Printable Charts to 12x12
• Multiplication Math Games

Here you will find a range of Free Printable Multiplication Games.

The following games develop the Math skill of multiplying in a fun and motivating way.

• learn their multiplication facts;
• practice and improve their multiplication table recall;
• develop their strategic thinking skills.

Our quizzes have been created using Google Forms.

At the end of the quiz, you will get the chance to see your results by clicking 'See Score'.

This will take you to a new webpage where your results will be shown. You can print a copy of your results from this page, either as a pdf or as a paper copy.

For incorrect responses, we have added some helpful learning points to explain which answer was correct and why.

We do not collect any personal data from our quizzes, except in the 'First Name' and 'Group/Class' fields which are both optional and only used for teachers to identify students within their educational setting.

We also collect the results from the quizzes which we use to help us to develop our resources and give us insight into future resources to create.

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This quick quiz tests your knowledge and skill at solving multiplication word problems.

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Solve problems involving addition, subtraction, multiplication and division and a combination of these, including understanding the meaning of the equals sign

New Maths Curriculum (2014): Year 5 objectives.

Related Worksheets

Y5 number - multiplication and division - interactive quiz.

An interactive quiz which test your understanding of all the Multiplication and Division objectives in the Year 5 curriculum. Choose one objective or multiple objectives. You can save, or print, your test results as a pdf at the end of the quiz. Ideal for formative or summative assessment. 

Alternatively, you can use the  Interactive Maths Quiz  which includes objectives from all strands of the Year 5 curriculum.

This quiz tests the following objectives:

• Online in HTML5 format

Round up or down after division

Answer division word problems using a calculator and decide whether the answer needs to be rounded up or down depending on the context of the question

For more calculator resources click here.

Formal Written Division - Round Up or Down?

Answer a word problem by using either long division or short division and then round your answer according to the context.

This game is now part of the 'Calculations' collection, which includes the following 17 games and resources: Column Addition, Expanded Addition, Expanded Addition - Place Value Counters, Number Bonds(2), Addition - Digit Drag, Missing Symbols, Column Subtraction, Column Subtraction using Place Value Counters, Counting on to find difference on a beadstring, Multiplication Grid Method, Multiplication Written Method, Ratio and Scaling Numbers, Representing Multiplication, Division by Chunking Up, Division by Chunking Down, Formal Written Division - Round Up or Down?, Short Division Writen Formal Method.

The Calculations app is available on Google Play and the App Store.

Missing Symbols

Drag the symbols to the correct position to make the number sentence correct. Lots of levels to play. See if you can improve your score over time.

For more multiplication and division resources click here.

For more addition and subtraction resources click here.

Thinking of a Number

Use inverse operations to work backwards and find a missing number.

Balancing Sums

An excellent tool for reinforcing an understanding of the role of the equals sign. Choose one objective, or many. You can choose all the objectives for a single year group (or multiple year groups). You can also choose to balance between different types of calculations (eg addition and multiplication). 

Archery Arithmetic - Multiplication

Choose which times tables to practise. Answer questions as quickly as possible and you will get more time to aim your arrow. Play on your own or against a partner. Who can score the most points in your class?

For more multiplication games click here.

Tommy's Trek - Division

A fun platformer game that helps develop addition skills. Lots of choice over level. This game will work on any device.

Use the arrow keys to move Tommy. The ctrl key makes him jump and the up arrow selects a number.

Ratio and Scaling Numbers Word Problems

Designed to help children to visualise the relationships between numbers when answering word problems involving ratio and scaling quantities. Children should see that sometimes it is easier to scale numbers in one 'direction' than the other, and that they can choose the most efficient method.

For more resources involving word problems click here.

Click on 'Multiplication' and then 'Ratio and Scaling Numbers'.

Solve 2 Step Money Problems

Read and interpret 2 step word problems using a variety of contexts. This version will work on any device.

For more money resources click here.

Adding Money - Digit Drag

A challenging game in which you have to drag the digit cards to make the correct amount of money. Great for developing problem solving skills, generating 'rules' and encouraging trial and improvement.

Balancing scales - convert between g and kg

The scale is loaded with between 1 and 4 weights. The user then has to balance the scales, converting between g and kg in the process. There is a great deal of control over the level.

A great starter to explore addition, subtraction / difference and partitioning as well as the relationship between grams and kilog More… rams. Levels include converting whole kg to g (and vice versa), converting fractions of kg (eg 1/4kg = 250g) and weights in kg to 1 and 2 decimal places.

For more measuring resources click here.

Balancing Scales g and kg

Add weights to either pan until they are balanced. Can be used to teach addition, subtraction, partitioning, and mass, as well as developing an understanding of the equals sign.

Choose from grams, whole kilograms or kilograms with either 1 or 2 decimal places.

For more partitioning resources click here.

Balance scales - add, move or remove a weight

Balance the scales by adding, moving or removing a weight. A great mental starter to develop addition and subtraction skills.

Maths Race - Division

Solving Multi-Step Addition and Subtraction Word Problems

Solving multi-step addition and subtraction word problems can help students see the relationship between operations, as well as practice their calculations.  When I teach subtraction, I also like to show how students can use addition to check their work!  Students often think that they are done when they arrive at the answer, but it is my mission to show them that they can do just one more step to check and make sure their work is correct.  Addition and subtraction work together to help each other out.

The Art of Check Writing

Although the art of writing a check and balancing a checkbook is going away, those processes still have great value for practicing addition and subtraction.  I like to present math problems with money, budgets and purchasing items, but these days students tell me just to “charge it to a card”.  I have been tempted to find my old carbon copy paper receipts and bring them into class.  Just the other week, we had our lawnmower repaired.  The man ran a small, old business and he wrote out all of the receipts and transactions by hand, using the box with carbon copy receipts.  He wrote down the price of the parts, added the taxes and charges, and summed it up – BY HAND!  The only thing missing for a good math problem was writing the word form onto a check.

Set up Shop in Your Classroom

So I thought, why not create a shop in my classroom!  Students can be the shopkeepers and the customers, but no charge cards allowed!  Shopkeepers would have to write the receipt, and the customers would have to write a check for the exact amount.  The students would have to read scenarios and decide what to buy.  I could even give them a budget or a sales goal and they would have to periodically check the difference by subtracting.  It would be a lesson embedded in real life addition and subtraction word problems.

Set up for Solving Multi-Step Addition and Subtraction Word Problems

• Print or create price tags from page 6 of the worksheet for the list of items located pages 7-10.
• Set up “items” and pictures around the room. There are pictures provided on pages 7-10 on the worksheet , but you can always supplement or creatively add items of your own.
• Set up 5 shopkeeper stations with blank copies of “Receipts.” (See worksheet page 3)
• Set up each shop with a Sales Record to keep track of sales and income. Each shop has a distinct sales goal, such as $99, $250, $400, $505, $888.
• $99 – Bargain Basement
• $250 – Quarter Millie’s Party Store
• $400 – Four Double Zero
• $505 – 505 Faves
• $888 – Great Eights
• Provide each student with 6 blank checks and 1 balance sheet. (See worksheet pages 4-5.)
• Provide each student with a copy of the Addition and Subtraction Worksheet .
• If you have little bells or signs, you can set up the shopkeeper with things like an “Open/Closed” sign or “Ring bell for service”.

Launch the Solving Multi-Step Addition and Subtraction Word Problems

If you have time, set up some open/closed shop signs to create the feel of a store in the classroom.  You can even have students create a checkbook with their checks.

Introduce the activity by saying “Today we are going to go shopping!”  Explain that everyone will rotate through and get a chance to run a shop as well as go shopping for items on the worksheet.  The students will not actually pick up items and keep them, but you can have things on display or just have photos and a price list.

Explain the vocabulary and show an example of each: (All are found on the worksheet .)

this is used to write an exact amount of money.  It takes the money out of your bank account and gives it to the store or person it is written to when cashed at the bank.

Balance Sheet

This is a record of how much money you have and how much you spend.  The balance is the amount of money people have to spend.  Use the balance sheet to keep track of the money.  Tell the students that everyone starts with a balance of $445.00. The shoppers can write a brief description of what they buy in the white rows and then subtract the amount spent from the balance.  Explain that the students can do the subtraction vertically in the balance column.  The shaded rows show the running balance, and the white rows show how much is spent each time.  Subtract any money spent to get the new balance.Tell your students “When you reach zero – you are out of money!”

This is a set amount of money made from selling goods that sales people try to reach.  If you have a sales goal, you can subtract any sales to see how much more money you need to reach the goal.

This lists the items being purchased and how much they each cost.  The receipt also gives the total cost of the purchase.  It is a record to show the sale.

Use this scenario to model the work students will do:

“You walk into the store Four Double Zero and  decide to buy headphones for $125 and a case for $34.  What is your purchase price?”

Together, work through the addition.  Remind the students that BOTH the customer and the shopkeeper must do the addition.  Then the shopkeeper tells the customer what to pay.  (There are no taxes or charges unless indicated in the problem on the worksheet).

Pass Out Student Checkbooks

Give every student their checks or checkbook.  Use the first blank check to write the demo check together.  Every check must have:

• The name of the store or person you are writing the check to
• The purchase amount in word form
• The purchase amount in standard form
• The Memo line – items purchased

Then demonstrate the following:

• The customer must subtract: $455 – $159
• The Storekeeper must subtract it from the Sales Goal $400- $159

After the demo, space the students out among the stores.  Assign 1-2 students as shopkeepers at each shop and have the remaining students walk around and shop using the worksheet scenarios.

You can provide 5-10 minutes of shopping and then switch roles, or you can walk around and switch out shopkeepers as you see fit.

• Each student has a Budget of $455 for their entire shopping trip.
• The scenarios on the worksheet ask students to buy at least 2 items from each shop. Some of the word problems are set, others have a choice of items.
• Customers indicate the items and say the prices. The shopkeeper writes the prices and adds them up.  The Customer also adds them up and they check.
• Once the price is determined, the customer writes a check using standard and word form and the name of the store, then signs and dates it.
• The customer then subtracts the purchase from his/her running balance.
• The shopkeeper cashes the check, adds it to the sales list and subtracts it from the sales goal for that shop.
• Customers must check their balance after each purchase. Customers will keep a running balance throughout their shopping experience.
• Shopkeepers must check their Sales goals and how much more money they need after each customer. Shopkeepers can keep a running total while they are in charge of that shop.  However, when a new shopkeeper comes in, they must start with the original sales goal when subtracting.
• Depending on the class size, start each shop with 1-2 shopkeepers. Then have the remaining students go around to the available shopkeepers and work through their transactions.  Switch roles when appropriate.

Reflection on Solving Multi-Step Addition and Subtraction Word Problems

Walk around and help the shopkeepers and the customers as they work through the purchasing process.

At the end, share what people decided to buy and talk about the processes of making a transaction.

Ask questions such as:

• Which store did you like the best?
• Did you ever add up the numbers and get a different sum than the shopkeeper/customer? How did you resolve the price issue?
• Why do you need to subtract your purchases from the budget?
• How did you keep track of your purchases?
• Did any shopkeepers reach their sales goal before leaving their “shift”?
• In the end, how could you check the total amount of purchases you made today? Did you stay under budget?
• Did anyone go over their budget?
• Have students create a price list of items and write 4 of their own word problems. Give them a sales goal and come up with 4 different ways to reach that goal.  Can they hit the goal exactly?
• Research prices of items, but make sure the students find whole numbers.  (Many times ticket prices tend to be whole numbers).  Ask how much money the school would need to budget if the class went on a trip to that theater, sports game, or show.

FREE Solving Multi-Step Addition and Subtraction Word Problems Worksheets and Resources

These are all PDF Files. They will open and print easily. The Student Edition Files are labeled SE and the Teacher Editions Files are labeled TE. Click the links below to download the different resources.

2-4 Assignment SE – Solving Multi-Step Addition and Subtraction Word Problems 2-4 Assignment TE – Solving Multi-Step Addition and Subtraction Word Problems( Members Only ) 2-4 Bell Work SE – Solving Multi-Step Addition and Subtraction Word Problems (PDF) 2-4 Bell Work TE – Solving Multi-Step Addition and Subtraction Word Problems ( Members Only ) 2-4 Exit Quiz SE – Solving Multi-Step Addition and Subtraction Word Problems  (PDF) 2-4 Exit Quiz TE – Solving Multi-Step Addition and Subtraction Word Problems ( Members Only ) 2-4 Exit Quiz SE – Solving Multi-Step Addition and Subtraction Word Problems  (PDF) 2-4 Guided Notes TE – Solving Multi-Step Addition and Subtraction Word Problems ( Members Only ) 2-4 Interactive Notebook – Solving Multi-Step Addition and Subtraction Word Problems  (PDF) 2-4 Lesson Plan – Solving Multi-Step Addition and Subtraction Word Problems  (PDF) 2-4 Online Activities – Solving Multi-Step Addition and Subtraction Word Problems ( Members Only ) 2-4 Slide Show – Solving Multi-Step Addition and Subtraction Word Problems  (PDF)

Solving Multi-Step Addition and Subtraction Word Problems Worksheets and Resources

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Don’t Forget to Pin this Activity on Multi-Step Addition and Subtraction Word Problems…

Want to see the rest of the activities for unit 2 – multi-digit whole number addition and subtraction.

• 2-1  Multi-Digit Whole Number Addition
• 2-2  Solving Two-Step Addition Word Problems
• 2-3  Multi-Digit Whole Number Subtraction
• 2-4  Solving Two-Step Subtraction Word Problems
• 2-5  Solving Multi-Step Addition and Subtraction Word Problems
• 2-6  Creating Word Problems Involving Addition and Subtraction

If you enjoyed the read, don’t forget to check out more of our blogs on PreCalculusCoach , GeometryCoach , Algebra2Coach , Algebra1Coach , and PreAlgebraCoach !

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Problem Solving Addition Subtraction and Multiplication

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Roselyn Bonimy

Problem Solving Mixed Operations (Addition Subtraction and Multiplication)

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7.2: Addition and Multiplication Principles

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• Harris Kwong
• State University of New York at Fredonia via OpenSUNY

Addition Principle

Preliminaries.

Recall that the cardinality of a finite set \(A\), denoted \(|A|\), is the number of elements it contains.

Example \(\PageIndex{1}\)

If \(A=\{-1,0,2\}\), then \(|A|=3\). Also, \[\begin{aligned} |\{2\}| &=& 1, \\ |\{2,5,-1,-3\}| &=& 4, \\ |\{x\in\mathbb{R}\mid x^2=1\}| &=& 2. \end{aligned}\] Notice that \(|\emptyset|=0\), because an empty set does not contain any element.

It becomes more interesting when we consider the cardinality* of a union or an intersection of two or more sets.

*This cardinality link will take you to a not completely edited, but interesting dive into cardinality of infinite sets.

Example \(\PageIndex{2}\)

Determine \(|A \cup B|\) and \(|A \cap B|\) if \(A=\{2,5\}\) and \(B=\{7,9,10\}\).

Since \(A\cup B = \{2,5,7,9,10\}\), and \(A\cap B = \emptyset\), it is clear that \(|A\cup B|=5\), and \(|A\cap B|=0\).

Example \(\PageIndex{3}\)

Determine \(|A \cup B|\) and \(|A \cap B|\) if \(A=\{2,5\}\) and \(B=\{5,9,10\}\).

Since \(A\cup B = \{2,5,9,10\}\), and \(A\cap B = \{5\}\), it is clear that \(|A\cup B|=4\), and \(|A\cap B|=1\).

hands-on exercise \(\PageIndex{1}\label{he:addmult-01}\)

Let \(A=\{n\in\mathbb{Z} \mid -5\leq n\leq3\}\), and \(B=\{n\in\mathbb{Z} \mid -3\leq n\leq5\}\). Evaluate \(|A\cap B|\) and \(|A\cup B|\).

The difference between the last two examples is whether the two sets \(A\) and \(B\) have a nonempty intersection. Two sets \(A\) and \(B\) are disjoint if \(A \cap B = \emptyset\). A collection of sets \(A_1, A_2, \ldots, A_n\) is said to be pairwise disjoint if \(A_i \cap A_j = \emptyset\) whenever \(i\neq j\). 

Example \(\PageIndex{4}\)

Let \(A=\{1,0,-1\}\), \(B=\{-2,0,2\}\), \(C=\{-2,2\}\) and \(D=\{3,4,5\}\). Then \(A\), \(C\), and \(D\) are pairwise disjoint, so are \(B\) and \(D\), but \(A\), \(B\), and \(C\) are not.

Theorem \(\PageIndex{1}\): Addition Principle

If the finite sets \(A_1, A_2, \dots, A_n\) are pairwise disjoint, then \[|A_1 \cup A_2 \cup \cdots \cup A_n| = |A_1| + |A_2| + \cdots + |A_n|.\]

Use the addition principle if we can break down the problems into cases , and count how many items or choices we have in each case. The total number is the sum of these individual counts. The idea is, instead of counting a large set, we divide it up into several smaller subsets, and count the size of each of them. The cardinality of the original set is the sum of the cardinalities of the smaller subsets. This divide-and-conquer approach works perfectly only when the sets are pairwise disjoint.

Example \(\PageIndex{5}\)

To find the number of students present at a lecture, the teacher counts how many students there are in each row, then adds up the numbers to obtain the total count.

When the sets are not disjoint, the addition principle does not give us the right answer because the elements belonging to the intersection are counted more than once. We have to compensate the over-counting by subtracting the number of times these elements are over-counted. The simplest case covers two sets.

Theorem \(\PageIndex{2}\label{pie}\): Principle of Inclusion-Exclusion (PIE)

For any finite sets \(A\) and \(B\), we have \[|A\cup B| = |A|+|B|-|A\cap B|.\]

Observe that \(A\cup B\) is the disjoint union of three sets \[A \cup B = (A-B) \cup (A \cap B) \cup (B-A).\] It is clear that \(|A-B| = |A|-|A\cap B|\), and \(|B-A|=|B|-|A\cap B|\). Therefore, \[\begin{aligned} |A \cup B| &=& |A-B| + |A\cap B| + |B-A| \\ &=& (|A|-|A \cap B|) + |A\cap B| + (|B|-|A \cap B|) \\ &=& |A| + |B| - |A \cap B|, \end{aligned}\] which is what we have to prove.

The principle of inclusion-exclusion also works if \(A\) and \(B\) are disjoint, because in such an event, \(|A\cap B|=0\), reducing PIE to the addition principle.

Example \(\PageIndex{6}\label{eg:pie}\)

Assume the current enrollment at a college is 4689, with 60 students taking MATH 210, 42 taking CSIT 260, and 24 taking both. Together, how many different students are taking these two courses? In other words, determine the number of students who are taking either MATH 210 or CSIT 260.

Let \(A\) be the set of students taking MATH 210, and \(B\) the set of students taking CSIT 260, Then, \(|A|=60\), \(|B|=42\), and \(|A\cap B|=24\). We want to find \(|A\cup B|\). According to PIE, \[|A\cup B| = |A| + |B| - |A\cap B| = 60+42-24 = 78.\] Therefore, 78 students are taking either MATH 210 or CSIT 260.

Example \(\PageIndex{7}\)

Among 4689 students, 2112 of them have earned at least 60 credit hours and 2678 of them have earned at most 60 credit hours. How many students are there who have accumulated exactly 60 hours?

Let \(A\) be the set of students who have earned at least 60 credit hours, and \(B\) be the set of students who have earned at most 60 credit hours. We want to find \(|A\cap B|\). According to PIE, \[4689 = |A\cup B| = |A|+|B|-|A\cap B| = 2112+2678-|A\cap B|.\] Hence, \[|A\cap B| = (2112+2678)-4689 = 101.\] There are 101 students who have accumulated exactly 60 credit hours.

hands-on exercise \(\PageIndex{2}\label{he:addmult-02}\)

The attendance at two consecutive college football games was 72397 and 69211 respectively. If 45713 people attended both games, how many different people have watched the games?

hands-on exercise \(\PageIndex{3}\label{he:addmult-03}\)

The attendance at two consecutive college football games was 72397 and 69211 respectively. If 93478 different individuals attended these two games, how many have gone to both?

Sometimes, it is easy to work with the complement of a set.

Lemma \(\PageIndex{3}\)

For any finite set \(S\), we have \[|\overline{S}| = |{\cal U}| - |S|,\] where \({\cal U}\) is the universal set containing \(S\).

Example \(\PageIndex{8}\)

In Example 6 since there are 78 students taking either MATH 210 or CSIT 260, the number of students taking neither is \(4689-78=4611\).

The principle of inclusion-exclusion can be extended to any number of sets. The situation is more complicated, because some elements may be double-counted, some triple-counted, etc. To give you a taste of the general result, here is the principle of inclusion-exclusion for three sets.

Theorem \(\PageIndex{1}\)

For any three finite sets \(A\), \(B\) and \(C\), \[|A \cup B \cup B| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|. \nonumber\]

The union \(A \cup B \cup C\) is the disjoint union of seven subsets: \[\displaylines{ A-(B\cup C), \quad B-(C\cup A), \quad C-(A\cup B), \quad (A\cap B)-(A\cap B\cap C), \cr (B\cap C)-(A\cap B\cap C), \quad (C\cap A)-(A\cap B\cap C), \quad \mbox{and}\quad A\cap B\cap C. \cr} \nonumber\] We can apply an argument similar to the one used in the union of two sets to complete the proof. We leave the details as an exercise.

hands-on exercise \(\PageIndex{4}\label{he:addmult-04}\)

A group of students claims that each of them had seen at least one part of the Back to the Future trilogy. A quick show of hands reveals that

• 47 had watched Part I;
• 43 had watched Part II;
• 32 had watched Part III;
• 33 had watched both Parts I and II;
• 27 had watched both Parts I and III;
• 25 had watched both Parts II and III;
• 22 had watched all three parts.
• How many students are there in the group? Hint: A Venn Diagram can be helpful with this exercise.

Multiplication Principle

Another useful counting technique is the multiplication principle.

Theorem \(\PageIndex{5}\) (Multiplication Principle)

For any finite sets \(A\) and \(B\), we have \[|A\times B| = |A|\cdot|B|.\]

Clearly, this can be extended to an \(n\)-fold Cartesian product.

Theorem \(\PageIndex{6}\)

For any finite sets \(A_1, A_2, \ldots, A_n\), we have \[|A_1 \times A_2 \times \cdots \times A_n| = |A_1| \cdot |A_2| \cdot \cdots \cdot |A_n|. \nonumber\]

In many applications, it may be helpful to use an equivalent form.

Theorem \(\PageIndex{7}\) (Multiplication Principle: Alternate Form)

If a task consists of \(k\) steps, and if there are \(n_i\) ways to finish step \(i\), then the entire job can be completed in \(n_1 n_2 \ldots n_k\) different ways.

Now that we have two counting techniques, the addition principle and the multiplication principle, which one should we use? The major difference between them is whether

• the jobs can be divided into cases , groups , or categories ; or
• each job can be broken up into steps .

In practice, it helps to draw a picture of the configurations that we are counting.

Example \(\PageIndex{9}\)

How many different license plates are there if a standard license plate consists of three letters followed by three digits?

We need to decide how many choices we have in each position. Draw a picture to show the configuration. Draw six lines to represent the six positions. Above each line, describe briefly the possible candidates for that position, and under each line, write the the number of choices.

This left-to-right configuration suggests that the multiplication principle should be used. The answer is \(26\cdot 26\cdot 26\cdot 10\cdot 10\cdot 10 = 260^3\).

As you become more experienced, you can argue directly, as follows. There are 26 choices for each of the three letters, and 10 choices for each digit. So there are \(26\cdot 26\cdot 26\cdot 10\cdot 10\cdot 10 = 260^3\) different license plates.

Example \(\PageIndex{10}\)

Find the number of positive integers not exceeding 999 that end with 7.

The integers can have one, two, or three digits, so we have to analyze three cases.

Case 1 . There is only one integer with one digit, namely, the integer 7.

Case 2 . If there are two digits, the first could be any digit between 1 and 9, and the last digit must be 7.

This gives us nine choices.

Case 3 . If there are three digits, the first digit could be any digit between 1 and 9, the second any digit between 0 and 9, and the last digit must be 7.

Hence, there are 90 integers in this case. Combining the three cases, we have a total of \(1+9+90=100\) integers that meet the requirements.

The integers could be written as three-digit integers if we allow 0 as the leading digits. For instance, 7 can be written as \(007\), and \(34\) as \(034\). Under this agreement, we have to fill three positions where the last one is always occupied by the digit 7. The first two digits are \(0, 1, 2, \ldots, 8\), or 9, so there are 10 choices for each position.

Together, there are \(10\cdot10=100\) such integers.

hands-on exercise \(\PageIndex{5}\label{he:addmult-05}\)

How many natural numbers less than 1000000 are there that end with the digit 3?

hands-on exercise \(\PageIndex{6}\label{he:addmult-06}\)

How many natural numbers less than 10000 are there that end with the digit 0?

Example \(\PageIndex{11}\)

Determine the number of four-digit positive integers without repeated digits.

We want to determine how many choices there are for each place value. The first digit has nine choices because it cannot be 0. Once the first digit is chosen, there are nine choices left for the second digit; and then eight choices for the next digit, and seven choices for the last digit. Together, we have \(9\cdot 9\cdot 8\cdot 7 =4536\) four-digit positive integers that do not contain any repeated digits. Question: Can we start counting from the last digit?

hands-on exercise \(\PageIndex{7}\label{he:addmult-07}\)

How many six-digit natural numbers are there that do not have any repeated digit?

Example \(\PageIndex{12}\)

How many two-digit positive integers do not have consecutive 5s?

There are three disjoint cases:

both digits are not 5,

only the first digit is 5, and

only the last digit is 5.

There are \(8\cdot9+9+8=89\) integers that meet the requirement.

An easier solution is to consider the complement of the problem. There is only one integer with consecutive 5s, namely, the integer 55. There are 90 two-digit integers, hence \(90-1=89\) of them do not have consecutive 5s.

hands-on exercise \(\PageIndex{8}\label{he:addmult-08}\)

How many three-digit natural numbers are there that do not have consecutive 4s?

Example \(\PageIndex{13}\)

In how many ways can we draw a sequence of three cards from a standard deck of 52 cards?

This is a trick question! The answer depends on whether we can return a drawn card to the deck. With replacement, the answer is \(52^3\); without replacement, it is \(52 \cdot 51 \cdot 50\).

Example \(\PageIndex{14}\)

A standard New York State license plate consists of three letters followed by four digits. Determine the number of standard New York State license plates with K as the first letter or 8 as the first digit.

The keyword “or” suggests that we are looking at a union, hence, we have to apply PIE. We need to analyze three possibilities:

• There are \(26^2\cdot10^4\) license plates with K as the first letter.
• There are \(26^3\cdot10^3\) license plates with 8 as the first digit.
• There are \(26^2\cdot10^3\) license plates with K as the first letter and 8 as the first digit.

The answer is \(26^2\cdot10^4+26^3\cdot10^3-26^2\cdot10^3\).

hands-on exercise \(\PageIndex{9}\label{he:addmult-09}\)

To access personal account information, a customer could log in to the bank’s web site with a PIN consisting of two letters followed by

exactly four digits,

at most six digits,

at least two but at most 6 digits.

How many different PINs are there in each case?

Summary and Review

• Use the addition principle if the problem can be divided into cases. Make sure the cases do not overlap.
• If the cases overlap, the number of objects belonging to the overlapping cases must be subtracted from the total to obtain the correct count.
• In particular, the principle of inclusion-exclusion states that \(|A\cup B|=|A|+|B|-|A\cap B|\).
• Use the multiplication principle if the problem can be solved in several steps.
• How can we get started? Imagine you want to list all the possibilities, what is a systematic way of doing so? Follow the steps, and count how many objects you would end up with.
• It may be helpful to use a schematic diagram. Draw one line for each step. Above the lines, write the choices. Below the lines, write the number of choices. Apply the multiplication principle to finish the problem.
• If there are other cases involved, repeat, and add the results from all the possible cases.

Exercises 

exercise \(\PageIndex{1}\label{ex:addmult-01}\)

A professor surveyed the 98 students in her class to count how many of them had watched at least one of the three films in The Lord of the Rings trilogy. This is what she found:

• 74 had watched Part I;
• 57 had watched Part II;
• 66 had watched Part III;
• 52 had watched both Parts I and II;
• 51 had watched both Parts I and III;
• 45 had watched both Parts II and III;
• 43 had watched all three parts.

How many students did not watch any one of these three movies?

(Hint: A Venn Diagram may be helpful.)

exercise \(\PageIndex{2}\label{ex:addmult-02}\)

Forty-six students in a film class told the professor that they had watched at least one of the three films in The Godfather trilogy. Further inquiry led to the following data:

• 41 had watched Part I;
• 37 had watched Part II;
• 33 had watched Part III;
• 30 had watched both Parts I and III;
• 29 had watched both Parts II and III.
• How many students had watched all three films?
• How many students had watched only Part I?
• How many students had watched only Part II?
• How many students had watched only Part III?

exercise \(\PageIndex{3}\label{ex:addmult-03}\)

Joe has 10 dress shirts and seven bow ties. In how many ways can he match the shirts with bow ties?

exercise \(\PageIndex{4}\label{ex:addmult-04}\)

A social security number is a sequence of nine digits. Determine the number of social security numbers that satisfy the following conditions:

• There are no restrictions.
• The digit 8 is never used.
• The sequence does not begin or end with 8.
• No digit is used more than once.

exercise \(\PageIndex{5}\label{ex:addmult-05}\)

A professor has seven books on discrete mathematics, five on number theory, and four on abstract algebra. In how many ways can a student borrow two books not both on the same subject?

Which two subjects would the student choose?

\(7\cdot5+7\cdot4+5\cdot4\)

exercise \(\PageIndex{6}\label{ex:addmult-06}\)

How many different collections of cans can be formed from five identical Cola-Cola cans, four identical Seven-Up cans, and seven identical Mountain Dew cans?

How many cans of Cola-Cola, Seven-Up, and Mountain Dew would you pick?

exercise \(\PageIndex{7}\label{ex:addmult-07}\)

How many five-letter words (technically, we should call them strings, because we do not care if they make sense) can be formed using the letters A, B, C, and D, with repetitions allowed. How many of them do not contain the substring BAD?

For the second question, consider using a complement.

\(4^5\), \(4^5-3\cdot4^2\)

exercise \(\PageIndex{8}\label{ex:addmult-08}\)

How many different five-digit integers can be formed using the digits 1, 3, 3, 3, 5?

The three digits 3 are identical, so we cannot tell the difference between them. Consequently, what really matters is where we put the digits 1 and 5. Once we place the digits 1 and 5, the remaining three positions must be occupied by the digits 3.

exercise \(\PageIndex{9}\label{ex:addmult-09}\)

Four cards are chosen at random from a standard deck of 52 playing cards, with replacement allowed. This means after choosing a card, the card is return to the deck, and the deck is reshuffled before another card is selected at random. Determine the number of such four-card sequences if

• None of the cards can be spades.
• All four cards are from the same suit.
• The first card is an ace and the second card is not a king.
• At least one of the four cards is an ace.

(a) \(52^4\) (b) \(39^4\) (c) \(4\cdot13^4\) (d) \(4\cdot48\cdot52^2\) (e) \(52^4-48^4\)

exercise \(\PageIndex{10}\label{ex:addmult-10}\)

Three different mathematics final examinations and two different computer science final examinations are to be scheduled during a five-day period. Determine the number of ways to schedule these final examinations from 11 AM to 1 PM if (are these one-hour exams?)

• No two examinations can be scheduled on the same day.
• No two examinations from the same department can be scheduled on the same day.
• Each mathematics examination must be the only examination for the day on which it is scheduled.

exercise \(\PageIndex{11}\label{ex:addmult-11}\)

Determine the number of four-digit positive integers that satisfy the following conditions:

• No integer contains the digit 8.
• Every integer contains the digit 8 at least once.
• Every integer is a palindrome (A positive integer is a palindrome if it remains the same when read backward, for example, 3773 and 47874).

(a) \(9\cdot10^3\) (b) \(8\cdot9^3\) (c) \(9\cdot10^3-8\cdot9^3\) (d) \(9\cdot10\)

exercise \(\PageIndex{12}\label{ex:addmult-12}\)

A box contains 12 distinct colored balls (for instance, we could label them as 1, 2, …, 12 to distinguish them). Three of them are red, four are yellow, and five are green. Three balls are selected at random from the box, with replacement. Determine the number of sequences that satisfy the following conditions:

• The first ball is red, the second is yellow, and the third is green.
• The first ball is red, and the second and third balls are green.
• Exactly two balls are yellow.
• All three balls are green.
• All three balls are the same color.
• At least one of the three balls is red.

exercise \(\PageIndex{13}\label{ex:addmult-13}\)

Let \(A=\{a,b,c,d,e,f\}\) and \(B=\{1,2,3,4,5,6,7,8\}\). Determine the number of functions \(f:A \to B\) that satisfy the following conditions:

• \(f\) is one-to-one.
• \(f\) is onto.
• \(f(x)\) is odd for at least one \(x\) in \(A\).
• \(f(a)=3\) and \(f(b)\) is odd.
• \(f^{-1}(4)=\{a\}\).

(a) \(8^6\) (b) \(8\cdot7\cdot6\cdot5\cdot4\cdot3\) (c) 0 (d) \(8^6-4^6\) (e) \(4\cdot8^4\) (f) \(7^5\)

exercise \(\PageIndex{14}\label{ex:addmult-14}\)

How many onto functions are there from an \(n\)-element set \(A\) to \(\{a,b\}\)?

Word Problems Involving Addition and Subtraction

Word problems involving addition and subtraction are discussed here step by step.

There are no magic rules to make problem solving easy, but a systematic approach can help to the problems easily. 

Word problems based on addition are broadly of two types: (a) When objects of two or more collections are put together.  For example:

Amy has 20 lemon sweets and 14 orange sweets. What is the total number of sweets Amy has?  (b) When an increase in number takes place. 

For example:

Victor has 14 stamps. His friend gave him 23 stamps. How many stamps does Victor have in all? The key words used in problems involving addition are:

s um; total; in all; all together.

Word problems based on subtraction are of several types: (a) Partitioning : Take away, remove, given away.

(b) Reducing : Find out how much has been given away or how much remains.

(c) Comparison : More than / less than.

(d) Inverse of addition : How much more to be added. The key words to look out for in a problem sum involving subtraction are: take away; how many more ; how many less ; how many left ; greater ; smaller.

1. The girls had 3 weeks to sell tickets for their play. In the first week, they sold 75 tickets. In the second week they sold 108 tickets and in the third week they sold 210 tickets. How may tickets did they sell in all? Tickets sold in the first week = 75

Tickets sold in the second week = 108

Tickets sold in the third week = 210

Total number of tickets sold = 75 + 108 + 210 = 393

Answer: 393 tickets were sold in all.

2. Mr. Bose spent $450 for petrol on Wednesday. He spent $125 more than that on Thursday. How much did he spend on petrol on those two days. This problem has to be solved in two steps.

Step 1: Money spent for petrol on Thursday

450 + 125 = $575 Step 2: Money spent for petrol on both days

450 + 575 = $1025

Examples on word problems on addition and subtraction:

1. What is the sum of 4373, 4191 and 3127? Solution: The numbers are arranged in columns and added.

Therefore, sum =11,691

2. What is the difference of 3867 and 1298?

Solution: The numbers are arranged in columns and subtracted:

Therefore, difference = 2569

3. Subtract 4358 from the sum of 5632 and 1324. Solution: Sum of 5632 and 1324

Difference of 6956 and 4358

Therefore, 2598 is the answer.

4. Find the number, which is

(i) 1240 greater than 3267.

(ii) 1353 smaller than 5292. Solution: (i) The number is 1240 more than 3267

Therefore, the number = 3267 + 1240 or = 4507 (ii) The number is 1353 less than 5292

= 5292 – 1353 or

5. The population of a town is 16732. If there are 9569 males then find the number of females in the town. Solution:

6. In a factory there are 35,675 workers. 10,750 workers come in the first shift, 12,650 workers in the second shift and the rest come in the third shift. How many workers come in the third shift? Solution: Number of workers coming in the first and second shift

= 10750 + 12650 = 23400

Therefore, number of workers coming in the third shift = 35675 - 23400 = 12275

Related Concept

● Word Problems on Addition

● Subtraction

● Check for Subtraction and Addition

● Word Problems Involving Addition and Subtraction

● Estimating Sums and Differences

● Find the Missing Digits

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● Multiply a Number by a 2-Digit Number

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● Word Problems on Multiplication

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● Terms Used in Division

● Division of Two-Digit by a One-Digit Numbers

● Division of Four-Digit by a One-Digit Numbers

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● Estimating the Quotient

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Multiplication within 20.

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Solving Word Problems involving Addition and Subtraction of numbers within 120 1st Grade Math Worksheets

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Solving word problems both rely the development of reading and language skills. Addition means “putting together” groups of objects and finding how many they are in total while subtraction tells “how many are left” or “how many more or less”.

Steps in Problem Solving:

• Identify the Problem. Understand what is asked.
• Encircle important numbers.
• Underline the keywords. Analyze if it is for addition or subtraction .
• Solve the problem.
• Present the answer.

Addition: in all, sum, total, more than, plus, altogether, increased by add

Subtraction: fewer, left, less than, take away, minus, difference, remain, decreased

There are 6 surfboards and the surfer bought another 8 pieces. How many surfboards are there in all?

There were 8 beach balls but 5 of them were damaged. How many beach balls were left?

8 – 5 = 3

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