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## Math Problems and Solutions on Integers

Problems related to integer numbers in mathematics are presented along with their solutions.

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## How to Solve Integers and Their Properties

Last Updated: April 6, 2024

This article was reviewed by Joseph Meyer . Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University. This article has been viewed 30,242 times.

An integer is a set of natural numbers, their negatives, and zero. However, some integers are natural numbers, including 1, 2, 3, and so on. Their negative values are, -1, -2, -3, and so on. So integers are the set of numbers including (…-3, -2, -1, 0, 1, 2, 3,…). An integer is never a fraction, decimal, or percentage, it can only be a whole number. To solve integers and use their properties, learn to use addition and subtraction properties and use multiplication properties.

## Using Addition and Subtraction Properties

- a + b = c (where both a and b are positive numbers the sum c is also positive)
- For example: 2 + 2 = 4

- -a + -b = -c (where both a and b are negative, you get the absolute value of the numbers then you proceed to add, and use the negative sign for the sum)
- For example: -2+ (-2)=-4

- a + (-b) = c (when your terms are of different signs, determine the larger number's value, then get the absolute value of both terms and subtract the lesser value from the larger value. Use the sign of the larger number for the answer.)
- For example: 5 + (-1) = 4

- -a +b = c (get the absolute value of the numbers and again, proceed to subtract the lesser value from the larger value and assume the sign of the larger value)
- For example: -5 + 2 = -3

- An example of the additive identity is: a + 0 = a
- Mathematically, the additive identity looks like: 2 + 0 = 2 or 6 + 0 = 6

- The additive inverse is when a number is added to the negative equivalent of itself.
- For example: a + (-b) = 0, where b is equal to a
- Mathematically, the additive inverse looks like: 5 + -5 = 0

- For example: (5+3) +1 = 9 has the same sum as 5+ (3+1) = 9

## Using Multiplication Properties

- When a and b are positive numbers and not equal to zero: +a * + b = +c
- When a and b are both negative numbers and not equal to zero: -a*-b = +c

- However, understand that any number multiplied by zero, equals zero.

- For example: a(b+c) = ab + ac
- Mathematically, this looks like: 5(2+3) = 5(2) + 5(3)
- Note that there is no inverse property for multiplication because the inverse of a whole number is a fraction, and fractions are not an element of integer.

Joseph Meyer

The distributive property helps you avoid repetitive calculations. You can use the distributive property to solve equations where you must multiply a number by a sum or difference. It simplifies calculations, enables expression manipulation (like factoring), and forms the basis for solving many equations.

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## Integers Worksheets

Welcome to the integers worksheets page at Math-Drills.com where you may have a negative experience, but in the world of integers, that's a good thing! This page includes Integers worksheets for comparing and ordering integers, adding, subtracting, multiplying and dividing integers and order of operations with integers.

If you've ever spent time in Canada in January, you've most likely experienced a negative integer first hand. Banks like you to keep negative balances in your accounts, so they can charge you loads of interest. Deep sea divers spend all sorts of time in negative integer territory. There are many reasons why a knowledge of integers is helpful even if you are not going to pursue an accounting or deep sea diving career. One hugely important reason is that there are many high school mathematics topics that will rely on a strong knowledge of integers and the rules associated with them.

We've included a few hundred integers worksheets on this page to help support your students in their pursuit of knowledge. You may also want to get one of those giant integer number lines to post if you are a teacher, or print off a few of our integer number lines. You can also project them on your whiteboard or make an overhead transparency. For homeschoolers or those with only one or a few students, the paper versions should do. The other thing that we highly recommend are integer chips a.k.a. two-color counters. Read more about them below.

## Most Popular Integers Worksheets this Week

## Integer Resources

Coordinate graph paper can be very useful when studying integers. Coordinate geometry is a practical application of integers and can give students practice with using integers while learning another related skill. Coordinate graph paper can be found on the Graph Paper page:

Coordinate Graph Paper

Integer number lines can be used for various math activities including operations with integers, counting, comparing, ordering, etc.

- Integer Number Lines Integers Number Lines from -10 to 10 Integers Number Lines from -15 to 15 Integers Number Lines from -20 to 20 Integers Number Lines from -25 to 25 OLD Integer Number Lines

## Comparing and Ordering Integers

For students who are just starting with integers, it is very helpful if they can use an integer number line to compare integers and to see how the placement of integers works. They should quickly realize that negative numbers are counter-intuitive because they are probably quite used to larger absolute values meaning larger numbers. The reverse is the case, of course, with negative numbers. Students should be able to recognize easily that a positive number is always greater than a negative number and that between two negative integers, the one with the lesser absolute value is actually the greater number. Have students practice with these integers worksheets and follow up with the close proximity comparing integers worksheets.

- Comparing Integers Worksheets Comparing Positive and Negative Integers (-9 to +9) Comparing Positive and Negative Integers (-15 to +15) Comparing Positive and Negative Integers (-25 to +25) Comparing Positive and Negative Integers (-50 to +50) Comparing Positive and Negative Integers (-99 to +99) Comparing Negative Integers (-15 to -1)

By close proximity, we mean that the integers being compared differ very little in value. Depending on the range, we have allowed various differences between the two integers being compared. In the first set where the range is -9 to 9, the difference between the two numbers is always 1. With the largest range, a difference of up to 5 is allowed. These worksheets will help students further hone their ability to visualize and conceptualize the idea of negative numbers and will serve as a foundation for all the other worksheets on this page.

- Comparing Integers in Close Proximity Comparing Positive and Negative Integers (-9 to +9) in Close Proximity Comparing Positive and Negative Integers (-15 to +15) in Close Proximity Comparing Positive and Negative Integers (-25 to +25) in Close Proximity Comparing Positive and Negative Integers (-50 to +50) in Close Proximity Comparing Positive and Negative Integers (-99 to +99) in Close Proximity
- Ordering Integers Worksheets Ordering Integers on a Number Line Ordering Integers (range -9 to 9) Ordering Integers (range -20 to 20) Ordering Integers (range -50 to 50) Ordering Integers (range -99 to 99) Ordering Integers (range -999 to 999) Ordering Negative Integers (range -9 to -1) Ordering Negative Integers (range -99 to -10) Ordering Negative Integers (range -999 to -100)

## Adding and Subtracting Integers

Two-color counters are fantastic manipulatives for teaching and learning about integer addition. Two-color counters are usually plastic chips that come with yellow on one side and red on the other side. They might be available in other colors, so you'll have to substitute your own colors in the following description.

Adding with two-color counters is actually quite easy. You model the first number with a pile of chips flipped to the correct side and you also model the second number with a pile of chips flipped to the correct side; then you mash them all together, take out the zeros (if any) and behold, you have your answer! Need further elaboration? Read on!

The correct side means using red to model negative numbers and yellow to model positive numbers. You would model —5 with five red chips and 7 with seven yellow chips. Mashing them together should be straight forward although, you'll want to caution your students to be less exuberant than usual, so none of the chips get flipped. Taking out the zeros means removing as many pairs of yellow and red chips as you can. You can do this because —1 and 1 when added together equals zero (this is called the zero principle). If you remove the zeros, you don't affect the answer. The benefit of removing the zeros, however, is that you always end up with only one color and as a consequence, the answer to the integer question. If you have no chips left at the end, the answer is zero!

- Adding Integers Worksheets with 75 Questions Per Page (Some Parentheses) Adding Integers from -9 to 9 (75 Questions) ✎ Adding Integers from -12 to 12 (75 Questions) ✎ Adding Integers from -15 to 15 (75 Questions) ✎ Adding Integers from -20 to 20 (75 Questions) ✎ Adding Integers from -25 to 25 (75 Questions) ✎ Adding Integers from -50 to 50 (75 Questions) ✎ Adding Integers from -99 to 99 (75 Questions) ✎
- Adding Integers Worksheets with 75 Questions Per Page (All Parentheses) Adding Integers from (-9) to (+9) All Parentheses (75 Questions) ✎ Adding Integers from (-12) to (+12) All Parentheses (75 Questions) ✎ Adding Integers from (-15) to (+15) All Parentheses (75 Questions) ✎ Adding Integers from (-20) to (+20) All Parentheses (75 Questions) ✎ Adding Integers from (-25) to (+25) All Parentheses (75 Questions) ✎ Adding Integers from (-50) to (+50) All Parentheses (75 Questions) ✎ Adding Integers from (-99) to (+99) All Parentheses (75 Questions) ✎
- Adding Integers Worksheets with 75 Questions Per Page (No Parentheses) Adding Integers from -9 to 9 No Parentheses (75 Questions) ✎ Adding Integers from -12 to 12 No Parentheses (75 Questions) ✎ Adding Integers from -15 to 15 No Parentheses (75 Questions) ✎ Adding Integers from -20 to 20 No Parentheses (75 Questions) ✎ Adding Integers from -25 to 25 No Parentheses (75 Questions) ✎ Adding Integers from -50 to 50 No Parentheses (75 Questions) ✎ Adding Integers from -99 to 99 No Parentheses (75 Questions) ✎
- Adding Integers Worksheets with 50 Questions Per Page (Some Parentheses) Adding Integers from -9 to 9 (50 Questions) ✎ Adding Integers from -12 to 12 (50 Questions) ✎ Adding Integers from -15 to 15 (50 Questions) ✎ Adding Integers from -20 to 20 (50 Questions) ✎ Adding Integers from -25 to 25 (50 Questions) ✎ Adding Integers from -50 to 50 (50 Questions) ✎ Adding Integers from -99 to 99 (50 Questions) ✎
- Adding Integers Worksheets with 50 Questions Per Page (All Parentheses) Adding Integers from (-9) to (+9) All Parentheses (50 Questions) ✎ Adding Integers from (-12) to (+12) All Parentheses (50 Questions) ✎ Adding Integers from (-15) to (+15) All Parentheses (50 Questions) ✎ Adding Integers from (-20) to (+20) All Parentheses (50 Questions) ✎ Adding Integers from (-25) to (+25) All Parentheses (50 Questions) ✎ Adding Integers from (-50) to (+50) All Parentheses (50 Questions) ✎ Adding Integers from (-99) to (+99) All Parentheses (50 Questions) ✎
- Adding Integers Worksheets with 50 Questions Per Page (No Parentheses) Adding Integers from -9 to 9 No Parentheses (50 Questions) ✎ Adding Integers from -12 to 12 No Parentheses (50 Questions) ✎ Adding Integers from -15 to 15 No Parentheses (50 Questions) ✎ Adding Integers from -20 to 20 No Parentheses (50 Questions) ✎ Adding Integers from -25 to 25 No Parentheses (50 Questions) ✎ Adding Integers from -50 to 50 No Parentheses (50 Questions) ✎ Adding Integers from -99 to 99 No Parentheses (50 Questions) ✎
- Adding Integers Worksheets with 25 Large Print Questions Per Page (Some Parentheses) Adding Integers from -9 to 9 (Large Print; 25 Questions) ✎ Adding Integers from -12 to 12 (Large Print; 25 Questions) ✎ Adding Integers from -15 to 15 (Large Print; 25 Questions) ✎ Adding Integers from -20 to 20 (Large Print; 25 Questions) ✎ Adding Integers from -25 to 25 (Large Print; 25 Questions) ✎ Adding Integers from -50 to 50 (Large Print; 25 Questions) ✎ Adding Integers from -99 to 99 (Large Print; 25 Questions) ✎
- Adding Integers Worksheets with 25 Large Print Questions Per Page (All Parentheses) Adding Integers from (-9) to (+9) All Parentheses (Large Print; 25 Questions) ✎ Adding Integers from (-12) to (+12) All Parentheses (Large Print; 25 Questions) ✎ Adding Integers from (-15) to (+15) All Parentheses (Large Print; 25 Questions) ✎ Adding Integers from (-20) to (+20) All Parentheses (Large Print; 25 Questions) ✎ Adding Integers from (-25) to (+25) All Parentheses (Large Print; 25 Questions) ✎ Adding Integers from (-50) to (+50) All Parentheses (Large Print; 25 Questions) ✎ Adding Integers from (-99) to (+99) All Parentheses (Large Print; 25 Questions) ✎
- Adding Integers Worksheets with 25 Large Print Questions Per Page (No Parentheses) Adding Integers from -9 to 9 No Parentheses (Large Print; 25 Questions) ✎ Adding Integers from -12 to 12 No Parentheses (Large Print; 25 Questions) ✎ Adding Integers from -15 to 15 No Parentheses (Large Print; 25 Questions) ✎ Adding Integers from -20 to 20 No Parentheses (Large Print; 25 Questions) ✎ Adding Integers from -25 to 25 No Parentheses (Large Print; 25 Questions) ✎ Adding Integers from -50 to 50 No Parentheses (Large Print; 25 Questions) ✎ Adding Integers from -99 to 99 No Parentheses (Large Print; 25 Questions) ✎
- Vertically Arranged Integer Addition Worksheets 3-Digit Integer Addition (Vertically Arranged) 3-Digit Positive Plus a Negative Integer Addition (Vertically Arranged) 3-Digit Negative Plus a Positive Integer Addition (Vertically Arranged) 3-Digit Negative Plus a Negative Integer Addition (Vertically Arranged)

Subtracting with integer chips is a little different. Integer subtraction can be thought of as removing. To subtract with integer chips, begin by modeling the first number (the minuend) with integer chips. Next, remove the chips that would represent the second number from your pile and you will have your answer. Unfortunately, that isn't all there is to it. This works beautifully if you have enough of the right color chip to remove, but often times you don't. For example, 5 - (-5), would require five yellow chips to start and would also require the removal of five red chips, but there aren't any red chips! Thank goodness, we have the zero principle. Adding or subtracting zero (a red chip and a yellow chip) has no effect on the original number, so we could add as many zeros as we wanted to the pile, and the number would still be the same. All that is needed then is to add as many zeros (pairs of red and yellow chips) as needed until there are enough of the correct color chip to remove. In our example 5 - (-5), you would add 5 zeros, so that you could remove five red chips. You would then be left with 10 yellow chips (or +10) which is the answer to the question.

- Subtracting Integers Worksheets with 75 Questions Per Page (Some Parentheses) Subtracting Integers from -9 to 9 (75 Questions) ✎ Subtracting Integers from -12 to 12 (75 Questions) ✎ Subtracting Integers from -15 to 15 (75 Questions) ✎ Subtracting Integers from -20 to 20 (75 Questions) ✎ Subtracting Integers from -25 to 25 (75 Questions) ✎ Subtracting Integers from -50 to 50 (75 Questions) ✎ Subtracting Integers from -99 to 99 (75 Questions) ✎
- Subtracting Integers Worksheets with 75 Questions Per Page (All Parentheses) Subtracting Integers from (-9) to (+9) All Parentheses (75 Questions) ✎ Subtracting Integers from (-12) to (+12) All Parentheses (75 Questions) ✎ Subtracting Integers from (-15) to (+15) All Parentheses (75 Questions) ✎ Subtracting Integers from (-20) to (+20) All Parentheses (75 Questions) ✎ Subtracting Integers from (-25) to (+25) All Parentheses (75 Questions) ✎ Subtracting Integers from (-50) to (+50) All Parentheses (75 Questions) ✎ Subtracting Integers from (-99) to (+99) All Parentheses (75 Questions) ✎
- Subtracting Integers Worksheets with 75 Questions Per Page (No Parentheses) Subtracting Integers from -9 to 9 No Parentheses (75 Questions) ✎ Subtracting Integers from -12 to 12 No Parentheses (75 Questions) ✎ Subtracting Integers from -15 to 15 No Parentheses (75 Questions) ✎ Subtracting Integers from -20 to 20 No Parentheses (75 Questions) ✎ Subtracting Integers from -25 to 25 No Parentheses (75 Questions) ✎ Subtracting Integers from -50 to 50 No Parentheses (75 Questions) ✎ Subtracting Integers from -99 to 99 No Parentheses (75 Questions) ✎
- Subtracting Integers Worksheets with 50 Questions Per Page (Some Parentheses) Subtracting Integers from -9 to 9 (50 Questions) ✎ Subtracting Integers from -12 to 12 (50 Questions) ✎ Subtracting Integers from -15 to 15 (50 Questions) ✎ Subtracting Integers from -20 to 20 (50 Questions) ✎ Subtracting Integers from -25 to 25 (50 Questions) ✎ Subtracting Integers from -50 to 50 (50 Questions) ✎ Subtracting Integers from -99 to 99 (50 Questions) ✎
- Subtracting Integers Worksheets with 50 Questions Per Page (All Parentheses) Subtracting Integers from (-9) to (+9) All Parentheses (50 Questions) ✎ Subtracting Integers from (-12) to (+12) All Parentheses (50 Questions) ✎ Subtracting Integers from (-15) to (+15) All Parentheses (50 Questions) ✎ Subtracting Integers from (-20) to (+20) All Parentheses (50 Questions) ✎ Subtracting Integers from (-25) to (+25) All Parentheses (50 Questions) ✎ Subtracting Integers from (-50) to (+50) All Parentheses (50 Questions) ✎ Subtracting Integers from (-99) to (+99) All Parentheses (50 Questions) ✎
- Subtracting Integers Worksheets with 50 Questions Per Page (No Parentheses) Subtracting Integers from -9 to 9 No Parentheses (50 Questions) ✎ Subtracting Integers from -12 to 12 No Parentheses (50 Questions) ✎ Subtracting Integers from -15 to 15 No Parentheses (50 Questions) ✎ Subtracting Integers from -20 to 20 No Parentheses (50 Questions) ✎ Subtracting Integers from -25 to 25 No Parentheses (50 Questions) ✎ Subtracting Integers from -50 to 50 No Parentheses (50 Questions) ✎ Subtracting Integers from -99 to 99 No Parentheses (50 Questions) ✎
- Subtracting Integers Worksheets with 25 Large Print Questions Per Page (Some Parentheses) Subtracting Integers from -9 to 9 (Large Print; 25 Questions) ✎ Subtracting Integers from -12 to 12 (Large Print; 25 Questions) ✎ Subtracting Integers from -15 to 15 (Large Print; 25 Questions) ✎ Subtracting Integers from -20 to 20 (Large Print; 25 Questions) ✎ Subtracting Integers from -25 to 25 (Large Print; 25 Questions) ✎ Subtracting Integers from -50 to 50 (Large Print; 25 Questions) ✎ Subtracting Integers from -99 to 99 (Large Print; 25 Questions) ✎
- Subtracting Integers Worksheets with 25 Large Print Questions Per Page (All Parentheses) Subtracting Integers from (-9) to (+9) All Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-12) to (+12) All Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-15) to (+15) All Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-20) to (+20) All Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-25) to (+25) All Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-50) to (+50) All Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-99) to (+99) All Parentheses (Large Print; 25 Questions) ✎
- Subtracting Integers Worksheets with 25 Large Print Questions Per Page (No Parentheses) Subtracting Integers from (-9) to 9 No Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-12) to 12 No Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-15) to 15 No Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-20) to 20 No Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-25) to 25 No Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-50) to 50 No Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-99) to 99 No Parentheses (Large Print; 25 Questions) ✎
- Vertically Arranged Integer Subtraction Worksheets 3-Digit Integer Subtraction (Vertically Arranged) 3-Digit Positive Minus a Positive Integer Subtraction (Vertically Arranged) 3-Digit Positive Minus a Negative Integer Subtraction (Vertically Arranged) 3-Digit Negative Minus a Positive Integer Subtraction (Vertically Arranged) 3-Digit Negative Minus a Negative Integer Subtraction (Vertically Arranged)

The worksheets in this section include addition and subtraction on the same page. Students will have to pay close attention to the signs and apply their knowledge of integer addition and subtraction to each question. The use of counters or number lines could be helpful to some students.

- Adding and Subtracting Integers Worksheets with 75 Questions Per Page (Some Parentheses) Adding & Subtracting Integers from -9 to 9 (75 Questions) ✎ Adding & Subtracting Integers from -10 to 10 (75 Questions) ✎ Adding & Subtracting Integers from -12 to 12 (75 Questions) ✎ Adding & Subtracting Integers from -15 to 15 (75 Questions) ✎ Adding & Subtracting Integers from -20 to 20 (75 Questions) ✎ Adding & Subtracting Integers from -25 to 25 (75 Questions) ✎ Adding & Subtracting Integers from -50 to 50 (75 Questions) ✎ Adding & Subtracting Integers from -99 to 99 (75 Questions) ✎
- Adding and Subtracting Integers Worksheets with 75 Questions Per Page (All Parentheses) Adding & Subtracting Integers from (-5) to (+5) All Parentheses (75 Questions) ✎ Adding & Subtracting Integers from (-9) to (+9) All Parentheses (75 Questions) ✎ Adding & Subtracting Integers from (-12) to (+12) All Parentheses (75 Questions) ✎ Adding & Subtracting Integers from (-15) to (+15) All Parentheses (75 Questions) ✎ Adding & Subtracting Integers from (-20) to (+20) All Parentheses (75 Questions) ✎ Adding & Subtracting Integers from (-25) to (+25) All Parentheses (75 Questions) ✎ Adding & Subtracting Integers from (-50) to (+50) All Parentheses (75 Questions) ✎ Adding & Subtracting Integers from (-99) to (+99) All Parentheses (75 Questions) ✎
- Adding and Subtracting Integers Worksheets with 75 Questions Per Page (No Parentheses) Adding & Subtracting Integers from -9 to 9 No Parentheses (75 Questions) ✎ Adding & Subtracting Integers from -12 to 12 No Parentheses (75 Questions) ✎ Adding & Subtracting Integers from -15 to 15 No Parentheses (75 Questions) ✎ Adding & Subtracting Integers from -20 to 20 No Parentheses (75 Questions) ✎ Adding & Subtracting Integers from -25 to 25 No Parentheses (75 Questions) ✎ Adding & Subtracting Integers from -50 to 50 No Parentheses (75 Questions) ✎ Adding & Subtracting Integers from -99 to 99 No Parentheses (75 Questions) ✎
- Adding and Subtracting Integers Worksheets with 50 Questions Per Page (Some Parentheses) Adding & Subtracting Integers from -9 to 9 (50 Questions) ✎ Adding & Subtracting Integers from -12 to 12 (50 Questions) ✎ Adding & Subtracting Integers from -15 to 15 (50 Questions) ✎ Adding & Subtracting Integers from -20 to 20 (50 Questions) ✎ Adding & Subtracting Integers from -25 to 25 (50 Questions) ✎ Adding & Subtracting Integers from -50 to 50 (50 Questions) ✎ Adding & Subtracting Integers from -99 to 99 (50 Questions) ✎
- Adding and Subtracting Integers Worksheets with 50 Questions Per Page (All Parentheses) Adding & Subtracting Integers from (-9) to (+9) All Parentheses (50 Questions) ✎ Adding & Subtracting Integers from (-12) to (+12) All Parentheses (50 Questions) ✎ Adding & Subtracting Integers from (-15) to (+15) All Parentheses (50 Questions) ✎ Adding & Subtracting Integers from (-20) to (+20) All Parentheses (50 Questions) ✎ Adding & Subtracting Integers from (-25) to (+25) All Parentheses (50 Questions) ✎ Adding & Subtracting Integers from (-50) to (+50) All Parentheses (50 Questions) ✎ Adding & Subtracting Integers from (-99) to (+99) All Parentheses (50 Questions) ✎
- Adding and Subtracting Integers Worksheets with 50 Questions Per Page (No Parentheses) Adding & Subtracting Integers from -9 to 9 No Parentheses (50 Questions) ✎ Adding & Subtracting Integers from -12 to 12 No Parentheses (50 Questions) ✎ Adding & Subtracting Integers from -15 to 15 No Parentheses (50 Questions) ✎ Adding & Subtracting Integers from -20 to 20 No Parentheses (50 Questions) ✎ Adding & Subtracting Integers from -25 to 25 No Parentheses (50 Questions) ✎ Adding & Subtracting Integers from -50 to 50 No Parentheses (50 Questions) ✎ Adding & Subtracting Integers from -99 to 99 No Parentheses (50 Questions) ✎
- Adding and Subtracting Integers Worksheets with 25 Large Print Questions Per Page (Some Parentheses) Adding & Subtracting Integers from -9 to 9 (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from -12 to 12 (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from -15 to 15 (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from -20 to 20 (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from -25 to 25 (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from -50 to 50 (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from -99 to 99 (Large Print; 25 Questions) ✎
- Adding and Subtracting Integers Worksheets with 25 Large Print Questions Per Page (All Parentheses) Adding & Subtracting Integers from (-9) to (+9) All Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-12) to (+12) All Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-15) to (+15) All Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-20) to (+20) All Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-25) to (+25) All Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-50) to (+50) All Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-99) to (+99) All Parentheses (Large Print; 25 Questions) ✎
- Adding and Subtracting Integers Worksheets with 25 Large Print Questions Per Page (No Parentheses) Adding & Subtracting Integers from (-9) to 9 No Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-12) to 12 No Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-15) to 15 No Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-20) to 20 No Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-25) to 25 No Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-50) to 50 No Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-99) to 99 No Parentheses (Large Print; 25 Questions) ✎

These worksheets include groups of questions that all result in positive or negative sums or differences. They can be used to help students see more clearly how certain integer questions end up with positive and negative results. In the case of addition of negative and positive integers, some people suggest looking for the "heavier" value to determine whether the sum will be positive of negative. More technically, it would be the integer with the greater absolute value. For example, in the question (−2) + 5, the absolute value of the positive integer is greater, so the sum will be positive.

In subtraction questions, the focus is on the subtrahend (the value being subtracted). In positive minus positive questions, if the subtrahend is greater than the minuend, the answer will be negative. In negative minus negative questions, if the subtrahend has a greater absolute value, the answer will be positive. Vice-versa for both situations. Alternatively, students can always convert subtraction questions to addition questions by changing the signs (e.g. (−5) − (−7) is the same as (−5) + 7; 3 − 5 is the same as 3 + (−5)).

- Scaffolded Integer Addition and Subtraction Positive Plus Negative Integer Addition (Scaffolded) ✎ Negative Plus Positive Integer Addition (Scaffolded) ✎ Mixed Integer Addition (Scaffolded) ✎ Positive Minus Positive Integer Subtraction (Scaffolded) ✎ Negative Minus Negative Integer Subtraction (Scaffolded) ✎

## Multiplying and Dividing Integers

Multiplying integers is very similar to multiplication facts except students need to learn the rules for the negative and positive signs. In short, they are:

In words, multiplying two positives or two negatives together results in a positive product, and multiplying a negative and a positive in either order results in a negative product. So, -8 × 8, 8 × (-8), -8 × (-8) and 8 × 8 all result in an absolute value of 64, but in two cases, the answer is positive (64) and in two cases the answer is negative (-64).

Should you wish to develop some "real-world" examples of integer multiplication, it might be a stretch due to the abstract nature of negative numbers. Sure, you could come up with some scenario about owing a debt and removing the debt in previous months, but this may only result in confusion. For now students can learn the rules of multiplying integers and worry about the analogies later!

- Multiplying Integers with 100 Questions Per Page Multiplying Mixed Integers from -9 to 9 (100 Questions) ✎ Multiplying Positive by Negative Integers from -9 to 9 (100 Questions) ✎ Multiplying Negative by Positive Integers from -9 to 9 (100 Questions) ✎ Multiplying Negative by Negative Integers from -9 to 9 (100 Questions) ✎ Multiplying Mixed Integers from -12 to 12 (100 Questions) ✎ Multiplying Positive by Negative Integers from -12 to 12 (100 Questions) ✎ Multiplying Negative by Positive Integers from -12 to 12 (100 Questions) ✎ Multiplying Negative by Negative Integers from -12 to 12 (100 Questions) ✎ Multiplying Mixed Integers from -20 to 20 (100 Questions) ✎ Multiplying Mixed Integers from -50 to 50 (100 Questions) ✎
- Multiplying Integers with 50 Questions Per Page Multiplying Mixed Integers from -9 to 9 (50 Questions) ✎ Multiplying Positive by Negative Integers from -9 to 9 (50 Questions) ✎ Multiplying Negative by Positive Integers from -9 to 9 (50 Questions) ✎ Multiplying Negative by Negative Integers from -9 to 9 (50 Questions) ✎ Multiplying Mixed Integers from -12 to 12 (50 Questions) ✎ Multiplying Positive by Negative Integers from -12 to 12 (50 Questions) ✎ Multiplying Negative by Positive Integers from -12 to 12 (50 Questions) ✎ Multiplying Negative by Negative Integers from -12 to 12 (50 Questions) ✎
- Multiplying Integers with 25 Large Print Questions Per Page Multiplying Mixed Integers from -9 to 9 (25 Questions; Large Print) ✎ Multiplying Positive by Negative Integers from -9 to 9 (25 Questions; Large Print) ✎ Multiplying Negative by Positive Integers from -9 to 9 (25 Questions; Large Print) ✎ Multiplying Negative by Negative Integers from -9 to 9 (25 Questions; Large Print) ✎ Multiplying Mixed Integers from -12 to 12 (25 Questions; Large Print) ✎ Multiplying Positive by Negative Integers from -12 to 12 (25 Questions; Large Print) ✎ Multiplying Negative by Positive Integers from -12 to 12 (25 Questions; Large Print) ✎ Multiplying Negative by Negative Integers from -12 to 12 (25 Questions; Large Print) ✎

Luckily (for your students), the rules of dividing integers are the same as the rules for multiplying:

Dividing a positive by a positive integer or a negative by a negative integer will result in a positive integer. Dividing a negative by a positive integer or a positive by a negative integer will result in a negative integer. A good grasp of division facts and a knowledge of the rules for multiplying and dividing integers will go a long way in helping your students master integer division. Use the worksheets in this section to guide students along.

- Dividing Integers with 100 Questions Per Page Dividing Mixed Integers from -9 to 9 (100 Questions) ✎ Dividing Positive by Negative Integers from -9 to 9 (100 Questions) ✎ Dividing Negative by Positive Integers from -9 to 9 (100 Questions) ✎ Dividing Negative by Negative Integers from -9 to 9 (100 Questions) ✎ Dividing Mixed Integers from -12 to 12 (100 Questions) ✎ Dividing Positive by Negative Integers from -12 to 12 (100 Questions) ✎ Dividing Negative by Positive Integers from -12 to 12 (100 Questions) ✎ Dividing Negative by Negative Integers from -12 to 12 (100 Questions) ✎
- Dividing Integers with 50 Questions Per Page Dividing Mixed Integers from -9 to 9 (50 Questions) ✎ Dividing Positive by Negative Integers from -9 to 9 (50 Questions) ✎ Dividing Negative by Positive Integers from -9 to 9 (50 Questions) ✎ Dividing Negative by Negative Integers from -9 to 9 (50 Questions) ✎ Dividing Mixed Integers from -12 to 12 (50 Questions) ✎ Dividing Positive by Negative Integers from -12 to 12 (50 Questions) ✎ Dividing Negative by Positive Integers from -12 to 12 (50 Questions) ✎ Dividing Negative by Negative Integers from -12 to 12 (50 Questions) ✎
- Dividing Integers with 25 Large Print Questions Per Page Dividing Mixed Integers from -9 to 9 (25 Questions; Large Print) ✎ Dividing Positive by Negative Integers from -9 to 9 (25 Questions; Large Print) ✎ Dividing Negative by Positive Integers from -9 to 9 (25 Questions; Large Print) ✎ Dividing Negative by Negative Integers from -9 to 9 (25 Questions; Large Print) ✎ Dividing Mixed Integers from -12 to 12 (25 Questions; Large Print) ✎ Dividing Positive by Negative Integers from -12 to 12 (25 Questions; Large Print) ✎ Dividing Negative by Positive Integers from -12 to 12 (25 Questions; Large Print) ✎ Dividing Negative by Negative Integers from -12 to 12 (25 Questions; Large Print) ✎

This section includes worksheets with both multiplying and dividing integers on the same page. As long as students know their facts and the integer rules for multiplying and dividing, their sole worry will be to pay attention to the operation signs.

- Multiplying and Dividing Integers with 100 Questions Per Page Multiplying and Dividing Mixed Integers from -9 to 9 (100 Questions) ✎ Multiplying and Dividing Positive and Negative Integers from -9 to 9 (100 Questions) ✎ Multiplying and Dividing Negative and Positive Integers from -9 to 9 (100 Questions) ✎ Multiplying and Dividing Negative and Negative Integers from -9 to 9 (100 Questions) ✎ Multiplying and Dividing Mixed Integers from -12 to 12 (100 Questions) ✎ Multiplying and Dividing Positive and Negative Integers from -12 to 12 (100 Questions) ✎ Multiplying and Dividing Negative and Positive Integers from -12 to 12 (100 Questions) ✎ Multiplying and Dividing Negative and Negative Integers from -12 to 12 (100 Questions) ✎
- Multiplying and Dividing Integers with 75 Questions Per Page Multiplying and Dividing Mixed Integers from -9 to 9 (75 Questions) ✎ Multiplying and Dividing Positive and Negative Integers from -9 to 9 (75 Questions) ✎ Multiplying and Dividing Negative and Positive Integers from -9 to 9 (75 Questions) ✎ Multiplying and Dividing Negative and Negative Integers from -9 to 9 (75 Questions) ✎ Multiplying and Dividing Mixed Integers from -12 to 12 (75 Questions) ✎ Multiplying and Dividing Positive and Negative Integers from -12 to 12 (75 Questions) ✎ Multiplying and Dividing Negative and Positive Integers from -12 to 12 (75 Questions) ✎ Multiplying and Dividing Negative and Negative Integers from -12 to 12 (75 Questions) ✎
- Multiplying and Dividing Integers with 50 Questions Per Page Multiplying and Dividing Mixed Integers from -9 to 9 (50 Questions) ✎ Multiplying and Dividing Positive and Negative Integers from -9 to 9 (50 Questions) ✎ Multiplying and Dividing Negative and Positive Integers from -9 to 9 (50 Questions) ✎ Multiplying and Dividing Negative and Negative Integers from -9 to 9 (50 Questions) ✎ Multiplying and Dividing Mixed Integers from -12 to 12 (50 Questions) ✎ Multiplying and Dividing Positive and Negative Integers from -12 to 12 (50 Questions) ✎ Multiplying and Dividing Negative and Positive Integers from -12 to 12 (50 Questions) ✎ Multiplying and Dividing Negative and Negative Integers from -12 to 12 (50 Questions) ✎
- Multiplying and Dividing Integers with 25 Large Print Questions Per Page Multiplying and Dividing Mixed Integers from -9 to 9 (25 Questions; Large Print) ✎ Multiplying and Dividing Positive and Negative Integers from -9 to 9 (25 Questions; Large Print) ✎ Multiplying and Dividing Negative and Positive Integers from -9 to 9 (25 Questions; Large Print) ✎ Multiplying and Dividing Negative and Negative Integers from -9 to 9 (25 Questions; Large Print) ✎ Multiplying and Dividing Mixed Integers from -12 to 12 (25 Questions; Large Print) ✎ Multiplying and Dividing Positive and Negative Integers from -12 to 12 (25 Questions; Large Print) ✎ Multiplying and Dividing Negative and Positive Integers from -12 to 12 (25 Questions; Large Print) ✎ Multiplying and Dividing Negative and Negative Integers from -12 to 12 (25 Questions; Large Print) ✎

## All Operations with Integers

In this section, the integers math worksheets include all of the operations. Students will need to pay attention to the operations and the signs and use mental math or another strategy to arrive at the correct answers. It should go without saying that students need to know their basic addition, subtraction, multiplication and division facts and rules regarding operations with integers before they should complete any of these worksheets independently. Of course, the worksheets can be used as a source of questions for lessons, tests or other learning activities.

- All Operations with Integers with 50 Questions Per Page (Some Parentheses) All operations with integers from -9 to 9 (50 Questions) ✎ All operations with integers from -12 to 12 (50 Questions) ✎ All operations with integers from -15 to 15 (50 Questions) ✎ All operations with integers from -20 to 20 (50 Questions) ✎ All operations with integers from -25 to 25 (50 Questions) ✎ All operations with integers from -50 to 50 (50 Questions) ✎ All operations with integers from -99 to 99 (50 Questions) ✎
- All Operations with Integers with 50 Questions Per Page (All Parentheses) All operations with integers from (-9) to (+9) All Parentheses (50 Questions) ✎ All operations with integers from (-12) to (+12) All Parentheses (50 Questions) ✎ All operations with integers from (-15) to (+15) All Parentheses (50 Questions) ✎ All operations with integers from (-20) to (+20) All Parentheses (50 Questions) ✎ All operations with integers from (-25) to (+25) All Parentheses (50 Questions) ✎ All operations with integers from (-50) to (+50) All Parentheses (50 Questions) ✎ All operations with integers from (-99) to (+99) All Parentheses (50 Questions) ✎
- All Operations with Integers with 50 Questions Per Page (No Parentheses) All operations with integers from -9 to 9 No Parentheses (50 Questions) ✎ All operations with integers from -12 to 12 No Parentheses (50 Questions) ✎ All operations with integers from -15 to 15 No Parentheses (50 Questions) ✎ All operations with integers from -20 to 20 No Parentheses (50 Questions) ✎ All operations with integers from -25 to 25 No Parentheses (50 Questions) ✎ All operations with integers from -50 to 50 No Parentheses (50 Questions) ✎ All operations with integers from -99 to 99 No Parentheses (50 Questions) ✎

Order of operations with integers can be found on the Order of Operations page:

Order of Operations with Integers

Copyright © 2005-2024 Math-Drills.com You may use the math worksheets on this website according to our Terms of Use to help students learn math.

## Integer Word Problems Worksheets

An integer is defined as a number that can be written without a fractional component. For example, 11, 8, 0, and −1908 are integers whereas √5, Π are not integers. The set of integers consists of zero, the positive natural numbers, and their additive inverses. Integers are closed under the operations of addition and multiplication . Integer word problems worksheets provide a variety of word problems associated with the use and properties of integers.

## Benefits of Integers Word Problems Worksheets

We use integers in our day-to-day life like measuring temperature, sea level, and speed limit. Translating verbal descriptions into expressions is an essential initial step in solving word problems. Deposits are normally represented by a positive sign and withdrawals are denoted by a negative sign. Negative numbers are used in weather forecasting to show the temperature of a region. Solving these integers word problems will help us relate the concept with practical applications.

## Download Integers Word Problems Worksheet PDFs

These math worksheets should be practiced regularly and are free to download in PDF formats.

## ☛ Check Grade wise Integers Word Problems Worksheets

- 6th Grade Integers Worksheets
- Integers Worksheets for Grade 7
- 8th Grade Integers Worksheets

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## 1.3: Integers

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By the end of this section, you will be able to:

- Simplify expressions with absolute value
- Add and subtract integers
- Multiply and divide integers
- Simplify expressions with integers
- Evaluate variable expressions with integers
- Translate phrases to expressions with integers
- Use integers in applications

A more thorough introduction to the topics covered in this section can be found in the Elementary Algebra chapter, Foundations.

## Simplify Expressions with Absolute Value

A negative number is a number less than 0. The negative numbers are to the left of zero on the number line ( Figure \(\PageIndex{1}\)).

You may have noticed that, on the number line, the negative numbers are a mirror image of the positive numbers, with zero in the middle. Because the numbers \(2\) and \(−2\) are the same distance from zero, each one is called the opposite of the other. The opposite of \(2\) is \(−2\), and the opposite of \(−2\) is \(2\).

The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.

Figure \(\PageIndex{2}\) illustrates the definition.

## OPPOSITE NOTATION

\[\begin{align} & -a \text{ means the opposite of the number }a \\ & \text{The notation} -a \text{ is read as “the opposite of }a \text{.”} \end{align} \]

We saw that numbers such as 3 and −3 are opposites because they are the same distance from 0 on the number line. They are both three units from 0. The distance between 0 and any number on the number line is called the absolute value of that number.

## Definition: ABSOLUTE VALUE

The absolute value of a number is its distance from 0 on the number line.

The absolute value of a number \(n\) is written as \(|n|\) and \(|n|≥0\) for all numbers.

Absolute values are always greater than or equal to zero.

For example,

\[\begin{align} & -5 \text{ is } 5 \text{ units away from 0, so } |-5|=5. \\ & 5 \text{ is }5\text{ units away from 0, so }|5|=5. \end{align}\]

Figure \(\PageIndex{3}\) illustrates this idea.

The absolute value of a number is never negative because distance cannot be negative. The only number with absolute value equal to zero is the number zero itself because the distance from 0 to 0 on the number line is zero units.

In the next example, we’ll order expressions with absolute values.

## EXAMPLE \(\PageIndex{1}\)

Fill in \(<,\,>,\) or \(=\) for each of the following pairs of numbers:

- \(\mathrm{|−5|}\_\_\mathrm{−|−5|}\_\_\mathrm{−|5|}\)
- \(\text{8__−|−8|}\)
- \(\text{−9__−|−9|}\)
- (\text{−(−16)__|−16|}\).

\(\begin{array}{lrcc} { \text{ } \\ \text{Simplify.} \\ \text{Order.} \\ \text{ } } & {|−5| \\ 5 \\ 5 \\ |−5|} & {\_\_ \\ \_\_ \\ > \\ >} & {−|−5| \\ −5 \\ −5 \\ −|−5|} \end{array}\)

\(\begin{array}{llcc} { \text{ } \\ \text{Simplify.} \\ \text{Order.} \\ \text{ } } & {8 \\ 8 \\ 8 \\ 8} & {\_\_ \\ \_\_ \\ > \\ >} & {−|−8| \\ −8 \\ −8 \\ −|−8|} \end{array}\)

\(\begin{array}{lrcc} { \text{ } \\ \text{Simplify.} \\ \text{Order.} \\ \text{ } } & {−9 \\ −9 \\ −9 \\ −9} & {\_\_ \\ \_\_ \\ = \\ =} & {−|−9| \\ −9 \\ −9 \\ −|−9|} \end{array}\)

\(\begin{array}{lrcc} { \text{ } \\ \text{Simplify.} \\ \text{Order.} \\ \text{ } } & {−(−16) \\ 16 \\ 16 \\ −(−16)} & {\_\_ \\ \_\_ \\ = \\ =} & {−|−16| \\ 16 \\ 16 \\ |−16|} \end{array}\)

## EXAMPLE \(\PageIndex{2}\)

ⓐ \(−9 \_\_−|−9|\) ⓑ \(2 \_\_−|−2|\) ⓒ \(−8 \_\_|−8|\) ⓓ \(−(−9) \_\_|−9|.\)

ⓐ \(>\) ⓑ \(>\) ⓒ \(<\)

ⓓ \(=\)

## EXAMPLE \(\PageIndex{3}\)

Fill in \(<,>,\) or \(=\) for each of the following pairs of numbers:

- \(7 \_\_ −|−7|\)
- \(−(−10) \_ \_|−10|\)
- \(|−4| \_\_ −|−4|\)
- \(−1 \_\_ |−1|.\)

ⓐ \(>\) ⓑ \(=\) ⓒ \(>\)

ⓓ \(<\)

We now add absolute value bars to our list of grouping symbols. When we use the order of operations, first we simplify inside the absolute value bars as much as possible, then we take the absolute value of the resulting number.

## GROUPING SYMBOLS

- \[\begin{array}{lclc} \text{Parentheses} & () & \text{Braces} & \{ \} \\ \text{Brackets} & [] & \text{Absolute value} & ||\end{array}\]

In the next example, we simplify the expressions inside absolute value bars first just like we do with parentheses.

## EXAMPLE \(\PageIndex{4}\)

Simplify: \(\mathrm{24−|19−3(6−2)|}\).

\(\begin{array}{lc} \text{} & 24−|19−3(6−2)| \\ \text{Work inside parentheses first:} & \text{} \\ \text{subtract 2 from 6.} & 24−|19−3(4)| \\ \text{Multiply 3(4).} & 24−|19−12| \\ \text{Subtract inside the absolute value bars.} & 24−|7| \\ \text{Take the absolute value.} & 24−7 \\ \text{Subtract.} & 17 \end{array}\)

## EXAMPLE \(\PageIndex{5}\)

Simplify: \(19−|11−4(3−1)|\).

## EXAMPLE \(\PageIndex{6}\)

Simplify: \(9−|8−4(7−5)|\).

## Add and Subtract Integers

So far in our examples, we have only used the counting numbers and the whole numbers.

\[\begin{array}{ll} \text{Counting numbers} & 1,2,3… \\ \text{Whole numbers} 0,1,2,3…. \end{array}\]

Our work with opposites gives us a way to define the integers . The whole numbers and their opposites are called the integers. The integers are the numbers \(…−3,−2,−1,0,1,2,3…\)

## Definition: INTEGERS

The whole numbers and their opposites are called the integers .

The integers are the numbers

\[…-3,-2,-1,0,1,2,3…,\]

Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more challenging.

We will use two color counters to model addition and subtraction of negatives so that you can visualize the procedures instead of memorizing the rules.

We let one color (blue) represent positive. The other color (red) will represent the negatives.

If we have one positive counter and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero.

We will use the counters to show how to add:

\[5+3 \; \; \; \; \; \; −5+(−3) \; \; \; \; \; \; −5+3 \; \; \; \; \; \; \; 5+(−3)\]

The first example, \(5+3,\) adds 5 positives and 3 positives—both positives.

The second example, \(−5+(−3),\) adds 5 negatives and 3 negatives—both negatives.

When the signs are the same, the counters are all the same color, and so we add them. In each case we get 8—either 8 positives or 8 negatives.

So what happens when the signs are different? Let’s add \(−5+3\) and \(5+(−3)\).

When we use counters to model addition of positive and negative integers, it is easy to see whether there are more positive or more negative counters. So we know whether the sum will be positive or negative.

## EXAMPLE \(\PageIndex{7}\)

Add: ⓐ \(−1+(−4)\) ⓑ \(−1+5\) ⓒ \(1+(−5)\).

## EXAMPLE \(\PageIndex{8}\)

Add: ⓐ \(−2+(−4)\) ⓑ \(−2+4\) ⓒ \(2+(−4)\).

ⓐ \(−6\) ⓑ \(2\) ⓒ \(−2\)

## EXAMPLE \(\PageIndex{9}\)

Add: ⓐ \(−2+(−5)\) ⓑ \(−2+5\) ⓒ \(2+(−5)\).

ⓐ \(−7\) ⓑ \(3\) ⓒ \(−3\)

We will continue to use counters to model the subtraction. Perhaps when you were younger, you read \(“5−3”\) as “5 take away 3.” When you use counters, you can think of subtraction the same way!

We will use the counters to show to subtract:

\[5−3 \; \; \; \; \; \; −5−(−3) \; \; \; \; \; \; −5−3 \; \; \; \; \; \; 5−(−3) \]

The first example, \(5−3\), we subtract 3 positives from 5 positives and end up with 2 positives.

In the second example, \(−5−(−3),\) we subtract 3 negatives from 5 negatives and end up with 2 negatives.

Each example used counters of only one color, and the “take away” model of subtraction was easy to apply.

What happens when we have to subtract one positive and one negative number? We’ll need to use both blue and red counters as well as some neutral pairs. If we don’t have the number of counters needed to take away, we add neutral pairs. Adding a neutral pair does not change the value. It is like changing quarters to nickels—the value is the same, but it looks different.

Let’s look at \(−5−3\) and \(5−(−3)\).

## EXAMPLE \(\PageIndex{10}\)

Subtract: ⓐ \(3−1\) ⓑ \(−3−(−1)\) ⓒ \(−3−1\) ⓓ \(3−(−1)\).

## EXAMPLE \(\PageIndex{11}\)

Subtract: ⓐ \(6−4\) ⓑ \(−6−(−4)\) ⓒ \(−6−4\) ⓓ \(6−(−4)\).

ⓐ \(2\) ⓑ \(−2\) ⓒ \(−10\) ⓓ \(10\)

## EXAMPLE \(\PageIndex{12}\)

Subtract: ⓐ \(7−4\) ⓑ \(−7−(−4)\) ⓒ \(−7−4\) ⓓ \(7−(−4)\).

ⓐ \(3\) ⓑ \(−3\) ⓒ \(−11\) ⓓ \(11\)

Have you noticed that subtraction of signed numbers can be done by adding the opposite ? In the last example, \(−3−1\) is the same as \(−3+(−1)\) and \(3−(−1)\) is the same as \(3+1\). You will often see this idea, the Subtraction Property, written as follows:

## Definition: SUBTRACTION PROPERTY

\[a−b=a+(−b)\]

Subtracting a number is the same as adding its opposite.

## EXAMPLE \(\PageIndex{13}\)

Simplify: ⓐ \(13−8\) and \(13+(−8)\) ⓑ \(−17−9\) and \(−17+(−9)\) ⓒ \(9−(−15)\) and \(9+15\) ⓓ \(−7−(−4)\) and \(−7+4\).

\(\begin{array}{lccc} \text{} & −17−9 & \text{and} & −17+(−9) \\ \text{Subtract.} & −26 & \text{} & −26 \end{array}\)

\(\begin{array}{lccc} \text{} & 9−(−15) & \text{and} & 9+15 \\ \text{Subtract.} & 24 & \text{} & 24 \end{array}\)

\(\begin{array}{lccc} \text{} & −7−(−4) & \text{and} & −7+4 \\ \text{Subtract.} & −3 & \text{} & −3 \end{array}\)

## EXAMPLE \(\PageIndex{14}\)

Simplify: ⓐ \(21−13\) and \(21+(−13)\) ⓑ \(−11−7\) and \(−11+(−7)\) ⓒ \(6−(−13)\) and \(6+13\) ⓓ \(−5−(−1)\) and \(−5+1\).

ⓐ \(8,8\) ⓑ \(−18,−18\)

ⓒ \(19,19\) ⓓ \(−4,−4\)

## EXAMPLE \(\PageIndex{15}\)

Simplify: ⓐ \(15−7\) and \(15+(−7)\) ⓑ \(−14−8\) and \(−14+(−8)\) ⓒ \(4−(−19)\) and \(4+19\) ⓓ \(−4−(−7)\) and \(−4+7\).

ⓐ \(8,8\) ⓑ \(−22,−22\)

ⓒ \(23,23\) ⓓ \(3,3\)

What happens when there are more than three integers? We just use the order of operations as usual.

## EXAMPLE \(\PageIndex{16}\)

Simplify: \(7−(−4−3)−9.\)

\(\begin{array}{lc} \text{} & 7−(−4−3)−9 \\ \text{Simplify inside the parentheses first.} & 7−(−7)−9 \\ \text{Subtract left to right.} & 14−9 \\ \text{Subtract.} & 5 \end{array}\)

Simplify: \(8−(−3−1)−9.\)

## EXAMPLE \(\PageIndex{18}\)

Simplify: \(12−(−9−6)−14.\)

## Multiply and Divide Integers

Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we are using the model just to help us discover the pattern.

We remember that a⋅ba·b means add a , b times .

The next two examples are more interesting. What does it mean to multiply 5 by −3? It means subtract 5,3 times. Looking at subtraction as “taking away”, it means to take away 5, 3 times. But there is nothing to take away, so we start by adding neutral pairs on the workspace.

In summary:

\[\begin{array}{ll} 5·3=15 & −5(3)=−15 \\ 5(−3)=−15 & (−5)(−3)=15 \end{array}\]

Notice that for multiplication of two signed numbers, when the

\[ \text{signs are the } \textbf{same} \text{, the product is } \textbf{positive.} \\ \text{signs are } \textbf{different} \text{, the product is } \textbf{negative.} \]

What about division? Division is the inverse operation of multiplication. So, \(15÷3=5\) because \(15·3=15\). In words, this expression says that 15 can be divided into 3 groups of 5 each because adding five three times gives 15. If you look at some examples of multiplying integers, you might figure out the rules for dividing integers.

\[\begin{array}{lclrccl} 5·3=15 & \text{so} & 15÷3=5 & \text{ } −5(3)=−15 & \text{so} & −15÷3=−5 \\ (−5)(−3)=15 & \text{so} & 15÷(−3)=−5 & \text{ } 5(−3)=−15 & \text{so} & −15÷(−3)=5 \end{array}\]

Division follows the same rules as multiplication with regard to signs.

## MULTIPLICATION AND DIVISION OF SIGNED NUMBERS

For multiplication and division of two signed numbers:

If the signs are the same, the result is positive.

If the signs are different, the result is negative.

## EXAMPLE \(\PageIndex{19}\)

Multiply or divide: ⓐ \(−100÷(−4)\) ⓑ \(7⋅6\) ⓒ \(4(−8)\) ⓓ \(−27÷3.\)

\(\begin{array}{lc} \text{} & −100÷(−4) \\ \text{Divide, with signs that are} \\ \text{the same the quotient is positive.} & 25 \end{array}\)

\(\begin{array} {lc} \text{} & 7·6 \\ \text{Multiply, with same signs.} & 42 \end{array}\)

\(\begin{array} {lc} \text{} & 4(−8) \\ \text{Multiply, with different signs.} & −32 \end{array}\)

\(\begin{array}{lc} \text{} & −27÷3 \\ \text{Divide, with different signs,} \\ \text{the quotient is negative.} & −9 \end{array}\)

## EXAMPLE \(\PageIndex{20}\)

Multiply or divide: ⓐ \(−115÷(−5)\) ⓑ \(5⋅12\) ⓒ \(9(−7)\) ⓓ\(−63÷7.\)

ⓐ 23 ⓑ 60 ⓒ −63 ⓓ −9

Multiply or divide: ⓐ \(−117÷(−3)\) ⓑ \(3⋅13\) ⓒ \(7(−4)\) ⓓ\(−42÷6\).

ⓐ 39 ⓑ 39 ⓒ −28 ⓓ −7

When we multiply a number by 1, the result is the same number. Each time we multiply a number by −1, we get its opposite!

## MULTIPLICATION BY −1

\[−1a=−a\]

Multiplying a number by \(−1\) gives its opposite.

## Simplify Expressions with Integers

What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember Please Excuse My Dear Aunt Sally?

Let’s try some examples. We’ll simplify expressions that use all four operations with integers—addition, subtraction, multiplication, and division. Remember to follow the order of operations.

## EXAMPLE \(\PageIndex{22}\)

Simplify: ⓐ \((−2)^4\) ⓑ \(−2^4\).

Notice the difference in parts (a) and (b). In part (a), the exponent means to raise what is in the parentheses, the −2 to the 4 th power. In part (b), the exponent means to raise just the 2 to the 4 th power and then take the opposite.

\(\begin{array}{lc} \text{} & −2^4 \\ \text{Write in expanded form.} & −(2·2·2·2) \\ \text{We are asked to find} & \text{} \\ \text{the opposite of }24. & \text{} \\ \text{Multiply.} & −(4·2·2) \\ \text{Multiply.} & −(8·2) \\ \text{Multiply.} & −16 \end{array}\)

Simplify: ⓐ \((−3)^4\) ⓑ \(−3^4\).

ⓐ 81 ⓑ −81

## EXAMPLE \(\PageIndex{24}\)

Simplify: ⓐ \((−7)^2\) ⓑ \(−7^2\).

ⓐ 49 ⓑ −49

The last example showed us the difference between \((−2)^4\) and \(−2^4\). This distinction is important to prevent future errors. The next example reminds us to multiply and divide in order left to right.

## EXAMPLE \(\PageIndex{25}\)

Simplify: ⓐ \(8(−9)÷(−2)^3\) ⓑ \(−30÷2+(−3)(−7)\).

\(\begin{array}{lc} \text{} & 8(−9)÷(−2)^3 \\ \text{Exponents first.} & 8(−9)÷(−8) \\ \text{Multiply.} & −72÷(−8) \\ \text{Divide.} & 9 \end{array}\)

\(\begin{array}{lc} \text{} & −30÷2+(−3)(−7) \\ \text{Multiply and divide} \\ \text{left to right, so divide first.} & −15+(−3)(−7) \\ \text{Multiply.} & −15+21 \\ \text{Add.} & 6 \end{array}\)

Simplify: ⓐ \(12(−9)÷(−3)^3\) ⓑ \(−27÷3+(−5)(−6).\)

ⓐ 4 ⓑ 21

## EXAMPLE \(\PageIndex{27}\)

Simplify: ⓐ \(18(−4)÷(−2)^3\) ⓑ \(−32÷4+(−2)(−7).\)

ⓐ 9 ⓑ 6

## Evaluate Variable Expressions with Integers

Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

## EXAMPLE \(\PageIndex{28}\)

Evaluate \(4x^2−2xy+3y^2\) when \(x=2,y=−1\).

## EXAMPLE \(\PageIndex{29}\)

Evaluate: \(3x^2−2xy+6y^2\) when \(x=1,y=−2\).

## EXAMPLE \(\PageIndex{30}\)

Evaluate: \(4x^2−xy+5y^2\) when \(x=−2,y=3\).

## Translate Phrases to Expressions with Integers

Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

## EXAMPLE \(\PageIndex{31}\)

Translate and simplify: the sum of 8 and −12, increased by 3.

\(\begin{array}{lc} \text{} & \text{the } \textbf{sum } \underline{\text{of}} \; –8 \; \underline{\text{and}} −12 \text{ increased by } 3 \\ \text{Translate.} & [8+(−12)]+3 \\ \text{Simplify. Be careful not to confuse the} \; \; \; \; \; \; \; \; \; \; & (−4)+3 \\ \text{brackets with an absolute value sign.} \\ \text{Add.} & −1 \end{array}\)

## EXAMPLE \(\PageIndex{32}\)

Translate and simplify the sum of 9 and −16, increased by 4.

\((9+(−16))+4;−3\)

## EXAMPLE \(\PageIndex{33}\)

Translate and simplify the sum of −8 and −12, increased by 7.

\((−8+(−12))+7;−13\)

## Use Integers in Applications

We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

## EXAMPLE \(\PageIndex{34}\): How to Solve Application Problems Using Integers

The temperature in Kendallville, Indiana one morning was 11 degrees. By mid-afternoon, the temperature had dropped to −9−9degrees. What was the difference in the morning and afternoon temperatures?

## EXAMPLE \(\PageIndex{35}\)

The temperature in Anchorage, Alaska one morning was 15 degrees. By mid-afternoon the temperature had dropped to 30 degrees below zero. What was the difference in the morning and afternoon temperatures?

The difference in temperatures was 45 degrees Fahrenheit.

## EXAMPLE \(\PageIndex{36}\)

The temperature in Denver was −6 degrees at lunchtime. By sunset the temperature had dropped to −15 degrees. What was the difference in the lunchtime and sunset temperatures?

The difference in temperatures was 9 degrees.

## USE INTEGERS IN APPLICATIONS.

- Read the problem. Make sure all the words and ideas are understood.
- Identify what we are asked to find.
- Write a phrase that gives the information to find it.
- Translate the phrase to an expression.
- Simplify the expression.
- Answer the question with a complete sentence.

Access this online resource for additional instruction and practice with integers.

- Subtracting Integers with Counters

## Key Concepts

- \[\begin{align} & −a \text{ means the opposite of the number }a \\ & \text{The notation} −a \text{ is read as “the opposite of }a \text{.”} \end{align} \]

The absolute value of a number n is written as \(|n|\) and \(|n|≥0\) for all numbers.

- Subtraction Property \(a−b=a+(−b)\) Subtracting a number is the same as adding its opposite.

\(−1a=−a\)

- Read the problem. Make sure all the words and ideas are understood

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## Unit 4: Integers

Negative numbers.

- Negative symbol as opposite (Opens a modal)
- Intro to negative numbers (Opens a modal)
- Number opposites challenge Get 3 of 4 questions to level up!
- Interpreting negative numbers (temperature and elevation) Get 3 of 4 questions to level up!

## Comparing integers

- Ordering negative numbers (Opens a modal)
- Negative numbers, variables, number line (Opens a modal)
- Ordering rational numbers Get 3 of 4 questions to level up!
- Compare rational numbers using a number line Get 3 of 4 questions to level up!

## Adding and subtracting

- Adding numbers with different signs (Opens a modal)
- Adding & subtracting negative numbers (Opens a modal)
- Adding & subtracting negative numbers Get 5 of 7 questions to level up!
- Addition & subtraction: find the missing value Get 3 of 4 questions to level up!

## Algebra: Consecutive Integer Problems

In these lessons, we will learn how to solve

- consecutive integer word problems
- consecutive even integer word problems
- consecutive odd integer word problems

Related Pages Integers Word Problems Consecutive Integer Word Problems Consecutive Integers 2 Consecutive Even Integer Problems Consecutive Odd Integer Problems More Algebra Word Problems

## What Are Consecutive Integers?

Consecutive integers are integers that follow in sequence, each number being 1 more than the previous number, represented by n, n + 1, n + 2, n + 3, …, where n is any integer. For example: 23, 24, 25, …

If we start with an even number and each number in the sequence is 2 more than the previous number then we will get consecutive even integers . For example: 16,18, 20, …

If we start with an odd number and each number in the sequence is 2 more than the previous number then we will get consecutive odd integers . For example: 33, 35, 37, …

The following diagram shows an example of a consecutive integer problem. Scroll down the page for more examples and solutions on consecutive integer problems.

Consecutive Integer Problems

Consecutive integer problems are word problems that involve consecutive integers .

The following are common examples of consecutive integer problems.

Example: The sum of the least and greatest of 3 consecutive integers is 60. What are the values of the 3 integers?

Solution: Step 1 : Assign variables: Let x = least integer x + 1 = middle integer x + 2 = greatest integer

Translate sentence into an equation. Sentence: The sum of the least and greatest is 60. Rewrite sentence: x + (x + 2) = 60

Step 2: Solve the equation Combine like terms 2x + 2 = 60

Step 3: Check your answer 29 + 29 + 2 = 60 The question wants all the 3 consecutive numbers: 29, 30 and 31

Answer: The 3 consecutive numbers are 29, 30 and 31.

Consecutive Odd Integers

Example 2: The lengths of the sides of a triangle are consecutive odd numbers. What is the length of the longest side if the perimeter is 45?

Solution: Step 1: Being consecutive odd numbers we need to add 2 to the previous number. Assign variables: Let x = length of shortest side x + 2 = length of medium side x + 4 = length of longest side

Step 2: Write out the formula for perimeter of triangle . P = sum of the three sides

Step 3: Plug in the values from the question and from the sketch. 45 = x + x + 2 + x + 4

Combine like terms 45 = 3x + 6

Isolate variable x 3x = 45 – 6 3x = 39 x =13

Step 3: Check your answer 13 + 13 + 2 + 13 + 4 = 45

Be careful! The question requires the length of the longest side. The length of longest = 13 + 4 =17

Answer: The length of longest side is 17

Consecutive Even Integers

Example 3: John has a board that is 5 feet long. He plans to use it to make 4 shelves whose lengths are to be a series of consecutive even numbers. How long should each shelf be in inches?

Solution: Step 1: Being consecutive even numbers we need to add 2 to the previous number. Assign variables: Let x = length of first shelf x + 2 = length of second shelf x + 4 = length of third shelf x + 6 = length of fourth shelf

Step 2: Convert 5 feet to inches 5 × 12 = 60

Step 3: Sum of the 4 shelves is 60 x + x + 2 + x + 4 + x + 6 = 60

Combine like terms 4x + 12 = 60

Isolate variable x 4x = 60 – 12 4x = 48 x = 12

Step 3: Check your answer 12 + 12 + 2 + 12 + 4 + 12 + 6 = 60

The lengths of the shelves should be 12, 14, 16 and 18.

Answer: The lengths of the shelves in inches should be 12, 14, 16 and 18.

How to find consecutive integers, consecutive odd integers, or consecutive even integers that add up to a given number

- The sum of three consecutive integers is 657; find the integers.
- The sum of two consecutive integers is 519; find the integers.
- The sum of three consecutive even integers is 528; find the integers.
- The sum of three consecutive odd integers is 597; find the integers.

The following video shows how to solve the integer word problems.

- The sum of two consecutive integers is 99. Find the value of the smaller integer.
- The sum of two consecutive odd integers is 40. What are the integers?
- The sum of three consecutive even integers is 30. Find the integers.

How to solve consecutive integer word problems?

Example: The sum of three consecutive integers is 24. Find the integers.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

## Integers Questions

Integer questions for practice with solutions are provided here for students to help them in their examinations. All questions are important from the exam point of view, these questions are in accordance with the latest CBSE/ICSE syllabus for class 6 and 7 maths. It is recommended for all students to attempt these questions for a final touch-up for their preparations for examination.

For more resources: Integers Worksheets for Classes 6 and 7

Integers are a set of positive and negative whole numbers along with zero. The set of integers is denoted by Z .

Learn more about Integers .

## Integers Questions with Solutions

Practice these questions on integers to improve your understanding, skill and speed of solving questions.

Question 1: Evaluate the following:

(i) 22 – ( –87)

(ii) 198 + ( –12)

(iii) –16.87 – 30

(iv) – 19 + 34 – 34

(i) 22 – ( –87) = 22 + 87 = 109

(ii) 198 + ( –12) = 198 – 12 = 186

(iii) –16.87 – 30 = – (16.87 + 30) = – 46.87

(iv) – 19 + 34 – 34 = –19 + 0 = –19

Question 2: Find the additive inverse of the following:

Additive inverse of 23 is –23 such that;

23 + (–23) = 0 = (–23) + 23

Additive inverse of 108 is –108 such that;

108 + (–108) = 0 = (–108) + 108

Additive inverse of 476 is –476 such that;

476 + (–476) = 0 = (–476) + 476

Additive inverse of –39 is –(–39) = 39 such that;

–39 + 39 = 0 = 39 + (–39)

Question 3: Verify a + (b + c) = (a + b) + c for the following:

(i) a = 2, b = 0, c = –9

(ii) a = 34, b = 90, c = –1

L.H.S = a + (b + c) = 2 + (0 + 9)

= 2 + 9 = 11

R.H.S = (a + b) + c = (2 + 0) + 9

∴ L.H.S = R.H.S

L.H.S = a + (b + c) = 34 + {90 + (–1)}

34 + {90 – 1}

= 34 + 89 = 123

R.H.S = (a + b) + c = (34 + 90) + (–1)

= 124 + (–1)

= 124 – 1 = 123

- Number System
- Ratio and Proportion
- Real Numbers

Question 4: Evaluate using properties:

(i) 89 – 58 + 28 – (–32)

(ii) 193 + 208 – {29 – (367)}

(iii) 56 – 34 + 235 – (123)

(iv) (84 – 34) × (84 + 45)

= 89 – 58 + 28 + 32

= 89 – 58 + (28 + 32) {associativity}

= 89 – 58 + 60

= (89 + 60) – 58 {commutativity and associativity}

= 149 – 58 = 91

= 193 + 208 – {29 – 367}

= 193 + 208 – (–338)

= 193 + 208 + 338

= (193 + 208) + 338 {associativity}

= 401 + 338 = 739.

= 56 – 34 + 235 – 123

= (56 + 235) – (34 + 123) {commutativity and associativity}

= 291 – 157

= {84 + (–34)} × (84 + 45)

= 84 × (–34 + 45) {distributivity of multiplication over addition}

Question 5: Write ‘true’ or ‘false’ for the following:

(i) Zero is the smallest integer.

(ii) –10 is smaller than –7.

(iii) 1 is the smallest positive integer.

(iv) –1 is the smallest negative integer.

(v) Sum of two negative integers is a positive integer.

(i) Zero is the smallest integer. (False)

(ii) –10 is smaller than –7. (True)

(iii) 1 is the smallest positive integer. (True)

(iv) –1 is the smallest negative integer. (False)

(v) Sum of two negative integers is a positive integer. (False)

Question 6: Evaluate:

(i) (–4) × (–15) × (–33)

(ii) (–1) × (–1) × (–1) × … 100 times

(iii) (–7) × (–4) × 5

= – (4 × 15 × 33)

= (– 1) 100

= 1 {∵ 100 is an even number}

= (7 × 4) × 5

Question 7: In a class test containing 20 questions, 5 marks are awarded for each correct answer and 2 marks is deducted for each wrong answer. If Riya get 15 correct answers out of all the questions attempted. What is her total score?

Number of questions = 20

Marks awarded for corrected answer = 5

Marks awarded for wrong answer = –2

Number questions attempted correctly = 15

Number questions not attempted correctly = 5

Her total score = (15 × 5) + (–2 × 5) = 75 – 10 = 65

Question 8: An elevator descends at the rate of 4 m/min. If the elevator start descending 25 m above the ground level, how long will it take to cover –450 m? ‘

Given, the elevator is at 25 m above the ground level and have to reach 450 m below the ground level.

Distance have to be covered = 25 – (–450) = 475 m

Speed of the elevator = 4 m/min

Time taken = 475/4 = 118.75 min.

Question 9: What number should be added to the sum of 345 and 67 to make it equal to the smallest 3-digit number?

Smallest 3-digit number = 100

Now, 345 + 67 = 412

412 – 100 = 312

Thus, 412 + (–312) = 100

Question 10: During summer, the temperature within a room is 37 o C. If an air conditioner cools the room by 5 o C/min. What will be the temperature of the room after 5 minutes of switching on the air conditioner?

Temperature of the room before switching on the air conditioner = 37 o C

Rate of cooling = –5 o c/min

Temperature of the room after switching on the air conditioner for 5 min = 37 – (5 × 5)

= 37 – 25 = 12 o C.

## Video Lesson on Number Sytem

## Related Articles

- Linear Equations Questions
- Real Numbers Questions
- Compound Interest Questions
- Polynomials Questions

## Practice Questions on Integers

1. Evaluate the following:

(i) 34 – 56 + (–18)

(ii) [32 – 17] × [32 + 90]

(iii) 12 – 25 + (–35) + 2.9

(iv) 34 – 56 + 13 – ( –37)

2. A dishonest shopekeeper uses a weighing machine which 900 g as 1kg. If cost of per kg sugar is ₹ 40. How more money did the shopekeeper earned by selling 3 kg sugar to the customer.

3. An air conditioner cools a room by 4 o C/min. If the temperature of the room is 45 o C before switching on the air conditioner. Find temperature of the room after switching on the air contioner for 6 min.

4. Write true or fallse for the following statements:

(i) Zero is a positive integer.

(ii) –1 is greatest negative integer.

(iii) Sum of a positive integer and a negative integer is always a positive integer.

(iv) Division of integers is always an integer.

(v) Multiplication of a positive integer and a negative integer is always a negative integer.

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## Rhody Today

Commencement 2024: for biological sciences graduate, problem solving is a core skill.

KINGSTON, R.I. – May 13, 2024 – For most of us, a botched do-it-yourself plumbing job that results in a flooded cellar would send us scrambling to call a pro. But for Alberto Paz, the would-be disaster only bolstered his confidence in his problem-solving skills.

Paz, born and raised in Providence, first came to the University of Rhode Island while still in Classical High School. He spent two weeks on campus in a program that had him doing experiments with chemicals, including some that could create fireworks. When he graduated high school, he became a full-time student in biological sciences in URI’s College of the Environment and Life Sciences.

He’s currently working with associate professor Ying Zhang in using metabolic modeling, meta-omics, and other data-intensive approaches to answering research questions in bioengineering and environmental microbiology. His research group develops open-source software and models for simulation of biological systems to achieve mechanistic understandings at molecular, organismal, and ecosystems scales.

Making information easier to access for everyone is important to Paz. That is why he focuses on bioinformatics, an interdisciplinary field of science that develops methods and software tools for understanding biological data. Professor Zhang allowed him to choose a project that interested him. He’s been working on it since summer 2023 and feels confident that if he continues on the path he’s taken, the work will eventually be published.

“The goal of any single type of lab, whether it be cancer, aquatics, anything, is to get your research across so it can convince other people as well,” Paz said. “I’m also interested in coding, with a focus on biology. I’d like to make information available for everyone, not just academics.”

He’s currently researching an organism called foraminifera, which is only found in the first centimeter of water in salt marshes. Paz hopes to be the first to sequence its genetic code, upload it, and determine its behaviors.

Paz says his life experiences have made him even more certain that he has a knack for problem solving. “From painting houses to tarring streets, to plumbing. I feel that having grown up with a proclivity toward fixing things and taking the unconventional route makes me a better researcher. I would like to encourage people to have a variety of experiences instead of boxing themselves into one area.”

He recalls a time when he and his father were renovating their house, including the plumbing. Paz looked over the project and, confident in his problem-solving abilities, persuaded his dad to let him take over one particular chore.

Things were going just fine, except for one thing: he had forgotten to turn off a valve. It wasn’t until the basement was flooded that he realized his mistake. Lucky for him his father didn’t blow his top. He just calmly told Alberto that he would have to clean the mess. Once that was completed, Paz returned to the task, this time fixing the problem as he said he would. The process took hours, but he’s still proud of the way it came out. “That one will always stick with me,” he said.

When he’s not doing home renovations, Paz likes going to the gym and playing volleyball. He also enjoys being with his sister Naomi when he goes home on the weekend. That’s also a time when he’s likely to enjoy a plate of mangú, a Dominican dish often eaten as a breakfast. “I grew up with it; it’s my comfort food.”

For Alberto Paz, challenging himself to excel at problem solving is a way of life. “Most people don’t know how hard I work, especially behind the scenes in research, setting things up to be successful. But I really like the freedom that challenges an evolutionary thought. You can’t be complacent. I like things that are very difficult.”

This story was written by Hugh Markey.

## Into all problem-solving, a little dissent must fall

Events of the past several years have reiterated for executives the importance of collaboration and of welcoming diverse perspectives when trying to solve complicated workplace problems. Companies weren’t fully prepared for the onset of a global pandemic, for instance, and all that it engendered—including supply chain snarls and the resulting Great Attrition and shift to remote (and now hybrid) work, which required employers to fundamentally rethink their talent strategies . But in most cases leaders have been able to collaborate their way through the uncertainty, engage in rigorous debate and analyses about the best steps to take, and work with employees, suppliers, partners, and other critical stakeholders to react and, ultimately, recover.

And It’s not just COVID-19: many organisations have had to rethink their business strategies and practices in the wake of environmental concerns, the war in Ukraine, and social movements sparked by racial injustice, sexual misconduct, and widespread economic inequity . Ours are fast-moving, complex times, rich not just in worrisome challenges but also in exciting potential—organisations that enable innovation will find ample opportunities to thrive. So now more than ever, decision makers can’t act alone; they must bring diverse perspectives to the table and ensure that those voices are fully heard . 1 Sundiatu Dixon-Fyle, Kevin Dolan, Vivian Hunt, and Sara Prince, “ Diversity wins: How inclusion matters ,” McKinsey, May 19, 2020.

But while many leaders say they welcome dissent, their reactions often change when they actually get some. They may feel defensive. They may question their own judgment. They may resent having to take time to revisit the decision-making process. These are natural responses, of course; employees’ loyalty and affirmation are more reassuring to leaders than robust challenges from the group. There is discomfort, too, for potential dissenters; it is much safer to keep your thoughts to yourself and conform than to risk expulsion from the group. 2 Derived from this work on the evolutionary origins of social and political behavior: Christopher Boehm, Hierarchy in the Forest: The Evolution of Egalitarian Behavior , Cambridge, Massachusetts: Harvard University Press, 2001.

What’s missing in many companies, in our experience, is the use of “contributory dissent” or the capabilities required to engage in healthy if divergent discussions about critical business problems. Contributory dissent allows individuals and groups to air their differences in a way that moves the discussion toward a positive outcome and doesn’t undermine leadership or group cohesion . 3 McKinsey itself has established obligation to dissent as one of its core values alongside those focused on client service and talent development. For more, see Bill Taylor, “True leaders believe dissent is an obligation,” Harvard Business Review , January 12, 2017.

McKinsey’s research and experience in the field point to several steps leaders can take to engage in healthy dissent and build a culture where constructive feedback is expected and where communication is forthright. These include modeling “open” behaviors, embedding psychological safety and robust debate into decision-making processes, and equipping employees with the communication skills that will allow them to contribute dissenting opinions effectively.

In this article we outline the steps leaders can take to encourage healthy dissent, and the actions teams and individuals can take to share their voices and perspectives most effectively. It takes both sides, after all, to engage in robust debate, find the right solutions, and enable lasting, positive change.

## How leaders can encourage contributory dissent

Senior leaders in an organisation play a central role in ensuring that individuals and teams see contributory dissent as a normal part of any discussion. They can signal the importance of dissent by taking a series of steps to institutionalise the practice within an organisation and empower employees to share their ideas freely and productively. Specifically, senior leaders should strive to inspire rather than direct employees to collaborate, explicitly demand dissent and, taking that one step further, actively engage with naysayers (see sidebar “How to encourage healthy dissent”). 4 Leaders can also draw on McKinsey’s “influence model” for changing mindsets and behaviors: role modeling, fostering understanding and conviction, reinforcing with formal mechanisms, and developing talent and skills. For more, see Tessa Basford and Bill Schaninger, “ The four building blocks of change ,” McKinsey Quarterly , April 11, 2016.

## Inspire, don’t direct

How to encourage healthy dissent.

To encourage dissent through personal leadership:

Lead to inspire, not to direct:

- Empower the group to come up with ideas: “None of us knows the answer yet, but we can work it out together if we harness the best of everyone’s thinking.”

Foster dissent by actively seeking it:

- Explicitly seek dissent; give people permission and encouragement.
- Consider including dissent as a stated organisational value.
- Make provision for open discussion in the buildup to decisions.

Welcome open discussion when it comes:

- Listen to dissenters and naysayers, and thank them for their insights.
- Recognise this as a usefully unfiltered channel for understanding the organisation’s perceptions on issues.
- Seek to bring dissenters along the decision journey, so they become positive influencers later during implementation.
- Employ deliberate techniques such as red teaming and pre-mortems to widen the debate and mitigate groupthink.

As the inspirational speaker Simon Sinek put it, “The role of a leader is not to come up with all the great ideas. The role of a leader is to create an environment in which great ideas can happen.” 5 Simon Sinek, Start with Why: How Great Leaders Inspire Everyone to Take Action , New York, NY: Portfolio, 2009. That is especially important for fostering an atmosphere of collaboration and contributory dissent. Rather than immediately jump into a discussion about solutions, one senior leader in an international organisation addressed his team’s anxiety in the wake of a crisis. “Let me guess,” he said, “you’re all feeling confused and uncertain about the way ahead. Terrific. I’m so glad we are of one mind and that we all understand our situation correctly! I’m sure that we can work it out together, but it’s going to require the best of everyone’s thinking. Let’s get started.” His authenticity and understated humor allowed him to connect with the group and inspired them to keep calm, carry on, and generate solutions that the leader alone couldn’t have come up with. Harvard professor Ron Heifetz describes this as creating a holding environment, a key element of adaptive leadership. 6 Ronald A. Heifetz and Mary Linksy, Leadership on the Line: Staying Alive through the Dangers of Leading , Boston, MA: Harvard Business School Press, 2002; Ronald Heifetz, Alexander Grashow, and Marty Linksy, The Practice of Adaptive Leadership: Tools and Tactics for Changing Your Organization and the World , Boston, MA: Harvard Business Press, 2009.

## Explicitly demand dissent

It’s not enough for leaders to give people permission to dissent; they must demand it of people. In many companies, individuals and teams may (understandably) default to collegiality, not realizing that there are ways to challenge ideas while still respecting colleagues’ roles and intellect. It’s on senior leaders, then, to help employees understand where the boundaries are. In World War 1, Australia’s General Sir John Monash was determined to develop better tactics to overcome the catastrophic impasse of trench warfare. He knew there were answers to be found from the experience of soldiers in the trenches, but he needed to loosen the military discipline of blind obedience: “I don’t care a damn for your loyal service when you think I am right; when I really want it most is when you think I am wrong.” Monash scheduled open battle planning sessions and pulled in advice from whoever offered it. In doing so, he built ownership of and confidence in his plans among all ranks. The resulting orchestration of tanks, artillery, aircraft, and troops led to rapid advances along the Somme Valley, and Monash garnered respect and appreciation from his troops, whose chances of survival and ultimate victory had increased markedly.

## Actively engage with naysayers

Taking the demand imperative one step further, it’s beneficial for leaders to actively seek out the views of vocal naysayers , who can turn into influential champions just by being part of the conversation. They can immediately improve the nature of business debate and may boost the quality of the final decision, although engaging with naysayers can be tough. Some dissenting opinions can be ill-informed or uncomfortable to hear. The objective for senior leaders, then, is to put their discomfort aside and listen for signs of cognitive dissonance within an organisation. As an example, front-line employees may say things like “We’re not considered strategic thinkers,” or “The company doesn’t put people first,” while senior management may actually feel as though they have made strides in both of those areas. Still, leaders need to absorb such comments, treat them as useful data points, assess their validity, and engage in what may be a challenging discussion. They may want to use red teams and premortems , in which teams at the outset anticipate all the ways a project could fail, to frame up dissenting opinions, mitigate groupthink, and find a positive resolution. These behaviours also serve to enhance organizational agility and resilience .

## How leaders can establish psychological safety

Senior leaders need to establish a work environment in which it is safe to offer dissenting views. The McKinsey Health Institute’s work on employee well-being points to a strong correlation between leadership behaviors, collaborative culture, and resistance to mental health problems and burnout : only 15 percent of employees in environments with low inclusivity and low support for personal growth are highly engaged, compared with 38 percent in high-scoring environments. 7 “ Addressing employee burnout: Are you solving the right problem? ,” McKinsey, May 27, 2022. Leaders can build psychological safety (where team members feel they can take interpersonal risks and remain respected and accepted) and set the conditions for contributory dissent by rethinking how they engage in debate—both the dynamics and the choreography of it.

## The dynamics of debate

The poet and playwright Oscar Wilde described a healthy debating culture as one in which people are “playing gracefully with ideas”— listening to, and even nourishing, opposing points of view in a measured and respectful way. 8 The Complete Works of Oscar Wilde, Volume 2: De Profundis, “Epistola: In Carcere et Vinculis,” Oxford, United Kingdom: Clarendon Press, 2005. Indeed, the best ideas can emerge at the intersection of cultures and opinions. In 15th century Florence, for instance, the Medici family attracted and funded creators from across the arts and sciences to establish an epicenter of innovative thinking that sparked the Renaissance. 9 Frans Johansson, The Medici Effect: Breakthrough Insights at the Intersection of Ideas, Concepts, and Culture , Boston, MA: Harvard Business School Press, 2004. Closer to this century, we have seen cross-discipline innovations like the application of biologists’ research on ant colonies to solve problems in telecommunications routing. And in the business world, extraordinary innovations have been achieved by open-minded leaders bringing together smart people and creating the conditions for playful exploration.

To achieve a state of “graceful play,” senior leaders must carefully manage group dynamics during debates. Rather than lead with their own opinions, for instance, which might immediately carry outsize weight in the group and stifle discussion, senior leaders can hold back and let others lead the discussion . They can lean in to show genuine curiosity or to explicitly recognise when a dissenting view has changed their thinking. But by letting other, more junior voices carry the agenda and work through ideas, however imperfect, senior leaders can establish a climate of psychological safety—and garner more respect from colleagues long term. 10 Amy C. Edmondson, The Fearless Organization: Creating Psychological Safety in the Workplace for Learning, Innovation, and Growth , Hoboken, NJ: John Wiley & Sons, 2019.

Leaders will also need to be aware of cultural differences that may crop up during debates. For example, many Australians speak candidly and are happy to address issues squarely. By contrast, the concept of “face” is so important in many Asian cultures that a more circumspect approach is taken. And the Pacific and Maori cultures emphasize displays of both strength and respect. 11 Erin Meyer, The Culture Map: Breaking through the Invisible Boundaries of Global Business , Philadelphia, PA: PublicAffairs, 2014. These differences in debate dynamics really matter. They can be a great source of hybrid vigour, 12 “Heterosis, also called hybrid vigour: the increase in such characteristics as size, growth rate, fertility, and yield of a hybrid organism over those of its parents. The first-generation offspring generally show, in greater measure, the desired characteristics of both parents.” Encyclopedia Britannica , accessed September 19, 2022. if sensitively managed, or a source of conflict and disenfranchisement if not. To approach these differences in a positive way, senior leaders could undertake a mapping exercise that identifies the different styles of the cultures present, thereby providing validation and enabling pragmatic measures to integrate them.

## Choreographing debate

Beyond just managing debate dynamics, business leaders must take a hand in choreographing the debate and, specifically, in helping to design collective-thinking processes so people know how best to play their part. Business leaders may adopt a structured approach to brainstorming, for instance, or plan strategic off-site schedules that combine deliberate thinking with “distracted” thinking—taking time to engage in a social activity, for instance—to take advantage of employees’ deep-thinking processes.

## How deliberate choices by the leader can optimise a decision-making process

A leader must consciously assess each new situation and design the collective-thinking process accordingly, then articulate this so that people know how best to play their part.

In doing so, the leader should consider an array of questions, the answers to which will determine the context, for example:

- What does success look like?
- Will the organisation underwrite initial failures in the interests of agility and innovation?
- How broad and freethinking an analysis is required?
- What are the explicit expectations for contributory dissent?
- Are any topics and behaviours out of bounds?
- Who will lead the discussion, and how will comments be captured?
- Does urgency mean that it’s better to be directive?
- Who will be consulted?
- Which decisions can be delegated, and to whom?
- Whose support needs to be built?
- What parameters and boundaries exist?
- Are there interim decisions and communications required?
- What form should the deliverable outcomes take?
- When are the deliverables required?
- Direction setting on these parameters by the leader focuses the team, while also creating space for creativity and iterative learning.

To create a sustainable structure for debate, business leaders will need to consider questions relating to team structure and rules of engagement: What does success look like when it comes to contributory dissent? What topics and behaviors are out of bounds? Who will lead the discussion, and how will comments be captured? Who has the final say on decisions, or which decisions can be delegated, and to whom? (For a more comprehensive explanation, see sidebar “How deliberate choices by the leader can optimise a decision-making process.”)

Having these parameters in place can free up the team to think more creatively about the issue at hand. Establishing such protocols can also make it easier to raise dissenting opinions. At one company, people are asked to call out their underlying values or potential biases when expressing a dissenting view. During meetings of the promotion committee, for instance, a statement like “I think we are making the wrong decision” would be rephrased as “I am someone who values experience over collaboration, and this decision would risk losing too much institutional knowledge.”

## How individuals and teams can engage and dissent

As we’ve shared, senior leaders can take steps to set conditions for robust discussion and problem-solving, but individuals and teams themselves must also have the right mindsets and skills for contributory dissent to work well (see sidebar “How teams and individuals can dissent effectively”). In particular, they must embrace the obligation to dissent, actively make space to analyse ideas that are different from their own, and then find ways to either iterate on others’ ideas or respectfully agree to disagree.

## Embrace the obligation to dissent

How teams and individuals can dissent effectively.

For dissent to be effective, its delivery requires courage and tactical skills underpinned by sincere respect and grace. Speaking up with respect is the right thing to do, and the responsibility to do so exists, even if there is uncertainty. The following guidelines are useful in enabling effective dissent:

Prepare a welcome for dissenting views:

- Understand the context and motivations of others, appreciate their views, and syndicate your own.
- Stop and strategise before wading into the conversations, establish a solid platform for agreement, and explicitly seek permission to dissent.

Play the long game:

- Be open minded and iterative. Don’t expect to succeed on the first try.
- Listen to others for what their views might add rather than to defend your own.

Withhold assent if you need to, but do it carefully:

- Withholding assent is a legitimate option if done judiciously.
- Minimise offense to and loss of face for the decision maker.
- If principles or legality is at stake, document your dissent.

Individuals and teams need to exhibit a certain amount of humility and confidence in order to speak truth to power with respect; they must be sure for themselves that doing so is the right thing to do. To build this confidence, individuals and teams should remember that the very act of dissent can be valuable, even if the contribution itself isn’t 100 percent baked. Others can react or build on the dissenting view—which, in itself, can be a satisfying process for a dissenter. If the ultimate decision isn’t what they proposed, they still helped shape it by offering and testing a worthy possibility.

## Make space to analyse different views

Individuals and teams may need time to determine their positions on an issue. During this period, it’s important to be (and seen to be) open-minded and respectful of others’ views. That means asking lots of questions, gathering information, assessing others’ motivations, and acknowledging their views before syndicating alternatives of your own. Much of this fact gathering can be done one-on-one, in a nonconfrontational way, in offline conversations rather than in a tension-filled meeting room. In these conversations, individuals could start by reaffirming a shared commitment to finding a solution to the issue at hand, their respect for the decision-making process and the group, and areas of broad agreement. They could also signal their possible intention to dissent and seek permission to do so rather than confronting people head-on. People will find it harder to refuse that permission, and will be less likely to get defensive, when approached with statements like “This is a great discussion, and I love the vision of where we are headed, but would it be OK for us to explore some alternatives for how to get there?”

## Agree to iterate …

Individuals and teams that decide to offer dissenting views should agree to iterate on other solutions, rather than digging in. Their dissenting opinions should be cogent, persuasive, and open-minded—but dissenters shouldn’t expect to change hearts and minds on the first try. They should plant seeds gently and bide their time; they might even see their idea come back as someone else’s. The critical skill required here is active, open listening: dissenters should listen carefully for others’ additive insights and find ways to build on them. In their contributory dissent, individuals and teams can take a moment to summarize what others have said and then use statements like “Can I offer another take?” and then allow the momentum of the conversation to take over.

## … or agree to disagree

But what happens if, after all the considered and tactful input, the dissenter still believes a decision is heading in the wrong direction? In our experience, withholding assent then becomes a legitimate option: people shouldn’t agree if they don’t agree. This is where all the careful, respectful groundwork the dissenter has done can pay dividends. In fact, a dissenting view gains even more power when an individual can say something like, “I still believe in my alternate solution, but I’m grateful for the opportunity to contribute to this process, and I respect that you have the final say.” In this case, the dissenter is supporting the leader while flagging that the open debate hasn’t convinced them to change their initial view.

Of course, withholding assent should be a relatively rare action, taken only after an individual or team has shown that they can accommodate other views and have aligned with the consensus when they believe it’s right to do so. Think of US Supreme Court associate justice Ruth Bader Ginsburg, who joined the consensus view on many decisions but who is especially celebrated for the positive changes that arose from her highly influential dissenting opinions on issues such as gender equity, human rights, and religious freedom.

Contributory dissent can help strengthen employee engagement, unlock hidden insights, and help organisations solve tough challenges. But putting it into practice takes courage and humility, and it won’t just happen by accident. Leaders need to be intentional about welcoming challenges to their plans and opinions, even when it’s uncomfortable to do so. They need to establish cultures and structures where respectful debate can occur and where individuals and teams feel free to bring innovative—and often better—alternative solutions to the table.

Ben Fletcher is a senior partner in McKinsey’s Sydney office, Chris Hartley is a partner in the Melbourne office, Rupe Hoskin is a senior expert in the Canberra office, and Dana Maor is a senior partner in the Tel Aviv office.

The authors wish to thank Jacqueline Brassey, Nikki Dines, Richard Fitzgerald, Sam Hemphill, Ayush Jain, Jemma King, and Martin Nimmo for their contributions to this article.

## Explore a career with us

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California Today

## California Leaders Try to Solve Another Big Budget Deficit

Gov. Gavin Newsom released his revised spending plan to close a $27.6 billion gap, touching off what will probably be weeks of negotiations.

By Shawn Hubler

May has many pleasures in California. Poppies . Jumping frog contests. Waterfalls in Yosemite National Park.

In the capital, though, May can be painful. It’s when the annual haggling over the state budget usually shifts into high gear. Governors fine-tune their fiscal proposals. Lobbyists ramp up the pressure. The people who rely on state dollars the most brace for relief or disappointment, and legislators strain to balance the books.

Like the economy and the population of California, the state budget is the nation’s largest, and lawmakers are legally required to pass a balanced one by June 15 each year or lose their pay and expense money. Even in flush years, it’s a fraught process.

And this year, California is not flush.

## Why is there a hole to fill?

California’s system of taxation is volatile by nature. The state relies heavily on taxes on personal income and capital gains. When the rich do well, the state government reaps a bonanza . But when the stock market slumps or initial public offerings dwindle, revenue plummets . For the past couple of years, as the Federal Reserve tried to rein in inflation with higher interest rates, state finance officials warned that California’s fiscal outlook was darkening.

A stock market decline in 2022 drove down receipts from capital gains taxes. Higher borrowing costs for businesses led to higher unemployment and fewer business startups.

Making matters worse, the state was late to recognize the full extent of the problem, because the severe winter storms in early 2023 prompted the state to extend the deadline for filing income tax returns to November. That meant that lawmakers had to pass the current budget before it was clear how much money would be coming in, and they guessed too high.

## How big is the problem?

By December, it was clear that California was in a hole and would not soon emerge. The state’s nonpartisan budget analyst pegged the deficit at about $68 billion for the 2024-25 fiscal year. A month later, the governor’s finance experts, using different methods, put the shortfall at a somewhat less dire $38 billion.

Gov. Gavin Newsom and state lawmakers got proactive as the deficit continued to grow.

Last month, the Legislature approved a package of early cuts, shifts, delays and other spending maneuvers that, by the governor’s estimate, whittled the shortfall for the current year to about $27.6 billion. This week, the legislative analyst called that figure “plausible” but “optimistic.”

## Can it be plugged?

It must be. California is constitutionally required to balance its budget. The questions are what spending will be cut, delayed or shifted; how deeply state lawmakers will dip into state reserve funds; and whether taxes will be raised.

Last week, the governor proposed a hodgepodge of solutions that included taking $21.5 billion out of state reserves, canceling more than $19 billion in planned spending and shrinking a range of state programs by nearly $14 billion.

His proposal would eliminate 10,000 vacant state government jobs, close down 4,600 prison beds and greatly shrink a popular scholarship program for middle-class college students. It would cancel spending on early education facilities, toxic waste cleanup and broadband internet expansion, among other items.

But Newsom essentially ruled out one idea. “I’m not prepared to increase taxes,” he said.

## What’s next?

The legislative analyst will weigh in on the governor’s revisions, and state legislators will propose their own budget ideas.

Lobbyists are already geared up: Health care interests, including some of the governor’s most formidable allies, want to preserve higher reimbursements for treating Medi-Cal patients. Counties are fighting a proposed cut of hundreds of millions of dollars in public health funding. Arts backers are pushing to ensure that money from a popular 2022 initiative for art and music education will be used to hire additional teachers.

So, make the most of that local wildflower patch, frog derby or trip to the park. It’s going to be a long month.

## If you read one story, make it this

The number of Americans living near the scenes of fatal gun violence rose sharply during the pandemic , according to a review by The New York Times. An interactive map shows which neighborhoods in California and across the nation were affected the most.

## The rest of the news

The state will receive close to $64 million in federal funding to support electric vehicle charging and infrastructure, Representative Zoe Lofgren announced on Tuesday.

A coalition of Republican state attorneys general is suing the E.P.A. and the state of California over the state’s emissions rules for trucks and heavy-duty vehicles, The Hill reports.

## Central California

The criminal trial of two Fresno entrepreneurs charged with conspiring to commit wire fraud has been delayed until July while lawyers try to negotiate a plea deal, The Fresno Bee reports.

## Northern California

Pro-Palestinian demonstrators at the University of California, Berkeley, began dismantling their encampment , The San Francisco Chronicle reports.

Gunshots were fired at an Oakland teen rowing competition last month, but it was not until after the race that anyone realized what had happened, The Los Angeles Times reports.

An environmental nonprofit group has sued Tesla, claiming the company had violated the federal Clean Air Act by allowing its Fremont plant to emit harmful pollutants since 2021, Reuters reports. The news agency said Tesla did not respond to its request for comment.

## And before you go, some good news

The Los Angeles Times has put together a list of seven public gardens across the Los Angeles area where you can literally stop and smell the roses this spring.

Southern California, with its plentiful sunshine and mild weather, has an ideal climate for these beautiful flowers to thrive. And, according to Karen Haney, who teaches horticultural therapy at U.C.L.A. Extension, roses can be easy on the mind, too.

“Roses give us a wide range of sensory experience that we can use to improve our mood,” Haney told the news outlet, adding, “When you are experiencing that scent, the mindfulness that you’re giving to that moment is providing a restorative experience for your brain, a little bit of softness.”

You can check out the full list here .

Thanks for reading. We’ll be back tomorrow.

P.S. Here’s today’s Mini Crossword .

Soumya Karlamangla , Halina Bennet and Briana Scalia contributed to California Today. You can reach the team at [email protected].

Sign up here to get this newsletter in your inbox .

Shawn Hubler is based in Sacramento and covers California news, policy trends and personalities. She has been a journalist for more than four decades. More about Shawn Hubler

## Could Indiana’s Kel’el Ware solve the Sixers’ problem at backup center? The draft prospect thinks so.

Ware welcomes the thought of playing backup to Sixers center Joel Embiid: "That would be great to learn from [Embiid], especially since he’s a vet and has been in the league for a while."

CHICAGO — Standing at 6-foot-11 ¾ with a 230-pound frame, Kel’el Ware has the ideal size to back up Joel Embiid .

But his stature isn’t the only reason the Indiana center thinks he’s a solid fit for the 76ers or any other NBA team.

“I’m able to space the floor,” Ware said Tuesday at the NBA Draft Combine. “The NBA game is getting to that level where the bigs are not just back to the basket. I’m able to pick and pop. I’m able to be a lob threat. I’m able to move my feet. I’m able to defend on the perimeter. That’s why I feel I’m ready.”

» READ MORE: At the South Philly sports complex, hopes that a new music hall can carry a different tune

There’s a chance the Sixers may trade their first-round pick in June’s NBA draft. However, multiple mock drafts have them selecting the 20-year-old if they keep the No. 16 selection. And that would be fine for the Arkansas native.

“That would be great to learn from [Embiid], especially since he’s a vet and has been in the league for a while,” Ware said of the possibility of backing up the 2023 MVP. “So if that did happen, I wouldn’t be opposed to that.”

Ware averaged 15.9 points, 9.9 rebounds, and 1.9 blocks during his lone season with the Hoosiers. That came after he averaged 6.6 points, 4.1 rebounds, and 1.3 blocks as a freshman at Oregon during the 2022-23 season. His ability to shoot the ball from the outside was one of his biggest improvements since transferring.

He shot 42.5% on college three-pointers this past season after making just 27.3% with the Ducks. Ware credits Indiana coach Mike Woodson for his improved shooting.

“Coach Woodson allowed me to play,” he said, “and trusting me on the court and playing through my mistakes.”

On Monday, Ware displayed his athleticism during the combine testing, finishing the three-quarter-court sprint in 3.29 seconds. He also made 10 of 25 shots in the three-point star drill. The 2022 McDonald’s All American will not compete in the five-on-five scrimmages here. He will, however, have meetings with various NBA front office executives and coaches this week in the Windy City.

“It’s just a blessing to be here, especially the hard work I put in to get here,” he said.

## Edey’s mission

Zach Edey is out to prove that his game translates to the NBA.

Despite being the two-time national college player of the year at Purdue, the 7-4 center isn’t projected to be a lottery pick.

» READ MORE: No, the Sixers didn’t miss on the ‘Nova Knicks. They never fit in Philly.

Critics have questioned his speed and believe his back-to-the basket playing style is outdated. But Edey made 14 of 25 three-pointers to finish tied for second in the three-point star drill. He also finished the three-quarter-court sprint in 3.51 seconds.

“I think it’s a tough thing, obviously, when people want to take down your game when you play a certain way,” he said. “But at the end of the day, I think teams are going to value what I do. …

“It doesn’t matter what people say. Teams put stock into rebounding. Teams put stock into having strength in the paint, strength and length, all that stuff. People are going to say what they are going to say. But I know who I am, and I know what I’m good at.”

Edey averaged 25.2 points, 12.2 rebounds, and 2.2 blocks this past season as a senior.

## Rehoming wild horses won't solve the brumby problem, but it transforms lives for horses and owners

At 14, Ruby Wild would rather demonstrate her connection with horses than explain it.

Whether she's prompting the more than 500-kilogram animal to lie down or waiting for the mare to be still enough to stand on her back, the bond between the teenager and her horses is undeniable.

"If you don't have a strong connection, it's hard to do anything," Ruby said.

But her herd of four aren't regular domestic horses, they were once wild brumbies.

"We grew up together because she was only little. She was only two when we got her," she said.

Her mum Kylie Wild said, while Ruby was supervised, her young daughter had a natural intuition with horses.

"Ruby and Gidget built this connection together and it never really crossed my mind that she was a wild brumby," Ms Wild said.

Ruby's love of brumbies is one she shares with her cousin, 17-year-old Brooke Wild.

"Brumbies, they're definitely underestimated, that's for sure," Brooke said.

"Brumbies are just so, so versatile, anything you put them to: jumps, cattle, whatever. They'll just pick it up like that and they just learn so quickly."

## Where did brumbies come from?

Horses came to Australia on the First Fleet in 1788. Escaped animals become the country's first brumbies.

Over time, other domestic breeds have gone feral and now descendants of thoroughbreds, stock, quarter and heavy horses can be found in the wild.

It is estimated there are more than 400,000 brumbies in Australia and when concentrated in high numbers, their hard hooves pose a risk to the environment and native species.

Horse trainer Anna Uhrig runs a brumby rehoming camp in south-east Queensland, using horses trapped in local forests and from central Queensland.

"We can't solve the brumby problem but we're doing what we can," Ms Uhrig said.

"It's a very small part of the puzzle: the amount of horses we rehome. It's not thousands but I think the work adds quality of life to those horses."

Ecologist Dave Berman has been managing brumby populations for 40 years across the country.

For most of the past decade, he has managed the brumby herds of Tuan and Toolara state forests, which consist of pine plantations and native species north of Noosa.

The growing population has brought wild horses close to roads, putting drivers at risk.

"We catch them and find homes for them to reduce the risk of collisions between horses and people," Dr Berman said.

"Usually it's the horses that get killed, but eventually, you know, there will be people killed.

"Originally in about 2009 there were about four horses killed per year, we removed all the horses regularly crossing the road and then there were no collisions, so we showed that worked."

## What's a brumby rehoming camp?

Ms Uhrig's 10-day intensive camp attracts participants from all over the country to choose, train and adopt a brumby.

"Yarding, drafting and then coming to these yards is totally new for the brumbies so it can be quite scary for them," Ms Uhrig said.

In less than two weeks, some of the horses go from not letting a human within 100 metres to being ridden.

"I just love seeing the transformation and I love giving people the opportunity to learn those skills," Ms Uhrig said.

"I could break in a brumby continuously and I still wouldn't be able to rehome as many as we can through the camps."

Melissa Teunis's brumby Valour will join her herd of domestic horses used for equine therapy.

"I've loved horses forever and I just wanted the opportunity to start [training] my own [horse]," she said.

"It's the perfect place to come. Anna is amazing and just knows the process and talks and walks you through every step."

But Ms Uhrig said rehoming brumbies wouldn't work everywhere.

"In terms of the whole management debate, it's not a closed book. We don't have the answers necessarily," Ms Uhrig said.

"There's been some research done in some targeted areas and we think we have the answers and we can apply those but generally it's a nationwide issue and the management implications are different in each area."

## How are the horses trapped?

Dr Berman said the trapping process was slow and targeted brumbies who grazed near or regularly crossed roads.

"We use electric fences and we build a really large electric fence area that can be up to 4 kilometres long," he said.

"We'll build that around the horses and gradually make it smaller and then get them into a laneway and then into hessian panels, and then into panel yards."

He said patience and a respect for the animals were required.

"If you put too much pressure on, they go. They'll go through the fence anyway," he said.

Giving the horses a second chance can take weeks and sometimes months.

It's an emotional process for Dr Berman who creates a strong connection with the brumbies during the process.

"They're wonderful animals," he said.

## A national problem

Brumby management differs across the country.

Dr Berman said rehoming was becoming more common, but wasn't the solution alone.

"In a lot of other parts of Australia where populations are getting too much, horses can be shot from the air or the ground," he said.

In New South Wales's largest national park, Kosciuszko, the government is culling horses in the hundreds .

In 2022, the NSW government estimated about 18,000 horses lived in the park, with a population range between 14,501 and 23,535 horses .

Under the state government management plan, more than 15,000 brumbies could be killed or rehomed to reduce the population to 3,000 by mid-2027, to the despair of some of the locals.

Meanwhile, a Kosciuszko brumby rehoming program has been suspended after hundreds of horse carcasses were found on a property near Wagga Wagga .

Further north in NSW, the officers from the Local Land Services have a different approach, working with landholders and rescue groups to trap and rehome wild horses near Grafton.

"It's not just about safety for people. It's the safety of the horses as a number of them have been hit on the road," senior biosecurity officer Tiffany Felton said.

"But it's not just that, they're a hard-hoofed animals so they do, and especially in these dry times, start affecting the waterways and in environmentally sensitive areas."

In the past three years more than 130 have been captured and rehomed under the state government's Biosecurity Act.

"Previously, there have been probably far harsher approaches to wild horse management, and that's not what the community wanted," said Louise Orr of North Coast Local Land Services.

## Untapped potential

After just seven days at brumby camp, some of the horses are almost unrecognisable for participants, including Melissa Teunis.

"It's amazing to think that when we got here last Friday, he'd never been touched," Ms Teunis said.

"It shows you how resilient they are and just how willing — oh this will make me cry — how willing they are to try for us and, and give us all that they have."

Lilly Anderson, 14, could put a saddle on her horse Rain by day four of the course and three days later, she was able to ride the animal.

"Like, for me, that's incredible. That's like just gone so quickly. But she's also so just amazing," Lilly said.

The camp is the first stage of the horses transitioning to domestic life with more time and effort required after they leave.

"Once you give them a chance, they'll do anything for you," Lilly said.

"She's just so smart to catch on and she'll do anything for lucerne as well. Loves a bit of food."

Past participants of the camp said the biggest challenge with rehoming a brumby was fighting off the temptation to buy another.

"It's been hard but it gets better. The highs are really high," Ava Cloherty said.

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Nsw suspends brumby rehoming after discovery of hundreds of dead horses.

## Brumby re-homing program under investigation after mass horse carcass find on farm

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Problems with Solutions. Problem 1: Find two consecutive integers whose sum is equal 129. Solution to Problem 1: Let x and x + 1 (consecutive integers differ by 1) be the two numbers. Use the fact that their sum is equal to 129 to write the equation x + (x + 1) = 129 Solve for x to obtain x = 64 The two numbers are x = 64 and x + 1 = 65 We can see that the sum of the two numbers is 129.

In the following exercises, evaluate. 35 − a when a = −4. (−2r) 2 when r = 3. 3m − 2n when m = 6, n = −8. −|−y| when y = 17. In the following exercises, translate each phrase into an algebraic expression and then simplify, if possible. the difference of −7 and −4. the quotient of 25 and the sum of m and n.

Challenge Exercises Integer Word Problems. Directions: Read each question below. Click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR. Each answer should be given as a positive or ...

For example: 5 + (-1) = 4. 4. Use the commutative property when a is negative and b is positive. Do the addition as follows: -a +b = c (get the absolute value of the numbers and again, proceed to subtract the lesser value from the larger value and assume the sign of the larger value) For example: -5 + 2 = -3. 5.

Dividing a negative by a positive integer or a positive by a negative integer will result in a negative integer. A good grasp of division facts and a knowledge of the rules for multiplying and dividing integers will go a long way in helping your students master integer division. Use the worksheets in this section to guide students along.

Integer Word Problems Worksheets. An integer is defined as a number that can be written without a fractional component. For example, 11, 8, 0, and −1908 are integers whereas √5, Π are not integers. The set of integers consists of zero, the positive natural numbers, and their additive inverses.

Let us now see how various arithmetical operations can be performed on integers with the help of a few word problems. Solve the following word problems using various rules of operations of integers. Word problems on integers Examples: Example 1: Shyak has overdrawn his checking account by Rs.38. The bank debited him Rs.20 for an overdraft fee.

Solution: Step 1: Assign variables: Let x = red marbles. Sentence: Initially, blue marbles = red marbles = x, then John took out 5 blue marbles. Step 2: Solve the equation. x = 2 ( x -5) Answer: There are 10 red marbles in the bag. The following videos give more examples of integer word problems.

Figure 1.3.3 1.3. 3: The numbers 5 and −5 are 5 units away from 0. The absolute value of a number is never negative because distance cannot be negative. The only number with absolute value equal to zero is the number zero itself because the distance from 0 to 0 on the number line is zero units.

The first is five times the second and the sum of the first and third is 9. Find the numbers. Advanced Consecutive Integer Problems. Example: (1) Find three consecutive positive integers such that the sum of the two smaller integers exceed the largest integer by 5. (2) The sum of a number and three times its additive inverse is 16.

12. The temperature was -3o C last night. It is now -4o C. What was the change in temperature? 13. While watching a football game, Lin Chow decided to list yardage gained as positive integers and yardage lost as negative integers. After these plays, Lin recorded 14, -7, and 9.

What is an Integer? Integers Lesson and Integer Examples. Use the following examples and interactive exercises to learn about Integers, also can be called numbers. Problem: The highest elevation in North America is Mt. McKinley, which is 20,320 feet above sea level. The lowest elevation is Death Valley, which is 282 feet below sea level.

Learn how to use integers to represent positive and negative numbers, and how to perform operations with them. This unit covers the concepts of addition, subtraction, multiplication, and division of integers, as well as the properties of these operations. You will also practice solving word problems involving integers and applying them to real-world situations.

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If we were to take the rules for multiplication and change the multiplication signs to division signs, we would have an accurate set of rules for division. Here are three examples: Example 1: -9 ÷ 3 = -3. Example 2: 20 ÷ (-4) = -5. Example 3: -18 ÷ (-3) = 6. uizmaster: Dividing Integers.

The following diagram shows an example of a consecutive integer problem. Scroll down the page for more examples and solutions on consecutive integer problems. ... The following video shows how to solve the integer word problems. Examples: The sum of two consecutive integers is 99. Find the value of the smaller integer.

Below is a quick summary for the rules of adding integers. Problem 1:Add the integers: [latex]2 + 7[/latex] Answer. [latex]9[/latex] Explanation:The two integers are both positive that means they have the same sign. It implies that we should add their absolute values and copy the common sign which is positive.

Visit BYJU'S to solve integer questions. Login. Study Materials. NCERT Solutions. NCERT Solutions For Class 12. NCERT Solutions For Class 12 Physics; NCERT Solutions For Class 12 Chemistry; ... Sum of two negative integers is a positive integer. Solution: (i) Zero is the smallest integer. (False) (ii) -10 is smaller than -7. (True) (iii ...

Integer. An integer is one of the numbers obtained in counting the natural numbers ( ), zero ( ), or the negatives of the natural numbers ( ). If and are integers, then their sum , their difference , and their product are all integers (that is, the integers are closed under addition and multiplication), but their quotient may or may not be an ...

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An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers.In many settings the term refers to integer linear programming (ILP), in which the objective function and the constraints (other than the integer constraints) are linear.. Integer programming is NP-complete.

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An age problem is a type of word problem in math that involves calculating the age of one or more people at a specific point in time. These problems often use phrases such as 'x years ago,' 'in y years,' or 'y years later,' which indicate that the problem is related to time and age. Show more

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