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Subtracting integers word problems

Here are four great examples about subtracting integers word problems.

Problem #1:

The record high temperature for Massachusetts is 104 degrees Fahrenheit. The record low is -18 degrees Fahrenheit. What is the difference between high and low? Solution The problem has 1 important component. It is the phrase " difference between high and low ." Difference between high and low is the same as subtracting the smaller number from the bigger number.

104 - -18 = 104 + 18 = 122

The difference between high and low is 122 degrees Fahrenheit.

More subtracting integers word problems 

Problem #2:

Sylvia checked the balance in her bank account early in the morning and saw that she had 98 dollars. In the afternoon, she noticed that there was an overdraft of 6 dollars in the account for unpaid bills. How much were the unpaid bills? Solution The problem has 1 important component. It is the word " Overdraft ." Overdraft occurs when the bank gives you money to cover amount you did not have at the time the bills were paid. In this case, what you did not have is 6 dollars. Since the bank gave it to you, you owe it to the bank. Therefore, you owe 6 dollars to the bank and this can be expressed as -6.

The amount of unpaid bills is the difference between what you had that morning (98) and what you have this afternoon (-6). 98 - -6 = 98 + 6 = 104 dollars

The amount of unpaid bills is 104 dollars

Problem #3:

You are 5 dollars in debt. You borrow 12 dollars more. What is the total amount of your debt? Solution The problem has 2 important components shown in bold below. You are 5 dollars in debt . You borrow 12 dollars more . What is the total amount of your debt? 5 dollars in debt can be represented by -5 Borrow 12 more can be represented by - 12 We get -5 - 12 = -5 + -12 = -17 The amount of your debt is 17 dollars.

Problem #4:

An airplane takes off and then climbs 2500 feet. After 20 minutes, the airplane descends 150 feet.  What is the airplane's current height? Solution The problem has 2 important components shown in bold below. An airplane takes off and then climbs 2500 feet . After 20 minutes, the airplane descends 150 feet .  What is the airplane's current height? Climbs 2500 feet can be represented with +2500 Descends 150 feet can be represented with - 150 We get +2500 - 150 = +2500 + -150 = 2350

The airplane is now at a height of 2350 feet.

Take a look also at the problem below

Adding integers word problems

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Integer Subtraction

Subtracting integers.

If you know how to add integers , I’m sure that you can also subtract integers. The key step is to transform an integer subtraction problem into an integer addition problem. The process is very simple. Here’s how:

Steps on How to Subtract Integers

Step 1 : Transform the subtraction of integers problem into addition of integers problem. Here’s how:

• First, keep the first number (known as the minuend).
• Second, change the operation from subtraction to addition.
• Third, get the opposite sign of the second number (known as the subtrahend)
• Finally, proceed with the regular addition of integers.

Step 2 : Proceed with the regular addition of the integers.

Note that you will eventually add integers. So for your convenience, here’s a quick summary of the rules on how to add integers.

• Case 1 :  Adding two integers having the same sign

Add their absolute values then keep the common sign.

• Case 2 :  Adding two integers with different signs

Subtract their absolute values (larger absolute value minus smaller absolute value) then take the sign of the number with the larger absolute value.

Examples of Integer Subtractions

Example 1 : Subtract the integers below.

We will need to transform the problem from subtraction to addition. To do that, we keep the first number which is –13, change the operation from subtraction to addition, then switch the sign of +  4 to – 4.

The final step is to proceed with regular addition. Add their absolute values. Then we determine the sign of the final answer. Since we are adding integers with the same sign, we will keep the common sign which in this case is negative.

Example 2 : Subtract the integers below.

Just like before, convert a subtraction problem to an addition problem. Positive 9 remains, switch the operation from “minus” to “plus” then get the opposite sign of the subtrahend (second number) from negative to positive.

Now, let’s add them. We are adding two positive integers so we expect the answer to be positive as well because the common sign is positive.

Example 3 : Find the difference of the two integers.

I hope you are already getting the hang of it. Let’s make this an addition of integers problem first then proceed with regular addition of integers with different signs.

So we subtract their absolute values first then get the sign of the number with larger absolute value.

Subtracting the absolute values, we have 24 minus 19 which gives us +5. But the final answer is –  5 because the sign comes from – 24.

Take a quiz:

Subtracting Integers Quiz

You may also be interested in these related math lessons or tutorials:

Subtracting Integers Practice Problems with Answers

Adding Integers

Multiplying Integers

Dividing Integers

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A and S Integers

Adding and subtracting integers

Here you will learn strategies on how to add and subtract integers, including using visual models as well as the number line.

Students will first learn about integers in 6th grade math as part of their work with the number system and expand that knowledge to operations with integers in the 7th grade.

Every week, we teach lessons on adding and subtracting integers to students in schools and districts across the US as part of our online one-on-one math tutoring programs. On this page we’ve broken down everything we’ve learnt about teaching this topic effectively.

What are adding and subtracting integers?

Adding and subtracting integers is when you add or subtract two or more positive or negative numbers together.

You can add and subtract integers using visual models or a number line.

Adding Integers \hspace{2cm}

Visual model with counters: 3 + 2 = \, ? There are no zero-pairs. There are 5 positive pieces. Answer: 5	Number line: 3 + 2 = \, ? Start at 3 and move 2 places in the positive direction. You land at 5. Answer: 5
Visual model with counters: -5 + 3 = \, ? There are 3 zero pairs There are two negative counters left. Answer: -5 + 3 = -2	Number line: -5 + 3 = \, ? Start at -5 and move three places in the positive direction. You land at -2. Answer: -5 + 3 = -2
Visual model with counters: 6 + (-2) = \, ? There are two zero pairs with four positive counters left. Answer: 6 + (-2) = 4	Number line: 6 + (-2) = \, ? Start at 6 and move two places in the negative directions. You land at 4. Answer: 6 + (-2) = 4
Visual model with counters: -3 + (-4) = \, ? There are no zero pairs. There are 7 negative counters all together. Answer: -3 + (-4) = -7	Number line: -3 + (-4)= \, ? Start at -3 and move 4 places in the negative direction. You land on -7. Answer: -3 + (-4) = -7

Subtracting Integers \hspace{2cm}

Visual model with counters: -3 - (-2) = \, ? -3 take away -2 There are three negative counters. Taking away two of them, leaves you with 1 negative counter. Answer: -3-(-2) = -1	Number line: -3-(-2) means from -2 to -3 on the number line. From -2 move left one unit to -3, which is -1 unit. Answer: -3-(-2) = -1
Visual model with counters: -3-(+2) = \, ? -3 take away +2. Add two zero pairs in order to remove +2. There are five negative counters left. Answer: -3-(+2) = -5	Number line: -3-(+2) means from +2 to -3 on the number line. From +2 move left 5 units to -3, which is -5 units. Answer: -3-(+2) = -5
Visual model with counters: 2-(+5) = \, ? 2 take away +5. Add three zero pairs in order to remove +5. There are three negative counters left. Answer: 2-(+5) = -3	Number line: 2-(+5) = ? 2-(+5) means from +5 to +2 on the number line. From +5 move left three units to +2, which is -3 units. Answer: 2-(+5) = -3

[FREE] Addition and Subtraction Worksheet (Grade 2 to 7)

Use this quiz to check your grade 2, 3, 4 and 7 students’ understanding of addition and subtraction. 15+ questions with answers covering a range of 2nd, 3rd, 4th and 7th grade addition and subtraction topics to identify areas of strength and support!

Common Core State Standards

How does this apply to 6th grade math and 7th grade math?

• Grade 6: Number System (6.NS.C.6) Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
• Grade 7: Number System (7.NS.A.1) Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

How to add and subtract integers?

In order to add and subtract integers using counters:

• Represent the problem with counters identifying zero pairs with addition or adding zero pairs when necessary for subtraction.

The answer is the leftover counters.

In order to add and subtract integers using a number line:

To add, start at the first number and move to the second number; to subtract, start from the second number and move to the first number.

Write your answer.

Adding and subtracting integers examples

Example 1: adding integers with different signs using counters.

Add: -2 + 7 = \, ?

Represent the problem with counters, identifying zero pairs with addition or adding zero pairs when necessary for subtraction.

There are two zero pairs with 5 positive counters left.

2 The answer is the leftover counters.

Answer: -2 + 7 = 5

Example 2: subtracting integers with counters

Subtract: -4-(-5) = \, ?

-4 remove -5. Add 1 zero pair in order to remove -5.

-4-(-5) = 1

Example 3: adding integers with a number line

Add: -8 + (-5) = \, ?

-8 + (-5) is addition. Start at -8 and move in the negative direction (left) 5 places. You land at -13.

-8 + (-5) = -13

Example 4: subtracting integers with a number line

Solve: 7-(+9) = \, ?

From positive 9 move in the negative direction until you get to 7. You move 2 places to the left, which is -2.

7-(+9) = -2

Teaching tips for adding and subtracting integers

• Adding and subtracting integers is a foundational skill for Algebra 1. Using counters and/or a number line helps students formulate conceptual understanding of the concept.
• Using actual hand-held counters help students to manipulate the zero pairs instead of using digital counters.
• Have students try and identify the patterns with adding and subtracting integers so that they can figure out the rules on their own.
• Although practice integer worksheets have their place, have students practice problems through digital games or scavenger hunts around the room to make it engaging.

Easy mistakes to make

Related addition and subtraction lessons

This adding and subtracting integers topic guide is part of our series on adding and subtracting. You may find it helpful to start with the main adding and subtracting topic guide for a summary of what to expect or use the step-by-step guides below for further detail on individual topics. Other topic guides in this series include:

• Addition and subtraction
• Standard algorithm addition
• Adding and subtracting negative numbers
• Adding and subtracting rational numbers
• Add and subtract within 100

Practice adding and subtracting integers

1. Look at the model below to add -7 + 6.

There are 6 zero pairs with one negative counter leftover.

-7 + 6 = -1

2. Subtract: -15-(9) = \, ?

Using the rule, change the sign of the second number, +9 becomes -9.

Then add the two numbers together.

-15 + (-9) = -24

From 9 move left 24 units, you get to -15.

So, -15-(9) = -24

3. Add: 8 + (-19) = \, ?

Using the rule, since the signs of the numbers are different, the difference between 8 and 19 is 11.

19 is the larger number, and it is negative, so the sum will be negative.

8 + (-19) = -11

You can also check your answer using a number line.

Start at 8 and move 19 places in the negative direction. You land at -11.

4. Subtract: -14-(-8) = \, ?

Using the rule for subtracting integers, change the sign of the second number.

-8 will become +8.

Then add the number to the first one, -14 + 8 = -6

You can also use a number line to check your answer.

From -8 move left 6 places until you get to -14.

So, -14-(-8) = -6

5. Add: -13 + (-12) = \, ?

Using the rule, the signs of the numbers are both negative, so add the numbers.

The sum is negative too.

-12 + (-13) = -25

6. On a February day in Chicago, the morning temperature was -3 degrees Fahrenheit. Later that day, the temperature increased by 4 degrees. What is the new temperature in degrees?

-3 increased by 4 degrees is -3 + 4.

The signs of the numbers are different.

The difference between 3 and 4 is 1.

4 is the larger number, so the sum is positive.

Adding and subtracting integers FAQs

Yes, using the number line when adding and subtracting integers will always work. However, it might not always be the fastest way to get the answer.

A zero pair is a number and its opposite. For example, 5 and -5 are a zero pair. The opposite of positive integers is negative integers.

The sum of zero pairs is an additive inverse because the sum is 0.

Addition of integers and subtraction of integers help when simplifying algebraic expressions and also when factoring algebraic expressions.

The positive sign does not necessarily need to be written in front of a number. For example, +5 is the same as 5. The positive sign is understood.

The next lessons are

• Multiplication and division
• Multiplying and dividing integers
• Types of numbers
• Rounding numbers
• Solving one step equations
• Order of operations

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Subtracting Integers – Definition, Rules, Steps, Examples, Facts

What is meant by subtracting integers in math, rules for subtracting integers, properties of subtraction of integers, solved examples on subtracting integers, practice problems on subtracting integers, frequently asked questions on subtracting integers.

Subtracting integers is the method of finding the difference between two integers. These two integers may have the same sign or different signs. The set of integers is represented by

Z={…,-3,-2,-1,0,1,2,3,…} .

If we subtract the integer b from the integer a, we write it as a – b. 

Example: Subtract 7 from 9.

9 – 7 = 2

We can also write every subtraction problem as an addition problem. Replace the – sign by + sign, and replace the second integer by its additive inverse . This can be written symbolically as 

a – b = a + (-b)

Finally, you can find the answer using the rules for adding integers.

Additive inverse of an integer is written by simply removing its sign, keeping only the numerical value.

Examples: 

Additive inverse of 7 is -7.

Additive inverse of -9 is 9.

Subtracting Integers: Definition

Subtracting integers is the method of finding the difference between two integers having the same or different signs. 

Subtraction of integers can be written as the addition of the first number and the additive inverse of the second number. 

Related Worksheets

Subtrahend and Minuend

When we subtract one number from another, the number being subtracted is called subtrahend , the first number is called minuend .

Rules for Adding Integers

Any addition fact can be written as a subtraction fact. Also, any subtraction fact can be written as an addition fact. So, the rules for adding integers and the rules for subtracting integers correspond to each other. 

Positive integer + Positive integer	(+a) + (+b)	+(a + b)	5 + 7 = 12
Negative integer + Negative integer	(-a) + (-b)	-(a + b)	(-7)+(-5)=-(7+5)=-12
Negative integer + Positive integer	(-a) + (+b)	+(b – a) or -(a – b)	(-7)+5=-(7-5)=-2
Positive integer + Negative integer	(+a) + (-b)	+(a – b) or -(b – a)	7+(-5)=7-5=2

	If we subtract 0 from any integer, the answer will be the integer itself.a – 0 = aIf we subtract any integer from 0, we will find the additive inverse or the opposite of the integer.            0 – a = -a
	If you subtract a smaller integer from a larger integer, the sign of the result will be positive.
	If you subtract a larger integer from a smaller integer, the sign of the answer will be negative.
:	Subtraction of two integers can be written as sum of the first integer and the additive inverse of the second integer.a – b = a + (-b)Use the rules for adding integers to find the answer.
	When subtracting two positive integers, first subtract the smaller number from the larger number. The sign of the final answer will bei) positive when subtracting a smaller number from a larger number ii) negative when subtracting a larger number from a smaller number
	When subtracting two negative integers, change the subtraction sign to addition and change the sign of the second number. Next, perform the addition using the rules for adding integers.  (-9)-(-3)=(-9)+3=-(9-3)=-6(-1)-(-4)=(-1)+4=4-1=3
	Change the middle – sign to + sign and replace the second number by its additive inverse. Next, perform the addition using the rules for adding integers. Here, we have two cases. – Here, the result will always have a negative sign since we are subtracting a larger number from a smaller number. (-5) -( 2) = (-5) + (-2) = -7 – Here, the result will always have a positive sign since we are subtracting a larger number from a smaller number. :

Table of Rules for Subtracting Integers

(+) – (+)	Subtract	Sign of the larger integer	9-4=54-7=-3
(-) – (-)	Subtract	Sign of the larger integer	(-9)-(-5)=-9+5=-4
(+) – (-)	Add	+	(7) – (-2)=7+2=9
(-) – (+)	Add	–	(-1) – (8)=(-1)+(-8)=-9

How to Subtract Integers

The most simple rule to remember when subtracting integers is to convert the problem as the addition problem and use the rules for adding integers. 

• Subtracting Integers with the Same Sign: Let’s understand how to subtract integers with the same signs.

i) Positive integer – Positive integer

Here, the order of the numbers is important to decide the sign of the answer. 

The answer of the result will be  positive when subtracting a smaller number from a larger number.

30 – 13 = 17

The answer of the result will be negative when subtracting a larger number from a smaller  number.

4 – 7 = -3

ii) Negative integer – Negative integer

Change the problem to an addition problem and follow the rules for addition of integers.

(-5)-(-1) = (-5)+1 =-4

(-2) – (-5) = -2 + 5 = 3

• Subtracting Integers with Different Signs: Now, let’s understand how to subtract integers with different signs with examples.

i) Positive Integer – Negative integer

It is clear that we are subtracting a smaller value from a larger value. So, the answer will always be positive in this case. 

5-(-1) = 5 + 1 = 6

2 – (-5) = 2 + 5 = 3

ii) Negative Integer – Positive Integer

It is clear that we are subtracting a larger value from a smaller value. So, the answer will always be negative in this case. 

(-1)-5 = (-1) + (-5) = -6

(-2) – (5) = (-2) + (-5) = -7

There are some properties related to the subtraction of integers. They are as follows: 

• Closure property: The subtraction of any two integers is an integer. For any two integers a and b, the difference a -b is also an integer.
• Commutative property: Subtraction of integers is not commutative. For any two integers a and b, we have a – b b – a. 

2 – 4 4 – 2  and -6 – (-3 ) -3 – (- 6)

• Associative property: The subtraction of integers is not associative. That is, for any three integers a, b and c, we have, a -(b – c) (a – b) – c

For example, 2 – ( 5 – 3) = 2 – 2 = 0

                    (2 – 5) – 3 = – 3 – 3 = – 6 , 

• Identity property: We have  a – 0 = a , for any integer a indicating that 0 is correct identity for subtraction.vIt’s important to note that 0 – a a is not the left identity

Subtracting Integers on a Number Line

• Every subtraction fact can be written as an addition fact.
• Adding a positive integer is done by moving towards the right side (or the positive side) of the number line .
• Adding a negative integer is done by moving towards the left side (or the negative side) of the number line.
• Any one of the given integers can be taken as the base point from where we start moving on the number line.

Example: Subtract 4 from 2.

We have to find 2 – 4. 

Express 2 – 4 as the addition fact.

 2 – 4 = 2 + (-4).

Locate integer with greater absolute value, 2.

Start from 2, take 4 jumps to the left side as we are subtracting 4 to 2.

Therefore, -2 is the required answer.

Facts about Subtracting Integers

• Integers are the negative numbers, zero, and positive numbers.
• Subtracting integers is the method of finding the difference between two integers with the same or different signs.
• To subtract two integers, add the opposite of the second integer to the first integer. This can be written symbolically as a 
• – b = a + (- b)
• Adding a negative number is just like subtracting a positive number. 3 + (- 4) = 3 -(+4)
• Subtracting a negative number is just like adding a positive number.
• 3 -(- 4) = 3 + 4

In this article, we learnt about subtraction of integers definition, rules, properties,steps and how to subtract integers on a number line. Let’s solve some examples and practice problems to understand the concept better.

Example 1: Subtract: – 46 from – 80.

Solution:  

We have to subtract two negative integers.

Here, we have to find (-80) – (-46). 

(-80) – (-46)

Example 2: Subtract: – 8 from 7.

Solution: 

Here, we have to subtract two integers with different signs. 

Let’s write it in the form of an expression, 

7 – (- 8 )

Thus, 7 – (- 8 ) = 15

Example 3: Find 1 – ( – 5 ) using a number line.

1 – (- 5 ) = 1 + 5

Start from 1 and take 5 jumps to the right.

Thus, 1 – (- 5 ) = 6

Subtracting Integers - Definition, Rules, Steps, Examples, Facts

Attend this quiz & Test your knowledge.

Subtraction of integers can be written as the ________ of the opposite number.

$1000 - (- 1)=$, when subtracting a positive integer from a negative integer, the answer will have, which subtraction fact does the given number line represent.

What are integers?

Integers include positive integers, negative integers, and zero. It is a number with no decimal or fraction part.

Set of integers = {…,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5,…}

What is a minuend and a subtrahend?

In a subtraction equation, minuend is the number from which another number would be subtracted. A subtrahend is the term that denotes the number being subtracted from another.

Does commutative property hold true for subtraction?

The commutative property does not hold for subtraction. It means for any  two integers a and b, a – b ≠ b – a. For example, 2 – 4 ≠ 4 – 2

2 – 4 = – 2 and 4 – 2 = 2 and  -2 ≠ 2 

Thus, commutative property does not hold true for subtraction.

Does Associative property hold true for subtraction?

The associative property does not hold for subtraction. It means for any three integers a, b, and c, a -(b – c) ≠ (a – b) – c

                     (2 – 5) – 3 = – 3 – 3 = – 6 , 

Thus, associative property does not hold true for subtraction.

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How to Add and Subtract Integers: Word Problems

Mastering the art of tackling word problems involving the addition and subtraction of integers is a vital skill in the mathematical universe. Integers, which include positive, negative, and zero, are more nuanced than their natural number counterparts. To craft a highly detailed, comprehensive, and sophisticated guide, let's dive into the labyrinth of integers, unraveling the mystery of word problems step-by-step.

A Step-by-step Guide to Solve Integers Addition and Subtraction: Word Problems

Here is a step-by-step guide to solving word problems of integers addition and subtraction:

Step 1: Decipher the Problem

The journey begins with an intensive reading of the word problem. Identify the integers involved, noting their signs (\(+\) or \(-\)), and the operations stated or implied (addition or subtraction). Understand the context and constraints of the problem to guide your strategy.

Step 2: Pinpoint the Unknowns

Next, determine what the problem demands you to find. This could be an unknown quantity or a relationship between different quantities. Assign variables to these unknowns, typically ‘\(x\)’, ‘\(y\)’, or ‘\(z\)’.

Step 3: Translate into Mathematical Language

Now, morph the word problem into an equivalent mathematical expression or equation. Expressions such as “increased by” or “more than” often signify addition, while “decreased by” or “less than” hint towards subtraction. This translation serves as a bridge between the narrative of the problem and the mathematical steps to solve it.

Step 4: Construct the Equation(s)

Based on your translation, construct an equation or a system of equations that encapsulate the conditions outlined in the problem. Be vigilant of the signs of the integers; a positive integer added to a negative integer can be treated as subtraction and vice versa.

Step 5: Resolve the Equation(s)

Once your equation(s) are set, deploy your arithmetic and algebraic skills to solve them. Remember the basic rules of integer arithmetic, such as the fact that subtracting a negative integer is equivalent to adding a positive one.

Step 6: Validate Your Solution

Substitute your solution back into the original equation(s) to verify its correctness. If it stands the test, your solution is accurate. If it fails, reexamine your steps to identify potential missteps or miscalculations.

Step 7: Answer the Query

The final act is responding to the original question asked in the problem. Ensure your answer aligns with the question and is phrased appropriately, incorporating units if necessary.

by: Effortless Math Team about 12 months ago (category: Articles )

Effortless Math Team

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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 a negative integer.  Do not enter commas in your answers.

Hints:  Always double check the sign of your answer. Does it make sense for the problem? When subtracting integers, be sure to subtract the smaller integer from the larger integer. The smaller integer is farther to the left on the number line.

RESULTS BOX:

14°F in the morning. If the temperature dropped 7°F, what is the temperature now? RESULTS BOX:

RESULTS BOX:

RESULTS BOX:

RESULTS BOX:

39°C. The freezing point of alcohol is 114°C. How much warmer is the melting point of mercury than the freezing point of alcohol? RESULTS BOX:

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.

Integers Word Problems Worksheet - 1
Integers Word Problems Worksheet - 2
Integers Word Problems Worksheet - 3
Integers Word Problems Worksheet - 4

☛ Check Grade wise Integers Word Problems Worksheets

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

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• Addition Subtraction Integers

Addition And Subtraction Of Integers

In addition and subtraction of integers, we will learn how to add and subtract integers with the same sign and different signs. We can also make use of the num ber line to add and subtract signed integers. There are certain rules for integers that have to be followed to perform operations on them.

Adding two positive integers results in positive integers, whereas adding two negative integers will result in the sum with a negative sign. But, the addition of two different signed integers will result in subtraction only and the sign of the result will be the same as the larger number has. See a few examples below:

• 2 + (-2) = 0
• -2 + (-2) = -4
• -2 – (-2) = 0

Addition and subtraction

Addition and subtraction are two primary arithmetic operations in Maths. Besides these two operations, multiplication and division are also two primary operations that we learn in basic Maths. 

The addition represents the values added to the existing value. For example, a basket has two balls, and if we add more than 2 balls to it, there will be four balls in total. Similarly, if there are four balls in a basket and if we take out two balls out of it, then the basket is left with only two balls, which shows subtraction.

Addition and subtraction are not only used for integers but also rational numbers and irrational numbers. Therefore, both the operations are applicable for all real numbers and complex numbers. Also, the addition and subtraction algebraic expressions are done based on the same rules while performing algebraic operations.  

Learn more about addition and subtraction here.

Also, read:

Rules to Add and Subtract

Integers are a special group of numbers that are positive, negative and zero, which are not fractions. Rules for addition and subtraction are the same for all.

Negative Sign and Positive Sign

The integers which we add or subtract could be positive or negative.  Hence, it is necessary to know the rules for positive and negative symbols.

Positive sign/symbol: (+)

Negative sign/symbol: (-)

Addition of Integers

The three main possibilities in the addition of integers are:

• Addition between two positive numbers
• Addition between two negative numbers
• Addition between a positive number and a negative number

Positive + Positive	Add	Positive (+)	10 + 15 = 25
Negative + Negative	Add	Negative (-)	(-10) + (-15) = -25
Positive + Negative*	Subtract	Positive (+)	(-10) + 15 =5
Negative + Positive*	Subtract	Negative (-)	10 + (-15)= -5

Whenever a positive number and a negative number are added, the sign of the greater number will decide the operation and sign of the result. In the above example 10 + (-15) = -5 and (-10) + 15 =5; here, without sign 15 is greater than 10 hence, numbers will be subtracted and the answer will give the sign of the greater number.

We know that the multiplication of a negative sign and a positive sign will result in a negative sign, therefore if we write 10 + (-5), it means the ‘+’ sign here is multiplied by ‘-’ inside the bracket. Therefore, the result becomes 10 – 5 = 5.

Alternatively, to find the sum of a positive and a negative integer, take the absolute value (“ absolute value ” means to remove any negative sign of a number, and make the number positive) of each integer and then subtract these values. Take the above example, 10 + (-15); absolute value of 10 is 10 and -15 is 15.

⇒ 10 – 15 = -5

Thus, we can conclude the above table as follow:

Note: The sum of an integer and its opposite is always zero. (For example, -5 + 5= 0)

Subtraction of Integers

Like in addition, the subtraction of integers also has three possibilities. They are:

• Subtraction between two positive numbers
• Subtraction between two negative numbers
• Subtraction between a positive number and a negative number

For ease of calculation, we need to renovate subtraction problems the addition problems. There are two steps to perform this and are given below.

• Convert the subtraction sign into an addition sign.
• After converting the sign, take the inverse of the number which comes after the sign.

Once the transformation is done, follow the rules of addition given above.

For example, finding the value of (-5) – (7)

Step 1: Change the subtraction sign into an addition sign

⇒ (-5) + (7)

Step 2: Take the inverse of the number which comes after the sign

⇒ – 5 + (-7)                                  (opposite of 7 is -7)

⇒ – 5 + (-7) = -12      [Add and put the sign of greater number]

Properties Of Addition Of Integers

The addition properties for whole numbers are valid for integers.

Closure Property:  The sum of any 2 integers results in an integer.

For instance, 12 + 3 = 15 and 15 is an integer.

In the same way, 17 + (- 20) = – 3 and -3 is an integer.

Commutative property: Even if the order of addition is changed, the total of any 2 integers is the same.

For instance, – 19 + 15 = 15 + (- 19) = – 4

Associative property: The grouping of the integers does not matter when the total of 3 or more integers is computed.

For example, – 13 + (- 15 + 16) = (- 13 + (- 15)) + 16 = – 12

Additive identity: When the sum of zero with any integer is taken, the resultant answer is an integer. The additive identity is the integer zero.

For instance, 0 + 15 = 15

Additive inverse: For each integer, when an integer is added to that integer results in 0. The two converse integers are termed additive inverse of one another.

For instance, 9 + (- 9) = 0.

Properties Of Subtraction Of Integers

Closure property: The difference between any two given integers results in an integer.

For instance, 13 – 17 = – 4 and – 4 is an integer. In the same way, – 5 – 8 = – 13 and – 13 is an integer.

Commutative property: The difference between any two given integers changes when the order is reversed.

For example, 6 – 3 = 3 but 3 – 6 = – 3.

So, 6 – 3 ≠ 3 – 6

Associative property: In the method of subtraction, there is a change in the result if the grouping of 3 or more integers changes.

For example, (80 – 30) – 60 = – 10 however [80 – (30 – 60)] = 110.

So, (80 – 30) – 60 ≠ [80 – (30 – 60)].

Multiplication of Integers

In addition and subtraction, the sign of the resulting integer depends on the sign of the largest value. For example, -7+4 = -3 but in the case of multiplication of integers, two signs are multiplied together. 

(+) × (+) = +	Plus x Plus = Plus
(+) x (-) = –	Plus x Minus = Minus
(-) × (+) = –	Minus x Plus = Minus
(-) × (-) = +	Minus x Minus = Plus

• When two positive integers are multiplied then the result is positive.
• When two negative integers are multiplied then also the result is positive.
• But when one positive and one negative integer is multiplied, then the result is negative.
• When there is no symbol, then the integer is positive.

Solved Examples

Example 1: Evaluate the following:

• (-1) – ( -2)
• (-5 )+ 9 = 4  [Subtract and put the sign of greater number]

(-1) – ( -2) = 1

Example 2: Add -10 and -19.

Solution: -10 and -19 are both negative numbers. So if we add them, we get the sum in negative, such as;

(-10)+(-19) = -10-19 = -29

Example 3: Subtract -19 from -10.

Solution: (-10) – (-19)

Here, the two minus symbols will become plus. So,

-10 + 19 = 19 -10 = 9

Example 4: Evaluate 9 – 10 +(-5) + 6

Solution: First open the brackets.

9 – 10 -5 + 6

Add the positive and negative integers separately.

= 9 + 6 – 10 -5

= 15 – 15

Integers Addition and Subtraction Worksheet

Perform the addition of integers given below: (i) -12 + 25 (ii) 0 + 11 (iii) 38 + (-22) + 19 (iv) (-40) + 33 (v) (-15) + (-27) Subtract the following integers: (i) 8 – 9 (ii) (- 5) – 9 (iii) 6 – (- 8) (iv) (- 4) – (- 6) (v) (- 2) – (- 4) – (- 6)

Practice Problems

• Add -5 and -10.
• Subtract 20 from 10.
• Find the sum of 12 and 13.
• Find the difference between 40 and 30.

Frequently Asked Questions – FAQs

What is the rule to add integers, what is the rule for the subtraction of integers, are the rules of addition and subtraction the same as rules for the multiplication of integers, give examples of the addition of integers., when two negative integers are added together, then what is the sign of resulted value.

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Integer Word Problems

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.

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.

Adding Integers 1. Rearrange the terms so that integers with the same sign are next to each other. 2. Add integers with like signs together. 3. Subtract the absolute values of integers with different signs. 4. The sign of the solution will be the sign of the larger integer.

Subtracting Integers 1. Rewrite the problem by changing the second term to its additive inverse. 2. Add the values.

Multiplying Integers 1. Multiply the absolute values of the integers. 2. If the two factors have the same sign, the product is positive. 3. If the two factors have different signs, the product is negative.

The price of one share of stock fell 4 dollars each day for 8 days. How much value did one share of the stock lose?

The stock price fell, so it is represented by -4. This happened for 8 days. -4 x 8 = -32 The stock’s value dropped by $32.

Practice Integer Word Problems

Practice Problem 1

Practice Problem 2

Practice Problem 3

Practice Problem 4

Integer – whole numbers and their opposites {…, -3, -2, -1, 0, 1, 2, 3, …}

Negative integer – any integer less than zero.

Positive integer – any integer greater than zero.

Absolute value – the positive distance that a number is from 0 on a number line.

Zero Pair – a yellow counter and a red counter that represents zero.

Opposites – two integers that are the same distance from 0 on a number line but in opposite directions, like -5 and 5.

Additive inverses – two integers that are opposites.

Pre-requisite Skills Multi-Step Word Problems II Order of Operations

Related Skills Evaluating Algebraic Expressions Evaluate Algebraic Expressions Add Linear Expressions Subtract Linear Expressions Solve Complex Equations

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Subtraction of Integers: Definition, Rules, Properties, Examples

Subtraction of Integers:  You are already familiar with whole-number addition and subtraction. Are you aware that whole numbers are included in integers? Whole numbers and their negatives are included in integers. An integer is any number on a number line that does not have a fractional part. But, like whole numbers, can integers be added or subtracted?

For example, suppose the temperature in your city was \(3\) degrees Celsius, and it drops by \(8\) degrees Celsius. What is the temperature in your city right now? Integer addition and subtraction are two operations that we use to increase or decrease the value of integers. Let us study more about these two fundamental integer operations.

Subtraction of Integers

Natural numbers, their negatives, and zero are all examples of integers. A complete entity is an integer. Integers are numbers with no fractional element that can be positive, negative, or zero (no decimals). Integers, like whole numbers, can be subtracted. Integer subtraction refers to performing subtraction operations on two or more integers using the subtraction operators. It is critical to understand an integer’s absolute value before diving deeper into the idea.

The absolute value of an integer is the number’s distance from \(0\) on a number line. Because distance is a scalar quantity, it has no direction. It is always a good thing.

Subtraction of Integers with Examples

Subtraction usually refers to lowering the value. However, when dealing with integers, subtraction may increase or decrease in the value of the provided number. When we subtract a negative integer from another integer, the value increases, and when we subtract a positive integer, the value decreases. Please take a look at the samples below and notice the operation we are applying on integers. A worker descends three stairs from the \({{\rm{4}}^{{\rm{th}}}}\) the step he is now working on:

Also, the temperature drops by \(3\) degrees Fahrenheit from \(-1\) degrees Fahrenheit: \(\left( { – 1 – 3 =  – 4} \right)\)

We employ the concept of integer subtraction in the examples above. When subtracting a positive number from a given number, we must go to the left or negative side of the number line to indicate the subtraction of integers. When we remove a negative number from a given number, on the other hand, we move towards the right side or positive side.

Let’s look at how we can subtract \(-3\) from \(5\) 

Thus, we get \(5 – ( – 3) = 5 + 3 = 8\).

As a result, we can subtract \(-3\) from \(5\) by adding the negative (or additive inverse) of \(-3.\) As mentioned below, this is the rule for subtracting integers.

Subtraction of Integers Rule

Addition and subtraction are inverse operations, as you must know. As a result, any subtraction problem can be represented as a problem of addition. Let us look at a few examples to see how this is done.

\(5 – 8 = 5 + ( – 8)\)

\(9 – 4 = 9 + ( – 4)\)

\( – 7 – 5 = – 7 + ( – 5)\)

We must put the subtraction sign inside the bracket and add the addition operator between the two terms when creating any subtraction issue. This is one method of resolving subtraction problems.

Let us also study the rules of subtraction to make our calculations easier when working with integers.

Rule 1 : Two positive numbers are subtracted.

That is, \(( + x) – ( + y) = x – y\)

When subtracting two positive numbers, we take the difference between their absolute values and attach the sign of the larger number to the result.

Example: \(5 – 7 = – 2\) and \(9 – 3 = 6\)

Rule 2 : A positive and a negative number are subtracted.

That is, \(x – ( – y) = x + y\)

\(( – x) – y = – (x + y)\)

We add the absolute values of both integers and append the sign of the minuend to the response when subtracting a positive and a negative number.

Example: \(4 – ( – 7) = 4 + 7 = 11\) and \(( – 3) – 6 = – 9\)

Rule 3 : Two negative numbers are subtracted.

That is, \(( – x) – ( – y) = – (x – y)\)

We only need to remember one rule when subtracting two negative numbers: if a negative sign appears outside the bracket, the sign of the phrase inside the bracket is changed. After that, we must subtract the absolute values of both numbers and attach the revised sign of the higher number to the solution.

Example: \(( – 3) – ( – 5) = 2\) and \(( – 13) – ( – 8) = – 13 + 8 = – 5\)

Subtraction of Integers Properties

Closure property:  The subtraction of any two integers is an integer, i. e., for any two integers \(a\) and \(b,\,a – b\) is an integer.

Commutative property:  Subtraction of integers is not commutative, i.e., for any two integers \(a\) and \(b,\,a – b = b – a\) 

For example,

\(2 – 3 \ne 3 – 2\) and \( – 7 – ( – 3) \ne – 3 – ( – 7)\) and so on.

Identity property:  We have \(a – 0 = a\) for any integer \(a\), indicating that \(0\) is the correct identity for subtraction.

It’s important to note that \(0 – a \ne a\) is not the left identity.

If \(a,\,b\) and \(c\) are integers and \(a > b,\,a – c > b – c\) is true.  We learned that integer addition is both commutative and associative. As a result of these two features of integer addition and subtraction, we can now use the following methods to find the values of expressions, including various terms with plus and minus signs: Step 1: Obtain the expression whose value is to be determined. Step 2: Put all terms containing plus signs together and add them. Step 3: Put all terms containing minus signs and add them. Step 4: Find the difference in the absolute values of the two sums obtained in Steps \(1\) and \(3\) . Step 5: Assign to the result of step \(4\), the sign of the sum having a larger absolute value.

Subtraction of Integers Formula

Add two integers with the same sign, add their absolute values and use the same sign for the result as the given integers. To combine two integers with different signs, we subtract their absolute values (in the sequence larger number minus smaller number) and apply the sign of the larger number to the result.

Subtraction is done in the same way as addition, except we use the rule \(a – b = a + ( – b)\) Integer addition and subtraction formulas are as follows:

1. \(( + ) + ( + ) = + \) 2. \(( – ) + ( – ) = \,- \) 3. \(( + ) + ( – ) = + \)  (A positive number has a greater absolute value.) 4. \(( – ) + ( + ) =\, – \)  (A negative number has a larger absolute value.)

Solved Examples – Subtraction of Integers

Q.1. One particular day, the temperature in Delhi was \({13^{\rm{o}}}{\rm{C}}\) at \(10\,{\rm{a}}.{\rm{m}}\), but by midnight, it had dropped to \({6^{\rm{o}}}{\rm{C}}\). The temperature in Chennai was \({18^{\rm{o}}}{\rm{C}}\) at \(10\,{\rm{a}}.{\rm{m}}\). On the same day, but it dropped to \({10^{\rm{o}}}{\rm{C}}\) by midnight. Which of the two falls is greater? Ans: We have, The temperature in Delhi in the fall \( = {13^{\rm{o}}}{\rm{C}} – {6^{\rm{o}}}{\rm{C}} = {7^{\rm{o}}}{\rm{C}}\)  The temperature in Chennai in the fall \( = {18^{\rm{o}}}{\rm{C}} – {10^{\rm{o}}}{\rm{C}} = {8^{\rm{o}}}{\rm{C}}\)  Clearly, \({8^{\rm{o}}}{\rm{C}} > {7^{\rm{o}}}{\rm{C}}\). Hence, the fall in temperature of Chennai is greater.

Q.2. Find the value of \( – 12 + ( – 98) – ( – 84) + ( – 7)\) Ans: We have \( – 12 + ( – 98) – ( – 84) + ( – 7)\) \( = – 12 – 98 + 84 – 7\) \( = ( – 12 – 98 – 7) + 84\) \( = – 117 + 84\) \( = – (117 – 84) = – 33\) Hence, the value of the given expression is \( – 33\).

Q.3. Subtract the first integer from the second in \( – 225,\, – 135\). Ans: We have, \( – 225,\, – 135\) We need to subtract the first integer from the second. That is, \( – 135 – ( – 225)\) \( = – 135 + 225 = 90\) Hence, the difference between the given integers is \(90\).

Q.4. Subtract the sum of \(-1250\) and \(1138\) from the sum of \(1136\) and \(-1272\) Ans: The sum of \(-1250\) and \(1138\) is   \( – 1250 + 1138 = – 112\) And the sum of \(1136\) and \(-1272\) is \(1136 + ( – 1272) = 1136 – 1272 = – 136\) Now, we have to subtract \(-112\) from \(-136\). That is, \( – 136 – ( – 112) = – 136 + 112 = – 24\) Hence, the difference in the numbers is \( – 24\).

Q.5. Subtract the sum of \(-5020\) and \(2320\) from \(-709\). Ans: The sum of \(-5020\) and \(2320\) is \( – 5020 + 2320 = – 2700\). We have to subtract \(-2700\) from \(-709\). That is, \( – 709 – ( – 2700) = – 709 + 2700 = 1991\) Hence, the difference in the numbers is \(1991\).

In this article, we learnt about subtraction of integers definition, subtraction of integers with examples, subtraction of integers rule, subtraction of integers properties, subtraction of integers formula, solved examples on subtraction of integers and FAQs on subtraction of integers.

The learning outcome of this article is how to simplify integers with the same and different signs and how integers are used in real life like to compute temperature differences in a day.

Q.1. What is the definition of subtraction of integers? Ans: Integer subtraction refers to performing subtraction operations on two or more integers using the subtraction operators. Example: \( – 9 – ( – 6) = – 3\)

Q.2. What is an example of subtracting integers? Ans: The following is the example to subtract two integers. Example: Subtract \(-39\) from \(66\). Solution: We have to subtract \(-39\) from \(66\). That is, \(66 – ( – 39) = 66 + 39 = 105\).

Q.3. What is the formula for subtracting integers? Ans : Subtraction is done in the same way as addition, except we use the rule \(a – b = a + ( – b)\) Integer addition and subtraction formulas are as follows: 1. \(( + ) + ( + ) = + \) 2. \(( – ) + ( – ) = \,- \) 3. \(( + ) + ( – ) = + \)  (A positive number has a greater absolute value.) 4. \(( – ) + ( + ) =\, – \)  (A negative number has a larger absolute value.)

Q.4. How do you subtract integers? Ans: The following are steps to subtract integers: 1. Keep the first number for now (known as the minuend). 2. Second, switch from subtraction to addition as the procedure. 3. Get the opposite sign of the second number as the third step (known as the subtrahend). 4. Finally, perform a standard addition of integers.

Q.5. What are the rules of integers of addition and subtraction? Ans: The rules for addition and subtraction of integers are as follows: 1. Subtract the two numbers and sign the larger number if the two numbers have different signs, such as positive and negative. 2. If two numbers have the same sign, either positive or negative, add them together to get the common sign.

Learn About Addition and Subtraction Of Integers

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{FREE} Add & Subtract Integers: Real Life Lesson

Help your students make sense of integer addition & subtraction in context with this real life add & subtract integers lesson. Using a budget and spending, students will see what happens when you add & subtract with integers.

There are so many ways we see and use integers in the real world. We even add and subtract integers all the time, though we might not realize it. So even though teaching how to add & subtract integers might sound scary to both teacher and student, there are actually ways to introduce this in real and meaningful ways . This post includes a real life lesson you can use to add & subtract integers and help kids see the “rules” before you actually teach integer rules.

* Please Note : This post contains affiliate links which support the work of this site. Read our full disclosure here .*

Teaching Integer Operations Through Problem Solving

To begin introducing integers, I prefer to start with real world examples . This can include common things like temperature, being above or below sea level, or other ideas that may or may not be familiar to students such as golf scores (above or below par).

In this lesson, the focus is on money , and balancing your budget . In this example, kids will add & subtract integers as money is earned (through working) or lost (by buying things).

What’s great about this lesson is that kids will actually use integers in a meaningful context and will see integer rules play out ( such as when you have a negative and subtract a positive you end up with a bigger negative ).

In addition, kids will see the commutative property , so that no matter what order they include the transactions in their ledger, they will always end up with the same amount of money in the bank.

How to Add & Subtract Integers

When you’re ready to set up this lesson, begin by printing a tracking page for each student. You will want to print this front and back so that students have enough space to keep track of each line item in the budget.

Students will then need a set of word problem cards .

I suggest putting students in small groups, and having them go through the cards in a different order . In this case, you can print just one set per group .

Just be sure students are careful not to mix the cards up or lose track of what items they’ve already calculated.

You want them to see that at the end, they all end up with the same final balance .

You could also assign this work individually , which means each student will need a page of word problems .

Then they could go through them in order , without cutting the cards out.

If you do it this way, every student s hould have the same answers all the way through the assignment.

This might be helpful if you’d rather everyone complete it in the same order so you can discuss various steps and compare how they worked them out and how they wrote their equations.

Finally, you will need to give everyone a starting balance .

If the starting balance is $0, they will end the assignment with $277 in the bank .

If you start with money already, however, you will end with: (starting value) + $277 . For instance, if the starting balance is $500, you will end with $777.

I hope this lesson provides a fun, real life math lesson for your students, and begins to build a foundation for how to work with integers.

Need more practice and visuals to teaching addition & subtraction with integers? Grab this complete lesson and games collection: Add & Subtract Integers Lessons & Games .

Extension and Follow Up Questions

To follow up, you’ll want to discuss why all students ended up with the same final total (if they didn’t, have them go back through their work to check first).

You could also discuss this specific situation and ask, “How could Bob have saved more money?” or “What was the best/worst choice he made with his money?”

You could also challenge kids to compare specific problems that are similar and estimate which is better. For example, what is better for Bob, working 5 hours at $15 an hour or working 10 hours for $10 an hour?

After discussing ideas, estimations and strategies, work out each problem.

Finally, you can then discuss what they noticed about integers and as them to explain in their own words how to add negative numbers , or how to subtract negative numbers .

When you’re ready to explore the integer rules more specifically, you might like this lesson to add & subtract integers . This shows what happens as you add & subtract integers using +/- tables .

Find more helpful pre-algebra lessons in this post .

{Click HERE to go to my shop and grab the FREE Add & Subtract Integers in Real Life Lesson}

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