DIVIDING INTEGERS WORD PROBLEMS

We can use integer division to solve real-world problems. For some problems, we may need to perform more than one step.

And also, we have to be sure to check that the sign of the quotient makes sense for the situation.

Problem 1 :

Jake answers questions in two different online Olympic trivia quizzes. In each quiz, he loses points when he gives an incorrect answer. The table shows the points lost for each wrong answer in each quiz and Jake’s total  points lost in each quiz. In which quiz did he have more wrong answers ?

dividing integers problem solving

Step 1 : 

To find the number of incorrect answers in the winter quiz, divide the total points lost by the number of points lost per wrong answer.

-33 ÷ (-3)  =  11

Find the number of incorrect answers Jake gave in the summer quiz. Divide the total points lost by the number of points lost per wrong answer.

-56 ÷ (-7)  =  8

Compare the numbers of wrong answers.

So, Jake had more wrong answers in the winter quiz.

Problem 2 :

A penalty in Meteor-Mania is -5 seconds. A penalty in Cosmic Calamity is -7 seconds. Yolanda had penalties totaling -25 seconds in a game of Meteor-Mania and -35 seconds in a game of Cosmic Calamity. In which game did Yolanda receive more penalties ? Justify your answer.

To find the number of times Yolanda received penalties in Meteor-Mania, divide the total penalties by penalties per time.

-25 ÷ (-5)  =  5

In the game of  Meteor-Mania, s he received penalties 5 times  

Step 2 : 

To find the number of times Yolanda received penalties in  Cosmic Calamity , divide the total penalties by penalties per time.

-35 ÷ (-7)  =  5

In the game of  Cosmic Calamity, s he received penalties 5 times.

Compare the number of times she received penalties in both the games. 

5  =  5

Yolanda received the same number of penalties in each game

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M and D Integers

Multiplying and dividing integers

Here you will learn strategies on how to multiply and divide integers, including using visual models as well as using 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.

What are multiplying and dividing integers?

Multiplying and dividing integers is when you multiply or divide two or more integers together to give a product or quotient that can be either positive or negative.

You can multiply and divide integers using visual models or a rule.

Multiplying and Dividing Integers image 1.1

Do you notice a pattern or rule?

Rule for multiplying integers:

  • If the integers have the same sign , the product will be positive .
  • If the integers have different signs , the product will be negative .

Multiplying and Dividing Integers FAQS image 3

Rule for dividing integers:

  • If the integers have the same sign , the quotient will be positive .
  • If the integers have different signs , the quotient will be negative .

What are multiplying and dividing integers?

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.2) Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

[FREE] Multiplication and Division Check for Understanding (Grade 4, 5 and 7)

[FREE] Multiplication and Division Check for Understanding (Grade 4, 5 and 7)

Use this quiz to check your grade 4, 5 and 7 students’ understanding of multiplication and division. 10+ questions with answers covering a range of 4, 5 and 7 grade multiplication and division topics to identify areas of strength and support!

How to multiply and divide integers?

In order to add and subtract integers using counters:

  • If the integers have the same sign, the product or quotient is positive.  If not, go to step 2.

If the integers have different signs, the product or quotient is negative.

Find the product or quotient.

Multiplying and dividing integers examples

Example 1: multiplying integers with the same sign.

Multiply: (-4) \times(-12)= \, ?

  • If the integers have the same sign, the product or quotient is positive. If they don’t go to step 2. 

-4 and -12 have the same sign so the product is positive.

2 If the integers have different signs, the product or quotient is negative.

Integers have the same sign.

3 Find the product or quotient.

(-4) \times(-12)=48

Example 2: multiplying integers with different signs

Multiply: (-13) \times 8= \, ?

If the integers have the same sign, the product or quotient is positive.  If not, go to step 2.

-13 and 8 do not have the same sign.

The integers have different signs so the product is negative.

Find the product or the quotient.

(-13) \times 8=-104

Example 3: dividing integers with the same sign

Divide: \cfrac{(-18)}{(-3)}= \, ?

-18 and -3 have the same sign so the quotient will be positive.

\cfrac{(-18)}{(-3)}=6

Example 4: dividing integers with different signs

Divide: -120 \div 3= \, ?

-120 and 3 do not have the same sign.

-120 and 3 have different signs so the quotient will be negative.

-120 \div 3=-40

Example 5: multiplying integers word problem

From sea level, a submarine descends 25 \, ft. per minute (-25 \, ft.).

After 6 minutes, the submarine’s distance can be modeled by (-25) \times 6 = \, d , where d is the submarine in relation to sea level.

How far below sea level is the submarine?

-25 and 6 do not have the same sign.

-25 and 6 have different signs so the product will be negative.

-25 \times 6=-150

Because the submarine is descending, after 6 minutes it will be 150 feet below sea level.

Example 6: dividing integer word problems

On a certain winter day, the temperature changed at a rate of -4 degrees Fahrenheit per hour.

After a specific amount of time, the change in temperature was -36 degrees Fahrenheit, which is modeled by (-36) \div (-4) = \, h, where h represents the amount of hours.

How long did it take for the change in temperature to be -36 degrees Fahrenheit?

-36 and -4 have the same sign so the quotient is positive.

The integers have the same sign.

(-36) \div(-4)=9

Teaching tips for multiplying and dividing integers

  • Multiplying and dividing integers are foundational skills for Algebra 1. Using manipulatives helps students formulate conceptual understanding.
  • Have students identify the patterns with multiplying and dividing 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.
  • Reinforce essential vocabulary such as dividend, divisor, quotient, factors,  and product.

Easy mistakes to make

  • Mixing up the rules for multiplication and division For example, when multiplying integers with the same sign, you get a negative product, and when dividing integers with the same sign, you get a negative quotient.
  • Mixing up the rules of multiplication and division with addition and subtraction For example, applying the rule of addition of integers to (15) \div(-3)=5 where the absolute value of 15 is greater than the absolute value of 3 so the answer must be positive.

Related multiplication and division lessons

This multiplying and dividing integers topic guide is part of our series on multiplication and division. You may find it helpful to start with the main multiplication and division 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:

  • Multiplication and division
  • Multiplying multi-digit numbers
  • Understanding multiplication
  • Multiplication and division within 100
  • Long division
  • Multiplying and dividing rational numbers
  • Understanding division
  • Dividing multi-digit numbers
  • Multiplicative comparison

Practice multiplying and dividing integers

1. Multiply: (6) \times(-2)=\text { ? }

GCSE Quiz False

Using the rule for multiplying integers, 6 and -2 have different signs, so the product is negative.

(6) \times(-2)=-12

You can check your answer with counters.

6 groups of -2 counters is -12 counters

Multiplying and Dividing Integers practice 1

2. Multiply: (-15) \times(-3)= \text { ? }

Using the rule for multiplying integers, -15 and -3 have the same sign, so the product is positive.

(-15) \times(-3)=45

3. Divide: (-52) \div(4)= \text { ? }

Using the rule for dividing integers, -52 and 4 have different signs, so the quotient is negative.

(-52) \div(4)=-13

4. Divide: \cfrac{(-72)}{(-9)}= \text { ? }

Using the rule for dividing integers, -72 and -9 have the same sign, so the quotient is positive.

\cfrac{(-72)}{(-9)}=8

5. Multiply: (-190) \times(-10)= \text { ? }

Using the rule for multiplying integers, -190 and -10 have the same sign, so the product is positive.

(-190) \times(-10)=1900

6. A deep sea diver descends at a rate of 10 feet per minute below sea level. The diver descends at this rate for 8 minutes, which can be modeled by 8 \times (-10) = d, where d is how far the diver is below sea level. After 8 minutes, how far did the diver descend?

Using the rule, the signs of the numbers are different, so the product is negative.

8 \times(-10)=-80

The diver descended to -80 feet.

Multiplying and dividing integers FAQs

Multiplying and Dividing Integers FAQS image 1

When it comes to sets of numbers, whole numbers are 0 and positive whole numbers. The set of integers includes all negative whole numbers, 0, and all positive whole numbers.

Multiplication and division of integers help when simplifying algebraic expressions and solving equations.

Yes, positive numbers are to the right of 0 and negative numbers are to the left of 0.

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

  • Types of numbers
  • Rounding numbers
  • Factors and multiples

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Dividing integers

Dividing integers

After you’ve learned how to add , subtract , and multiply integers, division is next up!

You’ve been asked to calculate how many times one number fits into another. This is when division comes in handy. Dividing just means subtracting the same number a specific number of times until you get $$0$$, to see how many times a number you are subtracting with fits into the number you started with. So instead of calculating $$12-4=8, ~8-4=4,~ 4-4=0$$, you can think of it as if you were calculating $$12\div4$$.

But what if one or both of the factors are negative? Well, that’s when we don’t divide whole numbers anymore – that’s when integers come into the picture!

What does it mean to divide integers?

Dividing integers means splitting integers into an equal number of parts.

An integer is any number from a set of whole numbers and their additive inverses – the numbers with the same absolute value but with an opposite sign:

‘’Wait… what is the absolute value of a number?!’’

Good question! The absolute value of a number is that same number, but only without the sign in front of it. Why? Because the absolute value actually shows what the distance is from that number to $$0$$ on the number line. For example, the absolute value of $$|-4|=4$$. That’s the distance from $$-4$$ to $$0$$ on the number line; $$4$$ units:

Dividing integers comes down to dividing their absolute values and determining the sign of the result by applying the following rules:

  • The quotient of two positives equals a positive
  • The quotient of two negatives equals a positive
  • The quotient of a negative and a positive equals a negative.

And that’s pretty much it!

Why is dividing integers so useful?

Besides the fact that multiplying integers is one of the basic concepts we learn in math, think of it this way: there are many real-life problems it can solve! For example, a person scored $$-24$$ points because of many penalties in a game. Each penalty is $$-8$$ points. How many penalties did the person get? Well, that problem can be solved by dividing integers. The number of points has to be divided by the penalty points. It all comes down to a simple math problem of dividing integers:

How to divide integers

Before we learn how to divide integers, let’s check out some rules for division.

If two positive or two negative numbers are divided, their quotient is a positive number.

If two numbers of different signs are divided, their quotient is a negative number.

Divide the integers:

Dividing two negatives equals a positive $$(-)\div(-)=+$$: $$6\div 3$$ Divide the numbers:

Dividing a negative and a positive equals a negative $$(-)\div(+)=(-)$$: $$-(20 \div4)$$ Divide the numbers:

That wasn’t so bad, right? Now that we’ve walked through detailed examples, let’s review the overall process so you can learn how to use it with any problem:

Study summary

  • Use the rules for division.
  • Divide the numbers.

Try it yourself!

Practicing math concepts like this one is a great way to prepare yourself for the math journey to come! So, when you’re ready, we’ve got some practice problems for you!

  • $$-35\div 7$$
  • $$42\div(-6)$$
  • $$-63\div (-9)$$
  • $$-120\div 20$$

If you’re struggling through the solving process, that’s totally okay! Stumbling a few times is good for the learning process. If you get stuck or lost, scan the problem using your Photomath app and we’ll walk you through it!

Here’s a sneak peek of what you’ll see:

Related Topics

  • Math Explained: Arithmetic
  • Arithmetic Operations

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Part 4: Division

4.2: Word problems for dividing integers

Word problems for the division problem, A ÷ B, can either use the “how-many-groups” interpretation of division or the “how-many-units-in-one-group” interpretation.

How many groups?

A ÷ B represents, “How many groups are there if A units are divided into groups of B units?” For example,

  • Arithmetic problem: Solve 12 ÷ 4.
  • Word problem: A pizza cut into 12 slices is to be divided among friends such that each friend gets 4 slices. How many friends can share the pizza?
  • 12 ÷ 4 = 3, so 3 friends can share the pizza.
  • Here, each friend is a “group,” the pizza slices are the units, and we’re finding the number of friends: 12 slices ÷ 4 slices per fried = 3 friends.
  • Equivalently, 3 friends times 4 slices per friend equals 12 slices in total.

How many units in one group?

A ÷ B represents, “How many units are in one group if A units are divided into B groups?” For example,

  • Word problem: A pizza cut into 12 slices is to be divided among 4 friends. How many slices do they each get?
  • 12 ÷ 4 = 3, so each friend gets 3 slices.
  • Again, each friend is a “group” and the pizza slices are the units, but now we’re finding the number of slices per friend: 12 slices ÷ 4 friends = 3 slices per friend.
  • Equivalently, 3 slices per friend times 4 friends equals 12 slices in total.

The video below works through some examples of word problems for dividing integers.

Practice Exercises

Do the following exercises to practice matching integer division problems and word problems.

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Integers: Multiplication and Division Worksheets

This page contains printable worksheets which emphasize integer multiplication and division to 6th grade, 7th grade, and 8th grade students. Practice pages here contain exercises on multiplication squares, in-out boxes, evaluating expressions, filling in missing integers, and more. Procure some of these worksheets for free!

Multiplying Integers

Multiplication of Integers

Multiplication of Integers

Multiply the integers to find the product. A total of 48 problems are given in these integer worksheets for practice.

pdf 1

Multiplication Squares | 2x2

Conceptualize the fundamentals of multiplying integers with this bath of interesting 2x2 squares. Multiply the integers in the rows and columns and write the products in the squares.

printable 1

Multiplication Squares | 3x3

Get students to multiply the positive and negative numbers in each row and column and fill in the empty boxes in each 3x3 square.

exercise 1

Multiplying 3 or 4 Integers

Find the product of the integers. Apply the multiplication sign rule. Each worksheet consists of ten problems.

worksheet 1

Dividing Integers

Integer Division

Integer Division

Perform the division operation on the integers to find the quotient in these three pdf worksheets.

practice 1

Multiplying and Dividing Integers - Mixed

Multiplying and Dividing Integers

Multiplying and Dividing Integers

Simplify the integer equations by performing multiplication and division operations.

Missing Integers: Multiplication and Division

Missing Integers: Multiplication and Division

Find the missing integer in each equation. There are 16 problems in each worksheet.

In-and-Out boxes: Multiply and Divide

In-and-Out boxes: Multiply and Divide

Fill in the in-and-out boxes according to the rule mentioned in each problem. There are six rules given in each worksheet.

Evaluate: Multiply and Divide

Evaluate: Multiply and Divide

Evaluate the expressions by substituting the values in the variables. Each printable worksheet for grade 7 and grade 8 contains four problems with three expressions each.

Multiplication Chart

Integer Multiplication Charts | Display Charts

Integer Multiplication Charts | Display Charts

Designed to assist 6th grade students with multiplication of integers, this array of charts focuses on multiplication of integers from -5 to +5 and -10 to +10. Explore these visual aids and comprehend the two rules of multiplication of integers.

chart 1

Integer Multiplication Charts | Blank Charts

Print this set of ready-to-print blank charts and practice multiplying integers from -10 to +10. It is a fairly straightforward process. The product of integers will be positive, if the signs are same and the answer will be negative, if the multiplying integers have different signs.

Related Worksheets

» Adding and Subtracting Integers

» Comparing and Ordering Integers

» Integers on a Number Line

» Multiplication and Division Fact Family

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Unit 1: Multiplication and division of integers

Multiplication of integers.

  • Multiplying positive & negative numbers (Opens a modal)
  • Why a negative times a negative makes sense (Opens a modal)
  • Why a negative times a negative is a positive (Opens a modal)
  • Multiplying negative numbers Get 3 of 4 questions to level up!

Division of integers

  • Dividing positive and negative numbers (Opens a modal)
  • Dividing negative numbers Get 3 of 4 questions to level up!

Multiplication and division of integers word problems

  • Interpreting multiplication & division of negative numbers (Opens a modal)
  • Word problems involving negative numbers Get 3 of 4 questions to level up!
  • Multiplying & dividing negative numbers word problems Get 3 of 4 questions to level up!

Division of integers Calculator

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Examples on Division of Integers | Dividing Integers Problems with Solutions

Get the complete practice test questions and worksheet here. Follow the step by step procedure to solve examples on the division of integers all the problems. Know the shortcuts, tricks, and steps involved in solving Division of Integers problems. Also, find the definitions, formulae before going to start the practice sessions. Go through the below sections to know the detailed information regarding formulas, definitions, and problems.

Integers Division Rules

Rule 1: The quotient value of a positive integer number and a negative integer number is negative.

Rule 2: The quotient value of two positive integer numbers is a positive number.

Rule 3: The quotient value of two negative integer numbers is a positive number.

Division of Integers Rules and Examples

Question 1: Find the value of ||-17|+17| / ||-25|-42|

||-17|+17| / ||-25|-42|

= |17+17| / |25–42|

= |34| / |-17|

Question 2: Simplify: {36 / (-9)} / {(-24) / 6}

{36 / (-9)} / {(-24) / 6}

= {36/-9} / {-24/6}

= – (36/9) / – (24/6)

Question 3: Find the value of [32 + 2 x 17 + (-6)] ÷ 15

[32+2 x 17+(-6)] / 15

= [32+34+(-6)] / 15

= (66-6) / 15

Question 4: Divide the absolute values of the two given integers?

The quotient of the absolute value of integer +24 and the absolute value of integer -8

From the rules given above, dividing the integers with different signs gives the final result as negative.

When positive(+) 24 is divided by negative(-) 8 results negative(-) 3, which can be defined as +24/(-8) = -3

Question 5: Prove that [((-8) / (-4)] ≠ =-8 / [(-4) / (-2)]

From the given question

[((-8)/(-4)]/(-2)] ≠ -8/[(-4)/(-2)]

First of all, we will consider the LHS part i.e., [((-8) / (-4)] / (-2)]

To solve the equation, first, we divide 8 by 4, we get the result as 2

As both numbers have a negative sign, it will be positive.

Then we divide the result (2) by -2, then the final result will be -1.

Therefore, the result of the LHS part is -1.

Now, we consider the RHS part i.e., -8 / [(-4) / (-2)]

First of all, we divide -4 by -2, the result will be 2.

As both the numbers have a negative sign, the result will be positive.

Now, divide -8 by the above result 2.

Hence, the result will be -4.

Therefore, the result of RHS is -4.

Hence, LHS ≠ RHS

The above given equation [((-8) / (-4)] / (-2)] ≠ -8 / [(-4) / (-2)] is thus satisfied.

Question 6: In a maths test containing 10 questions, 2 marks are given for every correct answer and (-1) marks are given for every wrong answer. Rohith attempts all the questions and 8 questions answers are correct. What is Rohith’s total score?

From the given question,

The marks given for every correct answer = 2 marks

Marks given for 8 correct answers = 2 * 8= 16 marks

Marks given for 1 incorrect answer = -1 marks

Marks given for 2 incorrect answers = -1 * 2 = -2 marks

Rohit’s total score = 16-2 = 14 marks

Therefore, the answer is 14 marks.

Question 7: Priya sells 20 pens and some pencils losing 2 Rs in all. If Priya gains 2 Rs on each pen and loses 1 Rs on each pencil. How many pencils does Priya sell?

Suppose that Priya sells x pencils.

Total gain on pens = 2*20 = 40 Rs

Total loss on pencils = 1x = x

Total loss on selling pens and pencils = (-2)

40-x = (-2)

Therefore, Priya sells 42 pencils.

Thus, the answer is 42 pencils.

Question 8: To make ice cream, the room temperature must be decreased from 45degree C at the rate of 5 degrees C per hour. What will be the room temperature 12 hours after the freezing point of the icecream?

As given in the question,

Temperature after 12 hours = 12*5 = 60degree C

Room temperature = 45degree C

Hence, the room temperature after freezing process of 12 hours = (45-60)degree C

= -15degree C

Question 9:

A car runs at a rate of 50km/hr. If the car starts at 5 km above the starting point, how long will it take to reach 2505km?

As per the question,

Total distance covered by car = (2505-5) km = 2500 km

Rate of car = 50km/hr

From the above values, Distance = 2500 km, speed = 50 km/hr

Therefore, the time required by the car = distance/speed

=2500/50 = 50 hours

Therefore, the car will take 50 hours to travel 2505 km.

Hence, The final solution is 50 hours.

Question 10: Jason borrowed $5 a day to buy launch. She now owes $65 dollars. How many days did Jane borrow $5?

Jason borrowed a launch at =  $5

Now she owes = $65

No of days = (-65) / (-5)

Therefore, Jane borrows $5 for 13 days.

Hence, The final solution is 13 days.

Division of Integers Word Problems

Question 11: Allen’s score in a video game was changed by -120 points because he missed some target points. He got -15 points for each of the missed targets. How many targets did he miss?

As per the given question,

Allen scored points in a video game = -120

He got points for missed targets = -15

No of targets he missed = -120/-15 = 8 targets.

Therefore, he missed 8 targets.

Question 12: Karthik made five of his truck payments late and was fined five late fees. The total change in his savings of late fees was -$30. What integer represents the one late fee?

As given in the question, Karthik has made five of his truck payments. Therefore, it is positive.

He was fined -$30 as the late fees.

To find one late fee, we have to divide the fine by no of payments he did.

Therefore, One Late fee = -$30/5

Thus, He paid $6 for each payment as late fees.

Hence, the final answer is -$6.

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dividing integers problem solving

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7th Grade Multiplying and Dividing Integers Worksheets

Integers are a set of whole numbers and their opposites. These seventh grade multiplying and dividing integers worksheet s are really helpful to enhance a student’s knowledge about operations on integers by modeling and solving problems based on real life scenarios. ...Read More Read Less

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Choose math worksheets by topic, 7th grade multiplying and dividing integers worksheets explained:.

The multiplying and dividing integers worksheets for grade 7 helps students to understand the procedure that supports the multiplication and division of integers. The concept of multiplying and dividing integers will follow the student through each grade in their journey through math across higher grades. In these worksheets a student will go through the concepts such as the rules of multiplication, multiplication of integers with same or a different sign, rules of division, finding quotients of integers with the same or different sign and use of integers to model real life questions. These seventh grade multiplying and dividing integers worksheets contain questions related to the following topics:

  • Multiplying integers with same or different signs: The product of two integers with the same sign is positive but the product of two integers with different signs is negative.
  • Dividing integers with same or different signs: The quotient of two integers with the same sign is positive and the quotient of two integers with different signs is negative.  
  • Applying multiplication and division on integers to model real life problems: The online free multiplying and dividing integers worksheets for grade 7 contain context based questions where integers are used to represent quantities and operations(multiplication and division) are applied to find the solution of problems.  

If you feel the need to refresh your understanding about multiplying and dividing integers as a concept, click on the following links:

  • Integers on number line and their comparison
  • Adding multiplying and dividing integers
  • Operations on multiplying and dividing integers using signs and inductive reasoning
  • Dividing operations on multiplying and dividing integers
  • Properties of multiplying and dividing integers

Benefits of 7th Grade multiplying and dividing integers worksheets:

  • Integers worksheets (Easy): The worksheets for multiplying and dividing integers in the seventh grade aid the learner in recalling multiple terms and operations associated with integers. The learner is attracted to learn concepts and use them in the classroom by solving the problems in the free printable or online worksheets.  
  • Integers worksheets (medium): The learner can use various integer operations to answer various problems by using the multiplying and dividing integers worksheets for grade 7. These worksheets are both printable and online. Additionally, it enhances the student's ability to quickly solve addition and subtraction problems involving integers. These worksheets also help students improve their aptitude.
  • Integers worksheets (Hard): These free worksheets for seventh grade that are available online and in print format explain specific types of problems whose solutions require two or more steps to complete, suggesting that more than two operations (addition and subtraction) on integers may be necessary to answer the problem. This equips students to develop solutions to such complex challenges. These worksheets also assist students in developing their problem solving abilities for upcoming grades.    

Printable PDFs & Online 7th Grade Worksheets:

Free printable worksheets for multiplying and dividing integers in the seventh grade are a great way to improve a student’s problem solving abilities and get them ready for assessments like standardized examinations and aptitude tests. Grade 7 multiplying and dividing integers worksheets help the student to gain a thorough understanding of how to apply concepts they employ while solving the worksheets. Attempting difficult questions that are found in higher grades is a bonus. The grade 7 multiplying and dividing integers worksheets include fundamental concepts like rules of multiplying or dividing integers, finding the product of two integers, finding the quotient after dividing two integers and using integers through simulating real world situations. Timed online worksheets also help the student to enhance time management skills, especially while solving particular types of questions. With these timed worksheets the student can also compare scores and time spent on the worksheets with classmates and challenge them as well.

Write down the general rules of multiplication.

Four rules of multiplying integers are: 

  • Positive × Positive = Positive
  • Positive × Negative = Negative
  • Negative × Positive = Negative
  • Negative × Negative = Positive

What will be the sign of product if two integers have different signs?

The product of two integers with different signs is always a negative number, so the sign of the product is ‘minus’.

How can hard level problems help a student in the journey through mathematics?

Hard level problems are context based and may have solutions with multiple steps. Students will have to apply the basic concepts as well as two or more operations to solve some problems. These hard level problems prepare a student to enhance aptitude skills and gear up for future grades.

What if the divisor is zero?

The quotient obtained by dividing an integer by zero is undefined.

How are 7th grade multiplying and dividing integers worksheets advantageous to students?

When students solve worksheets on their own, it promotes active learning as they feel a sense of accomplishment. It raises the curiosity level about multiplying and dividing integers in relation to specific topics, and in turn, enables them to take up more challenging questions. In the multiplying and dividing integers worksheets for grade 7 from BYJU’S math, students also learn the method of applying step-wise solutions that further strengthens their understanding about multiplying and dividing integers.

Why are these interactive worksheets necessary for students?

Free online grade 7 multiplying and dividing integers worksheets link learning multiple concepts with a fun element as they are interactive. These worksheets follow the common core math structure and assist students to be completely familiar with the methods and terms used in 7th grade.

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1.5: Multiply and Divide Integers

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Learning Objectives

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

  • Multiply integers
  • Divide integers
  • Simplify expressions with integers
  • Evaluate variable expressions with integers
  • Translate English phrases to algebraic expressions
  • Use integers in applications

A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers .

Multiply 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 will use the model just to help us discover the pattern.

We remember that \(a\cdot b\) means add \(a,\, b\) times. Here, we are using the model just to help us discover the pattern.

Two images are shown side-by-side. The image on the left has the equation five times three at the top. Below this it reads “add 5, 3 times.” Below this depicts three rows of blue counters, with five counters in each row. Under this, it says “15 positives.” Under thisis the equation“5 times 3 equals 15.” The image on the right reads “negative 5 times three. The three is in parentheses. Below this it reads, “add negative five, three times.” Under this are fifteen red counters in three rows of five. Below this it reads” “15 negatives”. Below this is the equation negative five times 3 equals negative 15.”

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. Then we take away \(5\) three times.

This figure has two columns. In the top row, the left column contains the expression 5 times negative 3. This means take away 5, three times. Below this, there are three groups of five red negative counters, and below each group of red counters is an identical group of five blue positive counters. What are left are fifteen negatives, represented by 15 red counters. Underneath the counters is the equation 5 times negative 3 equals negative 15. In the top row, the right column contains the expression negative 5 times negative 3. This means take away negative 5, three times. Below this, there are three groups of five blue positive counters, and below each group of blue counters is an identical group of five red negative counters. What are left are fifteen positives, represented by 15 blue counters. Underneath the blue counters is the equation negative 5 times negative 3 equals 15.

In summary:

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

Notice that for multiplication of two signed numbers, when the:

  • signs are the same , the product is positive .
  • signs are different , the product is negative .

We’ll put this all together in the chart below.

MULTIPLICATION OF SIGNED NUMBERS

For multiplication of two signed numbers:

Example \(\PageIndex{1}\)

  • \(-9\cdot 3\)
  • \(7\cdot 6\)
  • \[\begin{array} {ll} {} &{-9\cdot 3} \\ {\text{Multiply, noting that the signs are different, so the product is negative.}} &{-27} \end{array}\]
  • \[\begin{array} {ll} {} &{-2(-5)} \\ {\text{Multiply, noting that the signs are same, so the product is positive.}} &{10} \end{array}\]
  • \[\begin{array} {ll} {} &{4(-8)} \\ {\text{Multiply, with different signs.}} &{-32} \end{array}\]
  • \[\begin{array} {ll} {} &{7\cdot 6} \\ {\text{Multiply, with different signs.}} &{42} \end{array}\]

Try It \(\PageIndex{2}\)

  • \(-6\cdot 8\)
  • \(5\cdot 12\)

Try It \(\PageIndex{3}\)

  • \(-8\cdot 7\)
  • \(3\cdot 13\)

When we multiply a number by \(1\), the result is the same number. What happens when we multiply a number by \(−1\)? Let’s multiply a positive number and then a negative number by \(−1\) to see what we get.

\[\begin{array} {lll} {} &{-1\cdot 4} &{-1(-3)}\\ {\text{Multiply.}} &{-4} &{3} \\ {} &{-4\text{ is the opposite of 4.}} &{3\text{ is the opposite of } -3} \end{array}\] 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.

Example \(\PageIndex{4}\)

  • \(-1 \cdot 7\)
  • \(-1(-11)\)
  • \[\begin{array} {ll} {} &{-1\cdot 7} \\ {\text{Multiply, noting that the signs are different}} &{-7} \\ {\text{so the product is negative.}} &{-7\text{ is the opposite of 7.}} \end{array}\]
  • \[\begin{array} {ll} {} &{-1(-11)} \\ {\text{Multiply, noting that the signs are different}} &{11} \\ {\text{so the product is positive.}} &{11\text{ is the opposite of -11.}} \end{array}\]

Try It \(\PageIndex{5}\)

  • \(-1\cdot 9\)
  • \(-1\cdot(-17)\)

Try It \(\PageIndex{6}\)

  • \(-1\cdot 8\)
  • \(-1\cdot(-16)\)

Divide Integers

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

\[\begin{array} {ll} {5\cdot 3 = 15\text{ so }15\div 3 = 5} &{-5(3) = -15\text{ so }-15\div 3 = -5} \\ {(-5)(-3) = 15\text{ so }15\div (-3) = -5} &{5(-3) = -15\text{ so }-15\div (-3) = 5} \end{array}\]

Division follows the same rules as multiplication!

For division of two signed numbers, when the:

  • signs are the same , the quotient is positive .
  • signs are different , the quotient is negative .

And remember that we can always check the answer of a division problem by multiplying.

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 \(\PageIndex{7}\)

  • \(-27\div 3\)
  • \(-100\div (-4)\)
  • \[\begin{array} {ll} {} &{-27 \div 3} \\ {\text{Divide, with different signs, the quotient is}} &{-9} \\ {\text{negative.}} &{} \end{array}\]
  • \[\begin{array} {ll} {} &{-100 \div (-4)} \\ {\text{Divide, with signs that are the same the}} &{25} \\ {\text{ quotient is negative.}} &{} \end{array}\]

Try It \(\PageIndex{8}\)

  • \(-42\div 6\)
  • \(-117\div (-3)\)

Try It \(\PageIndex{9}\)

  • \(-63\div 7\)
  • \(-115\div (-5)\)

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 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 \(\PageIndex{10}\)

\(7(-2)+4(-7)-6\)

\[\begin{array} {ll} {} &{7(-2)+4(-7)-6} \\ {\text{Multiply first.}} &{-14+(-28)-6} \\ {\text{Add.}} &{-42-6} \\{\text{Subtract}} &{-48} \end{array}\]

Try It \(\PageIndex{11}\)

\(8(-3)+5(-7)-4\)

Try It \(\PageIndex{12}\)

\(9(-3)+7(-8)-1\)

Example \(\PageIndex{13}\)

  • \((-2)^{4}\)
  • \[\begin{array} {ll} {} &{(-2)^{4}} \\ {\text{Write in expanded form.}} &{(-2)(-2)(-2)(-2)} \\ {\text{Multiply}} &{4(-2)(-2)} \\{\text{Multiply}} &{-8(-2)} \\{\text{Multiply}} &{16} \end{array}\]
  • \[\begin{array} {ll} {} &{-2^{4}} \\ {\text{Write in expanded form. We are asked to find the opposite of }2^{4}.} &{-(2\cdot 2\cdot 2 \cdot 2)} \\ {\text{Multiply}} &{-(4\cdot 2\cdot 2)} \\{\text{Multiply}} &{-(8\cdot 2)} \\{\text{Multiply}} &{-16} \end{array}\]

Notice the difference in parts (1) and (2). In part (1), the exponent means to raise what is in the parentheses, the \((−2)\) to the \(4^{th}\) power. In part (2), the exponent means to raise just the \(2\) to the \(4^{th}\) power and then take the opposite.

Try It \(\PageIndex{14}\)

  • \((-3)^{4}\)

Try It \(\PageIndex{15}\)

  • \((-7)^{2}\)

The next example reminds us to simplify inside parentheses first.

Example \(\PageIndex{16}\)

\(12-3(9 - 12)\)

\[\begin{array} {llll} {} &{12-3(9 - 12)} \\ {\text{Subtract parentheses first}} &{12-3(-3)} \\ {\text{Multiply.}} &{12-(-9)} \\{\text{Multiply}} &{-(8\cdot 2)} \\{\text{Subtract}} &{21} \end{array}\]

Try It \(\PageIndex{17}\)

\(17 - 4(8 - 11)\)

Try It \(\PageIndex{18}\)

\(16 - 6(7 - 13)\)

Example \(\PageIndex{19}\)

\(8(-9)\div (-2)^{3}\)

\[\begin{array} {ll} {} &{8(-9)\div(-2)^{3}} \\ {\text{Exponents first}} &{8(-9)\div(-8)} \\ {\text{Multiply.}} &{-72\div (-8)} \\{\text{Divide}} &{9} \end{array}\]

Try It \(\PageIndex{20}\)

\(12(-9)\div (-3)^{3}\)

Try It \(\PageIndex{21}\)

\(18(-4)\div (-2)^{3}\)

Example \(\PageIndex{22}\)

\(-30\div 2 + (-3)(-7)\)

\[\begin{array} {ll} {} &{-30\div 2 + (-3)(-7)} \\ {\text{Multiply and divide left to right, so divide first.}} &{-15+(-3)(-7)} \\ {\text{Multiply.}} &{-15+ 21} \\{\text{Add}} &{6} \end{array}\]

Try It \(\PageIndex{23}\)

\(-27\div 3 + (-5)(-6)\)

Try It \(\PageIndex{24}\)

\(-32\div 4 + (-2)(-7)\)

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 \(\PageIndex{25}\)

When \(n=−5\), evaluate:

  • \(−n+1\).
  • \[\begin{array} {ll} {} &{n+ 1} \\ {\text{Substitute }{ \color{red}{-5}}\text{ for } n} &{\color{red}{-5}}+1 \\ {\text{Simplify.}} &{-4} \end{array}\]
  • \[\begin{array} {ll} {} &{n+ 1} \\ {\text{Substitute }{ \color{red}{-5}}\text{ for } n} &{- {\color{red}{(-5)}} +1} \\ {\text{Simplify.}} &{5+1} \\{\text{Add.}} &{6} \end{array}\]

Try It \(\PageIndex{26}\)

When \(n=−8\), evaluate:

  • \(−n+2\).

Try It \(\PageIndex{27}\)

When \(y=−9\), evaluate:

  • \(−y+8\).

Example \(\PageIndex{28}\)

Evaluate \((x+y)^{2}\) when \(x = -18\) and \(y = 24\).

\[\begin{array} {ll} {} &{(x+y)^{2}} \\ {\text{Substitute }-18\text{ for }x \text{ and } 24 \text{ for } y} &{(-18 + 24)^{2}} \\ {\text{Add inside parentheses}} &{(6)^{2}} \\{\text{Simplify.}} &{36} \end{array}\]

Try It \(\PageIndex{29}\)

Evaluate \((x+y)^{2}\) when \(x = -15\) and \(y = 29\).

Try It \(\PageIndex{30}\)

Evaluate \((x+y)^{3}\) when \(x = -8\) and \(y = 10\).

Example \(\PageIndex{31}\)

Evaluate \(20 -z \) when

  • \(z = -12\)
  • \[\begin{array} {ll} {} &{20 - z} \\ {\text{Substitute }12\text{ for }z.} &{20 - 12} \\ {\text{Subtract}} &{8} \end{array}\]
  • \[\begin{array} {ll} {} &{20 - z} \\ {\text{Substitute }-12\text{ for }z.} &{20 - (-12)} \\ {\text{Subtract}} &{32} \end{array}\]

Try It \(\PageIndex{32}\)

Evaluate \(17 - k\) when

  • \(k = -19\)

Try It \(\PageIndex{33}\)

Evaluate \(-5 - b\) when

  • \(b = -14\)

Example \(\PageIndex{34}\)

\(2x^{2} + 3x + 8\) when \(x = 4\).

Substitute \(4\) for \(x\). Use parentheses to show multiplication.

\[\begin{array} {ll} {} &{2x^{2} + 3x + 8} \\ {\text{Substitute }} &{2(4)^{2} + 3(4) + 8} \\ {\text{Evaluate exponents.}} &{2(16) + 3(4) + 8} \\ {\text{Multiply.}} &{32 + 12 + 8} \\{\text{Add.}} &{52} \end{array}\]

Try It \(\PageIndex{35}\)

\(3x^{2} - 2x + 6\) when \(x =-3\).

Try It \(\PageIndex{36}\)

\(4x^{2} - x - 5\) when \(x = -2\).

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 \(\PageIndex{37}\)

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

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

Try It \(\PageIndex{38}\)

Translate and simplify: the sum of \(9\) and \(−16\), increased by \(4\).

\((9 + (-16)) + 4 - 3\)

Try It \(\PageIndex{39}\)

Translate and simplify: the sum of \(-8\) and \(−12\), increased by \(7\).

\((-8 + (-12)) + 7 - 13\)

When we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.

Be careful to get a and b in the right order!

Example \(\PageIndex{40}\)

Translate and then simplify

  • the difference of \(13\) and \(−21\)
  • subtract \(24\) from \(−19\).
  • \[\begin{array} {ll} {} &{\text{the } \textbf{difference } \text{of 13 and -21}} \\ {\text{Translate.}} &{13 - (-21)} \\ {\text{Simplify.}} &{34} \end{array}\]
  • \[\begin{array} {ll} {} &\textbf{subtract }24 \textbf{ from }-19 \\ {\text{Translate.}} &{-19 - 24} \\ {\text{Remember, subtract b from a means }a - b} &{} \\{\text{Simplify.}} &{-43} \end{array}\]

Try It \(\PageIndex{41}\)

Translate and simplify

  • the difference of \(14\) and \(−23\)
  • subtract \(21\) from \(−17\).
  • \(14 - (-23); 37\)
  • \(-17 - 21; -38\)

Try It \(\PageIndex{42}\)

  • the difference of \(11\) and \(−19\)
  • subtract \(18\) from \(−11\).
  • \(11 - (-19); 30\)
  • \(-11 - 18; -29\)

Once again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is “ product ” and for division is “ quotient .”

Example \(\PageIndex{43}\)

Translate to an algebraic expression and simplify if possible: the product of \(−2\) and \(14\).

\[\begin{array} {ll} {} &{\text{the product of }-2 \text{ and } 14} \\ {\text{Translate.}} &{(-2)(14)} \\{\text{Simplify.}} &{-28} \end{array}\]

Try It \(\PageIndex{44}\)

Translate to an algebraic expression and simplify if possible: the product of \(−5\) and \(12\).

\(-5(12); -60\)

Try It \(\PageIndex{45}\)

Translate to an algebraic expression and simplify if possible: the product of \(8\) and \(-13\).

\(-8(13); -104\)

Example \(\PageIndex{46}\)

Translate to an algebraic expression and simplify if possible: the quotient of \(−56\) and \(−7\).

\[\begin{array} {ll} {} &{\text{the quotient of }-56 \text{ and } -7} \\ {\text{Translate.}} &{-56\div(-7)} \\{\text{Simplify.}} &{8} \end{array}\]

Try It \(\PageIndex{47}\)

Translate to an algebraic expression and simplify if possible: the quotient of \(−63\) and \(−9\).

\(-63\div (-9); 7\)

Try It \(\PageIndex{48}\)

Translate to an algebraic expression and simplify if possible: the quotient of \(−72\) and \(−9\).

\(-72\div (-9); 8\)

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.

How to Apply a Strategy to Solve Applications with Integers

Example \(\PageIndex{49}\)

The temperature in Urbana, Illinois one morning was \(11\) degrees. By mid-afternoon, the temperature had dropped to \(−9\) degrees. What was the difference of the morning and afternoon temperatures?

Try It \(\PageIndex{50}\)

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.

Try It \(\PageIndex{51}\)

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.

APPLY A STRATEGY TO SOLVE APPLICATIONS WITH INTEGERS.

  • 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.

Example \(\PageIndex{52}\)

The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?

Try It \(\PageIndex{53}\)

The Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of \(15\) yards. What is the number of yards lost due to penalties?

The Bears lost \(105\) yards.

Try It \(\PageIndex{54}\)

Bill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a $2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM?

A $16 fee was deducted from his checking account.

Key Concepts

  • Same signs—Product is positive
  • Different signs—Product is negative
  • Identify what you are asked to find.

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

dividing integers problem solving

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How to Do Long Division: Step-by-Step Instructions

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A long division problem on a blackboard

In math, few skills are as practical as knowing how to do long division . It's the art of breaking down complex problems into manageable steps, making it an essential tool for students and adults alike.

This operation has many practical uses in our daily lives. For instance, imagine you have a bag of 2,436 candies and want to share them equally among 4 friends. Long division helps you determine that each friend gets 609 candies, ensuring everyone gets their fair share.

Let's dive into the fundamentals of long division and learn about other everyday situations where we can put it to use.

What Is Long Division?

How to do long division in simple steps, long division method: an apple example, using long division in everyday life, how to divide a decimal point by a whole number, practice problems and answers.

Long division is a handy way to divide big numbers by smaller ones, helping us figure out how many times one number fits into another. It turns a tricky math problem into easier steps.

When we do long division, we work with four main parts:

  • the big number we want to divide (called the " dividend ")
  • the smaller number we're dividing by (the " divisor ")
  • the answer to our division (the " quotient ")
  • sometimes a little bit left over (the " remainder ")

Long Division vs. Short Division

Short and long division are both methods to divide numbers, but they differ in complexity. The short-division method is a quick way to find the answer when dividing simple numbers. For example, say you want to divide 36 by 6. You write it as 36 ÷ 6, using a division sign, and quickly get the answer, which is 6.

Long division is used for bigger, more complicated numbers, typically two or more digits. This method involves several steps, like writing out the numbers neatly and carefully.

Let's dive into long division with a clear example. We'll use 845 ÷ 3 to walk through this step-by-step process:

  • Set up the problem. Write the dividend (845) under the division bar and the divisor (3) outside the bar.
  • Divide. Look at the first digit of the dividend (8). How many times does 3 go into 8? Twice, because 3 x 2 = 6, and that's the closest we can get without going over. Write the 2 above the division bar, over the 8.
  • Multiply. Multiply the quotient (2) by the divisor (3). (2 x 3 = 6). Write 6 under the 8.
  • Subtract. Subtract 6 from 8 to get 2. Draw a line under the 6, subtract, and write 2 below the line.
  • Bring down the next digit. Now, bring down the next digit of the dividend, which is 4, to sit next to the 2, making 24.
  • Repeat the steps. 3 goes into 24 eight times (3 x 8 = 24), so write 8 above the bar next to the 2. Subtract 24 from 24 to get 0. Now, follow the same process you used in steps 1 through 5 and bring down the last digit, which is 5, to form 05. The number 3 goes into 5 once (3 x 1 = 3), leaving a remainder of 2. Write the 1 above the bar and the remainder 2 below after subtracting 3 from 5.
  • The final answer with a remainder. You've divided 845 by 3 to get a final answer of 281 with a remainder of 2.
  • Convert the remainder to decimal form. Depending on how far along you are in learning long division, this may be your final answer. If you've progressed to decimals, you will add .0 to 845 and put a decimal point above the division bar, right after the 1. Bring 0 down to form 20. The number 3 goes into 20 six times (3 x 6 = 18). Write 6 after the decimal point above the division bar. Normally, you would continue adding another 0 after 845. until there is no remainder, but since 20 – 18 = 2, you would be repeating this process infinitely because 3 does not divide evenly into 845. Instead, you will draw a horizontal line over the 6 in 281.6 to indicate that it is a repeating decimal. A calculator would show the answer as 281.666667 to indicate that the repeating decimal rounds up.

Now let's use a practical example to work through the long division process.

Imagine you just went apple picking and came home with a massive haul of delicious fruit. In your kitchen, you have 456 apples, and you want to share them equally among 3 baskets to give to your friends, so you're dividing 3 by 456 (456 ÷ 3).

To figure out how many apples go into each basket, you'd tackle the division problem step by step.

  • 3 goes into the first digit (4) once, so you write 1 above the division bar, above the 4 in 456. Then you show the subtraction: 4 – 3 = 1.
  • Bring down the next digit (5) to form 15. 3 goes into 15 five times (3 x 5 = 15), so you write 5 above the division bar, above the 5 in 456. Then you show the subtraction: 15 – 15 = 0.
  • Bring down the final digit (6) to form 06. 3 goes into 6 twice (3 x 2 = 6), so you write 2 above the division bar, above the 2 in 456. Then you show the subtraction: 6 – 6 = 0.
  • Since there is no remainder left to divide, you quotient is now written atop the division bar: 152. You will need to place 152 apples in each of the 3 baskets to evenly distribute the 456 apples.

Long division also pops up in real-life situations . Think about when you need to divide something, like pizza or cake, into equal parts.

Want to cut a large recipe in half or figure out how many days are left till summer vacation? Long division can help with that. It's a great way to help us figure out those splits and manage resources better.

And, of course, practicing long division sharpens our problem-solving skills . It teaches us to tackle big problems step by step, breaking them down into smaller, more manageable pieces. This approach is super helpful in math and figuring out all sorts of challenges we might face.

So, long division is more than just a bunch of steps we follow. It's a key that unlocks a lot of doors in the world of math and beyond, helping us understand and connect different concepts and apply them in all sorts of ways.

Dividing decimals by whole numbers is useful in our everyday lives. For instance, if you're splitting a sum of money equally among a certain number of people, you'll need to divide the total (a decimal) by the number of people (a whole number) to determine how much each person gets.

Dividing a decimal point (decimal number) by a whole number is similar to regular division, but you must be mindful of the placement of the decimal point. Here's how to do it:

Example : Divide 0.5 by 5.

  • Set up the problem. Begin by setting up the division, with 0.5 as the dividend (the number you're dividing, which will be under the division bar) and 5 as the divisor (the number you're dividing by, which will be to the left of the division bar).
  • Begin dividing. 5 goes into the first digit of the dividend 0 times, so you'll write 0 above the division bar, above the 0 in 0.5, and place a decimal point after the 0 you just wrote. It should be directly above the decimal point in the dividend.
  • Bring down the next digit. Bring down the 5 to form 05 (you do not bring the decimal down). 5 goes into 5 once (5 x 1 = 5), so you'll write 1 above the division bar, above the 5 in 0.5.
  • Show the final answer. When you show the subtraction (5 – 5 = 0), you'll have no remainder. This means the number above the division bar is your final answer: 0.1.

Let's put our long division skills to the test with some word problems. Tackle these problems one step at a time, and don't rush. If you get stuck, pause and review the steps. Remember, practice makes perfect, and every problem is an opportunity to improve your long-division skills.

1. Emma has 672 pieces of candy to share equally among her 4 friends. How many pieces of candy does each friend get?

Solution : To find out, divide 672 by 4. Start with the first part of 672, which is 6, and see how many times 4 can fit into it. It fits 1 time, leaving us with 2. Bringing down the 7 turns it into 27, which 4 fits into 6 times, leaving us with 3. Finally, bringing down the 2 to join the remaining 3 makes 32, which 4 divides into 8 times. So, each friend gets 168 pieces of candy.

2. A teacher has 945 stickers to distribute equally in 5 of her classes. How many stickers does each class get?

Solution : We'll divide 945 by 5. Looking at 9 first, 5 goes into it 1 time. With 4 leftover, we bring down the 4 from 945 to get 44, which 5 divides into 8 times with another 4 leftover. Lastly, bringing down the 5 to the remaining 4 makes 45, which 5 divides into 9 times. Therefore, each class receives 189 stickers.

3. A library has 2,310 books to be placed equally on 6 shelves. How many books will each shelf contain?

Solution : Divide 2,310 by 6. Starting with 23, 6 goes into it 3 times with 5 leftover. After subtracting, we bring down the 1 to get 51, which 6 divides into 8 times with 3 leftover. Bringing down the 0 to the remaining 3 gives us 30, which 6 divides into 5 times. So, each shelf will have 385 books.

This article was updated in conjunction with AI technology, then fact-checked and edited by a HowStuffWorks editor.

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  1. Dividing Integers Practice Problems With Answers

    Answer Problem 3: Divide the integers: [latex]36 \div \left ( { - 4} \right) [/latex] Answer Problem 4: Divide the integers: [latex]\left ( { - 54} \right) \div \left ( { - 9} \right) [/latex] Answer Problem 5: Divide the integers: [latex]\left ( { - 144} \right) \div 6 [/latex] Answer

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    Step 1 : To find the number of incorrect answers in the winter quiz, divide the total points lost by the number of points lost per wrong answer. -33 ÷ (-3) = 11 Step 2 : Find the number of incorrect answers Jake gave in the summer quiz. Divide the total points lost by the number of points lost per wrong answer. -56 ÷ (-7) = 8 Step 3 :

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    Step 1: Divide their absolute values. Step 2: Determine the sign of the final answer (known as a quotient) using the following conditions. Condition 1: If the signs of the two numbers are the same, the quotient is always a positive number. Condition 2: If the signs of the two numbers are different, the quotient is always a negative number.

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    Example 1: multiplying integers with the same sign Multiply: (-4) \times (-12)= \, ? (−4) × (−12) =? If the integers have the same sign, the product or quotient is positive. If they don't go to step 2. -4 −4 and -12 −12 have the same sign so the product is positive.

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    We can now use Rule 1 to solve the problem above arithmetically: -8,000 ÷ +4 = -2,000. Each of Mrs. Jenson's four children will pay $2,000. Let's look at some more examples of dividing integers using the above rules. Example 1: Find the quotient of each pair of integers. Example 2: Find the quotient of each pair of integers.

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    Arithmetic problem: Solve 12 ÷ 4. Word problem: A pizza cut into 12 slices is to be divided among 4 friends. How many slices do they each get? 12 ÷ 4 = 3, so each friend gets 3 slices.

  9. Math Practice Problems

    You can start playing for free! Integer Division - Sample Math Practice Problems The math problems below can be generated by MathScore.com, a math practice program for schools and individual families. References to complexity and mode refer to the overall difficulty of the problems as they appear in the main program.

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    Integers: Multiplication and Division Worksheets. This page contains printable worksheets which emphasize integer multiplication and division to 6th grade, 7th grade, and 8th grade students. Practice pages here contain exercises on multiplication squares, in-out boxes, evaluating expressions, filling in missing integers, and more.

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    Division is the inverse operation of multiplication. So, 15 ÷ 3 = 5 because 5 ⋅ 3 = 15. In words, this expression says that 15 can be divided into three groups of five each because adding five three times gives 15. Look at some examples of multiplying integers, to figure out the rules for dividing integers.

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    Rule 1: The quotient of the two integers, either both positive or both negative, is a positive integer equal to the quotient of the corresponding fundamental values of the integers. Thus, for dividing two integers with like signs, we divide their values regardless of their sign and give plus sign to the quotient.

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    Divide Integers. What about division? Division is the inverse operation of multiplication. So, \(15\div 3=5\) because \(5 \cdot 3 = 15\). In words, this expression says that \(15\) can be divided into three groups of five each because adding five three times gives \(15\). Look at some examples of multiplying integers, to figure out the rules ...

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  24. How to Do Long Division: Step-by-Step Instructions

    It turns a tricky math problem into easier steps. When we do long division, we work with four main parts: Advertisement. the big number we want to divide (called the "dividend") ... you'd tackle the division problem step by step. 3 goes into the first digit (4) once, so you write 1 above the division bar, above the 4 in 456. Then you show the ...