Division as Inverse Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Division as Inverse.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Understanding division as the reverse of multiplication: if $a \times b = c$ , then $c \div b = a$ . This inverse relationship lets you undo multiplication to find missing factors.

If $3 \times 4 = 12$ , then $12 \div 4 = 3$ . Division reverses the multiplication.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Inverse division reverses multiplication: if $a \times b = c$ , then $c \div b = a$ .

Common stuck point: The procedure for division as inverse is the easy part; the trap is dividing by the product instead of the known factor. Asking "Am I undoing a multiplication to find a factor that makes the product?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I undoing a multiplication to find a factor that makes the product?

Worked Examples

Example 1

easy

You know that

7 \times 8 = 56

. Use this fact to find

56 \div 7

and

56 \div 8

Answer

56 \div 7 = 8

and

56 \div 8 = 7

First step

From

7 \times 8 = 56

, division undoes multiplication.

Full solution

2
$56 \div 7 = 8$ (divide by 7 to get 8).
3
$56 \div 8 = 7$ (divide by 8 to get 7).
4
These are the two related division facts for the fact family.

Division is the inverse of multiplication. Every multiplication fact produces two division facts in the same fact family.

Example 2

medium

A number times 9 equals 81. Use division as the inverse to find the unknown number.

Example 3

easy

You know

7 \times 9 = 63

. Write the two related division facts.

Example 4

medium

a \times 7 = 91

, find

a

by using division as the inverse.

Example 5

medium

Use the fact

15 \times 4 = 60

to compute

60 \div 15

and

60 \div 4

without recomputing.

Example 6

hard

Explain why dividing

20

\tfrac{1}{4}

gives

80

using the inverse view.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Given

6 \times 9 = 54

, write the two related division facts.

Example 2

medium

Find the missing number:

? \times 7 = 42

. Use division.

Example 3

easy

3 \times 4 = 12

, what is

12 \div 4

Example 4

easy

5 \times 6 = 30

, what is

30 \div 6

Example 5

easy

What is

20 \div 5

Example 6

easy

Fill the blank:

\_ \times 7 = 28

Example 7

easy

8 \times 2 = 16

, what is

16 \div 8

Example 8

easy

What is

18 \div 3

Example 9

easy

Write the division fact that matches

9 \times 4 = 36

using

36 \div 4

Example 10

easy

What is

24 \div 6

Example 11

medium

A box has

42

eggs in rows of

6

. How many rows are there?

Example 12

medium

Solve for the missing factor:

\_ \times 8 = 56

Example 13

medium

Does dividing

10

\frac{1}{2}

give a number bigger or smaller than

10

? Compute it.

Example 14

medium

54 \div 9 = 6

, write the two multiplication facts that confirm it.

Example 15

medium

A baker uses

4

eggs per cake and used

32

eggs. How many cakes were made?

Example 16

medium

A number divided by

7

gives

9

. What is the number?

Example 17

medium

Fill in:

63 \div \_ = 9

. What divisor makes this true?

Example 18

challenge

From

a \div b = c

, which is always a correct multiplication fact:

c \times b = a

b \times c = a

? Explain.

Example 19

challenge

A rectangle has area

84

and one side

12

. Use division to find the other side, then the perimeter.

Example 20

challenge

A number is multiplied by

6

and then divided by

3

, giving

10

. What was the original number?

Example 21

medium

A theater has

48

seats in rows of

8

. How many rows?

Example 22

medium

Solve for the missing factor:

\_ \times 5 = 45

Example 23

easy

11 \times 6 = 66

, what is

66 \div 11

Example 24

easy

Fill in the blank:

\_ \times 9 = 72

Example 25

easy

What number times

6

equals

48

Example 26

easy

What is

40 \div 8

Example 27

medium

A rectangle has area

96

and one side

8

. What is the other side?

Example 28

medium

A bookshelf holds

144

books in stacks of

12

. How many stacks?

Example 29

medium

A number multiplied by

11

gives

132

. What is the number?

Example 30

medium

A class of

24

students splits into teams; each team has the same number, and each team has

6

students. How many teams?

Example 31

medium

A garden has

108

tomato plants in equal rows of

9

. How many rows?

Example 32

medium

Maya saved the same amount each week for

8

weeks and saved

\$96

total. How much per week?

Example 33

medium

6 \times m = 78

, what is

m

Example 34

medium

A number divided by

4

gives

14

. What is the number?

Example 35

hard

3 \times n = 2 \times 24

, find

n

using division as inverse.

Example 36

hard

A number is multiplied by

5

, then

3

is added, giving

48

. What was the original number?

Example 37

hard

A bag of

144

candies is shared so each kid gets

12

. The same total is later shared among twice as many kids. How many does each get?

Example 38

hard

If the area of a rectangle is

147

and the length is

3

times the width, find both side lengths.

Example 39

hard

A truck driver covers

462

miles at a steady

66

mph. How many hours did the trip take?

Example 40

challenge

Find all whole-number values of

k

such that

k \times (k+2) = 168

← Back to Division as Inverse Formula Explained Practice Mode

Related Concepts

Division Multiplication Inverse Operations Solving Linear Equations

Background Knowledge

These ideas may be useful before you work through the harder examples.

divisionmultiplication