Introduction

Let's consider the following division problem:

{\color{red}{10}} \div {\color{blue}{5}} = 2

This tells us that {\color{red}{10}} can be gathered into 2 groups of {\color{blue}{5}}{:}

We can use the same model to represent the following multiplication problem:

{\color{red}{10}} = {\color{blue}{5}} \times 2

The order in which we multiply numbers does not matter. So, we can swap the two numbers on the right-hand side to create another multiplication problem:

{\color{red}{10}} = 2\times {\color{blue}{5}}

Multiplication and division can be considered "opposite" operations. And since we can swap the order of multiplication, every division problem can be expressed as two equivalent multiplication problems.

We usually put the result on the right-hand side when writing down an equivalent multiplication problem. Let's see an example.

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Example: Constructing an Equivalent Multiplication Problem

What is 45\div 5 = 9 expressed as an equivalent multiplication problem?

EXPLANATION

Our division problem is as follows:

{\color{red}{45}}\div {\color{blue}{5}} = 9

There are two multiplication problems equivalent to the given division problem:

  • {\color{red}{45}} = {\color{blue}{5}} \times 9

  • {\color{red}{45}} = 9 \times {\color{blue}{5}}

When writing down a multiplication problem, we usually put the result on the right-hand side. Therefore, our equivalent problems are as follows:

  • 5 \times 9 = 45

  • 9 \times 5 = 45

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Practice: Constructing an Equivalent Multiplication Problem

What is $28\div 4 = 7$ expressed as an equivalent multiplication problem?

a
 $7\times 4 = 28$
b
 $28\times 4 = 7$
c
 $28\times 7 = 4$
d
 $7\times 28 = 4$
e
 $4\times 28 = 7$
Practice: Constructing an Equivalent Multiplication Problem

What is $88\div 11 = 8$ expressed as an equivalent multiplication problem?

a
 $88\times 8 = 11$
b
 $8\times 88 = 11$
c
 $11\times 88 = 8$
d
 $88\times 11 = 8$
e
 $11\times 8 = 88$
Example: Constructing Two Equivalent Multiplication Problems

Which of the following multiplication problems are equivalent to the division problem 120\div 12 = 10?

  1. 12\times 10 = 120
  2. 120\times 12 = 10
  3. 10\times 12 = 120
EXPLANATION

Our division problem is as follows:

{\color{red}{120}}\div {\color{blue}{12}} = 10

There are two multiplication problems equivalent to the given division problem:

  • {\color{red}{120}} = {\color{blue}{12}} \times 10

  • {\color{red}{120}} = 10 \times {\color{blue}{12}}

When writing down a multiplication problem, we usually put the result on the right-hand side. Therefore, our equivalent problems are as follows:

  • 12\times 10 = 120

  • 10\times 12 = 120

Therefore, the correct answer is "I and III only."

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Practice: Constructing Two Equivalent Multiplication Problems

Which of the following multiplication problems are equivalent to the division problem $27\div 9 = 3?$

  1. $3\times 27 = 9$
  2. $9\times 3 = 27$
  3. $27\times 9 = 3$
a
 I only
b
 I and II only
c
 II and III only
d
 II only
e
 III only
Practice: Constructing Two Equivalent Multiplication Problems

Which of the following multiplication problems are equivalent to the division problem $56\div 7 = 8?$

  1. $7\times 8 = 56$
  2. $7\times 56 = 8$
  3. $8\times 7 = 56$
a
 I only
b
 III only
c
 I and III only
d
 II only
e
 I and II only
Constructing Equivalent Division Problems

Similarly, every multiplication problem can be expressed as two equivalent division problems.

Let's consider the following multiplication problem:

{\color{red}{4}}\times {\color{blue}{3}} = 12

There are two division problems equivalent to this multiplication problem:

  • The first equivalent problem is {\color{red}{4}} = 12\div {\color{blue}{3}}. This tells us that 12 can be grouped into {\color{red}{4}} groups of {\color{blue}{3}}\mathbin{:}
  • The second equivalent problem is {\color{blue}{3}} = 12\div {\color{red}{4}}. This tells us that 12 can be grouped into {\color{blue}{3}} groups of {\color{red}{4}}\mathbin{:}

We usually put the result on the right-hand side when writing down a division problem. Therefore, our equivalent problems can also be written as follows:

  • 12\div 3 = 4

  • 12\div 4 = 3

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Example: Constructing an Equivalent Division Problem

What is 9\times 8 = 72 expressed as an equivalent division problem?

EXPLANATION

Our multiplication problem is as follows:

{\color{red}{9}}\times {\color{blue}{8}} = 72

There are two division problems equivalent to this multiplication problem:

  • {\color{red}{9}} = 72\div {\color{blue}{8}}

  • {\color{blue}{8}} = 72\div {\color{red}{9}}

When writing down a division problem, we usually put the result on the right-hand side. Therefore, our equivalent problems are as follows:

  • 72\div 8 = 9

  • 72\div 9 = 8

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Practice: Constructing an Equivalent Division Problem

What is $6\times 4 = 24$ expressed as an equivalent division problem?

a
 $24\div 4 = 6$
b
 $4\div 6 = 24$
c
 $6\div 24 = 4$
d
 $4\div 24 = 6$
e
 $6\div 4 = 24$
Practice: Constructing an Equivalent Division Problem

What is $11\times 12 = 132$ expressed as an equivalent division problem?

a
 $11\div 132 = 12$
b
 $12\div 132 = 11$
c
 $12\div 11 = 132$
d
 $11\div 12 = 132$
e
 $132\div 11 = 12$
Example: Constructing Two Equivalent Division Problems

Which of the following division problems are equivalent to the multiplication problem 9\times 6 = 54?

  1. 54\div 6 = 9
  2. 9\div 6 = 54
  3. 9\div 54 = 6
EXPLANATION

Our multiplication problem is as follows:

{\color{red}{9}}\times {\color{blue}{6}} = 54

There are two division problems equivalent to this multiplication problem:

  • {\color{red}{9}} = 54\div {\color{blue}{6}}

  • {\color{blue}{6}} = 54\div {\color{red}{9}}

When writing down a division problem, we usually put the result on the right-hand side. Therefore, our equivalent problems are as follows:

  • 54\div 6 = 9

  • 54\div 9 = 6

Therefore, the correct answer is "I only."

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Practice: Constructing Two Equivalent Division Problems

Which of the following division problems are equivalent to the multiplication problem $6\times 3 = 18?$

  1. $18\div 6 = 3$
  2. $18\div 3 = 6$
  3. $6\div 3 = 18$
a
 III only
b
 II only
c
 II and III only
d
 I only
e
 I and II only
Practice: Constructing Two Equivalent Division Problems

Which of the following division problems are equivalent to the multiplication problem $8\times 6 = 48?$

  1. $48\div 6 = 8$
  2. $48\div 8 = 6$
  3. $6\div 8 = 48$
a
 I only
b
 I and II only
c
 II only
d
 III only
e
 I, II, and III
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