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 inverse of multiplication—recovering the missing factor in a product.

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: Division and multiplication are inverse operations—each undoes the other.

Common stuck point: Dividing by a fraction means multiplying by its reciprocal: 6 \div \frac{1}{2} = 6 \times 2 = 12.

Sense of Study hint: Rewrite the division as a missing-factor problem: _ x 4 = 12, so 12 / 4 = _.

Worked Examples

Example 1

easy
You know that \(7 \times 8 = 56\). Use this fact to find \(56 \div 7\) and \(56 \div 8\).

Solution

  1. 1
    From \(7 \times 8 = 56\), division undoes multiplication.
  2. 2
    \(56 \div 7 = 8\) (divide by 7 to get 8).
  3. 3
    \(56 \div 8 = 7\) (divide by 8 to get 7).
  4. 4
    These are the two related division facts for the fact family.

Answer

\(56 \div 7 = 8\) and \(56 \div 8 = 7\)
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.

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.
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Background Knowledge

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

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