Dividing Fractions Formula

The Formula

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

When to use: Imagine you have 2 cups of flour and each serving of a recipe needs \frac{1}{3} cup. How many servings can you make? You are asking 'how many one-thirds fit into 2?'—that is 2 \div \frac{1}{3} = 6 servings. Division by a fraction counts how many pieces of that size fit inside the whole.

Quick Example

\frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = \frac{3}{2} = 1\frac{1}{2}

Notation

\frac{a}{b} \div \frac{c}{d} — 'keep, change, flip': keep \frac{a}{b}, change \div to \times, flip \frac{c}{d} to \frac{d}{c}

What This Formula Means

Dividing by a fraction by multiplying by its reciprocal (inverting the divisor and multiplying).

Imagine you have 2 cups of flour and each serving of a recipe needs \frac{1}{3} cup. How many servings can you make? You are asking 'how many one-thirds fit into 2?'—that is 2 \div \frac{1}{3} = 6 servings. Division by a fraction counts how many pieces of that size fit inside the whole.

Formal View

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc} where b, c, d \neq 0

Worked Examples

Example 1

easy
Divide \frac{3}{4} \div \frac{2}{5}.

Solution

  1. 1
    Take the reciprocal of the divisor: \frac{2}{5} becomes \frac{5}{2}.
  2. 2
    Multiply: \frac{3}{4} \times \frac{5}{2} = \frac{15}{8}.
  3. 3
    Convert to a mixed number if desired: \frac{15}{8} = 1\frac{7}{8}.

Answer

\frac{15}{8} = 1\frac{7}{8}
Dividing by a fraction is equivalent to multiplying by its reciprocal. This 'keep-change-flip' rule works because division asks 'how many groups of the divisor fit into the dividend.'

Example 2

medium
A ribbon is \frac{7}{8} of a metre long. It is cut into pieces that are each \frac{1}{4} of a metre. How many pieces are there?

Common Mistakes

  • Flipping the first fraction instead of the second
  • Forgetting to change division to multiplication after flipping
  • Dividing numerators and denominators separately: \frac{6}{8} \div \frac{2}{4} = \frac{3}{2} (works accidentally but wrong method)

Why This Formula Matters

Completes the four operations on fractions and is essential for solving equations involving fractions.

Frequently Asked Questions

What is the Dividing Fractions formula?

Dividing by a fraction by multiplying by its reciprocal (inverting the divisor and multiplying).

How do you use the Dividing Fractions formula?

Imagine you have 2 cups of flour and each serving of a recipe needs \frac{1}{3} cup. How many servings can you make? You are asking 'how many one-thirds fit into 2?'—that is 2 \div \frac{1}{3} = 6 servings. Division by a fraction counts how many pieces of that size fit inside the whole.

What do the symbols mean in the Dividing Fractions formula?

\frac{a}{b} \div \frac{c}{d} — 'keep, change, flip': keep \frac{a}{b}, change \div to \times, flip \frac{c}{d} to \frac{d}{c}

Why is the Dividing Fractions formula important in Math?

Completes the four operations on fractions and is essential for solving equations involving fractions.

What do students get wrong about Dividing Fractions?

Students flip the wrong fraction (the dividend instead of the divisor).

What should I learn before the Dividing Fractions formula?

Before studying the Dividing Fractions formula, you should understand: multiplying fractions, inverse operations.

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