Dividing Fractions Formula with Worked Examples

Key Points

• The dividing fractions formula is $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$, flipping only the second fraction.

• Dividing by any number is the same as multiplying by its reciprocal, which is why the rule works rather than being an arbitrary trick.

• Only the divisor is inverted, so flipping the first fraction instead returns the reciprocal of the correct answer.

• Dividing by a fraction smaller than 1 makes the answer larger, which is the opposite of what division usually does.

• Mixed numbers must become improper fractions before the formula is applied, since the rule needs a single numerator and denominator.

What Is the Dividing Fractions Formula?

The dividing fractions formula replaces a division with a multiplication:

$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$$

The second fraction is turned upside down and the operation becomes multiplication. The first fraction is left exactly as it is.

Why Multiplying by the Reciprocal Works

Division asks how many times one quantity fits into another. Asking how many halves fit into 3 gives 6, and $3 \times 2 = 6$, so dividing by $\frac{1}{2}$ did the same job as multiplying by 2.

That pattern is general. Dividing by $\frac{c}{d}$ undoes multiplication by $\frac{c}{d}$, and the number that undoes a multiplication is the reciprocal, because their product is 1.

So the rule is not a trick to be memorised. It is the statement that dividing and multiplying by the reciprocal are the same operation, which also explains why dividing by a fraction below 1 increases the result.

Variable Key

Examples of the Dividing Fractions Formula

Example 1

Compute $\frac{3}{4} \div \frac{2}{5}$.

$$\frac{3}{4} \times \frac{5}{2} = \frac{15}{8}$$

Final answer: $\frac{15}{8}$, or $1\frac{7}{8}$

Example 2

Compute $\frac{3}{4} \div \frac{2}{5}$ by flipping the first fraction.

The rule involves a flip, so turning over the fraction at the front looks plausible:

$$\frac{4}{3} \times \frac{2}{5} = \frac{8}{15}$$

That gives $\frac{8}{15}$, which is less than 1. But $\frac{2}{5}$ is smaller than $\frac{3}{4}$, so it must fit into it more than once, and the answer has to exceed 1.

The answer $\frac{8}{15}$ is in fact the reciprocal of the correct one, which is the signature of flipping the wrong fraction:

$$\frac{3}{4} \times \frac{5}{2} = \frac{15}{8}$$

Final answer: $\frac{15}{8}$. Only the divisor is inverted.

Example 3

Compute $\frac{2}{3} \div 4$.

Write the whole number as $\frac{4}{1}$, whose reciprocal is $\frac{1}{4}$:

$$\frac{2}{3} \times \frac{1}{4} = \frac{2}{12} = \frac{1}{6}$$

Final answer: $\frac{1}{6}$

Here the divisor is larger than 1, so the answer is smaller than the original, as ordinary division would suggest.

Example 4

Compute $2\frac{1}{2} \div 1\frac{1}{4}$.

Convert both mixed numbers to improper fractions first:

$$\frac{5}{2} \div \frac{5}{4} = \frac{5}{2} \times \frac{4}{5} = \frac{20}{10} = 2$$

Final answer: 2

Example 5

A recipe needs $\frac{1}{8}$ cup per portion. How many portions come from $\frac{3}{4}$ of a cup?

$$\frac{3}{4} \div \frac{1}{8} = \frac{3}{4} \times \frac{8}{1} = \frac{24}{4} = 6$$

Final answer: 6 portions

This is the how-many-fit reading of division, and it is the clearest way to see why the answer grew.

Don't Confuse Division With These Three

Division and multiplication need no common denominator, while addition and subtraction do. That is the opposite of what most students expect, since division feels like the harder operation.

Where Do Students Trip Up on Dividing Fractions?

Mistake 1: Flipping the first fraction

Where it slips in: The rule is remembered as "flip and multiply" without which fraction gets flipped.

Don't do this: $\frac{3}{4} \div \frac{2}{5} = \frac{4}{3} \times \frac{2}{5}$

The correct way: Flip only the divisor, giving $\frac{3}{4} \times \frac{5}{2}$. An answer that is the reciprocal of what you expected is the signature of this error.

Students who say the words "keep, change, flip" while pointing at the three parts in order rarely flip the wrong one, because the pointing ties each instruction to a specific place on the page. Saying it without pointing is where the pairing gets lost.

Mistake 2: Dividing numerators and denominators separately

Where it slips in: Multiplication works across, so division looks as though it should too.

Don't do this: $\frac{3}{4} \div \frac{2}{5} = \frac{3 \div 2}{4 \div 5}$

The correct way: Use the reciprocal. Dividing across happens to work when both divisions are exact, as in $\frac{8}{15} \div \frac{2}{5} = \frac{4}{3}$, but it fails in general and is not worth relying on.

Mistake 3: Applying the rule to mixed numbers directly

Where it slips in: A mixed number already looks like it contains a fraction.

Don't do this: $2\frac{1}{2} \div 1\frac{1}{4}$ computed by flipping $1\frac{1}{4}$ as it stands.

The correct way: Convert to $\frac{5}{2}$ and $\frac{5}{4}$ first. The formula needs a single numerator over a single denominator before the reciprocal means anything.

Key Takeaways

• The dividing fractions formula is $\frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc}$, flipping only the divisor.

• Dividing by a number and multiplying by its reciprocal are the same operation, which is why the rule is exact.

• Flipping the first fraction returns the reciprocal of the right answer, so the error is easy to spot.

• Mixed numbers become improper fractions before the rule applies.

• To work through fractions with a teacher, explore a 5th grade math tutor or math classes online.

Practice These Three Before Moving On

Compute $\frac{5}{6} \div \frac{2}{3}$ and $\frac{7}{8} \div 2$. Then work out how many $\frac{1}{3}$-cup scoops come from $\frac{5}{6}$ of a cup, and check whether each answer should be larger or smaller than where you started.

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Read More

• What Is a Reciprocal

• What Is an Improper Fraction

• What Is a Mixed Number

• Fractions, Decimals and Percentages

• Mental Math With Fractions