Why do we invert and multiply instead of just dividing the numerators and denominators?

Inverting and multiplying works because division by a fraction is the same as multiplication by its reciprocal. This is a mathematical rule that extends from how whole numbers divide. If you tried to divide numerators and denominators separately—say, 2/3 5/9 = 2 5/3 9—you'd get wrong results and end up with decimals, not fractions. The invert-and-multiply method always gives you the right answer in fraction form.

Do I have to simplify before multiplying, or can I simplify after?

You can do either. Simplifying before multiplying (canceling common factors across the numerators and denominators) makes the numbers smaller and easier to work with, so mistakes are less likely. Simplifying after multiplying also works but requires more arithmetic. Most students find it faster to cancel first. Either way, your final answer must be in lowest terms.

What if I get a whole number as my answer?

That's perfectly fine. For example, 8/3 4/9 = 8/3 × 9/4 = 72/12 = 6. You can write 6 as 6/1 if you want, but 6 is the simplest form. Always check: 6 × 4/9 = 24/9 = 8/3 ✓.

What's a reciprocal and how is it different from a divisor?

A reciprocal is what you get when you flip a fraction. The reciprocal of 5/7 is 7/5. A divisor is the number you are dividing by. In the problem 3/4 5/7, the divisor is 5/7, and its reciprocal is 7/5. When you divide by a fraction, you multiply by its reciprocal instead.

M6MATH-3.2

Dividing Fractions by Fractions

Learn to divide fractions by fractions using the invert-and-multiply rule: flip the second fraction and multiply, then simplify.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Dividing Fractions by Fractions, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

Dividing fractions might seem harder than multiplying them, but there's a clever trick that turns any fraction division problem into multiplication. Once you learn the invert-and-multiply algorithm, you'll see that

\frac{2}{3} \div \frac{5}{7}

isn't so different from

\frac{2}{3} \times \frac{7}{5}

. In this lesson, you'll master the step-by-step process, learn why it works, and practice checking your answers so you can divide fractions with confidence.

The Invert-and-Multiply Algorithm

To divide a fraction by a fraction, keep the first fraction exactly as it is, flip (invert) the second fraction by swapping its numerator and denominator, then multiply. Here's the process:

If you're dividing

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

, rewrite it as

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

and multiply normally.

For example,

\frac{2}{3} \div \frac{5}{7}

becomes

\frac{2}{3} \times \frac{7}{5} = \frac{2 \times 7}{3 \times 5} = \frac{14}{15}

.

The inverted fraction

\frac{d}{c}

is called the reciprocal of

\frac{c}{d}

. Two fractions are reciprocals if their product equals 1. For instance,

\frac{5}{7} \times \frac{7}{5} = 1

. This relationship is the key to why invert-and-multiply works: dividing by a number is the same as multiplying by its reciprocal.

Remember: only invert the second fraction (the divisor), never the first one. A common mistake is flipping both fractions or flipping the wrong one.

Simplifying Before and After Multiplying

After you rewrite the division as multiplication, you can simplify before multiplying—this makes the arithmetic easier and gives you a simpler final answer. Look for common factors in the numerators and denominators.

For example,

\frac{4}{9} \div \frac{8}{3}

becomes

\frac{4}{9} \times \frac{3}{8}

. Before multiplying, notice that 4 and 8 share a factor of 4, and 3 and 9 share a factor of 3:

\frac{4}{9} \times \frac{3}{8} = \frac{\cancel{4}^{\,1}}{\cancel{9}_{\,3}} \times \frac{\cancel{3}^{\,1}}{\cancel{8}_{\,2}} = \frac{1}{3} \times \frac{1}{2} = \frac{1}{6}

If you don't simplify first, you multiply to get

\frac{12}{72}

and then simplify to

\frac{1}{6}

anyway—you'll reach the same answer, but with more work. Always simplify in lowest terms before you consider the problem finished.

Checking Your Answer by Multiplying Back

A quick way to verify that your division answer is correct is to multiply your result by the divisor (the second fraction). You should get the dividend (the first fraction).

For instance, if you calculated

\frac{3}{5} \div \frac{2}{7} = \frac{21}{10}

, check by computing

\frac{21}{10} \times \frac{2}{7}

\frac{21}{10} \times \frac{2}{7} = \frac{\cancel{21}^{\,3}}{10} \times \frac{2}{\cancel{7}_{\,1}} = \frac{3}{10} \times \frac{2}{1} = \frac{6}{10} = \frac{3}{5}

✓

You recovered the original dividend, so your answer is right. This check only takes a few extra seconds and catches careless errors like inverting the wrong fraction or making a computational slip. Use this method every time until checking becomes automatic.

Step-by-Step Process Summary

Here's the complete algorithm in order:

Step	Action
1	Write down the division problem: $\frac{a}{b} \div \frac{c}{d}$
2	Invert (flip) the second fraction: $\frac{c}{d}$ becomes $\frac{d}{c}$
3	Rewrite as multiplication: $\frac{a}{b} \times \frac{d}{c}$
4	Cancel common factors across numerators and denominators
5	Multiply numerators and denominators
6	Write the result in lowest terms
7	Check: multiply your answer by the divisor to recover the dividend

Following this order ensures you don't skip steps and helps you catch mistakes before you finish.

Common Mistakes to Avoid

Students often make these errors when dividing fractions:

Inverting the first fraction instead of the second:

\frac{2}{5} \div \frac{3}{7}

is not the same as

\frac{5}{2} \times \frac{3}{7}

. Only flip the divisor (second fraction).

Forgetting to invert at all: Writing

\frac{2}{5} \times \frac{3}{7}

directly without inverting gives the wrong answer. Division and multiplication are different operations.

Forgetting to simplify: Leaving your answer as

\frac{12}{72}

instead of simplifying to

\frac{1}{6}

may be marked incomplete by your teacher.

Simplifying incorrectly: You can only cancel factors, not subtract.

\frac{8}{12}

simplifies to

\frac{2}{3}

(divide by 4), not

\frac{0}{8}

(wrong—you can't subtract the numerator from the denominator).

Test yourself: Does your check multiply correctly? If you multiply your answer by the divisor and don't get back the original dividend, redo the division.

Key terms

Invert (or reciprocal).: To flip a fraction by swapping its numerator and denominator. The reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$ . Any fraction times its reciprocal equals 1.
Dividend.: The number being divided. In $\frac{3}{5} \div \frac{2}{7}$ , the dividend is $\frac{3}{5}$ .
Divisor.: The number you are dividing by. In $\frac{3}{5} \div \frac{2}{7}$ , the divisor is $\frac{2}{7}$ .
Quotient.: The result of division. In $\frac{3}{5} \div \frac{2}{7} = \frac{21}{10}$ , the quotient is $\frac{21}{10}$ .
Simplify (or reduce to lowest terms).: Divide both the numerator and denominator by their greatest common factor so that no factor other than 1 divides both.
Common factor.: A number that divides evenly into two or more numbers. For example, 3 is a common factor of 9 and 12.

Worked example

Divide

\frac{5}{6} \div \frac{2}{9}

and check your answer.

Step 1: Write the division problem.

\frac{5}{6} \div \frac{2}{9}

Step 2 & 3: Invert the second fraction and rewrite as multiplication.

The second fraction is

\frac{2}{9}

. Its reciprocal is

\frac{9}{2}

. Now write:

\frac{5}{6} \times \frac{9}{2}

Step 4: Cancel common factors.

Look across numerators and denominators. The 6 in the denominator and the 9 in the numerator share a common factor of 3. Also, 9 ÷ 3 = 3 and 6 ÷ 3 = 2.

\frac{5}{\cancel{6}_{\,2}} \times \frac{\cancel{9}^{\,3}}{2} = \frac{5}{2} \times \frac{3}{2}

Step 5: Multiply.

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

Step 6: Check if in lowest terms.

The GCD of 15 and 4 is 1, so

\frac{15}{4}

is already in lowest terms.

Step 7: Check by multiplying your quotient by the divisor.

\frac{15}{4} \times \frac{2}{9} = \frac{15 \times 2}{4 \times 9} = \frac{30}{36}

Now simplify:

\frac{30}{36} = \frac{5}{6}

(divide numerator and denominator by 6).

This equals the original dividend

\frac{5}{6}

, so the answer is correct. ✓

Final answer:

\frac{15}{4}

Practice questions

Divide

\frac{7}{8} \div \frac{3}{4}

. Show all steps and simplify.

Answer: $\frac{7}{6}$

Rewrite as

\frac{7}{8} \times \frac{4}{3}

. Cancel the common factor of 4:

\frac{7}{\cancel{8}_{\,2}} \times \frac{\cancel{4}^{\,1}}{3} = \frac{7}{2} \times \frac{1}{3} = \frac{7}{6}

. Since GCD(7, 6) = 1, this is in lowest terms. Check:

\frac{7}{6} \times \frac{3}{4} = \frac{21}{24} = \frac{7}{8}

✓.

Which expression equals

\frac{4}{5} \div \frac{8}{15}

$\frac{4}{5} \times \frac{8}{15}$
$\frac{4}{5} \times \frac{15}{8}$
$\frac{5}{4} \times \frac{8}{15}$
$\frac{8}{15} \div \frac{4}{5}$

Answer: $\frac{4}{5} \times \frac{15}{8}$

When you divide by a fraction, you invert (flip) the second fraction and multiply. The divisor

\frac{8}{15}

inverts to

\frac{15}{8}

. The first fraction

\frac{4}{5}

stays the same. So

\frac{4}{5} \div \frac{8}{15} = \frac{4}{5} \times \frac{15}{8}

. The first choice keeps division instead of inverting; the third flips the first fraction instead of the second; the fourth reverses the order, which gives a different result.

A student solved

\frac{2}{3} \div \frac{5}{9}

and got

\frac{6}{15}

. What error did the student make? What is the correct answer?

Answer: The student multiplied instead of inverting and multiplying; the correct answer is $\frac{6}{5}$ .

The student wrote

\frac{2}{3} \times \frac{5}{9} = \frac{10}{27}

and then simplified incorrectly (or confused it with

\frac{6}{15}

). The correct method is to invert the divisor:

\frac{2}{3} \div \frac{5}{9} = \frac{2}{3} \times \frac{9}{5} = \frac{18}{15} = \frac{6}{5}

. A quick check:

\frac{6}{5} \times \frac{5}{9} = \frac{30}{45} = \frac{2}{3}

✓. The error happened because the student forgot to invert before multiplying.

FAQ

Why do we invert and multiply instead of just dividing the numerators and denominators?: Inverting and multiplying works because division by a fraction is the same as multiplication by its reciprocal. This is a mathematical rule that extends from how whole numbers divide. If you tried to divide numerators and denominators separately—say, $\frac{2}{3} \div \frac{5}{9} = \frac{2 \div 5}{3 \div 9}$ —you'd get wrong results and end up with decimals, not fractions. The invert-and-multiply method always gives you the right answer in fraction form.
Do I have to simplify before multiplying, or can I simplify after?: You can do either. Simplifying before multiplying (canceling common factors across the numerators and denominators) makes the numbers smaller and easier to work with, so mistakes are less likely. Simplifying after multiplying also works but requires more arithmetic. Most students find it faster to cancel first. Either way, your final answer must be in lowest terms.
What if I get a whole number as my answer?: That's perfectly fine. For example, $\frac{8}{3} \div \frac{4}{9} = \frac{8}{3} \times \frac{9}{4} = \frac{72}{12} = 6$ . You can write 6 as $\frac{6}{1}$ if you want, but 6 is the simplest form. Always check: $6 \times \frac{4}{9} = \frac{24}{9} = \frac{8}{3}$ ✓.
What's a reciprocal and how is it different from a divisor?: A reciprocal is what you get when you flip a fraction. The reciprocal of $\frac{5}{7}$ is $\frac{7}{5}$ . A divisor is the number you are dividing by. In the problem $\frac{3}{4} \div \frac{5}{7}$ , the divisor is $\frac{5}{7}$ , and its reciprocal is $\frac{7}{5}$ . When you divide by a fraction, you multiply by its reciprocal instead.

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