Dividing fractions

The whole rule#

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

The second fraction is flipped — the numerator and denominator swap places, and the result is called the reciprocal. The division sign becomes a multiplication sign. The first fraction is not touched at all. Many teachers call this "keep, change, flip".

Why it works. Dividing by $\frac{c}{d}$ means asking how many times $\frac{c}{d}$ fits into the dividend. It is easiest to see with whole numbers: $6 \div \frac{1}{2} = 12$, because there are exactly twelve halves in six whole things. Notice that the result is larger than the number you started with: dividing by a fraction less than one always makes a number bigger. If your answer came out smaller and you divided by a proper fraction, you lost the flip somewhere.

Example 1: a fraction by a fraction#

Work out $\frac{3}{4} \div \frac{2}{5}$.

Step 1. Flip the divisor: the reciprocal of $\frac{2}{5}$ is $\frac{5}{2}$. The dividend $\frac{3}{4}$ stays as it is.

Step 2. Change division to multiplication: $\frac{3}{4} \times \frac{5}{2}$.

Step 3. Multiply the numerators and the denominators separately: $\frac{3 \times 5}{4 \times 2} = \frac{15}{8}$.

Step 4. The numerator is larger than the denominator, so write it as a mixed number: eight goes into fifteen once with seven left over. The answer is $1\frac{7}{8}$.

Example 2: a fraction by a whole number#

Work out $\frac{5}{6} \div 10$.

Step 1. Write ten as a fraction: $10 = \frac{10}{1}$. This is the step people skip most often, and without it it is not clear what to flip.

Step 2. Flip it: the reciprocal of $\frac{10}{1}$ is $\frac{1}{10}$.

Step 3. Multiply: $\frac{5}{6} \times \frac{1}{10} = \frac{5}{60}$.

Step 4. Divide top and bottom by 5: $\frac{5}{60} = \frac{1}{12}$.

There is a shortcut for this case: to divide a fraction by a whole number, just multiply the denominator by that number. But use the shortcut only once the long way comes without thinking.

Example 3: mixed numbers#

Work out $2\frac{1}{3} \div 1\frac{1}{6}$.

Step 1. Turn both numbers into improper fractions. Multiply the whole part by the denominator and add the numerator: $2\frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3}$, and $1\frac{1}{6} = \frac{1 \times 6 + 1}{6} = \frac{7}{6}$.

Step 2. Flip the divisor: $\frac{6}{7}$.

Step 3. Multiply: $\frac{7}{3} \times \frac{6}{7} = \frac{42}{21}$.

Step 4. Simplify: $\frac{42}{21} = 2$. A whole-number answer happens, and it is not a sign of a mistake.

You cannot divide mixed numbers piece by piece — wholes by wholes, fractions by fractions. Test it on the same example: $2 \div 1 = 2$ and $\frac{1}{3} \div \frac{1}{6} = 2$, so the "answer" would be $2\frac{2}{1}$, which is not equal to two.

Common mistakes#

Flipping the first fraction instead of the second. Saying the order out loud helps: "keep the first, flip the second".

Dividing numerators and denominators separately, the way you multiply them. That rule does not exist: $\frac{3}{4} \div \frac{2}{5}$ is not $\frac{3 \div 2}{4 \div 5}$.

Forgetting to simplify. The fraction $\frac{42}{21}$ is technically correct, but only the fraction in lowest terms counts as the answer.

Mixing up division and multiplication in word problems. If the question is "how many pieces $\frac{1}{4}$ metre long can you cut from $\frac{3}{2}$ metres", that is division, and the answer is $\frac{3}{2} \div \frac{1}{4} = \frac{3}{2} \times 4 = 6$.

You do not need a common denominator to divide — that step belongs to adding and subtracting fractions with unlike denominators. If a problem mixes ordinary fractions and decimals, it is usually easier to convert everything to ordinary fractions; going back and forth is covered in operations with decimals.

How to check yourself#

Multiply your answer by the divisor — you should get the dividend back. For the first example: $\frac{15}{8} \times \frac{2}{5} = \frac{30}{40} = \frac{3}{4}$, which matches. This check catches flipping the wrong fraction and a missed simplification, and it takes half a line.

Step-by-step plan

1. Step 1 — the reciprocalPractise naming the reciprocal of 2/5, 7/3, 9 and 1/8 out loud, ten in a row.
2. Step 2 — fraction by fractionTen problems of the form a/b ÷ c/d, always simplifying the answer.
3. Step 3 — whole numbers and fractionsDivide a fraction by a whole number and a whole number by a fraction, writing the whole number over 1.
4. Step 4 — mixed numbersConvert to improper fractions, divide, then convert the answer back to a mixed number.
5. Step 5 — checkingAfter every problem, multiply the answer by the divisor to see that you get the dividend back.

Check yourself

Sources

• Khan Academy — ArithmeticPractice on multiplying and dividing fractions with a check at every step
  free
• OpenStax Prealgebra 2eFree textbook; the fractions chapter explains reciprocals with worked examples
  free

Was this helpful?

Yes No