Dividing a Fraction by a Fraction

Still a question about fitting

$\frac{2}{3} \div \frac{1}{6}$ asks how many sixths fit into $\frac{2}{3}$. Thirds and sixths are different sizes, so cut the thirds to match: cut each third into 2 equal pieces. The whole is now 6 sixths, and the 2 thirds are 4 of them, so $\frac{2}{3} = \frac{4}{6}$.

Four sixths make up $\frac{2}{3}$, so 4 sixths fit: $\frac{2}{3} \div \frac{1}{6} = 4$.

Cut every third into 2. The shaded $\frac{2}{3}$ is now 4 of the 6 sixths, so 4 sixths fit into $\frac{2}{3}$.

Eighths in three quarters

$\frac{3}{4} \div \frac{1}{8}$ asks how many eighths fit into $\frac{3}{4}$. Each quarter holds 2 eighths, so $\frac{3}{4} = \frac{6}{8}$, and 6 eighths fit: $\frac{3}{4} \div \frac{1}{8} = 6$.

Choose the $\frac{1}{8}$ piece and lay pieces in until they fill $\frac{3}{4}$. Exactly 6 fit, because $\frac{3}{4} = \frac{6}{8}$.

Flip the divisor and multiply

Counting pieces always works, and there is a shortcut that gives the same count. Each whole holds 6 sixths, so $\frac{2}{3}$ of a whole holds $\frac{2}{3} \times 6$ sixths. Dividing by $\frac{1}{6}$ is the same as multiplying by 6.

So turn the divisor, $\frac{1}{6}$, upside down to get $\frac{6}{1}$, and multiply: $\frac{2}{3} \div \frac{1}{6} = \frac{2}{3} \times \frac{6}{1} = \frac{12}{3} = 4$. That is the same 4 the bar counted. A fraction turned upside down is called its reciprocal: the reciprocal of $\frac{1}{6}$ is $\frac{6}{1}$, which is 6.

When less than one piece fits

The divisor does not have to be one piece. $\frac{1}{2} \div \frac{3}{4}$ asks how many pieces of size $\frac{3}{4}$ fit into $\frac{1}{2}$. Write both in quarters: $\frac{1}{2} = \frac{2}{4}$. One piece is 3 quarters, and there are only 2 quarters, so the whole piece does not fit. The 2 quarters fill 2 of the piece's 3 quarters, which is $\frac{2}{3}$ of a piece. So $\frac{1}{2} \div \frac{3}{4} = \frac{2}{3}$.

Flipping the divisor gives the same answer: $\frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \times \frac{4}{3} = \frac{4}{6} = \frac{2}{3}$.

Both bars are cut into quarters of the same size. The piece, $\frac{3}{4}$, is 3 quarters long, and $\frac{1}{2}$ is 2 quarters long, so $\frac{1}{2}$ fills $\frac{2}{3}$ of one piece.

Any fraction divides this way

Every division by a fraction works the same way: dividing $\frac{a}{b}$ by $\frac{c}{d}$ gives $\frac{a}{b} \times \frac{d}{c}$. Flip the second fraction, the divisor, and multiply across. The first fraction stays as it is.

Try $\frac{3}{4} \div \frac{2}{5}$. Flip the divisor and multiply: $\frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1\frac{7}{8}$. So $\frac{2}{5}$ fits into $\frac{3}{4}$ once, and then $\frac{7}{8}$ of the way into a second time.

Writing both fractions with the same denominator shows why. $\frac{3}{4} = \frac{15}{20}$ and $\frac{2}{5} = \frac{8}{20}$, so the question is how many 8 twentieths fit into 15 twentieths. One group of 8 fits, and the 7 left over are $\frac{7}{8}$ of another group: $15 \div 8 = 1\frac{7}{8}$.

In twentieths, $\frac{3}{4}$ is 15 and $\frac{2}{5}$ is 8. One piece of 8 fits into the 15, and the 7 left over are $\frac{7}{8}$ of another piece: $\frac{3}{4} \div \frac{2}{5} = 1\frac{7}{8}$.

Mixed numbers, and checking

A mixed number must become an improper fraction before you divide. $1\frac{1}{3} \div \frac{1}{3}$ becomes $\frac{4}{3} \div \frac{1}{3} = \frac{4}{3} \times \frac{3}{1} = \frac{12}{3} = 4$.

To check a division, multiply the answer by the fraction you divided by. For $\frac{2}{3} \div \frac{1}{6} = 4$, the check is $4 \times \frac{1}{6} = \frac{4}{6} = \frac{2}{3}$, the amount you started with.

The usual mistakes

Flipping the first fraction. Only the divisor is flipped. $\frac{2}{3} \div \frac{2}{9}$ is $\frac{2}{3} \times \frac{9}{2} = \frac{18}{6} = 3$, not $\frac{3}{2} \times \frac{2}{9} = \frac{1}{3}$.

Multiplying straight across without flipping. $\frac{2}{3} \times \frac{2}{9} = \frac{4}{27}$ multiplies the fractions. Division flips the second fraction first, then multiplies.