Dividing a Fraction by a Fraction

Still a question about fitting

$\frac{2}{3}÷\frac{1}{6}$2/3 ÷ 1/6 asks how many sixths fit into $\frac{2}{3}$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}$2/3 = 4/6.

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

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

Eighths in three quarters

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

Choose the $\frac{1}{8}$1/8 piece and lay pieces in until they fill $\frac{3}{4}$3/4. Exactly 6 fit, because $\frac{3}{4}=\frac{6}{8}$3/4 = 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}$2/3 of a whole holds $\frac{2}{3}\times 6$2/3 × 6 sixths. Dividing by $\frac{1}{6}$1/6 is the same as multiplying by 6.

So turn the divisor, $\frac{1}{6}$1/6, upside down to get $\frac{6}{1}$6/1, and multiply: $\frac{2}{3}÷\frac{1}{6}=\frac{2}{3}\times \frac{6}{1}=\frac{12}{3}=4$2/3 ÷ 1/6 = 2/3 × 6/1 = 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}$1/6 is $\frac{6}{1}$6/1, which is 6.

When less than one piece fits

The divisor does not have to be one piece. $\frac{1}{2}÷\frac{3}{4}$1/2 ÷ 3/4 asks how many pieces of size $\frac{3}{4}$3/4 fit into $\frac{1}{2}$1/2. Write both in quarters: $\frac{1}{2}=\frac{2}{4}$1/2 = 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}$2/3 of a piece. So $\frac{1}{2}÷\frac{3}{4}=\frac{2}{3}$1/2 ÷ 3/4 = 2/3.

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

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

Any fraction divides this way

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

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

Writing both fractions with the same denominator shows why. $\frac{3}{4}=\frac{15}{20}$3/4 = 15/20 and $\frac{2}{5}=\frac{8}{20}$2/5 = 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}$7/8 of another group: $15÷8=1\frac{7}{8}$15 ÷ 8 = 1 7/8.

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

Mixed numbers, and checking

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

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

The usual mistakes

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

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