Dividing a Fraction by a Whole Number

Sharing a piece

Half a cake is left, and 3 people share it equally. Each person gets $\frac{1}{2}÷3$1/2 ÷ 3 of the cake. Each share must be smaller than the half, because the half is being split up among them.

The whole cake is cut into 2 halves, and one half is left.

Cut the piece into 3

Cut the half into 3 equal slices, one for each person. To name one slice, cut the other half in the same way. The whole cake is now 2 × 3 = 6 equal slices, so one slice is $\frac{1}{6}$1/6 of the cake.

So $\frac{1}{2}÷3=\frac{1}{6}$1/2 ÷ 3 = 1/6. Each person gets $\frac{1}{6}$1/6 of the whole cake.

Cut every half into 3. The whole is now 6 equal pieces, and the half that is left is 3 of them: one piece of $\frac{1}{6}$1/6 for each of the 3 people.

One person's share is 1 of the 6 equal slices of the whole cake: $\frac{1}{6}$1/6.

The denominator multiplies

Sharing among 3 people is the same as taking a third: $\frac{1}{2}÷3$1/2 ÷ 3 is $\frac{1}{3}$1/3 of $\frac{1}{2}$1/2, which is $\frac{1}{3}\times \frac{1}{2}=\frac{1}{6}$1/3 × 1/2 = 1/6. So dividing a fraction by a whole number multiplies its denominator by that number: $\frac{1}{2}÷3=\frac{1}{\left(2\times 3\right)}=\frac{1}{6}$1/2 ÷ 3 = 1/(2 × 3) = 1/6.

The denominator gets bigger, so the pieces get smaller. It takes 3 times as many pieces to make the whole cake, so each piece is a third of the size.

Sharing more than one piece

Share $\frac{2}{3}$2/3 of a cake among 4 people. Cut each of the thirds into 4. The whole is now 3 × 4 = 12 pieces, and the 2 thirds are 2 × 4 = 8 of them. Each person gets 8 ÷ 4 = 2 pieces, so $\frac{2}{3}÷4=\frac{2}{12}$2/3 ÷ 4 = 2/12, which simplifies to $\frac{1}{6}$1/6.

Multiplying the denominator gives the same answer at once: $\frac{2}{3}÷4=\frac{2}{\left(3\times 4\right)}=\frac{2}{12}$2/3 ÷ 4 = 2/(3 × 4) = 2/12.

Cut every third into 4. The whole is 12 pieces, and $\frac{2}{3}$2/3 is 8 of them, so each of the 4 people gets 2 pieces: $\frac{2}{12}$2/12.

When the pieces share out

Sometimes the pieces share out without being cut. In $\frac{4}{5}÷2$4/5 ÷ 2 there are 4 fifths to share between 2 people, so each person gets 4 ÷ 2 = 2 fifths: $\frac{4}{5}÷2=\frac{2}{5}$4/5 ÷ 2 = 2/5. The size of a piece stays the same, and only the number of pieces is divided.

Multiplying the denominator still works: $\frac{4}{5}÷2=\frac{4}{10}$4/5 ÷ 2 = 4/10, and $\frac{4}{10}$4/10 simplifies to $\frac{2}{5}$2/5, the same amount.

The 4 fifths are shared between 2 people, so each person gets 2 of the fifths: $\frac{2}{5}$2/5.

The usual mistakes

Multiplying the numerator. $\frac{1}{2}÷3$1/2 ÷ 3 is not $\frac{3}{2}$3/2. Sharing makes each share smaller than the amount shared, and $\frac{3}{2}$3/2 is bigger than $\frac{1}{2}$1/2.

Adding to the denominator. $\frac{1}{2}÷3$1/2 ÷ 3 is not $\frac{1}{\left(2+3\right)}=\frac{1}{5}$1/(2 + 3) = 1/5. Cutting each of the 2 halves into 3 makes 2 × 3 = 6 pieces, so the denominator is 6.