A Fraction of an Amount

The denominator: find one share

Find $\frac{2}{3}$2/3 of 12. The denominator, 3, says to split the whole amount into 3 equal shares. Draw 12 as a bar and cut it into 3 equal parts. One part is 12 ÷ 3 = 4, so $\frac{1}{3}$1/3 of 12 is 4.

The bar of 12 is cut into 3 equal parts of 4. One part, $\frac{1}{3}$1/3 of 12, is colored.

The numerator: take that many shares

The numerator, 2, says how many of the shares to take. Each share is 4, so 2 shares are 2 × 4 = 8. $\frac{2}{3}$2/3 of 12 = 8.

So to find a fraction of an amount, divide the amount by the denominator, then multiply by the numerator.

2 of the 3 parts are colored. Each holds 4, so together they are 8.

Larger amounts

The amount does not have to be something you can count one by one. Find $\frac{3}{4}$3/4 of 60 minutes. Divide by the denominator: 60 ÷ 4 = 15, so a quarter of an hour is 15 minutes. Multiply by the numerator: 3 × 15 = 45. So $\frac{3}{4}$3/4 of 60 minutes is 45 minutes.

To check, add on the part you did not take. The quarter left over is 15 minutes, and 45 + 15 = 60, the whole amount again.

When the numerator is 1, dividing is the whole job: $\frac{1}{5}$1/5 of 35 is 35 ÷ 5 = 7.

The bar of 60 is cut into 4 equal parts of 15. Three of them make 45, and the fourth, 15, makes the total up to 60.

The button cuts the 60 into equal parts, each labeled with what it holds; the handle takes parts from the left. Quarters hold 15 each, and three of them make 45.

When the division is not exact

Find $\frac{2}{3}$2/3 of 10. Dividing first gives 10 ÷ 3, which does not come out exactly. Instead, multiply first: 2 × 10 = 20, then divide: $20÷3=\frac{20}{3}=6\frac{2}{3}$20 ÷ 3 = 20/3 = 6 2/3.

Either order gives the same answer. Dividing first, one third of 10 is $10÷3=3\frac{1}{3}$10 ÷ 3 = 3 1/3, and two thirds is $3\frac{1}{3}+3\frac{1}{3}=6\frac{2}{3}$3 1/3 + 3 1/3 = 6 2/3. Divide first when the division is exact, because the numbers stay small.

Cut 10 into 3 equal parts and each part is $3\frac{1}{3}$3 1/3. Two parts make $6\frac{2}{3}$6 2/3.

The usual mistakes

Taking the wrong part. In $\frac{2}{3}$2/3 of 12, the 2 shares taken are 8, and the share left over is 4. Answering 4 gives the part that was not taken.

Taking one share too many. The numerator says exactly how many shares to take: $\frac{2}{3}$2/3 is 2 shares of 4, which is 8, not 3 shares, which would be all 12.