0.5 Is a Half

A half in tenths

Cut a bar into 2 equal parts and color one of them. That is one half, $\frac{1}{2}$1/2.

Now cut each half into 5 equal pieces. The bar has 10 pieces, which are tenths, and the colored half covers 5 of them. So one half is 5 tenths: $\frac{1}{2}=\frac{5}{10}=0.5$1/2 = 5/10 = 0.5.

One of the 2 equal parts is colored: one half.

Each half is cut into 5 pieces. The colored half is 5 of the 10 pieces, so $\frac{1}{2}=0.5$1/2 = 0.5.

Fifths in tenths

Tenths also name fifths. Ten tenths share equally into 5 groups of 2, so each fifth is 2 tenths. $\frac{1}{5}=\frac{2}{10}=0.2$1/5 = 2/10 = 0.2, and $\frac{2}{5}$2/5 is 4 tenths, 0.4. Three fifths is 0.6 and four fifths is 0.8.

A bar cut into fifths, with each fifth cut in two. 2 fifths cover 4 of the 10 pieces, so $\frac{2}{5}=0.4$2/5 = 0.4.

Quarters need hundredths

A quarter does not fit into tenths. 10 pieces cannot be shared into 4 equal groups of whole pieces: 4 groups of 2 use only 8, and 4 groups of 3 need 12.

So cut the whole into 100 equal pieces instead. 100 ÷ 4 = 25, so a quarter is 25 hundredths: $\frac{1}{4}=\frac{25}{100}=0.25$1/4 = 25/100 = 0.25. Three quarters is 3 × 25 = 75 hundredths, 0.75. Two quarters make a half, 50 hundredths, and 0.50 is the same number as 0.5.

A square cut into 100 equal parts, with 25 shaded: 2 full rows and half of the next. That is one quarter of the square.

Three quarters of the square: 75 of the 100 parts, which is 0.75.

Each fraction shades its share of the 100 parts: a half is 50 of them, a quarter 25, three quarters 75 and a fifth 20. The number of hundredths is the decimal.

The ones to know by heart

These come up so often that they are worth knowing without working them out: $\frac{1}{2}=0.5$1/2 = 0.5, $\frac{1}{4}=0.25$1/4 = 0.25, $\frac{3}{4}=0.75$3/4 = 0.75, $\frac{1}{5}=0.2$1/5 = 0.2 and $\frac{1}{10}=0.1$1/10 = 0.1.

The digits of a fraction are not the digits of its decimal. $\frac{1}{4}$1/4 is not 0.14 or 0.4: it is 25 hundredths, 0.25. Going the other way, 0.25 is not $\frac{2}{5}$2/5, and 0.2 is not $\frac{1}{2}$1/2. 0.2 is two tenths, which is one fifth.