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, $\frac{1}{2}$.

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: $\frac{1}{2} = \frac{5}{10} = 0.5$.

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 $\frac{1}{2} = 0.5$.

Fifths in tenths

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

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: $\frac{1}{4} = \frac{25}{100} = 0.25$. 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: $\frac{1}{2} = 0.5$, $\frac{1}{4} = 0.25$, $\frac{3}{4} = 0.75$, $\frac{1}{5} = 0.2$ and $\frac{1}{10} = 0.1$.

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