Ratio, Fraction, Percent

One bar, three ways to describe it

A bar is cut into 4 equal parts. 1 part is colored and the other 3 parts are plain. There are three ways to describe this bar, and all three say the same thing.

The first is a ratio. For every 1 colored part there are 3 plain parts, so the ratio of colored to plain is 1 : 3. A ratio compares one share with the other share.

1 colored part and 3 plain parts: colored to plain is 1 : 3.

From a ratio to a fraction

A fraction compares a share with the whole. So first find the whole: add the parts in the ratio. 1 + 3 = 4, so the bar has 4 parts altogether, and the colored part is 1 out of 4. As a fraction, the colored share is $\frac{1}{4}$1/4 of the bar.

The denominator is the total number of parts, 4. It is not the other share, 3. The fraction $\frac{1}{3}$1/3 compares the colored part with the plain parts only, which is what the ratio 1 : 3 already says.

The same bar. The colored part is 1 of the 4 equal parts, so it is $\frac{1}{4}$1/4 of the whole bar.

From a fraction to a percent

A percent is a fraction out of 100. To change $\frac{1}{4}$1/4 into a percent, make its denominator 100: multiply the numerator and the denominator by 25. $\frac{1}{4}=\frac{25}{100}$1/4 = 25/100, which is 25%.

The plain share is 3 of the 4 parts, so it is $\frac{3}{4}=\frac{75}{100}=75$3/4 = 75/100 = 75%. The two percents add to 100%, because the two shares together make the whole bar.

The whole bar is 100%, so each of its 4 equal parts is 100% ÷ 4 = 25%. The colored part is 25% and the plain parts are 3 × 25% = 75%.

Another ratio

Red and blue counters are in the ratio 2 : 3. Add the parts first: 2 + 3 = 5 parts altogether. Red is 2 of the 5 parts, so red is $\frac{2}{5}$2/5 of the counters. Multiply the numerator and the denominator by 20: $\frac{2}{5}=\frac{40}{100}=40$2/5 = 40/100 = 40%. Blue is the other $\frac{3}{5}$3/5, which is 60%.

Equivalent ratios give the same fraction and the same percent. 2 : 3 and 4 : 6 describe the same mix, and $\frac{4}{10}=\frac{2}{5}$4/10 = 2/5, so red is 40% of the counters either way.

Drag the handles to change the numbers of colored and plain parts. Make the colored share $\frac{1}{4}$1/4. The ratio 1 : 3 does it, and so does 2 : 6: both color 25% of the bar.

The usual mistakes

Using the other share as the denominator. In 1 : 9, the 1 share is $\frac{1}{10}$1/10 of the whole, which is 10%. Reading the ratio as the fraction $\frac{1}{9}$1/9 gives about 11%, because it leaves the 1 share out of the whole.

Counting every part as 10%. A part is 10% only when there are 10 parts altogether. In 2 : 3 there are 5 parts, so each part is 100% ÷ 5 = 20%, and the 2 share is 40%, not 20%.

Answering with the wrong share. In 3 : 1, the 3 share is $\frac{3}{4}=75$3/4 = 75% and the 1 share is $\frac{1}{4}=25$1/4 = 25%. Check which share the question names before you answer.