What a ratio is

A class with 12 boys and 18 girls has a boy-to-girl ratio of 12 to 18, which simplifies to 2 to 3.

Shade two cells out of five

Before any definitions, use the bar below. Shade 2 of its 5 cells. Think of a class where boys and girls come in groups of 2 and 3; the shaded cells are the boys. The bar reads your shading as a fraction, a decimal and a percent, and you should see 2/5, 0.4 and 40%. Keep the 40% in mind. Further down it settles what "2 to 3" does and does not mean.

Six more girls, or 3 girls for every 2 boys?

Now the class itself. It has 12 boys and 18 girls. One way to compare them is subtraction: there are 6 more girls than boys. A ratio compares by division instead, so it says that for every 2 boys there are 3 girls. OpenStax defines a ratio as a comparison of two numbers or two quantities measured with the same unit. It writes the ratio of a to b in three ways: a to b, a/b, or a : b. For this class that gives 12 to 18, 12/18 and 12 : 18.

Which number is the yardstick?

In 12 : 18 the second number is the one the first is measured against. Korean primary textbooks call it the base amount and the first number the compared amount. Dividing compared by base gives the ratio's value: 12 ÷ 18 = 2/3. Read that carefully. It says the boys number two thirds of the girls. It does not say two thirds of the class.

Why 12 : 18 can shrink to 2 : 3

A ratio simplifies exactly like a fraction. Divide both parts by a common factor; 12 and 18 share 6, so 12 : 18 = 2 : 3. The tiles show why that is allowed. Split the class into 6 equal groups and each group is a small copy of the whole class, with boys and girls in the same balance. There is one difference from fractions. OpenStax leaves a ratio such as 4/1 as 4/1 rather than writing 4, so both parts of the comparison stay in sight.

Ratios with decimals become whole-number ratios when you multiply both parts by the same power of ten. 1.2 : 0.4 becomes 12 : 4 after multiplying by 10, and that simplifies to 3 : 1. Ratios with fractions work the same way with a common denominator: 1/2 : 1/3 becomes 3 : 2 after multiplying both by 6.

2 : 3 is not 3 : 2, and an inch is not a foot

The ratio of boys to girls is 2 : 3; the ratio of girls to boys is 3 : 2. These are different statements, so always read which quantity comes first. The two quantities also need the same unit. To compare 1 inch with 1 foot, change the foot to 12 inches first: the ratio is 1 : 12, not 1 : 1. OpenStax advises converting to the smaller unit, which avoids fractions.

Boys are 2 parts out of 5, not 2 out of 3

The class ratio 2 : 3 compares one part with another part. You can also compare a part with the whole. Each group holds 2 + 3 = 5 pupils, so boys are 2/5 of the class and girls 3/5. That is the bar you shaded at the top: 2/5, or 40%. A 2 : 3 ratio does not mean 2/3 of the class are boys. Two thirds would be 20 boys out of 30, and this class has 12.

Sharing 60 dollars in the ratio 2 : 3

Two friends share 60 dollars in the ratio 2 : 3. Add the parts first, 2 + 3 = 5, to count the equal shares. One share is 60 ÷ 5 = 12 dollars. The first friend gets 2 × 12 = 24 and the second 3 × 12 = 36. Check: 24 + 36 = 60, and 24 : 36 simplifies back to 2 : 3. A ratio can have three parts too. Cement, sand and gravel mixed 1 : 2 : 3 in a made-up 30-kilogram batch make 1 + 2 + 3 = 6 shares of 5 kg each, so 5 kg, 10 kg and 15 kg.