Three amounts at once
A ratio can compare more than two amounts. A fruit punch is made from 1 cup of lemon juice, 2 cups of orange juice and 3 cups of water. The ratio of lemon juice to orange juice to water is 1 : 2 : 3.
As with two amounts, the numbers are in the same order as the names. The ratio of water to orange juice to lemon juice is 3 : 2 : 1.
Every part is the same size: 1 part of lemon juice, 2 parts of orange juice and 3 parts of water.
Add all three to find the whole
Add all three numbers to find the whole: 1 + 2 + 3 = 6 parts. The lemon juice is 1 of the 6 parts, so it is of the punch. The orange juice is of the punch, and the water is . The three fractions add up to , the whole punch.
The whole punch is 6 equal parts. The lemon juice is 1 of them, of the punch.
Scaling and simplifying all three
Everything you know about ratios of two amounts works for three. Multiply all three numbers by the same number and you get an equivalent ratio: 1 : 2 : 3 = 2 : 4 : 6 is a double batch of the same punch.
So a jug of 12 cups of punch holds of 12 = 2 cups of lemon juice, of 12 = 4 cups of orange juice and of 12 = 6 cups of water. Check: 2 + 4 + 6 = 12.
To simplify, divide all three numbers by a common factor. In 4 : 6 : 10, all three numbers are even, so divide each by 2: 4 : 6 : 10 = 2 : 3 : 5. The only common factor of 2, 3 and 5 is 1, so 2 : 3 : 5 is in its simplest form.
The jug of 12 cups: 2 cups of lemon juice, 4 of orange juice and 6 of water. Each amount is double the first recipe, so the ratio is still 1 : 2 : 3.
The usual mistakes
Multiplying instead of adding. The parts of one whole are counted up, so they add. In 2 : 3 : 4 there are 2 + 3 + 4 = 9 parts altogether, not 2 × 3 × 4 = 24.
Counting only the largest share. The 4 in 2 : 3 : 4 is one share. The whole needs all three shares added: 9 parts.
Scaling only some of the numbers. A double batch of 1 : 2 : 3 is 2 : 4 : 6. Doubling two of the numbers and leaving the third, as in 2 : 4 : 3, changes the recipe.