Ratios and proportions: equivalent ratios and word problems

What a ratio tells you

A ratio compares amounts. If a recipe uses 2 cups of flour for every 3 cups of oats, the ratio of flour to oats is 2:3, read as 2 to 3. Order matters: oats to flour is 3:2. A ratio doesn't give the actual amounts, only how they compare, so 4 cups to 6 cups and 10 cups to 15 cups follow the same recipe.

Ratios can compare part to part or part to whole. In a class of 12 girls and 15 boys, girls to boys is 12:15, which simplifies to 4:5. Girls to the whole class is 12:27, or 4:9, so 4/9 of the class are girls.

Equivalent ratios and simplifying

Equivalent ratios are made by multiplying or dividing both parts by the same number, just like equivalent fractions. 3:4 is equivalent to 6:8, 9:12 and 30:40. To simplify a ratio, divide both parts by their greatest common factor: 18:24 becomes 3:4 after dividing by 6.

Adding the same number to both parts does not make an equivalent ratio. A drink mixed 1:2, one cup of juice to two of water, tastes different from one mixed 2:3. Juice is a third of the first drink but two fifths of the second.

A ratio table keeps track of equivalent ratios. If 3 notebooks cost $5, then 6 cost $10, 9 cost $15 and 30 cost $50. Each column is the first one multiplied by the same number.

The bar model for ratio word problems

Most ratio word problems are easiest with a bar model. Draw one bar for each quantity, split into equal boxes, one box per part. Find the value of one box and everything else follows. These three examples cover the usual kinds of question.

• You know one quantity. The ratio of cats to dogs is 3:5 and there are 12 cats. Three boxes make 12, so one box is 4, and there are 5 × 4 = 20 dogs.
• You know the total. Share 56 stickers in the ratio 3:4. There are 7 boxes, each worth 56 ÷ 7 = 8, so the shares are 24 and 32.
• You know the difference. Paint is mixed blue to yellow in the ratio 2:5, and there are 9 more liters of yellow than blue. The difference is 3 boxes, so one box is 3 liters, giving 6 liters of blue and 15 of yellow.

Solving proportions

A proportion says two ratios are equal, such as x/12 = 3/4. Look for the scale factor first: 4 × 3 = 12, so x = 3 × 3 = 9. When the numbers don't scale neatly, cross-multiply. For 5/8 = x/20, 8x = 5 × 20 = 100, so x = 12.5.

Many real proportion problems go through one unit. If a recipe for 4 people uses 200 g of flour, one person needs 200 ÷ 4 = 50 g, so 6 people need 6 × 50 = 300 g. In the US this is called finding the unit rate; in UK classrooms it is often called the unitary method.

How it is taught in school

In the US, Common Core introduces ratio language, unit rates and tables of equivalent ratios in grade 6 (6.RP.A.1 to 6.RP.A.3). In grade 7, students decide whether two quantities are in a proportional relationship, find the constant of proportionality and solve multistep ratio problems (7.RP.A.2 and 7.RP.A.3).

In England, Year 6 pupils solve problems about the relative sizes of two quantities and about unequal sharing and grouping. At Key Stage 3 the curriculum adds ratio notation, simplifying ratios, dividing a quantity in a given ratio, and direct and inverse proportion.

How parents can help

• Cook together. Doubling or halving a recipe is ratio work, and scaling a recipe for 6 people instead of 4 is a proportion.
• Ask comparison questions at the table: if there are 3 forks for every 2 plates, how many forks are needed for 8 plates?
• Encourage a quick bar model before any calculation. A sketch with equal boxes makes most ratio word problems clear.
• Ask your child to check that a pair of answers adds up to the total and simplifies back to the ratio.
• Keep sessions short. Print a fresh worksheet, let your child mark it with the answer key, and read the worked steps together for any they missed.