All skills Math

Learn Index · Ratios, Rates, and Proportions Grade 6–8

Part-to-part and part-to-whole relationships

Part-to-part compares two parts; part-to-whole compares one part with the total.

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Two kinds of comparison

A group of 20 students has 8 who play soccer and 12 who play hockey.

Ratio type Compares Example
Part-to-part ratio one part to another soccer : hockey = 8:12 = 2:3
Part-to-whole ratio one part to the whole soccer : all = 8:20 = 2:5

Linking the two

Add the terms of a part-to-part ratio to get the total parts.

  • Part-to-part 2:3 → total parts = 2 + 3 = 5
  • So the part-to-whole ratios are 2:5 and 3:5
  • As fractions: 2/5 play soccer, 3/5 play hockey.

A common error is reading 2:3 as "2/3 play soccer". It’s 2/5!

Sharing in a ratio

To share an amount in a given ratio:

  1. Add the parts to find the total parts.
  2. Divide the amount by the total to find one part.
  3. Multiply by each term.

Example: Share $60 between Ana and Ben in the ratio 2:3.

  • Total parts = 5
  • One part = 60 ÷ 5 = $12
  • Ana: 2 × 12 = $24; Ben: 3 × 12 = $36
  • Check: 24 + 36 = 60 ✓

Finding the whole from a part

Example: The ratio of cats to dogs at a shelter is 3:4. There are 15 cats. How many animals in total?

  • 3 parts = 15, so 1 part = 5
  • Total parts = 7 → 7 × 5 = 35 animals

Bar models

Draw the ratio as bars of equal-sized boxes. For 2:3:

Ana ■ ■
Ben ■ ■ ■

Each box is one part. If the total is $60, each of the 5 boxes is $12.

Percent from part-to-whole

Part-to-whole ratios convert to percents: 2:5 = 2/5 = 40%.

Worked examples

Example 1: A smoothie has 1 part yogurt to 4 parts fruit. What fraction is yogurt?

Total 5 parts → 1/5.

Example 2: Share 45 marbles in the ratio 4:5.

One part = 45 ÷ 9 = 5 → 20 and 25.

Example 3: Boys to girls is 5:7. There are 24 students. How many girls?

12 parts; 24 ÷ 12 = 2 → girls 7 × 2 = 14.

Common mistakes

  • Treating a part-to-part ratio as a fraction of the whole.
  • Dividing by one term instead of the total parts when sharing.
  • Forgetting to check that the shares add to the total.

Try it yourself

  1. Red to blue is 3:5. What fraction is red?
  2. Share $80 in the ratio 3:1.
  3. Girls to boys is 4:5. There are 20 girls. How many students?
  4. Write the part-to-whole ratio as a percent for 1:3 (part-to-part).
  5. A drink is concentrate:water = 2:7. How much concentrate in 1,800 mL?

Answers: 1) 3/8 2) $60 and $20 3) 45 4) 1/4 = 25% 5) 400 mL

Quick recap

  • Part-to-part ratios compare parts; part-to-whole ratios compare a part to the whole.
  • Add the terms to get total parts.
  • To share, divide by the total parts, then multiply.