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Ratios and proportions

A ratio compares two quantities; a proportion says two ratios are equal. Nearly every practical use of proportion is the same move — three of the four numbers are known and the fourth is not — and children who can spot that shape can solve scaling, recipes, maps and unit conversions with one method.

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Finding a missing term

3 is to 4 as 9 is to what?

  1. Compare the two first terms: 3 becomes 9, which is three times as much.
  2. Whatever happens to one side of a ratio happens to the other.
  3. So 4 becomes three times 4.
  4. The answer is 12.

Worth knowing. Scaling is quicker than cross-multiplying when the multiplier is obvious, and it shows what a proportion actually means. Cross-multiplication is the method for when it is not.

Cross-multiplying

3 : 4 = 9 : ?

  1. In a true proportion the diagonal products are equal: 3 x ? = 4 x 9.
  2. 4 x 9 is 36.
  3. So 3 x ? is 36, which means ? is 36 divided by 3.
  4. The answer is 12.

Worth knowing. The two methods agree because they are the same fact. Cross-multiplication always works; scaling is faster when the numbers are friendly.

Where it usually goes wrong

  • Multiplying across instead of diagonally, giving 3 x 9 and 4 x ?.
  • Adding the difference instead of multiplying — turning 3:4 into 9:10 rather than 9:12.
  • Writing the second ratio the other way round, so the terms no longer correspond.

The Professor writes the two ratios one above the other before he does anything. The numbers that multiply together are the ones diagonally opposite.

Practise it

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Questions teachers and parents ask

What is the difference between a ratio and a fraction?

A fraction compares a part to the whole; a ratio compares two parts to each other. In a class of 3 girls to 4 boys the ratio is 3:4, but girls are 3/7 of the class. Reading a ratio as a fraction is where a great many wrong answers start.

Why does cross-multiplication work?

Because a proportion is two equal fractions, and multiplying both sides of an equation by both denominators clears them. 3/4 = 9/12 becomes 3 x 12 = 4 x 9, which is 36 = 36.

When are ratios taught?

Ratio language and reasoning are grade 6 (6.RP.A.1, 6.RP.A.3), and proportional relationships — recognising them, representing them, using them — are grade 7 (7.RP.A.2).

Does the order of a ratio matter?

Yes, always. 3:4 and 4:3 describe opposite situations, and mixing them up is the most common error in the topic. Naming what each term counts — "three girls to four boys" — keeps it straight.

Where this shows up

Standards

  • 6.RP · Ratios and proportional relationships
  • 7.RP · Ratios and proportional relationships

Formats