Ratio, proportion and rates of change: understand and use proportion as equality of ratios

The idea

Two ratios are in proportion when they reduce to the same simplified ratio, so 3 : 4 and 9 : 12 are in proportion because both reduce to 3 : 4. That lets you find a missing value: if 3 : 4 and 9 : x are in proportion, the first part has been multiplied by 3, so x = 12. Recipe, best-value and scaling questions all run on this. If 4 pens cost £3, then one pen costs £0.75 and 10 pens cost £7.50; you could also scale straight from 4 to 10 by multiplying by 2.5. Writing the two ratios as equal fractions gives you a check, since 4/3 and 10/7.5 both come to the same value. This kind of proportion is direct, so doubling one quantity doubles the other.

What earns the marks

• Setting up two ratios that are equal, or an equation between two fractions, from the figures given.
• The scale factor between the ratios, even where it is afterwards applied to the wrong part.
• A unitary step such as the cost of one item, where that is the route taken.
• The missing value, to the accuracy the question demands.

Common mistakes and how to put them right

• Adding the same number to both parts instead of multiplying, so 3.

  Put it right: 4 is scaled to 6 : 7.

• Pairing the parts the wrong way round when the two ratios are written in different orders.

• Rounding a unit cost to the nearest penny early, so the scaled-up total is out by several pence.

• Scaling a quantity that does not scale, such as a fixed delivery charge added to a per-item price.

Exam tips

• Write the two ratios one above the other with matching parts lined up, then look across for the multiplier.
• Keep a unit value as an exact fraction or a full decimal until the last line of working.
• Check the finished pair by simplifying both ratios; they must reduce to the same thing.