Proportion and continued proportion

Learning outcome

Recognise a proportion, find a missing positive term, and explain how continued proportion connects three ordered quantities through the square of the middle term.

Understanding proportion

A proportion states that two ratios are equal. For positive a, b, c and d, writing a:b = c:d means a/b = c/d. Individual quantities may differ; their relative sizes must match.

Multiplying both fractions by b × d gives a × d = b × c. This is the cross-product test. The outside terms a and d are called extremes, while b and c are called means. The product of the extremes equals the product of the means.

This equation does not permit multiplying whichever numbers look convenient. Keep each term in its original position. Write the equality, then isolate the unknown by dividing by its positive coefficient. Substitute the answer into the original ratios to check it.

For measurements of the same kind, match units before testing equality. A numerical cross-product cannot repair a ratio formed using inconsistent units.

Understanding continued proportion

Three positive numbers a, b and c are in continued proportion, in that order, when a:b = b:c. The same middle number appears in both ratios. Cross-multiplication therefore gives b × b = a × c, or b² = a × c.

To find b, seek the positive number whose square equals a × c. This is the geometric mean of the endpoints, not their ordinary arithmetic average. Equivalently, b/a = c/b: moving between successive terms uses the same multiplication factor, not necessarily the same addition.

Worked example 1

Do 1.2 m:80 cm and 9:6 form a proportion?

Convert 1.2 m to 120 cm. Now 120:80 simplifies to 3:2, and 9:6 also simplifies to 3:2. Therefore the ratios form a proportion. Their relative comparisons match despite different original terms.

Worked example 2

Find positive x in 8:14 = x:35.

Cross-multiplication gives 8 × 35 = 14 × x, so 280 = 14 × x. Divide by 14: x = 20. Checking, 8:14 and 20:35 both simplify to 4:7.

Worked example 3

Find positive b when 4, b and 25 are in continued proportion.

Here b² = 4 × 25 = 100, so b = 10. Check: 4:10 and 10:25 both simplify to 2:5. Each successive term is multiplied by 2.5, confirming the relationship.

Common mistakes

Do not treat any three numbers as a continued proportion. Order matters, and the middle term must satisfy the squared relationship. Do not use the arithmetic average automatically or forget to convert units.

Practice questions

1. Do 18:24 and 27:36 form a proportion? Justify.
2. Find positive x if 7:12 = 21:x.
3. Find positive b if 9:b = b:49.
4. The numbers 6, 15 and positive c are in continued proportion, in that order. Find c.

Worked answers

1. Yes. Dividing 18:24 by 6 gives 3:4; dividing 27:36 by 9 also gives 3:4.
2. Cross-multiply: 7 × x = 12 × 21 = 252. Thus x = 36, and 21:36 reduces to 7:12.
3. The repeated middle term gives b² = 9 × 49 = 441. Therefore b = 21; both ratios reduce to 3:7.
4. Since 15² = 6 × c, c = 225/6 = 37.5. Checking, 6/15 = 0.4 and 15/37.5 = 0.4 exactly.