Proportion is the equality of two ratios; when two ratios are equal, they form a proportion.
## Core concept
A proportion states that two ratios are equal. If a : b = c : d, then a, b, c, d are said to be in proportion, written as a : b :: c : d (read as "a is to b as c is to d").
Key terminology: - Extremes: a and d (the first and last terms) - Means: b and c (the middle terms) - Third proportional: if a : b = b : c, then c is the third proportional to a and b - Mean proportional (Geometric Mean): if a : b = b : c, then b is the mean proportional to a and c
## Formula / rule
Fundamental property of proportion: - If a : b :: c : d, then ad = bc (product of extremes = product of means)
Finding the unknown term: - If a : b :: c : x, then x = (b × c) / a
Continued proportion: - If a : b :: b : c :: c : d, then a, b, c, d are in continued proportion - This means a/b = b/c = c/d = k (constant ratio)
Mean proportional: - If b is the mean proportional to a and c, then b² = ac, so b = √(ac)
## Common exam applications
Type 1: Finding the missing term
If 4 : 6 :: 8 : x, find x. - Using ad = bc: 4x = 6 × 8 - 4x = 48 → x = 12 - Verify: 4/6 = 8/12 = 2/3 ✓
Type 2: Mean proportional
Find the mean proportional to 9 and 16. - Let b be the mean proportional - b² = 9 × 16 = 144 - b = 12
Type 3: Third proportional
Find the third proportional to 5 and 10. - 5 : 10 = 10 : x - 5x = 100 → x = 20
Type 4: Proportion in word problems
If 12 workers can complete a job in 20 days, how many days will 15 workers take? - Workers and days are inversely proportional - 12 : 15 = x : 20 (inverse ratio) - 12 × 20 = 15 × x - x = 240/15 = 16 days
## Common mistakes
- Error 1: Confusing extremes and means. Remember: extremes are the first and fourth terms; means are the second and third.
- Error 2: Not recognising inverse proportion in word problems. Not all proportions are direct.
- Error 3: Writing proportion incorrectly. The sequence matters: a : b :: c : d means a/b = c/d, not a/c = b/d.
- Error 4: Forgetting to simplify ratios before setting up the proportion equation. Always reduce to simplest form first.
- Error 5: In continued proportion, assuming a : b = c : d directly. Verify each pair separately.
## Quick revision checklist
✓ Recognise proportion notation (a : b :: c : d) ✓ Apply ad = bc instantly ✓ Distinguish between mean proportional and third proportional ✓ Identify direct vs. inverse proportion from context ✓ Solve for unknown in any position