Ratio & Proportion • Topic 5 of 5

Mean & Third Proportion

These are special proportions worth memorising as direct formulas. The mean proportional (or geometric mean) between a and b is the value x with a:x = x:b, so x² = ab and x = √(ab). It is the middle term of a continued proportion a:x:b. The third proportional to a and b is the value c with a:b = b:c, so b² = ac and c = b²/a — here b is the repeated mean term. Do not confuse these with the fourth proportional to a, b, c (the d in a:b = c:d, equal to bc/a). A reliable CAT habit: write the proportion in the a:b = c:d frame, mark which term repeats, and apply the matching formula. Mean proportional questions love perfect-square products (e.g. 4 and 9 give √36 = 6), so scan for numbers whose product is a clean square before reaching for a calculator.

✅ Solved examples

1. Find the mean proportional between 4 and 16.
x = √(4×16) = √64 = 8.
2. Find the third proportional to 6 and 12.
Third proportional = b²/a = 12²/6 = 144/6 = 24.
3. The mean proportional between two numbers is 12 and one number is 8. Find the other.
√(8×b) = 12 ⇒ 8b = 144 ⇒ b = 18.
4. Find the third proportional to 9 and 15.
b²/a = 15²/9 = 225/9 = 25.

✏️ Practice — try these, take hints as needed

1. Mean proportional between 9 and 25?
x = √(ab).
√(9×25).
√225.
15
2. Third proportional to 4 and 8?
c = b²/a.
8²/4.
64/4.
16
3. Mean proportional between 3 and 27?
√(3×27).
√81.
Perfect square.
9
4. Third proportional to 5 and 10?
b²/a.
10²/5.
100/5.
20
5. The mean proportional between two numbers is 10 and one number is 4. Find the other.
√(4×b) = 10.
4b = 100.
b = 100/4.
25

📝 Topic test — 8 questions

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