Mean Proportion Formula — Definition, Examples and Steps

A mean proportion (also called the geometric mean) is the middle term x in the proportion:

a : x = x : b

This is a continued proportion where x appears as both the second term (consequent of the first ratio) and the first term (antecedent of the second ratio).

In fraction form: a/x = x/b

Cross-multiplying: x × x = a × b x² = ab x = √(ab)

So the mean proportion formula is: Mean Proportion = √(a × b)

Where a and b are the two given numbers (both must be positive for a real result).

Given: a : x = x : b

Step 1: Write as fractions: a/x = x/b

Step 2: Cross-multiply: a × b = x × x ab = x²

Step 3: Take the square root of both sides: x = √(ab)

This is why the mean proportion is also called the geometric mean — it is the geometric mean of the two extreme values a and b.

Note on signs: • If a and b are both positive, x is positive. • If a and b are both negative, ab is positive and x is real (but context rarely requires this). • If a and b have opposite signs, ab is negative and x is not a real number.

Example 1: Find the mean proportion between 4 and 16. Solution: Mean proportion x = √(4 × 16) = √64 = 8 Verification: 4:8 = 1:2 and 8:16 = 1:2 ✓ (4:8 = 8:16)

Example 2: Find the mean proportion between 9 and 25. x = √(9 × 25) = √225 = 15 Verification: 9:15 = 3:5 and 15:25 = 3:5 ✓

Example 3: Find the mean proportion between 2 and 8. x = √(2 × 8) = √16 = 4 Verification: 2:4 = 1:2 and 4:8 = 1:2 ✓

Example 4: Find the mean proportion between 3 and 27. x = √(3 × 27) = √81 = 9 Verification: 3:9 = 1:3 and 9:27 = 1:3 ✓

Example 5: Find the mean proportion between 5 and 20. x = √(5 × 20) = √100 = 10 Verification: 5:10 = 1:2 and 10:20 = 1:2 ✓

Mean proportion and third proportion are often confused. Here is the key difference:

Mean Proportion: • Given: a and b • Find x such that: a:x = x:b • Formula: x = √(ab) • x is the MIDDLE term of a continued proportion a, x, b

Third Proportion: • Given: a and b • Find c such that: a:b = b:c • Formula: c = b²/a • c is the THIRD term; the proportion is a, b, c

Comparison example (a = 4, b = 8): • Mean proportion: x = √(4 × 8) = √32 ≈ 5.66 • Third proportion: c = 8²/4 = 64/4 = 16 • Verify third: 4:8 = 8:16 = 1:2 ✓

Mnemonic: • Mean proportion → the MEAN (middle) → √(ab) • Third proportion → the THIRD number → b²/a

Mean proportion appears in many areas of mathematics and science:

1. Geometry: • The altitude drawn from the right angle of a right triangle to the hypotenuse is the mean proportion between the two segments it creates. • If altitude h divides hypotenuse into segments p and q: h = √(pq)

2. Similar triangles: • Corresponding sides of similar triangles form proportions; the mean proportional appears in the geometric mean theorem.

3. Finance: • Geometric mean return in compound interest calculations uses the same principle.

4. Pythagoras connection: • In a right triangle with legs a, b and hypotenuse c: b is the mean proportion between (c−a) and (c+a) only in special cases.

5. Golden ratio: • The golden ratio φ = 1.618... is related to a continued mean proportion in a specific rectangle.