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Mean Proportional — Definition, Formula & Examples

The mean proportional between two numbers aa and bb is the value xx such that aa, xx, and bb form a proportion where ax=xb\frac{a}{x} = \frac{x}{b}. It equals the square root of the product of the two numbers: x=abx = \sqrt{ab}.

Given two positive real numbers aa and bb, the mean proportional is the positive number xx satisfying the continued proportion a:x=x:ba : x = x : b, which is equivalent to x2=abx^2 = ab, so x=abx = \sqrt{ab}. This value is identical to the geometric mean of aa and bb.

Key Formula

x=abx = \sqrt{ab}
Where:
  • xx = The mean proportional between a and b
  • aa = The first number
  • bb = The second number

How It Works

To find the mean proportional between two numbers, multiply them together and take the square root. The idea comes from setting up a proportion with three terms: the unknown sits in the middle of both ratios. Cross-multiplying ax=xb\frac{a}{x} = \frac{x}{b} gives x2=abx^2 = ab, and solving yields x=abx = \sqrt{ab}. In geometry, this appears when an altitude is drawn to the hypotenuse of a right triangle — the altitude is the mean proportional between the two segments of the hypotenuse.

Worked Example

Problem: Find the mean proportional between 4 and 25.
Set up the proportion: You need xx such that 4x=x25\frac{4}{x} = \frac{x}{25}.
4x=x25\frac{4}{x} = \frac{x}{25}
Cross-multiply: Multiply both sides to get x2=4×25x^2 = 4 \times 25.
x2=100x^2 = 100
Solve: Take the positive square root.
x=100=10x = \sqrt{100} = 10
Answer: The mean proportional between 4 and 25 is 10.

Why It Matters

The mean proportional appears frequently in right-triangle geometry, where the altitude to the hypotenuse creates segments whose mean proportional equals the altitude length. It also shows up in similar-triangle proofs and in deriving the geometric mean used in finance and statistics.

Common Mistakes

Mistake: Using the arithmetic mean (average) instead of the geometric mean.
Correction: The mean proportional is ab\sqrt{ab}, not a+b2\frac{a+b}{2}. For example, between 4 and 25 the mean proportional is 10, while the arithmetic mean is 14.5.