Mean Proportional appearing in the central positions
The mean proportional of two numbers or quantities A and B is a value M that satisfies the proportion $$ A:M = M:B $$ The remaining terms are called the first term (A) and the third term (B).
Equivalently, in a proportion, a term is called a geometric mean if it appears twice in the middle positions, that is, as the mean terms of the proportion.
$$ A:\color{red}M = \color{red}M:B $$
This type of proportion is known as a continued proportion.
A practical example
In the following proportion, the geometric mean is 9.
$$ 27:9 = 9:3 $$
This is because the number 9 appears in both mean positions.
$$ 27:\color{red}9 = \color{red}9:3 $$
How to compute the geometric mean
In a proportion of the form $$ A:M = M:B $$ the geometric mean is given by the square root of the product of the extreme terms $$ M = \sqrt{AB} $$
This formula provides a quick and effective way to determine the geometric mean.
Alternatively, the same result can be obtained by following the algebraic steps in the proof.
Proof
Consider the proportion
$$ A:M = M:B $$
By the fundamental property of proportions, apply cross multiplication.
$$ A \cdot B = M \cdot M $$
$$ AB = M^2 $$
Using the invariant property of equations, take the square root of both sides.
$$ \sqrt{AB} = \sqrt{M^2} $$
Then simplify the radical expression using radical simplification, √M2=M.
$$ \sqrt{AB} = \sqrt[\not{2}]{M^{\not{2}}} $$
$$ \sqrt{AB} = M $$
Hence, the geometric mean is equal to the square root of the product of the extreme terms.
$$ M = \sqrt{AB} $$
This completes the proof.
Example 1. Consider the proportion $$ 2:x = x:32 $$ The geometric mean is the square root of the product of the extreme terms $$ x = \sqrt{2 \cdot 32} = \sqrt{64} = \pm 8 $$ Since we are dealing with positive quantities, the geometric mean is 8. $$ 2:8 = 8:32 $$ To verify, compute the cross product $$ 2 \cdot 32 = 8 \cdot 8 $$ $$ 64 = 64 $$ The identity holds, confirming that 2:8=8:32 is a valid proportion.
Example 2. Now compute the geometric mean by carrying out each step explicitly $$ 2:x = x:32 $$ By the fundamental property of proportions, compute the cross product of the terms $$ 2 \cdot 32 = x \cdot x $$ $$ 64 = x^2 $$ Using the invariant property of equations, take the square root of both sides $$ \sqrt{64} = \sqrt{x^2} $$ $$ x = \pm 8 $$ The geometric mean is 8. $$ 2:8 = 8:32 $$ This matches the result obtained using the direct formula.
Useful remarks
The following observations are often useful.
- Finding an extreme term
The formula for the geometric mean can also be used to determine the extreme term of a proportion of the form $$ M:A = B:M $$ First, swap the antecedent and consequent using the inversion property to obtain $$ A:M = M:B $$ then apply the formula $$ M = \sqrt{AB} $$ This yields the extreme term of the original proportion M:A=B:M.
Example. Find the extreme term of the proportion $$ x:2 = 32:x $$ First invert the proportion $$ 2:x = x:32 $$ Then compute the geometric mean $$ x = \sqrt{2 \cdot 32} = \sqrt{64} = \pm 8 $$ Since we consider positive quantities, the geometric mean is 8 $$ 2:8 = 8:32 $$ Finally, invert the terms again to obtain the desired result. The extreme term is 8 $$ \color{red}8:2 = 32:\color{red}8 $$ Alternatively, recall that the extreme term is equal to the square root of the product of the mean terms of the proportion.
And so on.
