Discussion
Last year the population of town was X and if it increases at the same rate, next year it will be Y. The present population of town is
Answer: B
The present population is \((XY)^{1/2}\), the geometric mean of X and Y.
Let the present population be P and the yearly growth factor be r. Then last year's population was \(X = P/r\) and next year's will be \(Y = Pr\). Multiplying:
\[ \begin{aligned} XY &= \frac{P}{r} \cdot Pr = P^{2} \\ \Rightarrow P &= \sqrt{XY} \end{aligned} \]
Growth at a constant rate is multiplicative, so the middle term of three is the geometric mean, not the arithmetic mean \((X+Y)/2\), which would apply to growth by a constant number each year. \(2XY/(X+Y)\) is the harmonic mean, and \(XY/2\) has the wrong units altogether.
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