Between 40% and 50%

Statement

In a choir, more than $40\%$ but less than $50\%$ of the singers are male. What is the smallest number of singers the choir can have?

Solution

We need the smallest $n$ for which there is an integer $k$ (the males) with $0.40 < \frac{k}{n} < 0.50.$ The mediant of $\dfrac{2}{5}$ and $\dfrac{1}{2}$ is $\dfrac{3}{7}$, giving $n=7$, $k=3$: $\frac{3}{7} \approx 0.4286 \in (0.40,\ 0.50).$ For every $n\le 6$ no integer $k$ satisfies the double inequality. Hence the minimum is $n = \boxed{7}.$