15. On an island, kangaroos are always either grey or red. One day, the number of grey kangaroos increased by $28 \%$ while the number of red kangaroos decreased by $28 \%$. The ratios of the two types of kangaroos were exactly reversed. By what percentage did the total number of kangaroos change?
SOLUTION
004

Let the initial populations of grey and red kangaroos be $G$ and $R$ respectively. After the change, the new populations are $1.28 G$ and $0.72 R$ respectively. As the ratios are reversed, we have $\frac{1.28 G}{0.72 R}=\frac{R}{G}$.
Therefore $R^{2}=\frac{128}{72} \times G^{2}=\frac{16 G^{2}}{9}$ and hence $R=\frac{4 G}{3}$.
The ratio of new population:old population is
$$
\begin{array}{l}
(1.28 G+0.72 R):(G+R)=\left(1.28+0.72 \frac{R}{G}\right):\left(1+\frac{R}{G}\right)=\left(1.28+0.72 \times \frac{4}{3}:\left(1+\frac{4}{3}\right)\right. \\
=(3.84+2.88):(4+3)=6.72: 7=0.96: 1
\end{array}
$$

Therefore there is a reduction of $4 \%$ in the population.