# Problem 3. (3 points)

All residents of the island are either blondes or brunettes with green or blue eyes. The proportion of brunettes among blue-eyed residents is $65 \%$. The proportion of blue-eyed residents among blondes is $70 \%$. Finally, the proportion of blondes among green-eyed residents is 10\%. What percentage of the island's population are green-eyed brunettes?
Answer: 54

## Solution:

Let the number of blue-eyed brunettes be $a$, blue-eyed blondes be $b$, green-eyed blondes be $c$, and green-eyed brunettes be $d$.

Then $\frac{a}{a+b}=0.65$, from which $\frac{a+b}{a}=\frac{20}{13}$ and $\frac{b}{a}=\frac{20}{13}-1=\frac{7}{13}$.

Similarly, $\frac{b}{b+c}=0.7$, from which $\frac{b+c}{b}=\frac{10}{7}$ and $\frac{c}{b}=\frac{10}{7}-1=\frac{3}{7}$, and $\frac{c}{a}=\frac{c}{b} \cdot \frac{b}{a}=\frac{7}{13} \cdot \frac{3}{7}=\frac{3}{13}$.

Finally, $\frac{c}{c+d}=0.1$, from which $\frac{c+d}{c}=10$ and $\frac{d}{c}=10-1=9$, and $\frac{d}{a}=\frac{d}{c} \cdot \frac{c}{a}=9 \cdot \frac{3}{13}=\frac{27}{13}$.

Then $\frac{a+b+c+d}{a}=1+\frac{b}{a}+\frac{c}{a}+\frac{d}{a}=1+\frac{7}{13}+\frac{3}{13}+\frac{27}{13}=\frac{50}{13}$.

Accordingly, $\frac{d}{a+b+c+d}=\frac{d}{a} \cdot \frac{a}{a+b+c+d}=\frac{27}{13} \cdot \frac{13}{50}=\frac{27}{50}=54\%$.