1. (7p) Determine the number $\overline{a b}$ for which $\frac{\overline{a,(b)}+\overline{b,(a)}}{a+b}=\frac{a+b}{3 a}$.

GM12/2015
1. $\overline{a,(b)}+\overline{b,(a)}=a+\frac{a}{9}+b+\frac{b}{9}=\frac{10(a+b)}{9}$ ..... (2p)
$\frac{10}{9}=\frac{a+b}{3a}$ ..... (1p)
$7a=3b$ ..... (2p)
$a$ and $b$ are digits, so $a=3$ and $b=7$, hence $\overline{ab}=37$ ..... (2p)