#### Answer

$m=\frac{7}{3}$

#### Work Step by Step

We start with the given equation: $\frac{2}{3}+\frac{3m}{5}=\frac{31}{15}$
We multiply by the least common denominator: $15(\frac{2}{3}+\frac{3m}{5}=\frac{31}{15})$
$10+9m=31$
We subtract $10$ from each side of the equation: $10-10+9m=31-10$
We combine like terms: $9m=21$
Finally, we divide by $9$ on each side of the equation: $m=\frac{21}{9}$
This fraction can be reduced to $m=\frac{7}{3}$