Problem (Number Theory):
The positive integers $a$ and $b$ have no common divisor larger than 1 . If the difference between $b$ and $a$ is 15 and $\frac{5}{9}<\frac{a}{b}<\frac{4}{7}$, what is the value of $\frac{a}{b}$ ?

Solution:
Since $\frac{a}{b}<\frac{4}{7}$ and $\frac{4}{7}<1$, then $\frac{a}{b}<1$.

Since $a$ and $b$ are positive integers, then $a<b$.

Since the difference between $a$ and $b$ is 15 and $a<b$, then $b=a+15$.

Therefore, we have $\frac{5}{9}<\frac{a}{a+15}<\frac{4}{7}$.

We multiply both sides of the left inequality by $9(a+15)$ (which is positive) to obtain $5(a+15)<9 a$ from which we get $5 a+75<9 a$ and so $4 a>75$.

From this, we see that $a>\frac{75}{4}=18.75$.

Since $a$ is an integer, then $a \geq 19$.

We multiply both sides of the right inequality by $7(a+15)$ (which is positive) to obtain $7 a<4(a+15)$ from which we get $7 a<4 a+60$ and so $3 a<60$.

From this, we see that $a<20$.

Since $a$ is an integer, then $a \leq 19$.

Since $a \geq 19$ and $a \leq 19$, then $a=19$, which means that $\frac{a}{b}=\frac{19}{34}$.

Final answer: $\frac{19}{34}$