The present ages of $A$ and $B$ are in the ratio $15:8$. After $8$ years their ages will be in the ratio $17:10$. What will be the ratio of the ages of $A$ and $B$ after $10$ years from now?

Let the present ages of $A$ and $B$ be $15x$ and $8x$ respectively.After $8$ years, their ages will be:
$$\frac{15x+8}{8x+8}=\frac{17}{10}$$
Cross-multiplying:
$$10(15x+8)=17(8x+8)$$
$$150x+80=136x+136$$
$$150x-136x=136-80$$
$$14x=56$$
$$x=4$$
Present age of $A=15\times 4=60\text{ years}$ Present age of $B=8\times 4=32\text{ years}$ Their ages after $10$ years from now will be:
$$\text{Age of }A=60+10=70\text{ years}$$
$$\text{Age of }B=32+10=42\text{ years}$$
Ratio of their ages after $10$ years:
$$\text{Ratio}=\frac{70}{42}=\frac{5}{3}$$