1. **State the problem:** We have two points and two lines: - Point $(p,r)$ lies on the line $y = x + b$. - Point $(2p,5r)$ lies on the line $y = 2x + b$. We need to find the value of $\frac{r}{p}$ given $p \neq 0$. 2. **Write the equations from the points on the lines:** - Since $(p,r)$ lies on $y = x + b$, substitute $x = p$ and $y = r$: $$r = p + b$$ - Since $(2p,5r)$ lies on $y = 2x + b$, substitute $x = 2p$ and $y = 5r$: $$5r = 2(2p) + b = 4p + b$$ 3. **Express $b$ from the first equation:** $$b = r - p$$ 4. **Substitute $b$ into the second equation:** $$5r = 4p + (r - p)$$ Simplify the right side: $$5r = 4p + r - p = r + 3p$$ 5. **Isolate terms to solve for $r$ and $p$ ratio:** Subtract $r$ from both sides: $$5r - r = 3p$$ $$4r = 3p$$ 6. **Divide both sides by $p$ (since $p \neq 0$):** $$4 \frac{r}{p} = 3$$ 7. **Solve for $\frac{r}{p}$:** $$\frac{r}{p} = \frac{3}{4}$$ **Final answer:** $\boxed{\frac{3}{4}}$ which corresponds to option B.