Time taken by the boat in river A to cover $(D + 120)$ km in downstream is 25% more than the time taken by the same boat to cover $(D - 80)$ km in upstream in river B. The speed of the stream in river B is 20% less than the speed of stream in river A. Ratio of the speed of the boat to the speed of the stream in river A is $9 : 5$. Find the value of $(D + 40)$ if the speed of the boat is constant.

### Concept & Boat and Stream Ratios The core formula for time in boat and stream problems is: $$ \text{Time} = \frac{\text{Distance}}{\text{Speed}} $$ Downstream Speed = $V_{\text{boat}} + V_{\text{stream}}$ Upstream Speed = $V_{\text{boat}} - V_{\text{stream}}$ ### Step-by-Step Solution * **Given Ratios & Speeds:** Let the speed of the boat be $V$ and speed of stream A be $S_A$. Ratio $V : S_A = 9 : 5$. Let $V = 9x$ and $S_A = 5x$. Speed of stream B ($S_B$) is 20% less than $S_A$. $$ S_B = S_A - 0.20(S_A) = 0.80 \times 5x = 4x $$ * **Formulating Downstream/Upstream Speeds:** Downstream speed in River A = $V + S_A = 9x + 5x = 14x$. Upstream speed in River B = $V - S_B = 9x - 4x = 5x$. * **Setting up the Time Equation:** Time for $(D+120)$ downstream A = $1.25 \times$ Time for $(D-80)$ upstream B. $$ \frac{D + 120}{14x} = 1.25 \times \frac{D - 80}{5x} $$ * **Solving for D:** Cancel $x$ from denominators and use $1.25 = \frac{5}{4}$: $$ \frac{D + 120}{14} = \frac{5}{4} \times \frac{D - 80}{5} $$ $$ \frac{D + 120}{14} = \frac{D - 80}{4} $$ Cross-multiply: $$ 4(D + 120) = 14(D - 80) $$ $$ 4D + 480 = 14D - 1120 $$ $$ 10D = 1600 \implies D = 160 $$ * **Finding Final Value:** We need to find $(D + 40)$. $160 + 40 = 200$. ### Exam Strategy & Shortcut Whenever dealing with proportional speeds (percentages or ratios), assign a variable like $x$ immediately. Notice how $x$ naturally cancels out when creating a time equivalence equation, meaning you don't even need the absolute speeds to find the distance $D$. ### Common Pitfall Forgetting that the question asks for $(D + 40)$ rather than just $D$. Always double-read the final query line before selecting an option! ### Final Answer Therefore, the correct answer is **200**.