Two boats P & Q are rowing in two different rivers X & Y respectively. Find the distance covered by boat Q in 15 hours downstream.

(i) Sum of speed of stream in X and Y is 12 km/hr.

(ii) Distance covered by boat Q in *Z* hours downstream is 120 km more than the distance covered by boat Q in *(Z + 2)* hours upstream.

(iii) Boat P can cover 420 km downstream in 8 hours while boat Q can cover 120 km upstream in six hours.

Which of the following statements are sufficient to find the distance covered by boat Q in 15 hours downstream?

Question

Two boats P & Q are rowing in two different rivers X & Y respectively. Find the distance covered by boat Q in 15 hours downstream.

(i) Sum of speed of stream in X and Y is 12 km/hr.

(ii) Distance covered by boat Q in *Z* hours downstream is 120 km more than the distance covered by boat Q in *(Z + 2)* hours upstream.

(iii) Boat P can cover 420 km downstream in 8 hours while boat Q can cover 120 km upstream in six hours.

Which of the following statements are sufficient to find the distance covered by boat Q in 15 hours downstream?

Accepted Answer

**Solution:**

1. Let the speed of boat P in still water be *p* km/hr and boat Q in still water be *q* km/hr.

2. Let the speed of the stream in river X be *X* km/hr and in river Y be *Y* km/hr.

3. From statement (i):
   $$X + Y = 12 \quad 	ext{...(a)}$$

4. From statement (ii):
   Distance covered by boat Q downstream in *Z* hours is 120 km more than distance covered upstream in *(Z + 2)* hours.

Downstream speed of Q = $q + Y$

Upstream speed of Q = $q - Y$

So,
   $$Z 	imes (q + Y) - (Z + 2) 	imes (q - Y) = 120 \quad 	ext{...(b)}$$

5. From statement (iii):
   Boat P covers 420 km downstream in 8 hours:
   $$\frac{420}{8} = p + X = 52.5 \quad 	ext{...(c)}$$

Boat Q covers 120 km upstream in 6 hours:
   $$\frac{120}{6} = q - Y = 20 \quad 	ext{...(d)}$$

6. We have 5 variables: *p, q, X, Y, Z* and 4 equations (a), (b), (c), (d).

7. The question asks for distance covered by boat Q in 15 hours downstream:
   $$	ext{Distance} = 15 	imes (q + Y)$$

8. From (d), we know $q - Y = 20$.

9. From (a), $X + Y = 12$ but this does not directly help find $q + Y$.

10. From (b), the equation involves *Z*, *q*, and *Y* but without a value for *Z*, we cannot solve for $q + Y$.

11. From (c), we know $p + X = 52.5$, but this does not help find $q + Y$.

12. Since there are 5 variables and only 4 equations, the system is underdetermined.

**Conclusion:**

- None of the combinations of the given statements provide enough information to find the distance covered by boat Q in 15 hours downstream.

**Answer:** (e) None

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