Boat A covers $z$ km upstream in 9 hours. Boat B's upstream speed is equal to Boat A's downstream speed, and Boat B covers $m$ km downstream in 1 hour. The speed of Boat B in still water is $(m - 4) ext{ km/hr}$, and the speed of Boat A in still water is $8 ext{ km/hr}$. Find the correct relation(s) between $z$ and $m$ from the following:

(i) $\frac{z}{m} = \frac{9}{5}$

(ii) $m + z$ is an even number

(iii) $z

Question

Boat A covers $z$ km upstream in 9 hours. Boat B's upstream speed is equal to Boat A's downstream speed, and Boat B covers $m$ km downstream in 1 hour. The speed of Boat B in still water is $(m - 4) 	ext{ km/hr}$, and the speed of Boat A in still water is $8 	ext{ km/hr}$. Find the correct relation(s) between $z$ and $m$ from the following:

(i) $\frac{z}{m} = \frac{9}{5}$

(ii) $m + z$ is an even number

(iii) $z < m$

Accepted Answer

**Solution:**

1. Let the speed of the stream be $s$ km/hr.

2. Given speed of Boat A in still water = $8$ km/hr.

3. Boat A covers $z$ km upstream in $9$ hours, so upstream speed of A:
   $$	ext{Upstream speed of A} = \frac{z}{9} 	ext{ km/hr}$$

4. Upstream speed of A is also equal to $(8 - s)$ km/hr, so:
   $$8 - s = \frac{z}{9} \implies s = 8 - \frac{z}{9}$$

5. Boat B's upstream speed equals Boat A's downstream speed.

6. Downstream speed of A = $(8 + s)$ km/hr.

7. Speed of B in still water = $(m - 4)$ km/hr.

8. Boat B covers $m$ km downstream in 1 hour, so downstream speed of B = $m$ km/hr.

9. Downstream speed of B = speed of B in still water + speed of stream:
   $$m = (m - 4) + s \implies s = 4$$

10. From step 4, $s = 8 - \frac{z}{9} = 4$:
    $$8 - \frac{z}{9} = 4 \implies \frac{z}{9} = 4 \implies z = 36$$

11. Since $s = 4$, speed of B in still water = $m - 4$, so:
    - Upstream speed of B = $(m - 4) - 4 = m - 8$
    - Given upstream speed of B = downstream speed of A = $8 + 4 = 12$

12. Equate upstream speed of B to 12:
    $$m - 8 = 12 \implies m = 20$$

13. Check the ratio $\frac{z}{m} = \frac{36}{20} = \frac{9}{5}$, so statement (i) is true.

14. Sum $m + z = 20 + 36 = 56$, which is an even number, so statement (ii) is true.

15. Check if $z < m$: $36 < 20$ is false, so statement (iii) is false.

**Answer:** Only statements (i) and (ii) are correct, so option (c) is the right choice.

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