The speed of the boat while travelling upstream is 24 km/hr. The boat can travel $X$ km in $Y$ hours while travelling upstream and 144 km in $Y$ hours while travelling downstream. The time taken (in hours) by the boat to travel 216 km upstream is $Z$ hours more than while travelling the same distance downstream.

Which of the statement/s is/are sufficient/necessary to find the ratio between speed of the boat in still water to speed of the stream?

(A) $X - Y^3 + Z^3 102$

(B) Time taken by boat to cover $Z$ km downstream is 18 hours.

Question

The speed of the boat while travelling upstream is 24 km/hr. The boat can travel $X$ km in $Y$ hours while travelling upstream and 144 km in $Y$ hours while travelling downstream. The time taken (in hours) by the boat to travel 216 km upstream is $Z$ hours more than while travelling the same distance downstream.

Which of the statement/s is/are sufficient/necessary to find the ratio between speed of the boat in still water to speed of the stream?

(A) $X - Y^3 + Z^3 > 102$

(B) Time taken by boat to cover $Z$ km downstream is 18 hours.

Accepted Answer

**Solution:**

1. Let the speed of the boat in still water be $a$ km/hr and the speed of the stream be $b$ km/hr.

2. Given upstream speed = $a - b = 24$ km/hr. (Equation i)

3. The boat travels $X$ km in $Y$ hours upstream, so:
   $$ (a - b) 	imes Y = X $$ (Equation ii)

4. The boat travels 144 km in $Y$ hours downstream, so:
   $$ (a + b) 	imes Y = 144 $$ (Equation iii)

5. Time taken to travel 216 km upstream is $Z$ hours more than downstream:
   - Time upstream = $\frac{216}{a - b}$
   - Time downstream = $\frac{216}{a + b}$
   - Given:
     $$ \frac{216}{a - b} - \frac{216}{a + b} = Z $$ (Equation iv)

6. From (i), $a - b = 24$, so upstream speed is known.

7. Using (iii), we can find $Y = \frac{144}{a + b}$.

8. Using (ii), $X = (a - b) Y = 24 Y$.

9. The statements (A) and (B) are:
   - (A) $X - Y^3 + Z^3 > 102$ (an inequality involving $X$, $Y$, and $Z$)
   - (B) Time taken by boat to cover $Z$ km downstream is 18 hours, i.e.,
     $$ \frac{Z}{a + b} = 18 $$

10. However, these statements do not provide direct or sufficient information to find the ratio $\frac{a}{b}$.

11. Since $a - b$ is known but $a + b$ is unknown, and the given statements do not allow us to solve for $a + b$ or $b$ individually, the ratio $\frac{a}{b}$ cannot be determined.

**Answer:** (c) None

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