A boat can row to a place 56 km away and come back in 22 hours. The time to row 21 km with the stream is same as the time to row 12 km against the stream. Find the speed of boat in still water?

Correct Answer: 5.5 kmph

Step-by-Step Explanation:

Step 1: Define the variables for speeds

Let the speed of the boat in still water be $u$ km/h and the speed of the stream be $v$ km/h.

The downstream speed (with the stream) is given by:

$\mathrm{Downstream\; Speed\; (D)}=u+v$

The upstream speed (against the stream) is given by:

$\mathrm{Upstream\; Speed\; (U)}=u-v$

Step 2: Use the relationship between downstream and upstream times

We are given that the time taken to row 21 km with the stream is the same as the time taken to row 12 km against the stream.

Since $\mathrm{Time}=\frac{\mathrm{Distance}}{\mathrm{Speed}}$, we can equate the two time expressions:

$\frac{21}{D}=\frac{12}{U}$

Cross-multiplying to find the ratio of downstream speed to upstream speed:

$\frac{D}{U}=\frac{21}{12}=\frac{7}{4}$

Thus, we get:

$D=\frac{7}{4}U$

Step 3: Set up the total time equation for the round trip

The boat rows a distance of 56 km to a destination and comes back (56 km upstream and 56 km downstream) in a total time of 22 hours.

$\frac{56}{D}+\frac{56}{U}=22$

Substitute $D=\frac{7}{4}U$ into the equation:

$\frac{56}{\frac{7}{4}U}+\frac{56}{U}=22$

Simplifying the first term:

$\frac{56\times 4}{7U}+\frac{56}{U}=22$

$\frac{32}{U}+\frac{56}{U}=22$

$\frac{88}{U}=22$

$U=\frac{88}{22}=4\text{km/h}$

Step 4: Calculate downstream speed and boat speed in still water

Now, calculate the downstream speed $D$:

$D=\frac{7}{4}\times 4=7\text{km/h}$

The speed of the boat in still water is the average of downstream and upstream speeds:

$u=\frac{D+U}{2}$

$u=\frac{7+4}{2}=\frac{11}{2}=5.5\text{km/h}$

Thus, the speed of the boat in still water is 5.5 kmph.