The ratio of speed of a boat in still water to the speed of boat in upstream 4 : 3. The boat covers 560 km in 7 hours in downstream. Find the time taken by boat to cover 480 km distance in upstream?

The correct answer is 10 hours.

Step 1: Understand the given ratios and values

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

The speed of the boat upstream is given by:

$v_{up}=u-v$

The ratio of the speed of the boat in still water ($u$) to its upstream speed ($v_{up}$) is given as 4 : 3.

Let $u=4x$ and $v_{up}=3x$.

Step 2: Find the speed of the stream and downstream speed in terms of $x$

Since $v_{up}=u-v$, we have:

$3x=4x-v$

$\Rightarrow v=4x-3x=x$

Now, the downstream speed ($v_{\mathrm{d}\mathrm{o}\mathrm{w}\mathrm{n}}$) is the sum of the speed in still water and the stream speed:

$v_{\mathrm{d}\mathrm{o}\mathrm{w}\mathrm{n}}=u+v=4x+x=5x$

Step 3: Calculate the actual downstream speed

The boat covers 560 km in 7 hours downstream.

$v_{\mathrm{d}\mathrm{o}\mathrm{w}\mathrm{n}}=\frac{560\text{km}}{7\text{hours}}=80\text{km/h}$

Step 4: Determine the value of $x$ and the upstream speed

Equating the downstream speed expressions:

$5x=80$

$\Rightarrow x=16$

Therefore, the upstream speed of the boat is:

$v_{up}=3x=3\times 16=48\text{km/h}$

Step 5: Calculate the time taken to cover 480 km upstream

$\text{Time}=\frac{\text{Distance}}{\text{Upstream Speed}}$

$\text{Time}=\frac{480}{48}=10\text{hours}$

Thus, the time taken by the boat to cover 480 km upstream is 10 hours.