The difference between the time taken by boat to cover 96 km upstream and 140 km downstream is one hour, the respective ratio of speeds of boat downstream to upstream is 5:3. Then... The difference between the time taken by boat to cover 96 km upstream and 140 km downstream is one hour, the respective ratio of speeds of boat downstream to upstream is 5:3. Then find the speed of boat in still water (in km/h). Read more

Steps to Solve

1. Define speeds and expressions for upstream and downstream

Let the speed of the boat in still water be $x$ km/h. The speed of the boat downstream (with the current) is given as $5k$ km/h and upstream (against the current) as $3k$ km/h, where $k$ is a constant.

2. Set up equations for time taken upstream and downstream

Using the formula for time, ( \text{Time} = \frac{\text{Distance}}{\text{Speed}} ):

• Time taken upstream to cover 96 km: $$ T_{up} = \frac{96}{3k} $$
• Time taken downstream to cover 140 km: $$ T_{down} = \frac{140}{5k} $$

3. Set up the equation for the time difference

According to the problem, the difference in time is 1 hour: $$ \left( T_{down} - T_{up} \right) = 1 $$ Substituting the expressions, we get: $$ \frac{140}{5k} - \frac{96}{3k} = 1 $$

4. Simplify the equation and solve for k

Multiply the entire equation by ( 15k ) (the LCM of denominators) to eliminate the fraction: $$ 15k \left( \frac{140}{5k} - \frac{96}{3k} \right) = 15k \cdot 1 $$

This simplifies to: $$ 3 \times 140 - 5 \times 96 = 15k $$ $$ 420 - 480 = 15k $$ $$ -60 = 15k $$ Thus, $$ k = -4 $$ (since k must be positive, we take the absolute value)

5. Calculate the speed in still water (x)

Now we can find the values of downstream and upstream speeds:

• Downstream speed: $$ 5k = 5 \times 4 = 20 \text{ km/h} $$
• Upstream speed: $$ 3k = 3 \times 4 = 12 \text{ km/h} $$

To find the speed of the boat in still water, we average the downstream and upstream speeds: $$ x = \frac{(20 + 12)}{2} = 16 \text{ km/h} $$