Boat Speed Upstream Downstream Time Calculation

This solution details the steps to calculate the total time a boat takes to travel both upstream and downstream, based on its speed in still water and the speed of the river current.

Understanding Boat and Stream Problem Parameters

Let's first identify the key values provided in the question:

• Speed of the boat in still water ($V_b$) = 15 km/h.
• Speed of the current ($V_c$) = $\frac{1}{3}$ of the boat's speed in still water.
• Distance to cover upstream ($D_{up}$) = 24 km.
• Distance to cover downstream ($D_{down}$) = 20 km.
• The objective is to find the total time required for the entire trip (upstream + downstream).

Calculating Effective Speeds for Travel

Before calculating time, we need to determine the boat's speed relative to the ground during upstream and downstream journeys.

1. Determining the Speed of the Current

The question states the current's speed is one-third of the boat's speed in still water. We calculate this as:

$V_c = \frac{1}{3} \times V_b$

Plugging in the boat's speed:

$V_c = \frac{1}{3} \times 15 \text{ km/h}$

$V_c = 5 \text{ km/h}$

So, the speed of the water current is 5 km/h.

2. Calculating Downstream Speed

When travelling downstream, the current helps the boat, so their speeds add up. The effective downstream speed ($V_{down}$) is:

$V_{down} = V_b + V_c$

Using the values we have:

$V_{down} = 15 \text{ km/h} + 5 \text{ km/h}$

$V_{down} = 20 \text{ km/h}$

The boat travels at 20 km/h downstream.

3. Calculating Upstream Speed

When travelling upstream, the current opposes the boat's motion. The effective upstream speed ($V_{up}$) is calculated by subtracting the current's speed from the boat's speed:

$V_{up} = V_b - V_c$

Substituting the known speeds:

$V_{up} = 15 \text{ km/h} - 5 \text{ km/h}$

$V_{up} = 10 \text{ km/h}$

The boat travels at 10 km/h upstream.

Calculating Time for Each Journey Segment

We use the fundamental relationship: Time = Distance / Speed.

1. Time for the Downstream Journey

The distance downstream is 20 km, and the speed is 20 km/h.

Time downstream ($T_{down}$) = $\frac{D_{down}}{V_{down}}$

$T_{down} = \frac{20 \text{ km}}{20 \text{ km/h}}$

$T_{down} = 1 \text{ hour}$

2. Time for the Upstream Journey

The distance upstream is 24 km, and the speed is 10 km/h.

Time upstream ($T_{up}$) = $\frac{D_{up}}{V_{up}}$

$T_{up} = \frac{24 \text{ km}}{10 \text{ km/h}}$

$T_{up} = 2.4 \text{ hours}$

Calculating Total Travel Time

The total time is the sum of the time spent travelling upstream and downstream.

Total Time ($T_{total}$) = $T_{down} + T_{up}$

$T_{total} = 1 \text{ hour} + 2.4 \text{ hours}$

$T_{total} = 3.4 \text{ hours}$

Converting Decimal Hours to Hours and Minutes

The total time calculated is 3.4 hours. To convert this into a more common format:

• The whole number part '3' represents 3 full hours.
• The decimal part '0.4' needs to be converted into minutes.
• Minutes = $0.4 \times 60$ (since there are 60 minutes in an hour).
• Minutes = 24 minutes.

Therefore, the total time taken for the boat to complete the upstream and downstream journeys is 3 hours and 24 minutes.