A boat covers 36 km in downstream in 4 hrs, if the speed of the current is `(1)/(3)rd` of its downstream speed, then in what time will it cover a distance of 78 km upstream?

Options

A. 30 hrs
B. 26 hrs
C. 28 hrs
D. 24 hrs

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Determine the downstream speed of the boat.
The boat covers 36 km downstream in 4 hours. We can calculate the downstream speed (x + y) using the formula:

\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]

Substituting the given values:

\[ \text{Downstream Speed} = \frac{36 \text{ km}}{4 \text{ hours}} = 9 \text{ km/h} \]

**Hint:** Remember that speed is calculated as distance divided by time.

### Step 2: Relate the speed of the current to the downstream speed.
According to the problem, the speed of the current (y) is \( \frac{1}{3} \) of the downstream speed. Therefore, we can write:

\[ y = \frac{1}{3}(x + y) \]

Substituting the downstream speed we found:

\[ y = \frac{1}{3}(9) = 3 \text{ km/h} \]

**Hint:** When relating the current speed to the downstream speed, express it as a fraction of the total downstream speed.

### Step 3: Find the speed of the boat in still water.
We know that the downstream speed is the sum of the speed of the boat in still water (x) and the speed of the current (y):

\[ x + y = 9 \]

Substituting the value of y:

\[ x + 3 = 9 \]

Solving for x:

\[ x = 9 - 3 = 6 \text{ km/h} \]

**Hint:** The speed of the boat in still water is the downstream speed minus the speed of the current.

### Step 4: Calculate the upstream speed.
The upstream speed is calculated as the speed of the boat in still water minus the speed of the current:

\[ \text{Upstream Speed} = x - y = 6 - 3 = 3 \text{ km/h} \]

**Hint:** For upstream speed, subtract the current speed from the boat's speed in still water.

### Step 5: Calculate the time taken to cover 78 km upstream.
Using the formula for time:

\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]

Substituting the values for distance and upstream speed:

\[ \text{Time} = \frac{78 \text{ km}}{3 \text{ km/h}} = 26 \text{ hours} \]

**Hint:** Time can be found by dividing the distance by the speed, ensuring both are in compatible units.

### Final Answer:
The time taken to cover a distance of 78 km upstream is **26 hours**.