A boat can cover 36 km downstream and 16 km upstream in 10 hours. It can cover 3 km downstream and 1.2 km upstream in 48 minutes. What is the speed (in km/hr of the boat when going upstream?

Options

A. 2
B. 2.5
C. 3.5
D. 4

Answer

To solve the problem step by step, we will first summarize the given information and then derive the required speeds.

### Step 1: Understand the Problem
We have two scenarios:
1. A boat covers 36 km downstream and 16 km upstream in a total of 10 hours.
2. The same boat covers 3 km downstream and 1.2 km upstream in 48 minutes.

We need to find the speed of the boat when going upstream.

### Step 2: Define Variables
Let:
- \( u \) = speed of the boat downstream (in km/hr)
- \( v \) = speed of the boat upstream (in km/hr)

### Step 3: Set Up the Equations
From the first scenario:
- Time taken to cover 36 km downstream = \( \frac{36}{u} \)
- Time taken to cover 16 km upstream = \( \frac{16}{v} \)

The total time for the first scenario is:
\[ \frac{36}{u} + \frac{16}{v} = 10 \quad \text{(Equation 1)} \]

From the second scenario:
- Time taken to cover 3 km downstream = \( \frac{3}{u} \)
- Time taken to cover 1.2 km upstream = \( \frac{1.2}{v} \)

The total time for the second scenario (in hours) is:
\[ \frac{3}{u} + \frac{1.2}{v} = \frac{48}{60} = 0.8 \quad \text{(Equation 2)} \]

### Step 4: Simplify the Equations
Now we will manipulate these equations to eliminate \( u \) and \( v \).

From Equation 1:
\[ 36 \cdot \frac{1}{u} + 16 \cdot \frac{1}{v} = 10 \]
Let \( x = \frac{1}{u} \) and \( y = \frac{1}{v} \):
\[ 36x + 16y = 10 \quad \text{(Equation 1')} \]

From Equation 2:
\[ 3 \cdot \frac{1}{u} + 1.2 \cdot \frac{1}{v} = 0.8 \]
This can be rewritten as:
\[ 3x + 1.2y = 0.8 \quad \text{(Equation 2')} \]

### Step 5: Eliminate One Variable
To eliminate \( y \), we can multiply Equation 2' by 10 to make calculations easier:
\[ 30x + 12y = 8 \quad \text{(Equation 2'')} \]

Now we have:
1. \( 36x + 16y = 10 \) (Equation 1')
2. \( 30x + 12y = 8 \) (Equation 2'')

### Step 6: Solve the System of Equations
We can multiply Equation 1' by 3 and Equation 2'' by 4 to align the coefficients of \( y \):
\[ 108x + 48y = 30 \quad \text{(Equation 1''')} \]
\[ 120x + 48y = 32 \quad \text{(Equation 2''')} \]

Now, subtract Equation 1''' from Equation 2''':
\[ (120x + 48y) - (108x + 48y) = 32 - 30 \]
\[ 12x = 2 \implies x = \frac{1}{6} \]

### Step 7: Find \( u \)
Since \( x = \frac{1}{u} \):
\[ u = 6 \text{ km/hr} \]

### Step 8: Substitute \( x \) to Find \( y \)
Now substitute \( x \) back into Equation 1':
\[ 36 \cdot \frac{1}{6} + 16y = 10 \]
\[ 6 + 16y = 10 \implies 16y = 4 \implies y = \frac{1}{4} \]

### Step 9: Find \( v \)
Since \( y = \frac{1}{v} \):
\[ v = 4 \text{ km/hr} \]

### Conclusion
The speed of the boat when going upstream is **4 km/hr**.