The time taken by a boat to cover a distance of D-56 km upstream is half of that taken by it to cover a distance of D, km downstream. The ratio of the speed of the boat downstream to that upstream is 5 :3. If the time taken to cover D-32 km upstream is 4 hours, what is the speed of water current? (in km/h)

Options

A. 5
B. 3
C. 4
D. 16

Answer

To solve the problem step by step, we will follow the given information systematically.

### Step 1: Understand the given information
We know:
- Distance covered upstream = D - 56 km
- Distance covered downstream = D km
- Time taken to cover upstream distance is half of that taken to cover downstream distance.
- The ratio of the speed of the boat downstream to upstream is 5:3.
- Time taken to cover D - 32 km upstream is 4 hours.

### Step 2: Define variables for speeds
Let:
- Speed of the boat downstream = 5x km/h
- Speed of the boat upstream = 3x km/h

### Step 3: Set up the time equations
From the problem, we know:
- Time taken to cover upstream distance (D - 56) = (D - 56) / (3x)
- Time taken to cover downstream distance D = D / (5x)

According to the problem, the time taken upstream is half of the time taken downstream:
\[ \frac{D - 56}{3x} = \frac{1}{2} \cdot \frac{D}{5x} \]

### Step 4: Simplify the equation
Cross-multiplying gives:
\[ 2(D - 56) = \frac{3D}{5} \]
Multiplying through by 5 to eliminate the fraction:
\[ 10(D - 56) = 3D \]
Expanding:
\[ 10D - 560 = 3D \]
Rearranging:
\[ 10D - 3D = 560 \]
\[ 7D = 560 \]
\[ D = \frac{560}{7} = 80 \text{ km} \]

### Step 5: Find the upstream distance
Now, we can find the upstream distance:
\[ D - 56 = 80 - 56 = 24 \text{ km} \]

### Step 6: Use the time taken to find speed
We know the time taken to cover D - 32 km upstream is 4 hours:
\[ D - 32 = 80 - 32 = 48 \text{ km} \]
Using the formula for time:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
For upstream speed:
\[ 4 = \frac{48}{3x} \]
Cross-multiplying gives:
\[ 4 \cdot 3x = 48 \]
\[ 12x = 48 \]
\[ x = \frac{48}{12} = 4 \]

### Step 7: Calculate speeds
Now we can find the speeds:
- Speed downstream = \(5x = 5 \cdot 4 = 20 \text{ km/h}\)
- Speed upstream = \(3x = 3 \cdot 4 = 12 \text{ km/h}\)

### Step 8: Find the speed of the water current
Let the speed of the boat in still water be \(A\) and the speed of the water current be \(B\).
From the downstream speed:
\[ A + B = 20 \]
From the upstream speed:
\[ A - B = 12 \]

### Step 9: Solve the equations
Adding the two equations:
\[ (A + B) + (A - B) = 20 + 12 \]
\[ 2A = 32 \implies A = 16 \text{ km/h} \]

Substituting \(A\) back into one of the equations:
\[ 16 + B = 20 \implies B = 20 - 16 = 4 \text{ km/h} \]

### Final Answer
The speed of the water current is **4 km/h**.