The speed of a boat downstream is 2.5 times the speed of the boat upstream . If the time taken by the boat for going 30 km downstream and the same distance upstream is 7 hours , then what is the speed ( in km/h) of the boat downstream ?

Options

A. 12.5
B. 9
C. 15
D. 7.5

Answer

To solve the problem step by step, let's denote the speed of the boat upstream as \( V \) km/h. According to the problem, the downstream speed is \( 2.5V \) km/h.

### Step 1: Define the speeds
- Let the upstream speed be \( V \) km/h.
- Therefore, the downstream speed will be \( 2.5V \) km/h.

### Step 2: Write the time equations
The time taken to travel downstream and upstream can be expressed using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
For downstream (30 km):
\[ \text{Time}_{\text{downstream}} = \frac{30}{2.5V} \]
For upstream (30 km):
\[ \text{Time}_{\text{upstream}} = \frac{30}{V} \]

### Step 3: Set up the equation for total time
According to the problem, the total time taken for both journeys is 7 hours:
\[ \frac{30}{2.5V} + \frac{30}{V} = 7 \]

### Step 4: Simplify the equation
To simplify, we can find a common denominator, which is \( 2.5V \):
\[ \frac{30}{2.5V} + \frac{30 \times 2.5}{2.5V} = 7 \]
This simplifies to:
\[ \frac{30 + 75}{2.5V} = 7 \]
\[ \frac{105}{2.5V} = 7 \]

### Step 5: Cross-multiply to solve for \( V \)
Cross-multiplying gives:
\[ 105 = 7 \times 2.5V \]
\[ 105 = 17.5V \]

### Step 6: Solve for \( V \)
Now, divide both sides by 17.5:
\[ V = \frac{105}{17.5} = 6 \text{ km/h} \]

### Step 7: Calculate the downstream speed
Now that we have \( V \), we can find the downstream speed:
\[ \text{Downstream speed} = 2.5V = 2.5 \times 6 = 15 \text{ km/h} \]

### Final Answer
The speed of the boat downstream is **15 km/h**.

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