If a boat goes upstream at a speed of 24 km/hr and comes back the same distance at 40 km/hr. What is the average speed (in km/hr) for the total journey.

Options

A. 32
B. 30
C. 31
D. 33

Answer

To find the average speed of the boat for the total journey, we can follow these steps:

### Step 1: Define the distance
Let the distance traveled upstream be \( x \) km. Since the boat travels the same distance downstream, the total distance for the round trip is:
\[ \text{Total Distance} = x + x = 2x \text{ km} \]

**Hint:** Define the distance for one leg of the journey to set up the total distance.

### Step 2: Calculate the time taken for each leg of the journey
- **Time taken to go upstream** at a speed of 24 km/hr:
\[ t_1 = \frac{x}{24} \text{ hours} \]
- **Time taken to come downstream** at a speed of 40 km/hr:
\[ t_2 = \frac{x}{40} \text{ hours} \]

**Hint:** Use the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \) to find the time for each leg.

### Step 3: Calculate the total time for the journey
The total time for the round trip is:
\[ \text{Total Time} = t_1 + t_2 = \frac{x}{24} + \frac{x}{40} \]

To add these fractions, find a common denominator. The least common multiple (LCM) of 24 and 40 is 120.

Rewriting the times:
\[ t_1 = \frac{x}{24} = \frac{5x}{120} \]
\[ t_2 = \frac{x}{40} = \frac{3x}{120} \]

Thus, the total time becomes:
\[ \text{Total Time} = \frac{5x}{120} + \frac{3x}{120} = \frac{8x}{120} = \frac{x}{15} \text{ hours} \]

**Hint:** When adding fractions, make sure to find a common denominator.

### Step 4: Calculate the average speed
Average speed is defined as:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]

Substituting the values we have:
\[ \text{Average Speed} = \frac{2x}{\frac{x}{15}} = 2x \times \frac{15}{x} = 30 \text{ km/hr} \]

**Hint:** Use the formula for average speed and substitute the total distance and total time calculated in previous steps.

### Final Answer
The average speed for the total journey is:
\[ \boxed{30 \text{ km/hr}} \]