Twice the speed of A is equal to thrice the speed of B. To travel a certain distance, A takes 42 minutes less than B to travel the same distance. What is the time (in minutes) taken by B to travel the same distance ?

Options

A. a)126
B. b)135
C. c)120
D. d)96

Answer

To solve the problem, we need to establish the relationship between the speeds of A and B, and then use that to find the time taken by B to travel a certain distance.

### Step-by-Step Solution:

1. **Establish the relationship between speeds:**
We know that twice the speed of A is equal to thrice the speed of B. We can express this relationship mathematically:
\[ 2 \cdot \text{Speed of A} = 3 \cdot \text{Speed of B} \]
This implies:
\[ \frac{\text{Speed of A}}{\text{Speed of B}} = \frac{3}{2} \]

2. **Set up the time relationship:**
Since speed and time are inversely related (for a constant distance), we can express the times taken by A and B in terms of their speeds. Let the time taken by A be \( t_A \) and the time taken by B be \( t_B \). The relationship based on their speeds will be:
\[ \frac{t_A}{t_B} = \frac{\text{Speed of B}}{\text{Speed of A}} = \frac{2}{3} \]
This implies:
\[ t_A = \frac{2}{3} t_B \]

3. **Use the information about the time difference:**
According to the problem, A takes 42 minutes less than B to travel the same distance. This can be expressed as:
\[ t_B - t_A = 42 \]
Substituting the expression for \( t_A \) from the previous step:
\[ t_B - \frac{2}{3} t_B = 42 \]

4. **Simplify the equation:**
Combine the terms on the left side:
\[ \frac{1}{3} t_B = 42 \]

5. **Solve for \( t_B \):**
Multiply both sides by 3 to find \( t_B \):
\[ t_B = 42 \times 3 = 126 \]

Thus, the time taken by B to travel the same distance is **126 minutes**.