A and B travel the same distance at speeds of 8 km/h and 12 km/h, respectively. If B takes 30 minutes less than that taken by A. what is the distance (in km) travelled by each one of them?

Options

A. 12
B. 10
C. 15
D. 8

Answer

To solve the problem step by step, let's denote the distance traveled by both A and B as \( D \) kilometers.

### Step 1: Define the time taken by A and B
- The time taken by A to travel distance \( D \) at a speed of 8 km/h is given by the formula:
\[ \text{Time taken by A} = \frac{D}{8} \]
- The time taken by B to travel the same distance \( D \) at a speed of 12 km/h is:
\[ \text{Time taken by B} = \frac{D}{12} \]

### Step 2: Set up the equation based on the time difference
According to the problem, B takes 30 minutes less than A. We need to convert 30 minutes into hours since the speeds are in km/h.
\[ 30 \text{ minutes} = \frac{30}{60} \text{ hours} = \frac{1}{2} \text{ hours} \]
Thus, we can set up the equation:
\[ \frac{D}{8} - \frac{D}{12} = \frac{1}{2} \]

### Step 3: Solve the equation
To solve the equation, we first find a common denominator for the fractions on the left side. The least common multiple of 8 and 12 is 24. Thus, we rewrite the equation:
\[ \frac{3D}{24} - \frac{2D}{24} = \frac{1}{2} \]
This simplifies to:
\[ \frac{D}{24} = \frac{1}{2} \]

### Step 4: Cross-multiply to find D
Cross-multiplying gives:
\[ D = 24 \times \frac{1}{2} = 12 \]

### Conclusion
The distance traveled by each one of them is \( D = 12 \) kilometers.