The speed of three cars in the ratio `3:4:5`. The ratio between time taken by them to travel the same distance is

Options

A. `3:4:5`
B. `5:4:3`
C. `2:3:4`
D. `20:15:12`

Answer

To solve the problem of finding the ratio of time taken by three cars with speeds in the ratio of 3:4:5, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between speed and time**:
- The time taken to travel a certain distance is inversely proportional to speed. This means that if speed increases, time decreases, and vice versa.

2. **Define the speeds of the cars**:
- Let the speeds of the three cars be represented as:
- Car A: \( 3x \)
- Car B: \( 4x \)
- Car C: \( 5x \)
- Here, \( x \) is a common multiplier.

3. **Use the formula for time**:
- The formula for time taken is:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
- Since the distance is the same for all cars, we can denote it as \( D \).

4. **Calculate the time taken by each car**:
- For Car A:
\[ \text{Time}_A = \frac{D}{3x} \]
- For Car B:
\[ \text{Time}_B = \frac{D}{4x} \]
- For Car C:
\[ \text{Time}_C = \frac{D}{5x} \]

5. **Express the times in a common format**:
- To find the ratio of times, we can express the times as:
- Time for Car A: \( \frac{D}{3x} \)
- Time for Car B: \( \frac{D}{4x} \)
- Time for Car C: \( \frac{D}{5x} \)

6. **Find a common denominator**:
- The least common multiple of the denominators (3, 4, 5) is 60. Thus, we can express the times as:
- Time for Car A: \( \frac{20D}{60x} \)
- Time for Car B: \( \frac{15D}{60x} \)
- Time for Car C: \( \frac{12D}{60x} \)

7. **Write the ratio of times**:
- The ratio of times taken by the three cars is:
\[ \text{Time}_A : \text{Time}_B : \text{Time}_C = 20 : 15 : 12 \]

8. **Simplify the ratio**:
- To simplify, we can divide each term by 3:
\[ 20 : 15 : 12 = \frac{20}{3} : \frac{15}{3} : \frac{12}{3} = 20 : 15 : 12 \]

### Final Answer:
The ratio of time taken by the three cars is \( 20 : 15 : 12 \).