The speeds of three cars are in the ratio 2:3 : 4. What is the ratio between the times taken by these cars to travel the same distance?

Options

A. ` 4 : 3: 2 `
B. ` 2 : 3: 4`
C. ` 4 : 9 : 16 `
D. ` 6 : 4: 3`

Answer

To solve the problem, we need to find the ratio of the times taken by the three cars to travel the same distance, given their speeds in the ratio of 2:3:4.

### Step-by-Step Solution:

1. **Understand the Relationship Between Speed and Time**:
- Speed is defined as the distance traveled per unit of time. The formula is:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
- From this formula, we can see that time is inversely proportional to speed when the distance is constant. This means:
\[ \text{Time} \propto \frac{1}{\text{Speed}} \]

2. **Set Up the Speed Ratios**:
- The speeds of the three cars are given in the ratio of 2:3:4. We can denote the speeds as:
- Speed of Car A = 2x
- Speed of Car B = 3x
- Speed of Car C = 4x
- Here, \(x\) is a common multiplier.

3. **Calculate the Time Taken by Each Car**:
- Since time is inversely proportional to speed, we can express the time taken by each car as:
- Time taken by Car A = \(\frac{D}{2x}\)
- Time taken by Car B = \(\frac{D}{3x}\)
- Time taken by Car C = \(\frac{D}{4x}\)
- Here, \(D\) is the distance traveled by each car.

4. **Find the Ratio of Times**:
- To find the ratio of the times taken by the three cars, we take the reciprocal of their speeds:
\[ \text{Time Ratio} = \frac{1}{\text{Speed of Car A}} : \frac{1}{\text{Speed of Car B}} : \frac{1}{\text{Speed of Car C}} \]
- This gives us:
\[ \text{Time Ratio} = \frac{1}{2} : \frac{1}{3} : \frac{1}{4} \]

5. **Simplify the Ratio**:
- To simplify the ratio \(\frac{1}{2} : \frac{1}{3} : \frac{1}{4}\), we can find a common denominator, which is 12:
- Convert each fraction:
- \(\frac{1}{2} = \frac{6}{12}\)
- \(\frac{1}{3} = \frac{4}{12}\)
- \(\frac{1}{4} = \frac{3}{12}\)
- Thus, the ratio becomes:
\[ 6 : 4 : 3 \]

6. **Final Ratio**:
- Therefore, the ratio of the times taken by the cars to travel the same distance is:
\[ 6 : 4 : 3 \]