The speeds of three cars are in the ratio of `1 : 3 : 5`. The ratio among the time taken by these cars to travel the same distance is

Options

A. `3:5:15`
B. `15:3:5`
C. `15:5:3`
D. `5:3:1`

Answer

To solve the problem of finding the ratio among the time taken by three cars to travel the same distance, given their speeds in the ratio of 1:3:5, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between speed, distance, and time**:
We know that:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
From this, we can derive that:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
This means that if the distance is constant, time is inversely proportional to speed.

2. **Set up the speeds of the cars**:
Let the speeds of the three cars be represented as:
- Speed of Car A = 1x
- Speed of Car B = 3x
- Speed of Car C = 5x
Here, \(x\) is a common multiplier.

3. **Calculate the time taken by each car**:
Since the distance is the same for all cars, we can denote the distance as \(D\). The time taken by each car can be calculated as follows:
- Time taken by Car A:
\[ T_A = \frac{D}{1x} = \frac{D}{x} \]
- Time taken by Car B:
\[ T_B = \frac{D}{3x} = \frac{D}{3x} \]
- Time taken by Car C:
\[ T_C = \frac{D}{5x} = \frac{D}{5x} \]

4. **Form the ratio of the times**:
Now, we can express the ratio of the times taken by the three cars:
\[ T_A : T_B : T_C = \frac{D}{x} : \frac{D}{3x} : \frac{D}{5x} \]
Since \(D\) and \(x\) are common in all terms, we can simplify this to:
\[ T_A : T_B : T_C = 1 : \frac{1}{3} : \frac{1}{5} \]

5. **Eliminate the fractions**:
To eliminate the fractions, we can find a common denominator, which is 15 in this case:
- Multiply each term by 15:
\[ 15 \times 1 : 15 \times \frac{1}{3} : 15 \times \frac{1}{5} = 15 : 5 : 3 \]

6. **Final ratio**:
Therefore, the ratio of the time taken by the three cars is:
\[ T_A : T_B : T_C = 15 : 5 : 3 \]

### Conclusion:
The ratio among the time taken by the three cars to travel the same distance is **15 : 5 : 3**.