Three cars travelled distance in the ratio `1,2:3`. If the ratio of the time of travel is `3:2:1`, then the ratio of their speed is

Options

A. `3:9:1`
B. `1:3:9`
C. `1:2:4`
D. `4:3:2`

Answer

To solve the problem, we need to find the ratio of the speeds of three cars given the ratios of their distances and times of travel.

### Step-by-Step Solution:

1. **Identify the Ratios**:
- The distances travelled by the three cars are in the ratio \(1:2:3\).
- The times taken by the three cars are in the ratio \(3:2:1\).

2. **Assign Variables**:
- Let the distances travelled by the three cars be \(d_1 = 1x\), \(d_2 = 2x\), and \(d_3 = 3x\) for some value \(x\).
- Let the times taken by the three cars be \(t_1 = 3y\), \(t_2 = 2y\), and \(t_3 = 1y\) for some value \(y\).

3. **Use the Formula for Speed**:
- Speed is defined as \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \).
- Therefore, the speeds of the three cars can be calculated as follows:
- For Car 1:
\[ s_1 = \frac{d_1}{t_1} = \frac{1x}{3y} = \frac{x}{3y} \]
- For Car 2:
\[ s_2 = \frac{d_2}{t_2} = \frac{2x}{2y} = \frac{x}{y} \]
- For Car 3:
\[ s_3 = \frac{d_3}{t_3} = \frac{3x}{1y} = \frac{3x}{y} \]

4. **Find the Ratio of Speeds**:
- Now we have:
- \(s_1 = \frac{x}{3y}\)
- \(s_2 = \frac{x}{y}\)
- \(s_3 = \frac{3x}{y}\)
- To find the ratio of speeds \(s_1 : s_2 : s_3\), we can express them in a common form:
\[ s_1 : s_2 : s_3 = \frac{x}{3y} : \frac{x}{y} : \frac{3x}{y} \]
- To simplify, we can multiply each term by \(3y\) (the least common multiple of the denominators):
\[ s_1 : s_2 : s_3 = \frac{x}{3y} \cdot 3y : \frac{x}{y} \cdot 3y : \frac{3x}{y} \cdot 3y = x : 3x : 9x \]
- This simplifies to:
\[ 1 : 3 : 9 \]

5. **Final Ratio of Speeds**:
- Therefore, the ratio of the speeds of the three cars is \(1:3:9\).