The speeds of scooter, car and train are in the ratio of 1 : 4 : 16. If all of them covers equal distance then the ratio of time taken/velocity for the each of the vehicle is :

Options

A. 16:4:1
B. 4:3:2
C. 6:5:4
D. 1:3:4

Answer

To solve the problem, we need to find the ratio of time taken by each vehicle (scooter, car, and train) when they cover the same distance, given their speeds are in the ratio of 1:4:16.

### Step-by-Step Solution:

1. **Define the Speeds**:
Let the speed of the scooter be \( S \), the speed of the car be \( C \), and the speed of the train be \( T \). According to the problem, the speeds are in the ratio:
\[ S : C : T = 1 : 4 : 16 \]
We can express the speeds in terms of a variable \( x \):
\[ S = 1x, \quad C = 4x, \quad T = 16x \]

2. **Use the Distance Formula**:
The formula for distance is given by:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
Since all vehicles cover the same distance, we can denote this distance as \( D \).

3. **Set Up the Equations**:
For the scooter:
\[ D = S \times t_1 \implies D = (1x) \times t_1 \implies t_1 = \frac{D}{x} \]
For the car:
\[ D = C \times t_2 \implies D = (4x) \times t_2 \implies t_2 = \frac{D}{4x} \]
For the train:
\[ D = T \times t_3 \implies D = (16x) \times t_3 \implies t_3 = \frac{D}{16x} \]

4. **Find the Ratios of Time Taken**:
Now we can express the times taken by each vehicle:
\[ t_1 = \frac{D}{x}, \quad t_2 = \frac{D}{4x}, \quad t_3 = \frac{D}{16x} \]
To find the ratio of the times \( t_1 : t_2 : t_3 \):
\[ t_1 : t_2 : t_3 = \frac{D}{x} : \frac{D}{4x} : \frac{D}{16x} \]
We can simplify this by canceling \( D \) and \( x \):
\[ t_1 : t_2 : t_3 = 1 : \frac{1}{4} : \frac{1}{16} \]

5. **Convert to a Common Denominator**:
To express this in a more standard ratio, we can multiply each term by 16 (the least common multiple of the denominators):
\[ 1 \times 16 : \frac{1}{4} \times 16 : \frac{1}{16} \times 16 = 16 : 4 : 1 \]

6. **Final Ratio**:
Thus, the ratio of time taken by the scooter, car, and train is:
\[ t_1 : t_2 : t_3 = 16 : 4 : 1 \]

### Final Answer:
The ratio of time taken by the scooter, car, and train is \( 16 : 4 : 1 \).