The ratio of the speed of P, Q and R is `10 : 12 : 15` respectively. What is the ratio of the time taken by P. Q and R respectively to cover the same distance?

Options

A. `10 : 12 : 15`
B. `15 : 12 : 10`
C. `6:5:4`
D. `4:5:6`

Answer

To find the ratio of the time taken by P, Q, and R to cover the same distance, we can follow these steps:

### Step 1: Understand the relationship between speed, time, and distance.
The relationship can be expressed as:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
From this, we can derive that:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]

### Step 2: Set the speeds of P, Q, and R.
Given the ratio of speeds:
- Speed of P = 10
- Speed of Q = 12
- Speed of R = 15

### Step 3: Assume a common distance.
Let's assume the distance to be \( x \) (any arbitrary value, since we are interested in the ratio).

### Step 4: Write the time taken by each person.
Using the formula for time:
- Time taken by P, \( T_1 \):
\[ T_1 = \frac{x}{10} \]
- Time taken by Q, \( T_2 \):
\[ T_2 = \frac{x}{12} \]
- Time taken by R, \( T_3 \):
\[ T_3 = \frac{x}{15} \]

### Step 5: Find the ratio of the times.
Now we need to find the ratio \( T_1 : T_2 : T_3 \):
\[ T_1 : T_2 : T_3 = \frac{x}{10} : \frac{x}{12} : \frac{x}{15} \]

### Step 6: Simplify the ratio.
We can simplify this by dividing each term by \( x \):
\[ T_1 : T_2 : T_3 = \frac{1}{10} : \frac{1}{12} : \frac{1}{15} \]

### Step 7: Find a common denominator.
To simplify further, we can find a common multiple for the denominators 10, 12, and 15. The least common multiple (LCM) of these numbers is 60.

### Step 8: Multiply each term by the LCM.
Now, multiply each term by 60 to eliminate the fractions:
- For \( \frac{1}{10} \):
\[ 60 \times \frac{1}{10} = 6 \]
- For \( \frac{1}{12} \):
\[ 60 \times \frac{1}{12} = 5 \]
- For \( \frac{1}{15} \):
\[ 60 \times \frac{1}{15} = 4 \]

### Step 9: Write the final ratio.
Thus, the ratio of the time taken by P, Q, and R is:
\[ T_1 : T_2 : T_3 = 6 : 5 : 4 \]

### Final Answer:
The ratio of the time taken by P, Q, and R to cover the same distance is \( 6 : 5 : 4 \).

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