If a car travels a distance with `20%` less speed, then it will reach 15 minutes late. What is the usual time (in minutes ) taken by the car to travel the same distance ?

Options

A. 80
B. 90
C. 75
D. 60

Answer

To solve the problem, we need to establish the relationship between speed, distance, and time. Let's break it down step by step.

### Step 1: Define Variables
Let the usual speed of the car be \( S \) km/h. The distance traveled by the car is \( D \) km.

### Step 2: Calculate Usual Time
The usual time taken to travel the distance \( D \) at speed \( S \) is given by the formula:
\[ \text{Usual Time} = \frac{D}{S} \]

### Step 3: Determine New Speed
If the car travels at a speed that is 20% less than the usual speed, the new speed \( S' \) can be calculated as:
\[ S' = S - 0.2S = 0.8S \]

### Step 4: Calculate New Time
The time taken to travel the same distance \( D \) at the reduced speed \( S' \) is:
\[ \text{New Time} = \frac{D}{S'} = \frac{D}{0.8S} \]

### Step 5: Set Up the Equation
According to the problem, the car reaches 15 minutes late when traveling at the reduced speed. Therefore, we can set up the equation:
\[ \text{New Time} - \text{Usual Time} = 15 \text{ minutes} \]
Substituting the expressions we derived:
\[ \frac{D}{0.8S} - \frac{D}{S} = 15 \]

### Step 6: Simplify the Equation
To simplify the left-hand side, we can find a common denominator:
\[ \frac{D}{0.8S} - \frac{D}{S} = \frac{D \cdot S - D \cdot 0.8S}{0.8S^2} = \frac{D(1 - 0.8)}{0.8S} = \frac{0.2D}{0.8S} = \frac{D}{4S} \]
Thus, we have:
\[ \frac{D}{4S} = 15 \]

### Step 7: Solve for Distance
From the equation above, we can express \( D \) in terms of \( S \):
\[ D = 15 \times 4S = 60S \]

### Step 8: Calculate Usual Time
Now, we can find the usual time using the expression for \( D \):
\[ \text{Usual Time} = \frac{D}{S} = \frac{60S}{S} = 60 \text{ minutes} \]

### Final Answer
The usual time taken by the car to travel the same distance is **60 minutes**.

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