A person reaches his destination 32 minutes late if his speed is 6 km/ h. and he reaches 15 minutes before time if his speed is 7 km/h. Find the distance of his destination from his starting point.

Options

A. 28 km
B. 30 km
C. 32 km
D. 35 km

Answer

To solve the problem, we need to find the distance of the destination from the starting point based on the given conditions about the person's travel times at different speeds. Let's break it down step by step.

### Step 1: Define Variables
Let:
- \( d \) = distance to the destination (in kilometers)
- \( t \) = time taken to reach the destination on time (in hours)

### Step 2: Set Up Equations Based on Given Conditions
1. When the person travels at 6 km/h, he is 32 minutes late:
- Time taken = \( t + \frac{32}{60} \) hours
- Using the formula \( \text{Distance} = \text{Speed} \times \text{Time} \):
\[ d = 6 \left( t + \frac{32}{60} \right) \]
Simplifying gives:
\[ d = 6t + 6 \times \frac{32}{60} = 6t + 3.2 \]
(Equation 1)

2. When the person travels at 7 km/h, he is 15 minutes early:
- Time taken = \( t - \frac{15}{60} \) hours
- Using the same formula:
\[ d = 7 \left( t - \frac{15}{60} \right) \]
Simplifying gives:
\[ d = 7t - 7 \times \frac{15}{60} = 7t - 1.75 \]
(Equation 2)

### Step 3: Set the Equations Equal to Each Other
Since both equations equal \( d \), we can set them equal to each other:
\[ 6t + 3.2 = 7t - 1.75 \]

### Step 4: Solve for \( t \)
Rearranging the equation:
\[ 3.2 + 1.75 = 7t - 6t \]
\[ 5.95 = t \]
So, \( t = 5.95 \) hours.

### Step 5: Substitute \( t \) Back to Find \( d \)
Now substitute \( t \) back into either equation to find \( d \). Using Equation 1:
\[ d = 6t + 3.2 \]
Substituting \( t = 5.95 \):
\[ d = 6 \times 5.95 + 3.2 \]
\[ d = 35.7 + 3.2 = 38.9 \text{ km} \]

### Step 6: Final Answer
The distance of his destination from his starting point is approximately **38.9 km**.

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