A man covers a certain distance by his own car had he moved 6 km/h faster he would have taken 4 hours less if had moved 4 km/h slower he would have taken 4 hours more.The distance (in km) is:

Options

A. 240 km
B. 640 km
C. 480 km
D. none of these

Answer

To solve the problem, we need to set up equations based on the information provided. Let's denote:

- \( d \) = distance (in km)
- \( s \) = original speed (in km/h)

From the problem, we have two scenarios:

1. If the man had moved 6 km/h faster, he would have taken 4 hours less.
2. If he had moved 4 km/h slower, he would have taken 4 hours more.

### Step 1: Set up the equations

From the first scenario:
- The time taken at original speed \( s \) is \( \frac{d}{s} \).
- The time taken at increased speed \( s + 6 \) is \( \frac{d}{s + 6} \).
- According to the problem, the difference in time is 4 hours:

\[ \frac{d}{s} - \frac{d}{s + 6} = 4 \]

From the second scenario:
- The time taken at decreased speed \( s - 4 \) is \( \frac{d}{s - 4} \).
- According to the problem, the difference in time is also 4 hours:

\[ \frac{d}{s - 4} - \frac{d}{s} = 4 \]

### Step 2: Simplify the equations

**For the first equation:**

\[ \frac{d}{s} - \frac{d}{s + 6} = 4 \]

Multiplying through by \( s(s + 6) \) to eliminate the denominators:

\[ d(s + 6) - ds = 4s(s + 6) \]

This simplifies to:

\[ 6d = 4s^2 + 24s \]

Thus, we have:

\[ d = \frac{2s^2 + 12s}{3} \quad \text{(Equation 1)} \]

**For the second equation:**

\[ \frac{d}{s - 4} - \frac{d}{s} = 4 \]

Multiplying through by \( s(s - 4) \):

\[ d(s) - d(s - 4) = 4s(s - 4) \]

This simplifies to:

\[ 4d = 4s^2 - 16s \]

Thus, we have:

\[ d = s^2 - 4s \quad \text{(Equation 2)} \]

### Step 3: Set the equations equal to each other

Now we have two expressions for \( d \):

From Equation 1:
\[ d = \frac{2s^2 + 12s}{3} \]

From Equation 2:
\[ d = s^2 - 4s \]

Setting them equal gives us:

\[ \frac{2s^2 + 12s}{3} = s^2 - 4s \]

### Step 4: Solve for \( s \)

Multiply through by 3 to eliminate the fraction:

\[ 2s^2 + 12s = 3s^2 - 12s \]

Rearranging gives:

\[ 0 = s^2 - 24s \]

Factoring out \( s \):

\[ s(s - 24) = 0 \]

Thus, \( s = 0 \) or \( s = 24 \). Since speed cannot be zero, we have:

\[ s = 24 \text{ km/h} \]

### Step 5: Find the distance \( d \)

Substituting \( s = 24 \) back into Equation 2:

\[ d = 24^2 - 4 \cdot 24 \]

Calculating gives:

\[ d = 576 - 96 = 480 \text{ km} \]

### Final Answer

The distance is \( \boxed{480} \) km.

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