A man covers a certain distance in 9 hours. If he covers half distance at a speed of 12 kmph and remained half at a speed of 15 kmph, then find the distance.

Options

A. 60 km
B. 110 km
C. 80 km
D. 120 km

Answer

To solve the problem step by step, we will use the information given about the speeds and the total time taken to cover the distance.

### Step 1: Define the total distance
Let the total distance be \( D \). Since the man covers half of the distance at one speed and the other half at another speed, we can express the distance as:
\[ D = 2x \]
where \( x \) is half of the total distance.

### Step 2: Determine the time taken for each half of the distance
The man covers the first half of the distance \( x \) at a speed of 12 km/h and the second half at a speed of 15 km/h.

The time taken to cover the first half is:
\[ t_1 = \frac{x}{12} \]

The time taken to cover the second half is:
\[ t_2 = \frac{x}{15} \]

### Step 3: Set up the equation for total time
According to the problem, the total time taken to cover the entire distance is 9 hours. Therefore, we can write:
\[ t_1 + t_2 = 9 \]
Substituting the expressions for \( t_1 \) and \( t_2 \):
\[ \frac{x}{12} + \frac{x}{15} = 9 \]

### Step 4: Find a common denominator and solve for \( x \)
To solve the equation, we need a common denominator for the fractions. The least common multiple of 12 and 15 is 60. We can rewrite the equation as:
\[ \frac{5x}{60} + \frac{4x}{60} = 9 \]
Combining the fractions gives:
\[ \frac{9x}{60} = 9 \]

Now, multiply both sides by 60 to eliminate the fraction:
\[ 9x = 540 \]

Now, divide by 9:
\[ x = 60 \]

### Step 5: Calculate the total distance
Since \( x \) is half of the total distance, we can find \( D \):
\[ D = 2x = 2 \times 60 = 120 \text{ km} \]

### Conclusion
The total distance covered by the man is:
\[ \boxed{120 \text{ km}} \]