A person, who can walk down a hill a the speed of `4 (1)/(2)` km/hr and up the hill at the rate of 3 km/hr, ascends and comes down to his starting point in 5 hours. How far did he ascent ?

Options

A. 13.5 km
B. 3 km
C. 15 km
D. 9 km

Answer

To solve the problem step-by-step, we need to find the distance the person ascended, given their speeds while walking up and down the hill and the total time taken for the journey.

### Step 1: Define the Variables
Let the distance ascended (and descended) be denoted as \( x \) kilometers.

### Step 2: Identify the Speeds
- Speed going uphill = 3 km/hr
- Speed going downhill = 4.5 km/hr

### Step 3: Calculate the Time Taken for Each Journey
Using the formula for time, which is:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
- Time taken to ascend (uphill) = \( \frac{x}{3} \) hours
- Time taken to descend (downhill) = \( \frac{x}{4.5} \) hours

### Step 4: Set Up the Equation for Total Time
According to the problem, the total time for both journeys is 5 hours. Therefore, we can set up the equation:
\[ \frac{x}{3} + \frac{x}{4.5} = 5 \]

### Step 5: Find a Common Denominator
To solve the equation, we need a common denominator for the fractions. The least common multiple of 3 and 4.5 is 13.5. We can rewrite the equation:
\[ \frac{4.5x}{13.5} + \frac{3x}{13.5} = 5 \]
This simplifies to:
\[ \frac{(4.5 + 3)x}{13.5} = 5 \]

### Step 6: Combine the Terms
Combine the terms in the numerator:
\[ \frac{7.5x}{13.5} = 5 \]

### Step 7: Cross-Multiply to Solve for \( x \)
Cross-multiplying gives:
\[ 7.5x = 5 \times 13.5 \]
Calculating the right side:
\[ 7.5x = 67.5 \]

### Step 8: Solve for \( x \)
Now, divide both sides by 7.5:
\[ x = \frac{67.5}{7.5} = 9 \]

### Conclusion
The distance ascended by the person is \( 9 \) kilometers.

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