In reaching Purnagiri a man took half as long again to climb the second third as he did to climb the first third and a quarter as long again for the last third as for the second third. He took altogether 5 h 50 minutes. Find the time he spent on the first third of the journey?

Options

A. 72 min
B. 80 min
C. 81 min
D. 88 min

Answer

To solve the problem step by step, we will denote the time spent on the first third of the journey as \( x \) minutes.

### Step 1: Define the time for each segment of the journey
- Let the time taken to climb the first third of the journey be \( x \) minutes.
- The time taken to climb the second third is half as long again as the first third. This means:
\[ \text{Time for second third} = x + \frac{1}{2}x = \frac{3}{2}x \text{ minutes} \]
- The time taken to climb the last third is a quarter as long again as the second third. Thus:
\[ \text{Time for last third} = \frac{3}{2}x + \frac{1}{4} \left(\frac{3}{2}x\right) = \frac{3}{2}x + \frac{3}{8}x = \frac{12}{8}x + \frac{3}{8}x = \frac{15}{8}x \text{ minutes} \]

### Step 2: Set up the equation for total time
The total time for the journey is given as 5 hours and 50 minutes. We first convert this to minutes:
\[ 5 \text{ hours} = 5 \times 60 = 300 \text{ minutes} \]
Adding the 50 minutes gives:
\[ 300 + 50 = 350 \text{ minutes} \]

Now, we can set up the equation:
\[ x + \frac{3}{2}x + \frac{15}{8}x = 350 \]

### Step 3: Find a common denominator and simplify
The common denominator for the fractions \( 1, \frac{3}{2}, \text{ and } \frac{15}{8} \) is 8. We convert each term:
- \( x = \frac{8}{8}x \)
- \( \frac{3}{2}x = \frac{12}{8}x \)
- \( \frac{15}{8}x = \frac{15}{8}x \)

Now substituting these into the equation gives:
\[ \frac{8}{8}x + \frac{12}{8}x + \frac{15}{8}x = 350 \]
Combining the left side:
\[ \frac{8 + 12 + 15}{8}x = 350 \]
\[ \frac{35}{8}x = 350 \]

### Step 4: Solve for \( x \)
To isolate \( x \), multiply both sides by \( \frac{8}{35} \):
\[ x = 350 \times \frac{8}{35} \]
Calculating this gives:
\[ x = 80 \text{ minutes} \]

### Conclusion
The time spent on the first third of the journey is \( \boxed{80} \) minutes.