Arun took part in a triathlon an athletic event .He had to swim run and bicycle to 10 km ,24 km and 30 km respectively and return the same way Arun's average speed for the triathelon is 4 km/h He took a total of 4 min for swimming and 20 min for bicycling in the triathelon.How much time he took while returning for the bicycle if the time taken for the return of each phase (i.e,each event)is 50% greater than that of taken initially:

Options

A. 12 min
B. 11 min
C. 11(1//9)min
D. none of these

Answer

To solve the problem step by step, we will break down the information given and calculate the required time for Arun's return bicycle phase.

### Step 1: Understand the Problem
Arun participated in a triathlon that involved swimming, running, and bicycling. The distances for each phase are:
- Swimming: 10 km
- Running: 24 km
- Bicycling: 30 km

We know:
- Average speed for the entire triathlon = 4 km/h
- Time taken for swimming = 4 minutes
- Time taken for bicycling (outward journey) = 20 minutes

### Step 2: Convert Time to Hours
Since speed is given in km/h, we need to convert the time taken for swimming and bicycling from minutes to hours.

- Time taken for swimming in hours:
\[ \text{Swimming time} = \frac{4 \text{ minutes}}{60} = \frac{1}{15} \text{ hours} \]

- Time taken for bicycling in hours:
\[ \text{Bicycling time} = \frac{20 \text{ minutes}}{60} = \frac{1}{3} \text{ hours} \]

### Step 3: Calculate Total Time for Outward Journey
Now, we can calculate the total time taken for the outward journey (swimming + running + bicycling).

Let \( t_r \) be the time taken for running (in hours).

Total time for the outward journey:
\[ \text{Total time} = \text{Swimming time} + \text{Running time} + \text{Bicycling time} \]
\[ \text{Total time} = \frac{1}{15} + t_r + \frac{1}{3} \]

### Step 4: Calculate Total Distance and Total Time
The total distance for the triathlon is:
\[ \text{Total distance} = 10 + 24 + 30 = 64 \text{ km} \]

Using the average speed:
\[ \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} \]
\[ 4 = \frac{64}{\text{Total time}} \]
\[ \text{Total time} = \frac{64}{4} = 16 \text{ hours} \]

### Step 5: Set Up the Equation
Now we can set up the equation:
\[ \frac{1}{15} + t_r + \frac{1}{3} = 16 \]

### Step 6: Solve for Running Time
Convert \(\frac{1}{3}\) to a fraction with a common denominator of 15:
\[ \frac{1}{3} = \frac{5}{15} \]
Now, substituting back into the equation:
\[ \frac{1}{15} + t_r + \frac{5}{15} = 16 \]
\[ \frac{6}{15} + t_r = 16 \]
\[ t_r = 16 - \frac{6}{15} \]
Convert 16 to a fraction:
\[ t_r = \frac{240}{15} - \frac{6}{15} = \frac{234}{15} \text{ hours} \]

### Step 7: Calculate Return Time for Bicycling
The return time for each phase is 50% greater than the initial time. The time taken for the outward bicycle journey is 20 minutes (or \(\frac{1}{3}\) hours).

Return time for bicycling:
\[ \text{Return time} = \text{Initial time} + 50\% \text{ of Initial time} \]
\[ \text{Return time} = \frac{1}{3} + 0.5 \times \frac{1}{3} = \frac{1}{3} + \frac{1}{6} = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \text{ hours} \]

Convert back to minutes:
\[ \text{Return time} = \frac{1}{2} \times 60 = 30 \text{ minutes} \]

### Final Answer
Arun took **30 minutes** while returning for the bicycle.

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