Sarthak completed a marathon in 4 hours and 35 minutes. The marathon consisted of a 10km run followed by 20km cycle ride and the remaining distance again a run. He ran the first stage at 6km/hr and then cycled at 16km/hr. How much distance did Sarthak cover in total, if his speed in the last run was just half that of his first run?

Options

A. 5km
B. 35km
C. 40km
D. 45km

Answer

To solve the problem step by step, we will break down the information given and calculate the total distance Sarthak covered during the marathon.

### Step 1: Convert the total time into hours
Sarthak completed the marathon in 4 hours and 35 minutes. We need to convert this time into hours.

\[ \text{Total time in hours} = 4 + \frac{35}{60} = 4 + \frac{7}{12} = \frac{48}{12} + \frac{7}{12} = \frac{55}{12} \text{ hours} \]

**Hint:** To convert minutes to hours, divide the number of minutes by 60.

### Step 2: Identify the distances and speeds
1. First run: 10 km at 6 km/hr
2. Cycle ride: 20 km at 16 km/hr
3. Last run: Let the distance be \( x \) km, and the speed is half of the first run speed, which is \( 3 \) km/hr.

### Step 3: Calculate the time taken for each segment
Using the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \):

1. Time for the first run:
\[ \text{Time}_1 = \frac{10}{6} \text{ hours} \]

2. Time for the cycle ride:
\[ \text{Time}_2 = \frac{20}{16} = \frac{5}{4} \text{ hours} \]

3. Time for the last run:
\[ \text{Time}_3 = \frac{x}{3} \text{ hours} \]

### Step 4: Set up the equation for total time
The total time taken for all three segments is equal to the total time Sarthak took to complete the marathon:

\[ \frac{10}{6} + \frac{5}{4} + \frac{x}{3} = \frac{55}{12} \]

### Step 5: Simplify the equation
To solve the equation, we need to find a common denominator for the fractions on the left side. The least common multiple (LCM) of 6, 4, and 3 is 12.

1. Convert each term:
- \( \frac{10}{6} = \frac{20}{12} \)
- \( \frac{5}{4} = \frac{15}{12} \)
- \( \frac{x}{3} = \frac{4x}{12} \)

Now, substituting back into the equation:

\[ \frac{20}{12} + \frac{15}{12} + \frac{4x}{12} = \frac{55}{12} \]

### Step 6: Combine the fractions
Combine the left side:

\[ \frac{20 + 15 + 4x}{12} = \frac{55}{12} \]

### Step 7: Eliminate the denominator
Multiply both sides by 12:

\[ 20 + 15 + 4x = 55 \]

### Step 8: Solve for \( x \)
Combine the constants:

\[ 35 + 4x = 55 \]

Subtract 35 from both sides:

\[ 4x = 20 \]

Divide by 4:

\[ x = 5 \]

### Step 9: Calculate the total distance
Now, we can find the total distance covered by Sarthak:

\[ \text{Total distance} = 10 \text{ km (first run)} + 20 \text{ km (cycle)} + 5 \text{ km (last run)} = 35 \text{ km} \]

### Final Answer
Sarthak covered a total distance of **35 kilometers**.

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