Rastogi and Gupta can fix an AC in 3 hours working together at their respective constant rates. If Rastogi doubles his speed, then both can fix the AC in 2 hours working together at their respective rates. How long does it take Rastogi to fix the AC alone?

Options

A. 3 hrs
B. 2 hrs
C. 6 hrs
D. 1/2 hr

Answer

To solve the problem, we will denote the time it takes Rastogi to fix the AC alone as \( R \) hours, and the time it takes Gupta to fix the AC alone as \( G \) hours.

### Step 1: Determine the rates of work for Rastogi and Gupta
- The rate of work for Rastogi is \( \frac{1}{R} \) (ACs per hour).
- The rate of work for Gupta is \( \frac{1}{G} \) (ACs per hour).

### Step 2: Set up the equation for working together
When Rastogi and Gupta work together, they can fix the AC in 3 hours. Therefore, their combined rate is:
\[ \frac{1}{R} + \frac{1}{G} = \frac{1}{3} \]

### Step 3: Determine the new rate when Rastogi doubles his speed
If Rastogi doubles his speed, his new rate becomes \( \frac{2}{R} \). Together, they can now fix the AC in 2 hours, so their new combined rate is:
\[ \frac{2}{R} + \frac{1}{G} = \frac{1}{2} \]

### Step 4: Solve the system of equations
We now have two equations:
1. \( \frac{1}{R} + \frac{1}{G} = \frac{1}{3} \) (Equation 1)
2. \( \frac{2}{R} + \frac{1}{G} = \frac{1}{2} \) (Equation 2)

From Equation 1, we can express \( \frac{1}{G} \):
\[ \frac{1}{G} = \frac{1}{3} - \frac{1}{R} \]

Substituting this into Equation 2:
\[ \frac{2}{R} + \left(\frac{1}{3} - \frac{1}{R}\right) = \frac{1}{2} \]

### Step 5: Simplify the equation
Combine the terms:
\[ \frac{2}{R} - \frac{1}{R} + \frac{1}{3} = \frac{1}{2} \]
\[ \frac{1}{R} + \frac{1}{3} = \frac{1}{2} \]

### Step 6: Isolate \( \frac{1}{R} \)
Subtract \( \frac{1}{3} \) from both sides:
\[ \frac{1}{R} = \frac{1}{2} - \frac{1}{3} \]

### Step 7: Find a common denominator
The common denominator for 2 and 3 is 6:
\[ \frac{1}{2} = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6} \]
Thus,
\[ \frac{1}{R} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6} \]

### Step 8: Solve for \( R \)
Taking the reciprocal gives:
\[ R = 6 \]

### Conclusion
Rastogi takes 6 hours to fix the AC alone.