Question

One day, Rahul started a work at 9 AM and Gautam joined him two hours later.They then worked together and completed the work at 5 PM the same day. If both had started at 9 AM and worked together, the work would have been completed 30 minutes earlier. Working alone, the time Rahul would have taken, in hours, to complete the work is

• A. 12
• B. 11.5
• C. 12.5
• D. 10

Answer 1

The Correct Option is D

Let's denote the work rate of Rahul as \(R\) work/hour and the work rate of Gautam as \(G\) work/hour. 

1) When both started together at 9 AM: 
The total time they worked together was 8 hours (from 9 AM to 5 PM). They would have finished the work in 7.5 hours (30 minutes earlier). So, their combined work rate when they started together would be: 
\(Total \ Work = (R + G) \times 7.5\)

 2) On the day Rahul started at 9 AM and Gautam joined him 2 hours later: 
Rahul worked for 2 hours alone, and then they worked together for the next 6 hours (from 11 AM to 5 PM). This gives: 
\(Total \ Work = 2R + 6(R + G)\)

Since the total work done in both scenarios is the same, we can equate the two expressions: 

\(2R + 6(R + G) = 7.5(R + G)\)

Expanding and simplifying: 
\(2R + 6R + 6G = 7.5R + 7.5G\)
\(8R + 6G = 7.5R + 7.5G\)
\(0.5R = 1.5G\)
\(R = 3G\)

Now, let's use the combined work rate from the first scenario: 
\(Total \ Work = (R + G) \times 7.5\)

Using the relationship \(R = 3G\), we get: 
\(Total \ Work = (3G + G) \times 7.5\)
\(Total \ Work = 4G \times 7.5\)
\(Total \ Work = 30G\)

Now, using this total work with Rahul's individual work rate for the time he worked alone: 
\(2R = 2(3G) = 6G\)

Subtracting this from the total work to get the work done by both together: 
\(30G - 6G = 24G\)

This means that both of them, working together for 6 hours, did \(24G\) of the work: 
\(6(R + G) = 24G\) Using \(R = 3G\): 
\(6(4G) = 24G\)

The relationship holds true. 

Now, to find the time Rahul would take to complete the entire work by himself: 

Using \(R = 3G\): 
\(Total \ Work = 30G\)

If Rahul does the entire work: 
\(Time \ for \ Rahul = \frac{Total \ Work}{R} = \frac{30G}{3G} = 10 \ hours\)

Rahul would take 10 hours to complete the work by himself.

Answer 2

Let R be the fraction of work Rahul completes in one hour, 
and G be the fraction of work Gautam completes in one hour.

Rahul works for 8 hours per day (from 9 AM to 5 PM),
while Gautam works for 6 hours per day (2 hours less).

So, in one full day, the work done is:
\(8R + 6G = 1\)         (Equation 1)

If they start together at 9 AM and finish by 4:30 PM, they work for 7.5 hours each.
So, the total work done is:
\(7.5R + 7.5G = 1\)         (Equation 2)

Equating Equation 1 and Equation 2:
\(8R + 6G = 7.5R + 7.5G\)
\(0.5R = 1.5G\)
\(R = 3G\)

So, Rahul is 3 times as efficient as Gautam.

Substituting \(R = 3G\) into Equation 1:
\(8(3G) + 6G = 24G + 6G = 30G = 1\)
\(G = \frac{1}{30}, \quad R = \frac{1}{10}\)

So, Rahul alone can do the work in:
\(\frac{1}{R} = \frac{1}{1/10} = 10\) hours.

∴ Rahul can complete the work alone in 10 hours.