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 R represent Rahul's work rate in work/hour and G represent Gautam's work rate in work/hour.

1) If both started at 9 AM:

They worked together for 8 hours (9 AM to 5 PM) and would have completed the work in 7.5 hours. Their combined work rate was:

\(Total \ Work = (R + G) \times 7.5\)

2) If Rahul started at 9 AM and Gautam joined 2 hours later:

Rahul worked alone for 2 hours. Then, they worked together for the next 6 hours (11 AM to 5 PM). The total work done is:

\(Total \ Work = 2R + 6(R + G)\)

Equating the total work from both scenarios:

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

Simplifying the equation:

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

\(8R + 6G = 7.5R + 7.5G\)

\(0.5R = 1.5G\)

\(R = 3G\)

Using the combined work rate from the first scenario and substituting R = 3G:

\(Total \ Work = (R + G) \times 7.5\)

\(Total \ Work = (3G + G) \times 7.5\)

\(Total \ Work = 4G \times 7.5\)

\(Total \ Work = 30G\)

Rahul's work in the first 2 hours of the second scenario:

\(2R = 2(3G) = 6G\)

Work done by both together in the second scenario:

\(30G - 6G = 24G\)

This implies that their combined work for 6 hours was:

\(6(R + G) = 24G\)

Substituting R = 3G confirms this:

\(6(4G) = 24G\)

To find the time Rahul would take to complete the entire work alone:

Using Total Work = 30G and R = 3G:

\(Time \ for \ Rahul = \frac{Total \ Work}{R} = \frac{30G}{3G} = 10 \ hours\)

Rahul would complete the work in 10 hours individually.