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

Answer: (D)

10

Answer 2

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.