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CAT 2021 Slot 3 — QA Question 4

Working in ShiftsEasy

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:

Answer & solution

  • 10

  • B

    12

  • C

    12.5

  • D

    11.5

Solution

Easy

Compare two scenarios that finish the same job. In the real scenario Rahul works longer than Gautam; in the hypothetical both start together and finish 3030 min sooner. Equating the total work in the two scenarios gives the ratio of their rates, which then yields Rahul's solo time.

1

Set up the two scenarios. Let Rahul's rate be rr and Gautam's rate be gg (units/hour). Actual: Rahul 9:00→17:009{:}00\to17{:}00 is 88 h, Gautam joins at 11:0011{:}00 so works 66 h. Hypothetical: both work together and finish 3030 min earlier, i.e. by 16:3016{:}30, which is 7.57.5 h each.

Actual work=8r+6g...(1)Hypothetical work=7.5r+7.5g...(2)\begin{aligned} &\text{Actual work}=8r+6g \quad\text{...(1)}\\ &\text{Hypothetical work}=7.5r+7.5g \quad\text{...(2)} \end{aligned}
2

Equate the work and find the rate ratio. Both scenarios complete the identical job, so (1)==(2).

8r+6g=7.5r+7.5g[(1)=(2)]⇒ 0.5r=1.5g⇒ r=3g\begin{aligned} &8r+6g=7.5r+7.5g \quad\text{[(1)=(2)]}\\ &\Rightarrow\ 0.5r=1.5g\\ &\Rightarrow\ r=3g \end{aligned}
3

Total work, then Rahul's solo time. Substitute g=r3g=\tfrac{r}{3} into the actual work, then divide total work by Rahul's rate.

Total work=8r+6g=8r+6⋅r3=8r+2r=10r(from step 2)⇒ Rahul’s time=10rr=10 hours\begin{aligned} &\text{Total work}=8r+6g=8r+6\cdot\tfrac{r}{3}=8r+2r=10r \quad\text{(from step 2)}\\ &\Rightarrow\ \text{Rahul's time}=\frac{10r}{r}=10\text{ hours} \end{aligned}
Rahul alone=10 hours\text{Rahul alone}=10\text{ hours}
CAT 2021 Slot 3 QA Q4: One day, Rahul started a work at 9 AM and Gautam joined him two hours later. They then worked together and com — Solution | TheCATExam