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CAT 2018 Slot 1 — QA Question 10

Man Days (multiple groups of people)Easy

Humans and robots can both perform a job but at different efficiencies. Fifteen humans and five robots working together take thirty days to finish the job, whereas five humans and fifteen robots working together take sixty days to finish it. How many days will fifteen humans working together (without any robot) take to finish it?

Answer & solution

Correct answer: 32

  • A

    36

  • 32

  • C

    45

  • D

    40

Solution

Easy

Let one human do hh and one robot do rr units of work per day. The total job is the same in both scenarios, so equate 30(15h+5r)30(15h+5r) with 60(5h+15r)60(5h+15r) to relate hh and rr. Then find the days yy for 1515 humans alone.

1

Two expressions for the job.

W=30(15h+5r)W=60(5h+15r)\begin{aligned} &W = 30(15h + 5r)\\ &W = 60(5h + 15r) \end{aligned}
2

Relate hh and rr. Equate the two and simplify.

30(15h+5r)=60(5h+15r)⇒ 15h+5r=10h+30r(divide by 30)⇒ 5h=25r⇒ h=5r\begin{aligned} &30(15h+5r) = 60(5h+15r)\\ &\Rightarrow\ 15h + 5r = 10h + 30r \quad\text{(divide by 30)}\\ &\Rightarrow\ 5h = 25r\\ &\Rightarrow\ h = 5r \end{aligned}
3

Days for 15 humans alone. Their work in yy days equals the whole job WW.

y (15h)=30(15h+5r)⇒ y (15h)=30(15h+h)(5r=h)⇒ y (15h)=30(16h)\begin{aligned} &y\,(15h) = 30(15h + 5r)\\ &\Rightarrow\ y\,(15h) = 30(15h + h) \quad\text{(}5r = h\text{)}\\ &\Rightarrow\ y\,(15h) = 30(16h) \end{aligned}
4

Solve for yy. Cancel hh.

15y=480⇒ y=32 days\begin{aligned} &15y = 480\\ &\Rightarrow\ y = 32 \text{ days} \end{aligned}
32 days  ⇒  option (b)\boxed{32 \text{ days} \;\Rightarrow\; \text{option (b)}}

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CAT 2018 Slot 1 QA Q10: Humans and robots can both perform a job but at different efficiencies. Fifteen humans and five robots working — Solution | TheCATExam