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CAT 2023 Slot 3 — QA Question 10

Working in ShiftsEasy

Rahul, Rakshita and Gurmeet, working together, would have taken more than 7 days to finish a job. On the other hand, Rahul and Gurmeet, working together would have taken less than 15 days to finish the job. However, they all worked together for 6 days, followed by Rakshita, who worked alone for 3 more days to finish the job. If Rakshita had worked alone on the job then the number of days she would have taken to finish the job, cannot be

Answer & solution

  • 21

  • B

    17

  • C

    16

  • D

    20

Solution

Easy

Turn the two "more than / less than" clues into an inequality on Rakshita's daily output. Pick a total-work value, write the finishing equation for the actual 6+3 day schedule, and use the bound on the trio's combined rate to bound Rakshita's solo time. The option lying outside that bound is the one she cannot take.

Let total work =lcm⁡(7,15,6)=630=\operatorname{lcm}(7,15,6)=630 units. "Cannot be" means: find the value that is impossible given the constraints.

1

Use the "more than 7 days" clue for the three together. Slower (more days) ⇒\Rightarrow smaller combined rate:

rate(R+Rk+G)<6307=90 units/day\text{rate}(R{+}Rk{+}G)\lt \frac{630}{7}=90 \ \text{units/day}
2

Write the finishing equation for the real schedule: all three for 6 days, then Rakshita alone for 3 days.

630=rate(all 3)×6+rate(Rk)×3⇒ rate(Rk)=210−2 rate(all 3)\begin{aligned} &630=\text{rate}(\text{all }3)\times 6+\text{rate}(Rk)\times 3\\ &\Rightarrow\ \text{rate}(Rk)=210-2\,\text{rate}(\text{all }3) \end{aligned}
3

Substitute the bound from step 1. Since rate(all 3)<90\text{rate}(\text{all }3)\lt 90:

rate(Rk)>210−2(90)⇒ rate(Rk)>30 units/day\begin{aligned} &\text{rate}(Rk)\gt 210-2(90)\\ &\Rightarrow\ \text{rate}(Rk)\gt 30 \ \text{units/day} \end{aligned}
4

Convert the rate bound to a time bound for Rakshita working alone:

days(Rk)=630rate(Rk)<63030=21\text{days}(Rk)=\frac{630}{\text{rate}(Rk)}\lt \frac{630}{30}=21

So she finishes in strictly fewer than 21 days. Hence 17,16,2017,16,20 are all possible, but 2121 is not.

Rakshita's solo time must be <21\lt 21 days, so it cannot be 21.

CAT 2023 Slot 3 QA Q10: Rahul, Rakshita and Gurmeet, working together, would have taken more than 7 days to finish a job. On the other — Solution | TheCATExam