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CAT 2024 Slot 3 — QA Question 21

Working AlternatelyEasy

Sam can complete a job in 20 days when working alone. Mohit is twice as fast as Sam and thrice as fast as Ayna in the same job. They undertake a job with an arrangement where Sam and Mohit work together on the first day, Sam and Ayna on the second day, Mohit and Ayna on the third day, and this three-day pattern is repeated till the work gets completed. Then, the fraction of total work done by Sam is

Answer & solution

  • A

    1/5

  • 3/10

  • C

    1/20

  • D

    3/20

Solution

Hard

Convert everyone's rate to "units per day" on a common total of work. Compute how much is done in one full three-day cycle, find how many cycles complete the job, then total only Sam's contribution and divide by the whole job.

1

Rates. Take total work =60=60 units (LCM-friendly). Sam does the job in 2020 days, so Sam =3=3/day. Mohit is twice Sam =6=6/day, and Mohit is thrice Ayna, so Ayna =2=2/day.

Sam=6020=3,Mohit=2×3=6,Ayna=63=2\begin{aligned} &\text{Sam}=\frac{60}{20}=3,\quad \text{Mohit}=2\times3=6,\quad \text{Ayna}=\frac{6}{3}=2 \end{aligned}
2

One cycle (3 days). Day1 Sam+Mohit, Day2 Sam+Ayna, Day3 Mohit+Ayna.

Day1=3+6=9, Day2=3+2=5, Day3=6+2=8⇒ per cycle=9+5+8=22 units\begin{aligned} &\text{Day1}=3+6=9,\ \text{Day2}=3+2=5,\ \text{Day3}=6+2=8\\ &\Rightarrow\ \text{per cycle}=9+5+8=22\ \text{units} \end{aligned}
3

How the 60 units finish. Two full cycles do 4444; 1616 remain. Continue: Day7 (Sam+Mohit) 9⇒539\Rightarrow53; Day8 (Sam+Ayna) 5⇒585\Rightarrow58; Day9 (Mohit+Ayna) needs only 22 of its 88 to reach 6060.

after 2 cycles=44, remaining=16⇒ Day7=9→53, Day8=5→58, Day9 (partial)=2→60\begin{aligned} &\text{after 2 cycles}=44,\ \text{remaining}=16\\ &\Rightarrow\ \text{Day7}=9\to 53,\ \text{Day8}=5\to 58,\ \text{Day9 (partial)}=2\to 60 \end{aligned}
4

Sam's total work. Sam works on Day1, Day2 of each cycle (and Day7, Day8). In each of the first 2 cycles Sam contributes 3+3=63+3=6; in cycle 3 Sam works Day7 (33) and Day8 (33); Day9 has no Sam.

Sam=2×6+(3+3)⇒ Sam=12+6=18 units⇒ fraction=1860=310\begin{aligned} &\text{Sam}=2\times 6+(3+3)\\ &\Rightarrow\ \text{Sam}=12+6=18\ \text{units}\\ &\Rightarrow\ \text{fraction}=\frac{18}{60}=\frac{3}{10} \end{aligned}
310\dfrac{3}{10}
CAT 2024 Slot 3 QA Q21: Sam can complete a job in 20 days when working alone. Mohit is twice as fast as Sam and thrice as fast as Ayna — Solution | TheCATExam