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CAT 2024 Slot 2 — QA Question 21

Working AlternatelyEasy

Amal and Vimal together can complete a task in 150 days, while Vimal and Sunil together can complete the same task in 100 days. Amal starts working on the task and works for 75 days, then Vimal takes over and works for 135 days. Finally, Sunil takes over and completes the remaining task in 45 days. If Amal had started the task alone and worked on all days, Vimal had worked on every second day, and Sunil had worked on every third day, then the number of days required to complete the task would have been

Answer & solution

Answer: 139

Solution

Hard

First find the three individual one-day work rates from the two pair-rates and the actual 75/135/45-day completion. Then simulate the new schedule (Amal daily, Vimal every 2nd day, Sunil every 3rd day) and find the day the work hits 1.

1

Pair rates. Let a,v,sa,v,s be daily fractions done by Amal, Vimal, Sunil.

a+v=1150,v+s=1100\begin{aligned} &a+v=\tfrac{1}{150},\qquad v+s=\tfrac{1}{100} \end{aligned}
2

Use the actual completion. 75a+135v+45s=175a+135v+45s=1. Substitute a=1150−va=\tfrac{1}{150}-v and s=1100−vs=\tfrac{1}{100}-v.

75 ⁣(1150−v)+135v+45 ⁣(1100−v)=1⇒ 0.5−75v+135v+0.45−45v=1(evaluate constants)⇒ 0.95+15v=1 ⇒ v=1300\begin{aligned} &75\!\left(\tfrac{1}{150}-v\right)+135v+45\!\left(\tfrac{1}{100}-v\right)=1\\ &\Rightarrow\ 0.5-75v+135v+0.45-45v=1 \quad\text{(evaluate constants)}\\ &\Rightarrow\ 0.95+15v=1\ \Rightarrow\ v=\tfrac{1}{300} \end{aligned}
3

Recover all three rates.

a=1150−1300=1300s=1100−1300=2300=1150\begin{aligned} &a=\tfrac{1}{150}-\tfrac{1}{300}=\tfrac{1}{300}\\ &s=\tfrac{1}{100}-\tfrac{1}{300}=\tfrac{2}{300}=\tfrac{1}{150} \end{aligned}
4

Simulate the new schedule. In nn days Amal works all nn days, Vimal ⌊n/2⌋\lfloor n/2\rfloor days, Sunil ⌊n/3⌋\lfloor n/3\rfloor days. Work done =n300+⌊n/2⌋300+⌊n/3⌋150=1=\dfrac{n}{300}+\dfrac{\lfloor n/2\rfloor}{300}+\dfrac{\lfloor n/3\rfloor}{150}=1. Ignoring floors gives n300+n600+n450=1\tfrac{n}{300}+\tfrac{n}{600}+\tfrac{n}{450}=1, i.e. 13n1800=1⇒n≈138.5\tfrac{13n}{1800}=1\Rightarrow n\approx138.5, so test n=139n=139.

n=139: ⌊139/2⌋=69, ⌊139/3⌋=46⇒ 139300+69300+46150=139+69+92300=300300=1(exactly done)\begin{aligned} &n=139:\ \lfloor 139/2\rfloor=69,\ \lfloor 139/3\rfloor=46\\ &\Rightarrow\ \frac{139}{300}+\frac{69}{300}+\frac{46}{150}=\frac{139+69+92}{300}=\frac{300}{300}=1 \quad\text{(exactly done)} \end{aligned}
Days required=139\text{Days required}=139
CAT 2024 Slot 2 QA Q21: Amal and Vimal together can complete a task in 150 days, while Vimal and Sunil together can complete the same — Solution | TheCATExam