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CAT 2021 Slot 1 — QA Question 17

BasicsEasy

Amar, Akbar and Anthony are working on a project. Working together Amar and Akbar can complete the project in 1 year, Akbar and Anthony can complete in 16 months, Anthony and Amar can complete in 2 years. If the person who is neither the fastest nor the slowest works alone, the time in months he will take to complete the project is

Answer & solution

Answer: 32

Solution

Easy

Work in "units of work per month." Convert each pair-rate into combined efficiency, add the three pair-equations to get the total efficiency, then back out each individual rate. The middle worker (neither fastest nor slowest) gives the required time.

1

Set total work and rates. Let total work =LCM⁡(12,16,24)=48=\operatorname{LCM}(12,16,24)=48 units. Let Amar, Akbar, Anthony have efficiencies x,y,zx,y,z units/month. Convert each time to months: 11 year =12=12, 1616 months, 22 years =24=24.

x+y=4812=4(Amar+Akbar)(1)y+z=4816=3(Akbar+Anthony)(2)z+x=4824=2(Anthony+Amar)(3)\begin{aligned} &x+y=\tfrac{48}{12}=4\quad\text{(Amar+Akbar)}\quad(1)\\ &y+z=\tfrac{48}{16}=3\quad\text{(Akbar+Anthony)}\quad(2)\\ &z+x=\tfrac{48}{24}=2\quad\text{(Anthony+Amar)}\quad(3) \end{aligned}
2

Add all three equations.

2(x+y+z)=4+3+2=9⇒ x+y+z=4.5(4)\begin{aligned} &2(x+y+z)=4+3+2=9\\ &\Rightarrow\ x+y+z=4.5\quad(4) \end{aligned}
3

Isolate each rate by subtracting a pair-equation from (4).

z=4.5−(x+y)=4.5−4=0.5[(4)−(1)]x=4.5−(y+z)=4.5−3=1.5[(4)−(2)]y=4.5−(z+x)=4.5−2=2.5[(4)−(3)]\begin{aligned} &z=4.5-(x+y)=4.5-4=0.5\quad\text{[(4)}-(1)\text{]}\\ &x=4.5-(y+z)=4.5-3=1.5\quad\text{[(4)}-(2)\text{]}\\ &y=4.5-(z+x)=4.5-2=2.5\quad\text{[(4)}-(3)\text{]} \end{aligned}
4

Identify the middle worker and find his time. Efficiencies are Akbar 2.5>2.5> Amar 1.5>1.5> Anthony 0.50.5, so the one who is neither fastest nor slowest is Amar with x=1.5x=1.5 units/month.

Time=481.5=32 months\begin{aligned} &\text{Time}=\frac{48}{1.5}=32\ \text{months} \end{aligned}
32 months32\ \text{months}
CAT 2021 Slot 1 QA Q17: Amar, Akbar and Anthony are working on a project. Working together Amar and Akbar can complete the project in — Solution | TheCATExam