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

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Let’s draw the diagram for a better understanding.

$\begin{array}{llll} & \textbf{Amar + Akbar} & \textbf{Akbar + Anthony} & \textbf{Anthony + Amar} \\\hline \text{Time :} & 12\;\text{months} & 16\;\text{months} & 24\;\text{months} \\ \text{Total work :} & \text{LCM(12,16,24)} & = & 48\;\text{units} \\ \text{Efficiency :} & 4\;\text{units/month} & 3\;\text{units/month} & 2\;\text{units/month} \end{array}$

Let the efficiency of Amar, Akbar, and Anthony be $x, y$, and $z\;\text{units/month}.$

Now, $x+y+y+z+z+x = 4+3+2$

$\Rightarrow 2(x+y+z) = 9$

$\Rightarrow \boxed{x+y+z = 4.5}$

Now, we can calculate the individual efficiency.

- The efficiency of Amar $x = 4.5-3 = 1.5$
- The efficiency of Akbar $y = 4.5-2 = 2.5$
- The efficiency of Anthony $z = 4.5-4 = 0.5$

We can say that Anthony is the slowest, Akbar is the fastest.

$\therefore$ The Person who is neither the fastest nor the slowest (Amar) works alone, the time is taken by him to complete the project $= \frac{48\; \text{units}}{1.5\;\text{units/month}} = 32\; \text{months}.$

Correct Answer $: 32$