Question

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 1

Let the total work be $48$ units.
Let Amar do $m$ work, Akbar do $k$ work, and Anthony do $n$ units of work in a month.
Amar and Akbar complete the project in $12$ months.
Hence, in a month they do $\frac {48}{12}=4$ units of work.
$⇒m+k = 4$ ……… (i)
Similarly,
$k+n = 3$ ……….. (ii)
and $m+n = 2$ ……… (iii)
On solving these equations,
$m=\frac 32$

$k=\frac 52$

$n=\frac 12$
Here, Amar works neither the fastest not the slowest and he does 1.5 units of work in a month.
Hence, to complete the work, time taken,
$=\frac {48}{1.5}$

$= 32$ months

So, the answer is $32$ months.