Time and Work Pairwise Completion

Let the work rates of Amar, Akbar, and Anthony be $A, B, C$ parts/month respectively. 1 year $= 12$ months, 2 years $= 24$ months.

$A + B = \frac{1}{12}$ $B + C = \frac{1}{16}$ $C + A = \frac{1}{24}$

Sum of all three equations: $2(A + B + C) = \frac{1}{12} + \frac{1}{16} + \frac{1}{24} = \frac{4 + 3 + 2}{48} = \frac{9}{48} = \frac{3}{16}$ $A + B + C = \frac{3}{32}$

Now calculate individual rates:

• $C = (A + B + C) - (A + B) = \frac{3}{32} - \frac{1}{12} = \frac{9 - 8}{96} = \frac{1}{96}$
• $A = (A + B + C) - (B + C) = \frac{3}{32} - \frac{1}{16} = \frac{1}{32}$
• $B = (A + B + C) - (C + A) = \frac{3}{32} - \frac{1}{24} = \frac{9 - 4}{96} = \frac{5}{96}$

Comparing rates: $B = \frac{5}{96} > A = \frac{3}{96} > C = \frac{1}{96}$. The person who is neither fastest nor slowest is Amar ($A = \frac{1}{32}$). Time taken by Amar alone $= 32$ months.