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.

Step 1: Establishing the equations

Amar and Akbar complete the project in \( 12 \) months. Hence, in a month they do: \[ \frac{48}{12} = 4 \, \text{units of work}. \] Thus, we have the equation: \[ m + k = 4 \quad \text{…… (i)}. \] Similarly, for Akbar and Anthony working together: \[ k + n = 3 \quad \text{…… (ii)}. \] And for Amar and Anthony working together: \[ m + n = 2 \quad \text{…… (iii)}. \]

Step 2: Solving the system of equations

We have the following system of equations: \[ m + k = 4 \quad \text{(i)} \] \[ k + n = 3 \quad \text{(ii)} \] \[ m + n = 2 \quad \text{(iii)}. \] Solving these equations: - From equation (i), we have: \[ k = 4 - m. \] - Substituting this into equation (ii): \[ (4 - m) + n = 3 \quad \Rightarrow \quad n = 3 - (4 - m) = m - 1. \] - Now substitute \( n = m - 1 \) into equation (iii): \[ m + (m - 1) = 2 \quad \Rightarrow \quad 2m - 1 = 2 \quad \Rightarrow \quad 2m = 3 \quad \Rightarrow \quad m = \frac{3}{2}. \] - Substituting \( m = \frac{3}{2} \) into \( k = 4 - m \): \[ k = 4 - \frac{3}{2} = \frac{5}{2}. \] - Finally, substituting \( m = \frac{3}{2} \) into \( n = m - 1 \): \[ n = \frac{3}{2} - 1 = \frac{1}{2}. \]

Step 3: Time taken to complete the work

Amar works \( \frac{3}{2} \) units of work in a month. To complete the total work of 48 units, the time taken is: \[ \text{Time taken} = \frac{48}{\frac{3}{2}} = 32 \, \text{months}. \]

Final Answer:

The time taken to complete the work is \( \boxed{32} \) months.