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

Correct Answer: 32

We are given the following information about the efficiencies of Amar, Akbar, and Anthony:

• Amar and Akbar can complete the project together in 1 year (12 months).
• Akbar and Anthony can complete the project together in 16 months.
• Anthony and Amar can complete the project together in 2 years (24 months).

Step 1: Setting up the Equations

Let the efficiencies of Amar, Akbar, and Anthony be denoted as \(x\), \(y\), and \(z\), respectively.

The given information can be translated into the following equations:

• Amar and Akbar's combined efficiency: \[ x + y = 112 \quad \text{(Equation 1)} \]
• Akbar and Anthony's combined efficiency: \[ y + z = 116 \quad \text{(Equation 2)} \]
• Anthony and Amar's combined efficiency: \[ z + x = 124 \quad \text{(Equation 3)} \]

Step 2: Adding the Equations

We add all three equations to find a relationship between \(x\), \(y\), and \(z\):

\[ (x + y) + (y + z) + (z + x) = 112 + 116 + 124 \]

Simplifying this equation:

\[ 2(x + y + z) = 352 \]

So we have:

\[ x + y + z = \frac{352}{2} = 332 \]

Step 3: Solving for Individual Efficiencies

Now, we can solve for the individual efficiencies \(x\), \(y\), and \(z\) by using the above equation along with the original ones:

• From \(x + y = 112\) (Equation 1): \[ x = 332 - 116 = 132 \]
• From \(y + z = 116\) (Equation 2): \[ y = 332 - 124 = 208 \]
• From \(z + x = 124\) (Equation 3): \[ z = 332 - 112 = 220 \]

Step 4: Identifying the Worker Who is Neither the Fastest Nor the Slowest

Now we have the following efficiencies:

• Amar's efficiency \(x = 132\)
• Akbar's efficiency \(y = 208\)
• Anthony's efficiency \(z = 220\)

Clearly, Amar's efficiency (\(x = 132\)) is neither the fastest nor the slowest, so Amar is the worker who is neither the fastest nor the slowest.

Step 5: Time Taken by Amar to Complete the Project Alone

To find the time taken by Amar to complete the project alone, we use the formula for time:

\[ \text{Time} = \frac{1}{\text{Efficiency}} = \frac{1}{x} \]

Substitute \(x = 132\) into the formula:

\[ \text{Time} = \frac{1}{132} \quad \Rightarrow \quad \text{Time} = 32 \text{ months}. \]

Conclusion

Therefore, Amar will take 32 months to complete the project on his own.