A and B can do a piece of work in 72 days, B and C can do it in 120 days while A and C can do it in 90 days. In what time can A alone do it?

Options

A. 80 days
B. 100 days
C. 120 days
D. 150 days

Answer

To solve the problem, we need to determine how long A alone can complete the work given the combined work rates of A, B, and C.

**Step 1: Determine the work rates of A, B, and C.**

We know:
- A and B together can complete the work in 72 days.
- B and C together can complete the work in 120 days.
- A and C together can complete the work in 90 days.

Let's denote:
- The total work as W (which we can assume to be 360 units for convenience).
- The work done by A in one day as A, by B as B, and by C as C.

From the information given, we can write the following equations based on the work rates:

1. \( A + B = \frac{W}{72} = \frac{360}{72} = 5 \) units/day
2. \( B + C = \frac{W}{120} = \frac{360}{120} = 3 \) units/day
3. \( A + C = \frac{W}{90} = \frac{360}{90} = 4 \) units/day

**Step 2: Set up the equations.**

From the equations we have:
1. \( A + B = 5 \) (Equation 1)
2. \( B + C = 3 \) (Equation 2)
3. \( A + C = 4 \) (Equation 3)

**Step 3: Solve the equations.**

To find the individual work rates, we can manipulate these equations.

First, we can add all three equations together:
\[ (A + B) + (B + C) + (A + C) = 5 + 3 + 4 \]
This simplifies to:
\[ 2A + 2B + 2C = 12 \]
Dividing the entire equation by 2 gives:
\[ A + B + C = 6 \quad (Equation 4) \]

Now, we can use Equation 4 to find A, B, and C individually.

**Step 4: Substitute to find A.**

From Equation 4:
\[ C = 6 - (A + B) \]
Substituting \(A + B = 5\) from Equation 1 into this gives:
\[ C = 6 - 5 = 1 \]

Now substituting \(C = 1\) into Equation 2:
\[ B + 1 = 3 \implies B = 2 \]

Now substituting \(B = 2\) into Equation 1:
\[ A + 2 = 5 \implies A = 3 \]

**Step 5: Determine the time taken by A alone.**

Now that we have the work rates:
- A = 3 units/day
- B = 2 units/day
- C = 1 unit/day

To find out how long A will take to complete the work alone:
\[ \text{Time taken by A} = \frac{W}{A} = \frac{360}{3} = 120 \text{ days} \]

**Final Answer: A alone can complete the work in 120 days.**

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