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

Answer

To solve the problem step by step, we can use the concept of work rates. Let's denote the work done by A, B, and C in one day as \( A \), \( B \), and \( C \) respectively.

### Step 1: Set up the equations based on the given information.

1. A and B can complete the work in 72 days:
\[ A + B = \frac{1}{72} \quad \text{(work done in one day)} \]

2. B and C can complete the work in 120 days:
\[ B + C = \frac{1}{120} \quad \text{(work done in one day)} \]

3. A and C can complete the work in 90 days:
\[ A + C = \frac{1}{90} \quad \text{(work done in one day)} \]

### Step 2: Add the three equations.

Now, we will add all three equations:
\[ (A + B) + (B + C) + (A + C) = \frac{1}{72} + \frac{1}{120} + \frac{1}{90} \]
This simplifies to:
\[ 2A + 2B + 2C = \frac{1}{72} + \frac{1}{120} + \frac{1}{90} \]

### Step 3: Calculate the right-hand side.

To add the fractions, we need a common denominator. The least common multiple (LCM) of 72, 120, and 90 is 360. Now, we convert each fraction:
- For \( \frac{1}{72} \):
\[ \frac{1}{72} = \frac{5}{360} \]

- For \( \frac{1}{120} \):
\[ \frac{1}{120} = \frac{3}{360} \]

- For \( \frac{1}{90} \):
\[ \frac{1}{90} = \frac{4}{360} \]

Now, we can add them:
\[ \frac{5}{360} + \frac{3}{360} + \frac{4}{360} = \frac{12}{360} = \frac{1}{30} \]

### Step 4: Substitute back into the equation.

Now substituting back into our equation:
\[ 2A + 2B + 2C = \frac{1}{30} \]
Dividing the entire equation by 2 gives:
\[ A + B + C = \frac{1}{60} \]

### Step 5: Find A's work rate.

Now we can express A's work rate in terms of B and C:
\[ A = (A + B + C) - (B + C) \]
Substituting the values:
\[ A = \frac{1}{60} - \frac{1}{120} \]

### Step 6: Calculate A's work rate.

To subtract these fractions, we need a common denominator, which is 120:
\[ A = \frac{2}{120} - \frac{1}{120} = \frac{1}{120} \]

### Step 7: Find the time taken by A alone.

Since A can do \( \frac{1}{120} \) of the work in one day, the time taken by A to complete the work alone is:
\[ \text{Time} = \frac{1}{A} = 120 \text{ days} \]

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

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