A and B can do a work in 45 days, B and C in 36 days and A and C in 60 days. In how many days can A,B and C individually do the work?

Answer

To solve the problem of how many days A, B, and C can individually complete the work, we can follow these steps:

### Step 1: Determine the Work Rates
We know the following:
- A and B can complete the work in 45 days.
- B and C can complete the work in 36 days.
- A and C can complete the work in 60 days.

We can express the work done in one day by each pair:
- Work done by A and B in one day = \( \frac{1}{45} \)
- Work done by B and C in one day = \( \frac{1}{36} \)
- Work done by A and C in one day = \( \frac{1}{60} \)

### Step 2: Set Up the Equations
Let:
- \( A \) = work done by A in one day
- \( B \) = work done by B in one day
- \( C \) = work done by C in one day

From the information given, we can write the following equations:
1. \( A + B = \frac{1}{45} \) (Equation 1)
2. \( B + C = \frac{1}{36} \) (Equation 2)
3. \( A + C = \frac{1}{60} \) (Equation 3)

### Step 3: Add the Equations
Now, we will add all three equations together:
\[ (A + B) + (B + C) + (A + C) = \frac{1}{45} + \frac{1}{36} + \frac{1}{60} \]
This simplifies to:
\[ 2A + 2B + 2C = \frac{1}{45} + \frac{1}{36} + \frac{1}{60} \]

### Step 4: Calculate the Right Side
To combine the fractions on the right side, we need to find a common denominator. The least common multiple (LCM) of 45, 36, and 60 is 180. We convert each fraction:
- \( \frac{1}{45} = \frac{4}{180} \)
- \( \frac{1}{36} = \frac{5}{180} \)
- \( \frac{1}{60} = \frac{3}{180} \)

Adding these gives:
\[ \frac{4}{180} + \frac{5}{180} + \frac{3}{180} = \frac{12}{180} = \frac{1}{15} \]

### Step 5: Solve for A + B + C
Now we have:
\[ 2A + 2B + 2C = \frac{1}{15} \]
Dividing everything by 2:
\[ A + B + C = \frac{1}{30} \]

### Step 6: Find Individual Work Rates
Now we can find the individual work rates by substituting back into the equations.

#### Finding A:
From Equation 1:
\[ A + B = \frac{1}{45} \]
Substituting \( B = \frac{1}{36} - C \) from Equation 2 into Equation 1, we can express A in terms of C:
\[ A + \left(\frac{1}{36} - C\right) = \frac{1}{45} \]
This will require further substitution and simplification.

#### Finding B:
Using Equation 2:
\[ B + C = \frac{1}{36} \]
Substituting \( C = \frac{1}{60} - A \) from Equation 3 into Equation 2.

#### Finding C:
Using Equation 3:
\[ A + C = \frac{1}{60} \]
Substituting \( A = \frac{1}{30} - B \) from the total work rate.

### Final Calculation:
After substituting and solving all equations, we find:
- \( A = \frac{1}{180} \) → A can do the work in 180 days.
- \( B = \frac{1}{60} \) → B can do the work in 60 days.
- \( C = \frac{1}{90} \) → C can do the work in 90 days.

### Conclusion:
Thus, the individual work rates are:
- A can complete the work in **180 days**.
- B can complete the work in **60 days**.
- C can complete the work in **90 days**.