A,B and C can do a work in 20,45 and 120 days respectively.They started the work. A left 10 days before and B left 5 days before the completion of work. In how many days is the total work completed ?

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the work done by A, B, and C in one day.
- A can complete the work in 20 days, so A's work in one day = \( \frac{1}{20} \).
- B can complete the work in 45 days, so B's work in one day = \( \frac{1}{45} \).
- C can complete the work in 120 days, so C's work in one day = \( \frac{1}{120} \).

### Step 2: Calculate the total work done by A, B, and C together in one day.
- Total work done in one day by A, B, and C together = \( \frac{1}{20} + \frac{1}{45} + \frac{1}{120} \).

To add these fractions, we need a common denominator. The least common multiple (LCM) of 20, 45, and 120 is 180.

- Convert each fraction:
- \( \frac{1}{20} = \frac{9}{180} \)
- \( \frac{1}{45} = \frac{4}{180} \)
- \( \frac{1}{120} = \frac{1.5}{180} = \frac{3}{180} \)

Now, add them together:
- Total work done in one day = \( \frac{9}{180} + \frac{4}{180} + \frac{3}{180} = \frac{16}{180} = \frac{8}{90} = \frac{4}{45} \).

### Step 3: Determine the work done before A and B leave.
Let the total number of days taken to complete the work be \( x \).

- A works for \( x - 10 \) days (because A leaves 10 days before completion).
- B works for \( x - 5 \) days (because B leaves 5 days before completion).
- C works for all \( x \) days.

### Step 4: Set up the equation for total work done.
The total work done can be expressed as:
\[ \text{Work done by A} + \text{Work done by B} + \text{Work done by C} = 1 \]
This translates to:
\[ (x - 10) \cdot \frac{1}{20} + (x - 5) \cdot \frac{1}{45} + x \cdot \frac{1}{120} = 1 \]

### Step 5: Simplify the equation.
Substituting the values:
\[ \frac{x - 10}{20} + \frac{x - 5}{45} + \frac{x}{120} = 1 \]

To eliminate the fractions, multiply through by the LCM of 20, 45, and 120, which is 180:
\[ 180 \cdot \frac{x - 10}{20} + 180 \cdot \frac{x - 5}{45} + 180 \cdot \frac{x}{120} = 180 \]
This simplifies to:
\[ 9(x - 10) + 4(x - 5) + 1.5x = 180 \]
Expanding this gives:
\[ 9x - 90 + 4x - 20 + 1.5x = 180 \]
Combining like terms:
\[ 14.5x - 110 = 180 \]
Adding 110 to both sides:
\[ 14.5x = 290 \]
Dividing by 14.5:
\[ x = 20 \]

### Conclusion:
The total work is completed in **20 days**.

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