A can do a work in 40 days, B can do the same work in 60 days and C can do the same work in 80 days. They all together started the work and A left 11 days before also B left 8 days before the completion of whole task, find the number of days taken to complete the work.

Options

A. A)13 days
B. B)26 days
C. C)39 days
D. D)52 days

Answer

To solve the problem step by step, let's break it down:

### Step 1: Determine the work done by A, B, and C in one day.
- A can complete the work in 40 days, so A's work per day = \( \frac{1}{40} \) of the work.
- B can complete the work in 60 days, so B's work per day = \( \frac{1}{60} \) of the work.
- C can complete the work in 80 days, so C's work per day = \( \frac{1}{80} \) of the work.

### Step 2: Find the Least Common Multiple (LCM) to express total work in units.
- The LCM of 40, 60, and 80 is 240.
- Thus, we can assume the total work is 240 units.

### Step 3: Calculate the efficiency of A, B, and C in units.
- A's efficiency = \( \frac{240}{40} = 6 \) units/day
- B's efficiency = \( \frac{240}{60} = 4 \) units/day
- C's efficiency = \( \frac{240}{80} = 3 \) units/day

### Step 4: Calculate the total efficiency when all three are working together.
- Total efficiency = A's efficiency + B's efficiency + C's efficiency = \( 6 + 4 + 3 = 13 \) units/day.

### Step 5: Determine how many days A and B worked.
- Let \( x \) be the total number of days taken to complete the work.
- A left 11 days before completion, so A worked for \( x - 11 \) days.
- B left 8 days before completion, so B worked for \( x - 8 \) days.
- C worked for the entire duration, \( x \) days.

### Step 6: Set up the equation based on the work done.
- Work done by A = \( 6(x - 11) \)
- Work done by B = \( 4(x - 8) \)
- Work done by C = \( 3x \)

The total work done by A, B, and C should equal the total work (240 units):
\[ 6(x - 11) + 4(x - 8) + 3x = 240 \]

### Step 7: Simplify the equation.
Expanding the equation:
\[ 6x - 66 + 4x - 32 + 3x = 240 \]
Combine like terms:
\[ (6x + 4x + 3x) - 98 = 240 \]
\[ 13x - 98 = 240 \]

### Step 8: Solve for \( x \).
Add 98 to both sides:
\[ 13x = 240 + 98 \]
\[ 13x = 338 \]
Now divide by 13:
\[ x = \frac{338}{13} = 26 \]

### Conclusion:
The total number of days taken to complete the work is **26 days**.

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