A, B and C can do a piece of work in 24 days, 30 days and 40 days respectively. They began the work together but C left 4 days before the completion of the work. In how many days was the work completed?

Options

A. 13
B. 12
C. 14
D. 11

Answer

To solve the problem step by step, we will calculate the total work done by A, B, and C, and then determine how long it took to complete the work given that C left 4 days before the work was finished.

### Step 1: Determine the work done by A, B, and C in one day.
- A can complete the work in 24 days, so A's work in one day = 1/24 of the work.
- B can complete the work in 30 days, so B's work in one day = 1/30 of the work.
- C can complete the work in 40 days, so C's work in one day = 1/40 of the work.

### Step 2: Find the total work done by A, B, and C in one day.
To find the total work done by A, B, and C together in one day, we need to add their individual work rates:
\[ \text{Total work in one day} = \frac{1}{24} + \frac{1}{30} + \frac{1}{40} \]

### Step 3: Calculate the least common multiple (LCM) of the denominators.
The LCM of 24, 30, and 40 is 120. We will convert each fraction to have a common denominator of 120:
- A's work: \(\frac{1}{24} = \frac{5}{120}\)
- B's work: \(\frac{1}{30} = \frac{4}{120}\)
- C's work: \(\frac{1}{40} = \frac{3}{120}\)

Now, add these fractions:
\[ \text{Total work in one day} = \frac{5}{120} + \frac{4}{120} + \frac{3}{120} = \frac{12}{120} = \frac{1}{10} \]

### Step 4: Determine the work done in the last 4 days.
Since C left 4 days before the work was completed, only A and B worked during those last 4 days.
- A's work in one day = \(\frac{5}{120}\)
- B's work in one day = \(\frac{4}{120}\)

Together, A and B's work in one day:
\[ \text{A and B's work in one day} = \frac{5}{120} + \frac{4}{120} = \frac{9}{120} = \frac{3}{40} \]

In 4 days, A and B will complete:
\[ \text{Work done in 4 days} = 4 \times \frac{3}{40} = \frac{12}{40} = \frac{3}{10} \]

### Step 5: Calculate the total work and remaining work.
The total work is 1 (the whole work). If A and B completed \(\frac{3}{10}\) of the work in the last 4 days, then the remaining work before those 4 days was:
\[ \text{Remaining work} = 1 - \frac{3}{10} = \frac{7}{10} \]

### Step 6: Calculate the time taken to complete the remaining work.
To find out how many days it took A, B, and C to complete the remaining \(\frac{7}{10}\) of the work, we use their combined work rate:
\[ \text{Work rate of A, B, C} = \frac{1}{10} \text{ (from Step 3)} \]
Let \(x\) be the number of days they worked together:
\[ \frac{1}{10} \times x = \frac{7}{10} \]
Solving for \(x\):
\[ x = 7 \text{ days} \]

### Step 7: Calculate the total time taken to complete the work.
The total time taken to complete the work is the time they worked together plus the last 4 days:
\[ \text{Total time} = 7 + 4 = 11 \text{ days} \]

### Final Answer:
The work was completed in **11 days**.

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