A can complete a work in 24 days, B in 32 days and C in 64 days. They start together. A works for 6 days and leaves and B leaves 6 days before the work is finished. In how many days was the work finished?

Options

A. 20 days
B. 21 days
C. 22 days
D. 25 days

Answer

To solve the problem step by step, we will calculate the work done by A, B, and C, and determine how long it takes to finish the work under the given conditions.

### Step 1: Determine the work rates of A, B, and C
- A can complete the work in 24 days, so A's work rate is \( \frac{1}{24} \) of the work per day.
- B can complete the work in 32 days, so B's work rate is \( \frac{1}{32} \) of the work per day.
- C can complete the work in 64 days, so C's work rate is \( \frac{1}{64} \) of the work per day.

### Step 2: Calculate the combined work rate of A, B, and C
To find the combined work rate, we sum their individual rates:
\[ \text{Combined work rate} = \frac{1}{24} + \frac{1}{32} + \frac{1}{64} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 24, 32, and 64 is 192. We convert each fraction:
- \( \frac{1}{24} = \frac{8}{192} \)
- \( \frac{1}{32} = \frac{6}{192} \)
- \( \frac{1}{64} = \frac{3}{192} \)

Now, adding them together:
\[ \text{Combined work rate} = \frac{8 + 6 + 3}{192} = \frac{17}{192} \]
So, together they can complete \( \frac{17}{192} \) of the work in one day.

### Step 3: Calculate the work done in the first 6 days
A works for 6 days, so the work done by A, B, and C in those 6 days is:
\[ \text{Work done in 6 days} = 6 \times \frac{17}{192} = \frac{102}{192} \]
This simplifies to:
\[ \frac{102}{192} = \frac{51}{96} \]

### Step 4: Calculate the remaining work
The total work is considered as 1 (or 192 units). The remaining work after 6 days is:
\[ \text{Remaining work} = 1 - \frac{51}{96} = \frac{96 - 51}{96} = \frac{45}{96} = \frac{15}{32} \]

### Step 5: Determine who works after A leaves
After 6 days, A leaves, and B leaves 6 days before the work is finished. This means C will work alone for the last 6 days.

### Step 6: Calculate the work done by C in the last 6 days
C's work rate is \( \frac{1}{64} \) of the work per day. Therefore, in 6 days, C will do:
\[ \text{Work done by C in 6 days} = 6 \times \frac{1}{64} = \frac{6}{64} = \frac{3}{32} \]

### Step 7: Calculate the remaining work after C's contribution
The remaining work after C works for 6 days is:
\[ \text{Remaining work} = \frac{15}{32} - \frac{3}{32} = \frac{12}{32} = \frac{3}{8} \]

### Step 8: Determine how long B and C will work together to finish the remaining work
B and C will work together to complete the remaining \( \frac{3}{8} \) of the work. Their combined work rate is:
\[ \text{B's work rate} + \text{C's work rate} = \frac{1}{32} + \frac{1}{64} \]
Finding a common denominator (64):
\[ \frac{1}{32} = \frac{2}{64} \]
Thus,
\[ \text{Combined work rate of B and C} = \frac{2}{64} + \frac{1}{64} = \frac{3}{64} \]

### Step 9: Calculate the time taken by B and C to finish the remaining work
To find the time \( t \) taken to finish \( \frac{3}{8} \) of the work:
\[ \frac{3}{8} = t \times \frac{3}{64} \]
Solving for \( t \):
\[ t = \frac{3/8}{3/64} = \frac{3}{8} \times \frac{64}{3} = 8 \text{ days} \]

### Step 10: Calculate the total time taken to finish the work
The total time taken to complete the work is:
\[ \text{Total time} = 6 \text{ (days A worked)} + 8 \text{ (days B and C worked)} = 14 \text{ days} \]

### Final Answer
The work was finished in **14 days**.

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