A can do some work in 24 days, B can do it in 32 days and C can do it in 60 days. They start working together. A left after 6 days and B left after working for 8 days. How many more days are required to complete the whole work?
A)30
B)25
C)22
D)20

Options

A. 30
B. 25
C. 22
D. 20

Answer

To solve the problem step by step, we will calculate the work done by A, B, and C, and then determine how many more days are required to complete the work after A and B leave.

### Step 1: Calculate 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 = \( \frac{1}{24} \).
- B can complete the work in 32 days, so B's work in one day = \( \frac{1}{32} \).
- C can complete the work in 60 days, so C's work in one day = \( \frac{1}{60} \).

### Step 2: Find the combined 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 add their individual work rates:

\[ \text{Total work in one day} = \frac{1}{24} + \frac{1}{32} + \frac{1}{60} \]

To add these fractions, we need to find a common denominator. The least common multiple (LCM) of 24, 32, and 60 is 480.

Now, convert each fraction:

\[ \frac{1}{24} = \frac{20}{480}, \quad \frac{1}{32} = \frac{15}{480}, \quad \frac{1}{60} = \frac{8}{480} \]

Adding these together:

\[ \text{Total work in one day} = \frac{20 + 15 + 8}{480} = \frac{43}{480} \]

### Step 3: Calculate the work done in the first 6 days.

In 6 days, the total work done by A, B, and C is:

\[ \text{Work done in 6 days} = 6 \times \frac{43}{480} = \frac{258}{480} \]

### Step 4: Calculate the work done by A and B in the next 2 days.

After 6 days, A leaves, and B continues to work for 2 more days (totaling 8 days). In these 2 days, A and B work together:

\[ \text{Work done by A in 6 days} = 6 \times \frac{1}{24} = \frac{6}{24} = \frac{1}{4} \]
\[ \text{Work done by B in 8 days} = 8 \times \frac{1}{32} = \frac{8}{32} = \frac{1}{4} \]
\[ \text{Work done by C in 8 days} = 8 \times \frac{1}{60} = \frac{8}{60} = \frac{2}{15} \]

Now, we need to add the work done by A, B, and C in the first 8 days:

\[ \text{Total work done in 8 days} = \frac{1}{4} + \frac{1}{4} + \frac{2}{15} \]

Convert to a common denominator (LCM of 4 and 15 is 60):

\[ \frac{1}{4} = \frac{15}{60}, \quad \frac{1}{4} = \frac{15}{60}, \quad \frac{2}{15} = \frac{8}{60} \]

Adding these:

\[ \text{Total work done in 8 days} = \frac{15 + 15 + 8}{60} = \frac{38}{60} = \frac{19}{30} \]

### Step 5: Calculate the total work done after 8 days.

Now, we need to find the total work done after 8 days:

\[ \text{Total work done} = \frac{258}{480} + \frac{38}{60} \]

Convert \( \frac{38}{60} \) to a denominator of 480:

\[ \frac{38}{60} = \frac{38 \times 8}{60 \times 8} = \frac{304}{480} \]

Now add:

\[ \text{Total work done} = \frac{258 + 304}{480} = \frac{562}{480} \]

### Step 6: Calculate the remaining work.

The total work is 1 (or \( \frac{480}{480} \)). The remaining work is:

\[ \text{Remaining work} = 1 - \frac{562}{480} = \frac{480 - 562}{480} = \frac{-82}{480} \]

### Step 7: Calculate how many days C needs to finish the remaining work.

C will finish the remaining work alone. C's work rate is \( \frac{1}{60} \) per day.

To find how many days C needs to complete the remaining work:

\[ \text{Days required by C} = \frac{\text{Remaining work}}{\text{C's work rate}} = \frac{\frac{82}{480}}{\frac{1}{60}} = \frac{82 \times 60}{480} = \frac{4920}{480} = 10.25 \text{ days} \]

### Final Step: Total days required to complete the work.

After A and B left, C needs approximately 10.25 days to finish the work alone.

### Final Answer:

The total number of days required to complete the work after A and B left is approximately 10.25 days.