A, B and C can do a job working alone in 50, 75 and 20 days respectively. They all work together for 4 days, then C quits. How many days will A and B take to finish the rest of the job?

Options

A. A) 20
B. B) 30
C. C) 18
D. D) 24

Answer

To solve the problem step by step, we will first determine the work done by A, B, and C individually, then calculate how much work they complete together in 4 days, and finally find out how long A and B will take to finish the remaining work.

### Step 1: Calculate the work done by A, B, and C in one day.

- A can complete the job in 50 days, so the work done by A in one day is:
\[ \text{Work by A} = \frac{1}{50} \text{ of the job} \]

- B can complete the job in 75 days, so the work done by B in one day is:
\[ \text{Work by B} = \frac{1}{75} \text{ of the job} \]

- C can complete the job in 20 days, so the work done by C in one day is:
\[ \text{Work by C} = \frac{1}{20} \text{ of the job} \]

### 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 sum their individual contributions:
\[ \text{Total work in one day} = \frac{1}{50} + \frac{1}{75} + \frac{1}{20} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 50, 75, and 20 is 300. Now we convert each fraction:

- For A:
\[ \frac{1}{50} = \frac{6}{300} \]

- For B:
\[ \frac{1}{75} = \frac{4}{300} \]

- For C:
\[ \frac{1}{20} = \frac{15}{300} \]

Now, adding these together:
\[ \text{Total work in one day} = \frac{6 + 4 + 15}{300} = \frac{25}{300} = \frac{1}{12} \]

### Step 3: Calculate the total work done by A, B, and C in 4 days.

If they work together for 4 days, the total work done is:
\[ \text{Work done in 4 days} = 4 \times \frac{1}{12} = \frac{4}{12} = \frac{1}{3} \]

### Step 4: Calculate the remaining work.

The total work is considered as 1 (the whole job). Therefore, the remaining work after 4 days is:
\[ \text{Remaining work} = 1 - \frac{1}{3} = \frac{2}{3} \]

### Step 5: Calculate the work done by A and B together in one day.

Now, we need to find out how much work A and B can do together in one day:
\[ \text{Work by A and B in one day} = \frac{1}{50} + \frac{1}{75} \]

Using the common denominator of 150:
- For A:
\[ \frac{1}{50} = \frac{3}{150} \]

- For B:
\[ \frac{1}{75} = \frac{2}{150} \]

Adding these:
\[ \text{Total work by A and B in one day} = \frac{3 + 2}{150} = \frac{5}{150} = \frac{1}{30} \]

### Step 6: Calculate the number of days A and B will take to finish the remaining work.

Now we need to find how many days it will take A and B to complete the remaining work of \(\frac{2}{3}\):
\[ \text{Days required} = \frac{\text{Remaining work}}{\text{Work done by A and B in one day}} = \frac{\frac{2}{3}}{\frac{1}{30}} = \frac{2}{3} \times 30 = 20 \]

### Final Answer:
A and B will take **20 days** to finish the remaining work.

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