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. 20
B. 30
C. 18
D. 24

Answer

To solve the problem step by step, we'll first determine the work done by A, B, and C, and then calculate how long A and B will take to finish the remaining work after C quits.

### Step 1: Determine the work rates of A, B, and C.
- A can complete the job in 50 days, so A's work rate is:
\[ \text{Work rate of A} = \frac{1}{50} \text{ of the job per day} \]

- B can complete the job in 75 days, so B's work rate is:
\[ \text{Work rate of B} = \frac{1}{75} \text{ of the job per day} \]

- C can complete the job in 20 days, so C's work rate is:
\[ \text{Work rate of C} = \frac{1}{20} \text{ of the job per day} \]

### Step 2: Calculate the combined work rate of A, B, and C.
To find the combined work rate, we add their individual work rates:
\[ \text{Combined work rate} = \frac{1}{50} + \frac{1}{75} + \frac{1}{20} \]

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:
\[ \frac{1}{50} = \frac{6}{300}, \quad \frac{1}{75} = \frac{4}{300}, \quad \frac{1}{20} = \frac{15}{300} \]

Adding these together:
\[ \text{Combined work rate} = \frac{6 + 4 + 15}{300} = \frac{25}{300} = \frac{1}{12} \text{ of the job per day} \]

### Step 3: Calculate the total work done in 4 days.
Now, we calculate how much work A, B, and C can complete together in 4 days:
\[ \text{Work done in 4 days} = 4 \times \frac{1}{12} = \frac{4}{12} = \frac{1}{3} \text{ of the job} \]

### Step 4: Determine the remaining work.
Since they completed \(\frac{1}{3}\) of the job, the remaining work is:
\[ \text{Remaining work} = 1 - \frac{1}{3} = \frac{2}{3} \text{ of the job} \]

### Step 5: Calculate the work rate of A and B together.
Next, we find the combined work rate of A and B:
\[ \text{Work rate of A and B} = \frac{1}{50} + \frac{1}{75} \]

Using the same common denominator of 300:
\[ \frac{1}{50} = \frac{6}{300}, \quad \frac{1}{75} = \frac{4}{300} \]
\[ \text{Combined work rate of A and B} = \frac{6 + 4}{300} = \frac{10}{300} = \frac{1}{30} \text{ of the job per day} \]

### Step 6: Calculate the time taken by A and B to finish the remaining work.
To find the time \(T\) taken by A and B to complete the remaining \(\frac{2}{3}\) of the job:
\[ T = \frac{\text{Remaining work}}{\text{Work rate of A and B}} = \frac{\frac{2}{3}}{\frac{1}{30}} = \frac{2}{3} \times 30 = 20 \text{ days} \]

### Final Answer:
A and B will take **20 days** to finish the rest of the job.