Theere are 4 people whow can complete a work in 19 days. Individually. The work is started by one of people on the first day. Everyday one more person joins and starting from the 4th days, all of the 4 people work be completed?

Options

A. `6(1)/(4)` days
B. `6 (1)/(19)`days
C. 7 days
D. `7(1)/(19)` days

Explanation

To solve the problem step by step, we will follow the process of calculating the work done by each person and the total work completed over the days.

### Step-by-Step Solution:

1. **Understanding the Work Rate**:
Each of the 4 people can complete the work in 19 days. Therefore, the work done by one person in one day is:
\[ \text{Work done by one person in one day} = \frac{1}{19} \]

2. **Calculating Work Done Each Day**:
- **Day 1**: Only 1 person works:
\[ \text{Work done on Day 1} = \frac{1}{19} \]
- **Day 2**: 1 more person joins, so 2 people work:
\[ \text{Work done on Day 2} = 2 \times \frac{1}{19} = \frac{2}{19} \]
- **Day 3**: 1 more person joins, so 3 people work:
\[ \text{Work done on Day 3} = 3 \times \frac{1}{19} = \frac{3}{19} \]
- **Day 4**: All 4 people work:
\[ \text{Work done on Day 4} = 4 \times \frac{1}{19} = \frac{4}{19} \]

3. **Calculating Total Work Done in 4 Days**:
Now, we add the work done over the first 4 days:
\[ \text{Total work done in 4 days} = \frac{1}{19} + \frac{2}{19} + \frac{3}{19} + \frac{4}{19} = \frac{10}{19} \]

4. **Calculating Remaining Work**:
The total work is considered as 1 (the whole work). Therefore, the remaining work after 4 days is:
\[ \text{Remaining work} = 1 - \frac{10}{19} = \frac{19 - 10}{19} = \frac{9}{19} \]

5. **Calculating Work Done by 4 People Together**:
From Day 5 onwards, all 4 people work together. The work done by 4 people in one day is:
\[ \text{Work done by 4 people in one day} = 4 \times \frac{1}{19} = \frac{4}{19} \]

6. **Calculating Days Required to Complete Remaining Work**:
To find out how many days (let's call it \(X\)) are needed to complete the remaining work of \(\frac{9}{19}\):
\[ \frac{4}{19} \times X = \frac{9}{19} \]
Solving for \(X\):
\[ X = \frac{9}{19} \div \frac{4}{19} = \frac{9}{4} = 2.25 \text{ days} \]

7. **Total Days Taken**:
The total number of days taken to complete the work is:
\[ \text{Total days} = 4 \text{ (initial days)} + 2.25 \text{ (additional days)} = 6.25 \text{ days} \]

### Final Answer:
The total time taken to complete the work is **6.25 days**.