16 children can complete the work in 12 days, whereas 6 adults take 16 days to complete the same work. This work is started by 16 adults and after 3 days. 10 adults left the work, then, 4 children joined the work Accordingly, in how many days the remaining work get complete?

Options

A. 12 days
B. 15 days
C. 6 days
D. 9 days

Answer

To solve the problem step by step, let's break it down:

### Step 1: Determine the total work in terms of units
We know that 16 children can complete the work in 12 days.

- The total work done by 16 children in 12 days can be calculated as:
\[ \text{Total Work} = \text{Number of Children} \times \text{Days} = 16 \times 12 = 192 \text{ units} \]

### Step 2: Calculate the work done by 6 adults in 16 days
We also know that 6 adults can complete the same work in 16 days.

- The total work done by 6 adults in 16 days is also:
\[ \text{Total Work} = 6 \times 16 = 96 \text{ units} \]

Since both calculations represent the same total work, we can set them equal to each other to find the efficiency ratio between children and adults.

### Step 3: Find the efficiency ratio of children and adults
From the previous calculations, we can find the efficiency of one adult in terms of children.

- Since 16 children do 192 units of work, the work done by 1 child in 1 day is:
\[ \text{Work per Child per Day} = \frac{192}{16 \times 12} = 1 \text{ unit} \]

- For adults, since 6 adults do 192 units in 16 days, the work done by 1 adult in 1 day is:
\[ \text{Work per Adult per Day} = \frac{192}{6 \times 16} = 2 \text{ units} \]

Thus, the ratio of efficiency of children to adults is:
\[ \text{Efficiency Ratio} = 1 \text{ (child)} : 2 \text{ (adult)} \Rightarrow 2 : 1 \]

### Step 4: Work done by 16 adults in 3 days
Now, we need to calculate how much work is done by 16 adults in the first 3 days.

- Since 1 adult does 2 units of work per day, 16 adults will do:
\[ \text{Work done by 16 adults in 1 day} = 16 \times 2 = 32 \text{ units} \]

- Therefore, in 3 days, the work done is:
\[ \text{Total Work in 3 Days} = 32 \times 3 = 96 \text{ units} \]

### Step 5: Calculate the remaining work
Now, we subtract the work done in the first 3 days from the total work to find the remaining work.

- Remaining work:
\[ \text{Remaining Work} = 192 - 96 = 96 \text{ units} \]

### Step 6: Work done after 3 days
After 3 days, 10 adults leave, leaving 6 adults. Then, 4 children join.

- Now we have 6 adults and 4 children working together. The work done by them in one day is:
\[ \text{Work done by 6 adults} = 6 \times 2 = 12 \text{ units} \]
\[ \text{Work done by 4 children} = 4 \times 1 = 4 \text{ units} \]

- Total work done by 6 adults and 4 children in one day:
\[ \text{Total Work in 1 Day} = 12 + 4 = 16 \text{ units} \]

### Step 7: Calculate the number of days to complete the remaining work
Now we need to find out how many days it will take to complete the remaining 96 units of work.

- Number of days required:
\[ \text{Days} = \frac{\text{Remaining Work}}{\text{Work done in 1 Day}} = \frac{96}{16} = 6 \text{ days} \]

### Final Answer
Thus, the remaining work will be completed in **6 days**.

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