8 men and 14 women are working together in a field. After working for 3 days, 5 men and 8 women leave the work. How many more days will be required to complete the work?
I. 19 men and 12 women together can complete the work in 18 days.
II. 16 men complete two-third of the work in 16 days
III. In a day, the work done by three men is equal to the work done by four women.

Options

A. I only
B. II only
C. III only
D. I or II or III

Answer

To solve the problem step-by-step, we need to determine how many more days will be required to complete the work after some workers leave.

### Step 1: Understand the initial work done
We have:
- 8 men and 14 women working together for 3 days.

Let:
- The work done by 1 man in 1 day = x
- The work done by 1 woman in 1 day = y

The total work done by 8 men and 14 women in 3 days is:
\[ \text{Work done in 3 days} = 3 \times (8x + 14y) = 24x + 42y \]

### Step 2: Calculate remaining workers after 3 days
After 3 days, 5 men and 8 women leave:
- Remaining men = 8 - 5 = 3
- Remaining women = 14 - 8 = 6

### Step 3: Set up the equation for remaining work
Let D be the number of days required to complete the remaining work. The work done by the remaining workers in D days is:
\[ \text{Remaining work} = D \times (3x + 6y) \]

### Step 4: Total work equation
The total work done can be expressed as:
\[ \text{Total work} = \text{Work done in 3 days} + \text{Remaining work} \]
This gives us:
\[ 1 = 24x + 42y + D(3x + 6y) \]

### Step 5: Analyze the statements
We have three statements that can help us find the values of x and y.

**Statement I:** 19 men and 12 women can complete the work in 18 days.
\[ \text{Work} = 19x + 12y \text{ in 1 day} \implies 18(19x + 12y) = 1 \]

**Statement II:** 16 men complete two-thirds of the work in 16 days.
\[ \text{Work} = 16x \text{ in 1 day} \implies 16(16x) = \frac{2}{3} \implies 16x = \frac{2}{48} \implies x = \frac{1}{384} \]

**Statement III:** The work done by 3 men is equal to the work done by 4 women.
\[ 3x = 4y \implies y = \frac{3}{4}x \]

### Step 6: Solve the equations
Using Statement II, we can find x and substitute it into Statement III to find y:
- From Statement II, we have \( x = \frac{1}{384} \).
- Substitute \( x \) into Statement III to find \( y \):
\[ y = \frac{3}{4} \times \frac{1}{384} = \frac{3}{1536} = \frac{1}{512} \]

### Step 7: Substitute values back into the total work equation
Now substituting x and y back into the total work equation:
\[ 1 = 24\left(\frac{1}{384}\right) + 42\left(\frac{1}{512}\right) + D\left(3\left(\frac{1}{384}\right) + 6\left(\frac{1}{512}\right)\right) \]

### Step 8: Calculate the remaining work and solve for D
Calculate the left side:
\[ 1 = \frac{24}{384} + \frac{42}{512} + D\left(\frac{3}{384} + \frac{6}{512}\right) \]
Now, simplify and solve for D.

### Final Calculation
After calculating the fractions and simplifying, we find the value of D, which represents the number of additional days needed to complete the work.