The daily work of two men is equal to that of 3 women or that of 4 youngsters. By employing 14 men, 12 women, and 12 youngsters a certain work can be finished in 24 days.
If it is required to finish it in 14 days and as an additional labor, only men are available, how many of them will be required?

Options

A. A)20
B. B)30
C. C)25
D. D)15

Answer

To solve the problem step by step, we will first establish the efficiency of each type of labor (men, women, and youngsters) and then calculate the total work done. Finally, we will determine how many additional men are needed to complete the work in a shorter time frame.

### Step 1: Establish the efficiency of men, women, and youngsters
The problem states that the daily work of 2 men is equal to that of 3 women or that of 4 youngsters.

Let:
- The efficiency of 1 man = M
- The efficiency of 1 woman = W
- The efficiency of 1 youngster = Y

From the problem, we have:
- 2M = 3W → (1)
- 2M = 4Y → (2)

From (1), we can express W in terms of M:
\[ W = \frac{2M}{3} \]

From (2), we can express Y in terms of M:
\[ Y = \frac{2M}{4} = \frac{M}{2} \]

Now, we can express the efficiencies in terms of M:
- Efficiency of 1 man (M) = M
- Efficiency of 1 woman (W) = \( \frac{2M}{3} \)
- Efficiency of 1 youngster (Y) = \( \frac{M}{2} \)

### Step 2: Find the total efficiency of the workers employed
The problem states that 14 men, 12 women, and 12 youngsters are employed.

Total efficiency of the workers:
\[ \text{Total Efficiency} = (14 \times M) + (12 \times \frac{2M}{3}) + (12 \times \frac{M}{2}) \]

Calculating each term:
- Efficiency from men = \( 14M \)
- Efficiency from women = \( 12 \times \frac{2M}{3} = 8M \)
- Efficiency from youngsters = \( 12 \times \frac{M}{2} = 6M \)

Adding these together:
\[ \text{Total Efficiency} = 14M + 8M + 6M = 28M \]

### Step 3: Calculate the total work done
The work can be finished in 24 days, so the total work (W) can be calculated as:
\[ W = \text{Total Efficiency} \times \text{Number of Days} = 28M \times 24 = 672M \]

### Step 4: Determine the required work per day to finish in 14 days
If we want to finish the same work in 14 days, the required daily work (R) would be:
\[ R = \frac{W}{14} = \frac{672M}{14} = 48M \]

### Step 5: Calculate the number of additional men required
We already know the efficiency of 14 men is \( 14M \). To find out how many additional men (let's denote this number as \( x \)) are needed, we set up the equation:
\[ 14M + xM = 48M \]

Solving for \( x \):
\[ xM = 48M - 14M \]
\[ xM = 34M \]
\[ x = 34 \]

Thus, the number of additional men required is **34**.

### Summary of Steps:
1. Establish the efficiency of men, women, and youngsters.
2. Calculate the total efficiency of the employed workers.
3. Calculate the total work done in 24 days.
4. Determine the required daily work to finish in 14 days.
5. Calculate the number of additional men required.