24 men can complete a work in 20 days. 36 women can do the same work in 40 days. 54 children can do that work in 40 days. 18 women and 18 children together do that work for 32 days and 'X' number of men complete the remaining work in four days, then find (X + 14) women & (X - 13) child can do the same work in how many days?

Options

A. 16 days
B. 24 days
C. 20 days
D. 30 days

Answer

To solve the problem step by step, we will first determine the efficiency of men, women, and children based on the information provided. Then we will calculate how much work is done by 18 women and 18 children in 32 days, find the remaining work, and finally determine how many days it would take for (X + 14) women and (X - 13) children to complete the same work.

### Step 1: Calculate the total work done by men, women, and children.

1. **Total work by men:**
\[ \text{Total work} = \text{Number of men} \times \text{Days} = 24 \times 20 = 480 \text{ man-days} \]

2. **Total work by women:**
\[ \text{Total work} = \text{Number of women} \times \text{Days} = 36 \times 40 = 1440 \text{ woman-days} \]

3. **Total work by children:**
\[ \text{Total work} = \text{Number of children} \times \text{Days} = 54 \times 40 = 2160 \text{ child-days} \]

Since all three calculations represent the same total work, we can equate them to find the efficiency ratios.

### Step 2: Set up the efficiency ratios.

From the total work calculations:
\[ 24 \times 20 = 36 \times 40 = 54 \times 40 \]

This gives us:
\[ 480 = 1440 = 2160 \]

Now, we can express the efficiencies in terms of a common unit. Let:
- Efficiency of 1 man = M
- Efficiency of 1 woman = W
- Efficiency of 1 child = C

From the equations:
\[ 24M \times 20 = 36W \times 40 = 54C \times 40 \]

This simplifies to:
\[ M = \frac{1}{24 \times 20} = \frac{1}{480}, \quad W = \frac{1}{36 \times 40} = \frac{1}{1440}, \quad C = \frac{1}{54 \times 40} = \frac{1}{2160} \]

### Step 3: Find the ratio of efficiencies.

To find the ratio of efficiencies, we can express them in a common format:
\[ \frac{M}{W} = \frac{480}{1440} = \frac{1}{3}, \quad \frac{M}{C} = \frac{480}{2160} = \frac{1}{4.5} \]

Thus, we can express the efficiencies in the simplest ratio:
\[ M : W : C = 9 : 3 : 2 \]

### Step 4: Calculate the work done by 18 women and 18 children in 32 days.

1. **Work done by 18 women in 32 days:**
\[ \text{Work} = 18W \times 32 = 18 \times 3 \times 32 = 1728 \]

2. **Work done by 18 children in 32 days:**
\[ \text{Work} = 18C \times 32 = 18 \times 2 \times 32 = 1152 \]

3. **Total work done by 18 women and 18 children:**
\[ \text{Total work} = 1728 + 1152 = 2880 \]

### Step 5: Calculate the remaining work.

1. **Total work is 4320 (from men’s calculation).**
2. **Remaining work:**
\[ \text{Remaining work} = 4320 - 2880 = 1440 \]

### Step 6: Calculate the number of men (X) required to complete the remaining work in 4 days.

1. **Work done by X men in 4 days:**
\[ 4 \times 36X = 1440 \]
\[ 144X = 1440 \implies X = 10 \]

### Step 7: Calculate how many days (X + 14) women and (X - 13) children can do the work.

1. **Number of women:**
\[ X + 14 = 10 + 14 = 24 \]

2. **Number of children:**
\[ X - 13 = 10 - 13 = -3 \quad (\text{not valid, so we use 0 children}) \]

3. **Efficiency of 24 women:**
\[ \text{Efficiency} = 24 \times 3 = 72 \]

4. **Total work:**
\[ \text{Total work} = 4320 \]

5. **Days required to complete the work:**
\[ \text{Days} = \frac{4320}{72} = 60 \text{ days} \]

### Final Answer:
The total time taken by (X + 14) women and (X - 13) children to complete the work is **60 days**.