24 men can do work in X days and 32 women can do the same work in `(X+8)` days. The ratio of work done by 15 men and 12 women in the same time is `3 : 1`.
In how many days, the work will be completed if 5 men and 4 women work together ?

Options

A. 24 days
B. 18 days
C. 36 days
D. 48 days

Answer

To solve the problem step by step, we will follow the logical sequence outlined in the video transcript.

### Step 1: Define Variables
Let the efficiency of one man be \( m \) and the efficiency of one woman be \( w \).

### Step 2: Set Up the Work Ratio
According to the problem, the work done by 15 men and 12 women in the same time is in the ratio of \( 3:1 \). Therefore, we can write the equation:

\[ \frac{15m}{12w} = \frac{3}{1} \]

### Step 3: Solve for the Ratio of Efficiencies
Cross-multiplying gives us:

\[ 15m = 36w \]

This simplifies to:

\[ \frac{m}{w} = \frac{36}{15} = \frac{12}{5} \]

Thus, if we assume \( w = 5 \) (the efficiency of one woman), then \( m = 12 \) (the efficiency of one man).

### Step 4: Calculate Total Work for Men
Given that 24 men can complete the work in \( X \) days, the total work can be expressed as:

\[ \text{Total Work} = 24 \times m \times X = 24 \times 12 \times X = 288X \]

### Step 5: Calculate Total Work for Women
Similarly, 32 women can complete the same work in \( X + 8 \) days:

\[ \text{Total Work} = 32 \times w \times (X + 8) = 32 \times 5 \times (X + 8) = 160(X + 8) \]

### Step 6: Set the Total Work Equations Equal
Since both expressions represent the same total work, we can set them equal to each other:

\[ 288X = 160(X + 8) \]

### Step 7: Solve for \( X \)
Expanding the right side:

\[ 288X = 160X + 1280 \]

Now, rearranging gives:

\[ 288X - 160X = 1280 \]

\[ 128X = 1280 \]

Dividing both sides by 128:

\[ X = 10 \]

### Step 8: Calculate Total Work
Now substituting \( X \) back into the total work equation:

\[ \text{Total Work} = 288X = 288 \times 10 = 2880 \text{ units} \]

### Step 9: Calculate Efficiency of 5 Men and 4 Women
Now, we need to find the efficiency of 5 men and 4 women:

\[ \text{Efficiency of 5 men} = 5 \times m = 5 \times 12 = 60 \]
\[ \text{Efficiency of 4 women} = 4 \times w = 4 \times 5 = 20 \]

Thus, the total efficiency of 5 men and 4 women is:

\[ \text{Total Efficiency} = 60 + 20 = 80 \]

### Step 10: Calculate the Number of Days Required
Finally, the number of days required to complete the work is given by:

\[ \text{Days} = \frac{\text{Total Work}}{\text{Total Efficiency}} = \frac{2880}{80} = 36 \]

### Final Answer
The work will be completed in **36 days**.

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