12 men and 16 women can complete a jon in 5 days. 13 men and 24 women can complete the same job in 4 days. How long, in days, will 5 men and 10 women take to complete the same job?

Options

A. 5
B. 12
C. 10
D. 15

Answer

To solve the problem step by step, we will first find the work done by men and women, then calculate how long it will take for 5 men and 10 women to complete the same job.

### Step 1: Define Variables
Let the work done by 1 man in 1 day be \( m \) and the work done by 1 woman in 1 day be \( w \).

### Step 2: Set Up the Equations
From the problem, we have two scenarios:
1. **12 men and 16 women can complete the job in 5 days.**
\[ \text{Total work} = (12m + 16w) \times 5 \]
2. **13 men and 24 women can complete the job in 4 days.**
\[ \text{Total work} = (13m + 24w) \times 4 \]

Since both expressions represent the same total work, we can set them equal to each other:
\[ (12m + 16w) \times 5 = (13m + 24w) \times 4 \]

### Step 3: Simplify the Equation
Expanding both sides:
\[ 60m + 80w = 52m + 96w \]

Now, rearranging the equation:
\[ 60m - 52m = 96w - 80w \]
\[ 8m = 16w \]

### Step 4: Find the Ratio of Men to Women
Dividing both sides by 8:
\[ m = 2w \]
This means the work done by 1 man is equivalent to the work done by 2 women.

### Step 5: Calculate Total Work
Now, we can substitute \( m \) in terms of \( w \) into either equation to find the total work. Using the first equation:
\[ \text{Total work} = (12(2w) + 16w) \times 5 \]
\[ = (24w + 16w) \times 5 \]
\[ = 40w \times 5 = 200w \]

### Step 6: Find the Work Done by 5 Men and 10 Women
Now we need to find how long it will take for 5 men and 10 women to complete the same job:
\[ \text{Work done by 5 men and 10 women in 1 day} = 5m + 10w \]
Substituting \( m = 2w \):
\[ = 5(2w) + 10w = 10w + 10w = 20w \]

### Step 7: Calculate the Number of Days
Now, we know the total work is \( 200w \) and the work done per day by 5 men and 10 women is \( 20w \):
\[ \text{Number of days} = \frac{\text{Total work}}{\text{Work done per day}} = \frac{200w}{20w} = 10 \text{ days} \]

### Final Answer
Thus, it will take 5 men and 10 women **10 days** to complete the job.

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