12 men and 15 women can complete a work in 10 days. In how many days the same work will be completed by 7 men and 10 women?

Options

A. 12
B. 10
C. 9
D. 8

Answer

To solve the problem step by step, we will first determine the work done by 12 men and 15 women in one day, and then find out how many days it will take for 7 men and 10 women to complete the same work.

### Step 1: Calculate the total work done by 12 men and 15 women in one day.

Given that 12 men and 15 women can complete the work in 10 days, we can express the total work in terms of man-days and woman-days.

- Work done by 12 men in 10 days = 12 men × 10 days = 120 man-days.
- Work done by 15 women in 10 days = 15 women × 10 days = 150 woman-days.

Now, we need to find the work done by one man and one woman in one day.

### Step 2: Calculate the work done by one man and one woman in one day.

- If 12 men can do the work in 10 days, then the work done by one man in one day is:
\[ \text{Work done by 1 man in 1 day} = \frac{1}{120} \text{ of the work} \]

- If 15 women can do the work in 10 days, then the work done by one woman in one day is:
\[ \text{Work done by 1 woman in 1 day} = \frac{1}{150} \text{ of the work} \]

### Step 3: Calculate the work done by 7 men and 10 women in one day.

Now, we will find the total work done by 7 men and 10 women in one day.

- Work done by 7 men in one day:
\[ \text{Work done by 7 men} = 7 \times \frac{1}{120} = \frac{7}{120} \]

- Work done by 10 women in one day:
\[ \text{Work done by 10 women} = 10 \times \frac{1}{150} = \frac{10}{150} = \frac{1}{15} \]

### Step 4: Combine the work done by men and women.

Now, we will add the work done by 7 men and 10 women in one day:
\[ \text{Total work done in one day} = \frac{7}{120} + \frac{1}{15} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 120 and 15 is 120.

- Convert \(\frac{1}{15}\) to have a denominator of 120:
\[ \frac{1}{15} = \frac{8}{120} \]

Now, we can add:
\[ \text{Total work done in one day} = \frac{7}{120} + \frac{8}{120} = \frac{15}{120} = \frac{1}{8} \]

### Step 5: Calculate the number of days to complete the work.

If 7 men and 10 women can do \(\frac{1}{8}\) of the work in one day, the number of days required to complete the entire work is:
\[ \text{Number of days} = \frac{1}{\frac{1}{8}} = 8 \text{ days} \]

### Final Answer:
The same work will be completed by 7 men and 10 women in **8 days**.

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