Ten women can complete a picce of work in 15 days. Six men can complete the same piece of work in 10 days. In how many days can 5 women and six men together complete the piece of work?

Options

A. 15 days
B. 7.5 days
C. 9 days
D. 12.5 days

Answer

To solve the problem step by step, we will first determine the work rates of women and men, then find the combined work rate of 5 women and 6 men, and finally calculate how many days they will take to complete the work together.

### Step 1: Calculate the work rate of one woman.
Given that 10 women can complete the work in 15 days, we can find the work done by one woman in one day.

- Total work done by 10 women in 15 days = 1 (whole work)
- Work done by 10 women in one day = \( \frac{1}{15} \)
- Therefore, work done by 1 woman in one day = \( \frac{1}{15} \div 10 = \frac{1}{150} \)

### Step 2: Calculate the work rate of one man.
Given that 6 men can complete the work in 10 days, we can find the work done by one man in one day.

- Total work done by 6 men in 10 days = 1 (whole work)
- Work done by 6 men in one day = \( \frac{1}{10} \)
- Therefore, work done by 1 man in one day = \( \frac{1}{10} \div 6 = \frac{1}{60} \)

### Step 3: Calculate the combined work rate of 5 women and 6 men.
Now, we will find the combined work rate of 5 women and 6 men.

- Work done by 5 women in one day = \( 5 \times \frac{1}{150} = \frac{5}{150} = \frac{1}{30} \)
- Work done by 6 men in one day = \( 6 \times \frac{1}{60} = \frac{6}{60} = \frac{1}{10} \)

Now, we add the work rates of 5 women and 6 men together:

\[ \text{Combined work rate} = \frac{1}{30} + \frac{1}{10} \]

To add these fractions, we need a common denominator. The least common multiple of 30 and 10 is 30.

\[ \frac{1}{10} = \frac{3}{30} \]

So,

\[ \text{Combined work rate} = \frac{1}{30} + \frac{3}{30} = \frac{4}{30} = \frac{2}{15} \]

### Step 4: Calculate the time taken to complete the work.
Let \( N \) be the number of days taken by 5 women and 6 men to complete the work together.

Since their combined work rate is \( \frac{2}{15} \), we have:

\[ \text{Work done in } N \text{ days} = \text{Combined work rate} \times N = 1 \]

\[ \frac{2}{15} \times N = 1 \]

Now, solve for \( N \):

\[ N = \frac{1}{\frac{2}{15}} = \frac{15}{2} = 7.5 \text{ days} \]

### Final Answer:
5 women and 6 men together can complete the piece of work in **7.5 days**.

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