3 men can do a piece of work in 6 days. 5 women can do the same work in 18 days. If 4 men and 10 women work together, then how long will it take to finish the work?

Options

A. 3 days
B. 5 days
C. 2 days
D. 4 days

Answer

To solve the problem step by step, we will first determine the work done by one man and one woman in a day, and then calculate how long it will take for 4 men and 10 women to finish the work together.

### Step 1: Calculate the total work done by men
Given that 3 men can complete the work in 6 days, we can calculate the total work in terms of "man-days".

Total work = Number of men × Number of days
Total work = 3 men × 6 days = 18 man-days

### Step 2: Calculate the work done by one man in one day
Now, we can find the work done by one man in one day.

Work done by 1 man in 1 day = Total work / Total man-days
Work done by 1 man in 1 day = 1/18 of the total work (denote this as m)

### Step 3: Calculate the total work done by women
Next, we know that 5 women can complete the same work in 18 days.

Total work = Number of women × Number of days
Total work = 5 women × 18 days = 90 woman-days

### Step 4: Calculate the work done by one woman in one day
Now, we can find the work done by one woman in one day.

Work done by 1 woman in 1 day = Total work / Total woman-days
Work done by 1 woman in 1 day = 1/90 of the total work (denote this as w)

### Step 5: Calculate the combined work done by 4 men and 10 women in one day
Now we need to calculate how much work is done by 4 men and 10 women in one day.

Work done by 4 men in one day = 4 × (1/18) = 4/18 = 2/9 of the total work
Work done by 10 women in one day = 10 × (1/90) = 10/90 = 1/9 of the total work

### Step 6: Combine the work done by men and women
Now, we can combine the work done by 4 men and 10 women in one day.

Total work done in one day = Work done by 4 men + Work done by 10 women
Total work done in one day = (2/9) + (1/9) = 3/9 = 1/3 of the total work

### Step 7: Calculate the time taken to finish the work
If 4 men and 10 women together can complete 1/3 of the work in one day, we can find out how many days it will take to finish the entire work.

Let the number of days required be d.

Total work = Work done in one day × Number of days
Total work = (1/3) × d = 1 (the whole work)

Now, solving for d:
d = 1 / (1/3) = 3 days

### Final Answer
Thus, if 4 men and 10 women work together, they will finish the work in **3 days**.

---