Question

10 women can complete a piece of work in \(8\) days and 10 children can do the work in \(16\) days. In how many days can 5 women and 10 children complete the work?

• A. \(8\) days
• B. \(12\) days
• C. \(12.5\) days
• D. \(16\) days

Hint:

Find each group's one-day work rate, add them, then take the reciprocal.

Answer 1

The Correct Option is A

Step 1: Find the daily work rate of one woman.
If 10 women finish the whole job in 8 days, the total work can be measured as \(10 \times 8 = 80\) woman-days. This means one woman working alone would take 80 days to finish it, so in one day she completes \(\frac{1}{80}\) of the work.

Step 2: Find the daily work rate of one child.
Similarly, 10 children finish the job in 16 days, so the total work is \(10 \times 16 = 160\) child-days. One child alone would take 160 days, so in one day a child completes \(\frac{1}{160}\) of the work.

Step 3: Add up the daily rate of 5 women and 10 children.
5 women working together do \(5 \times \frac{1}{80} = \frac{5}{80} = \frac{1}{16}\) of the work per day.
10 children working together do \(10 \times \frac{1}{160} = \frac{10}{160} = \frac{1}{16}\) of the work per day.
Added together, the group finishes
\[ \frac{1}{16} + \frac{1}{16} = \frac{2}{16} = \frac{1}{8} \]
of the work in one day.

Step 4: Convert the rate into total days.
If the group finishes \(\frac{1}{8}\) of the work in a day, it needs
\[ \frac{1}{\frac{1}{8}} = 8 \text{ days} \]
to finish the whole job.

Step 5: Why the other options are wrong.
Option (b), 12 days, and option (c), 12.5 days, would come from only counting the children's contribution and forgetting to add the women's rate on top. Option (d), 16 days, is just the time children alone would take, ignoring the women completely. Since both groups work together, their rates must be added, not compared separately.

Final Answer:
5 women and 10 children together finish the work in 8 days. \[ \boxed{8 \text{ days}} \]