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: Convert women into an equivalent number of children.
From the given data, $10$ women $\times 8$ days $= 80$ women-days of total work, and $10$ children $\times 16$ days $= 160$ child-days of total work. Since both expressions equal the same job,
\[ 80 \text{ women-days} = 160 \text{ child-days} \]
which means 1 woman-day of work equals 2 child-days of work, so one woman works at the same rate as two children.

Step 2: Rewrite the group in terms of children only.
The group we care about is 5 women and 10 children. Using the equivalence above, 5 women are worth $5 \times 2 = 10$ children in terms of work rate. So the whole group behaves exactly like
\[ 10 \text{ (converted)} + 10 \text{ (actual)} = 20 \text{ children working together} \]

Step 3: Use the total work in child-days to find the time.
The total work is $160$ child-days (from Step 1). With 20 children working together, the number of days needed is
\[ \frac{160}{20} = 8 \text{ days} \]

Step 4: Confirm the result makes sense.
This matches what we would expect: since 5 women alone would take $80/5=16$ days, and 10 children alone take 16 days too, when both work side by side the finishing time should be noticeably less than 16 days, and 8 days fits that pattern well.

Final Answer:
Converting women into equivalent children also gives 8 days, matching option (a). \[ \boxed{8 \text{ days}} \]