Introduction / Context:
This time and work question involves converting different categories of workers into a single common unit of work. Men, women and children have different efficiencies, but you are asked to find how many women alone would be needed to do the same work in the same number of days. It tests ratio and equivalence concepts.
Given Data / Assumptions:
- 3 men, 4 women and 6 children can complete the work in 7 days.
- A woman does double the work of a man.
- A child does half the work of a man.
- All workers work at constant individual rates.
- We need the number of women alone who can complete the work in 7 days.
Concept / Approach:
Take the work done by one man per day as a basic unit. Express the daily work of women and children in terms of this unit. Then convert the entire mixed group into an equivalent number of men working per day. After finding the total work in man-units, convert that into women-units and calculate how many women are needed to finish the work in 7 days.
Step-by-Step Solution:
Let the daily work done by one man = 1 man-unit.
Given: a woman does double the work of a man, so 1 woman = 2 man-units per day.
A child does half the work of a man, so 1 child = 0.5 man-units per day.
Now compute daily work of 3 men, 4 women and 6 children.
3 men contribute 3 * 1 = 3 man-units.
4 women contribute 4 * 2 = 8 man-units.
6 children contribute 6 * 0.5 = 3 man-units.
Total daily work of the group = 3 + 8 + 3 = 14 man-units.
They complete the whole work in 7 days.
So total work W = 14 man-units/day * 7 days = 98 man-units.
Now let the required number of women be w.
Each woman works at 2 man-units per day.
So w women together do 2w man-units per day.
We want them to finish in 7 days: 2w * 7 = 98.
So 14w = 98, which gives w = 98 / 14 = 7.
Verification / Alternative check:
7 women in 7 days do 7 * 2 * 7 = 98 man-units, exactly equal to the total work computed.
Thus the conversion from mixed group to women-only group is correct.
Why Other Options Are Wrong:
6 women would do 6 * 2 * 7 = 84 man-units, less than needed.
9 women would do 9 * 2 * 7 = 126 man-units, more than needed.
5 women give 70 man-units, much less than the total work.
Common Pitfalls:
A common mistake is to average the numbers of men, women and children rather than converting properly to a base unit.
Some students may misapply the ratios and treat woman work and child work as ratios of days rather than per day rates.
Always select one type of worker as the base and express all others in terms of that base unit.
Final Answer:
The work can be completed in 7 days by 7 women working alone.
Discussion & Comments