A group of women working together at the same rate can build a wall in 45 hours. When the work started, all the women did not start working together. They joined the work over a period of time, one by one, at equal intervals. Once at work, each one stayed till the work was complete. If the first woman worked 5 times as many hours as the last woman, for how many hours did the first woman work?
Correct Answer: 75
Solution
Suppose there are $$n$$ women, the first works $$T$$ hours and the last works $$\frac{T}{5}$$ hours, the individual working times forming an arithmetic progression. The total labour is $$n \times \frac{T + \frac{T}{5}}{2} = \frac{3nT}{5}$$ woman hours, while the job needs $$45n$$ woman hours. Equating gives $$\frac{3nT}{5} = 45n$$, so $$T = 75$$ hours.
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