Problem:
In order to complete a large job, workers were hired, just enough to complete the job on schedule. All the workers stayed on the job while the first quarter of the work was done, so the first quarter of the work was completed on schedule. Then workers were laid off, so the second quarter of the work was completed behind schedule. Then an additional workers were laid off, so the third quarter of the work was completed still further behind schedule. Given that all workers work at the same rate, what is the minimum number of additional workers, beyond the workers still on the job at the end of the third quarter, that must be hired after three-quarters of the work has been completed so that the entire project can be completed on schedule or before?
Solution:
Choose a unit of time so that the job is scheduled to be completed in of these units. The first quarter was completed in time unit. For the second quarter of the work, there were only as many workers as in the first quarter, so it was completed in units. For the third quarter, there were only as many workers as in the first quarter, so it was completed in units. This leaves units to complete the final quarter. To finish the job on schedule, the number of workers that are needed is at least of the number of workers needed in the first quarter, or which is between and . There are workers at the end of the third quarter, so a minimum of additional workers must be hired.
The problems on this page are the property of the MAA's American Mathematics Competitions