2004 AIME II Problems/Problem 5

Problem

In order to complete a large job, 1000 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 100 workers were laid off, so the second quarter of the work was completed behind schedule. Then an additional 100 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 800 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

Assume that one worker does 1 unit of work per day, and that there are 1000 units of work to be done. It takes 1/4th of a day to do the first quarter of work. We then have 900 workers to complete the next 250 units of work, so it takes of a day to complete the remaining work. The third 250 units of work only have 800 workers, so it takes of a day to get this amount of work done. We now have one quarter of work to complete, but only days to complete it, so we have to work at of our original rate. Thus we need workers the last quarter-day. We thus need 1566 workers, so we must hire more workers.