EXAMPLE 21 Percent Effort

Faculty members at a certain university must state the percentage of their time spent in several activities. Professor Worktorule has requisitioned five stopwatches to keep track of her activities. Table 14.18 shows, in its left columns, what she recorded over the course of one week.

The professor is too busy to convert the data into percentages—which the university requires in whole numbers with sum 100%—so we’ll do it, using the Hill–Huntington method. (She requested this method because she wanted the result to display a nonzero percentage for each of her activities.) As with any percentage apportionment problem, the house size is 100, so the standard divisor—one percentage unit—is the population, 3600, divided by 100, or 36 minutes. Table 14.18 shows in its right columns, the quotas, the rounding points, and the tentative apportionment, obtained by rounding the quotas up or down, depending on whether the quota is above the rounding point or not.

Because too many “seats” were awarded, we’ll try divisors larger than the standard divisor, 36. The results are shown in Table 14.19. The first divisor we tried, 36.5, produced 98 “seats” too few. The left column under that divisor shows the corresponding apportionment quotients (AQ), obtained by dividing the minutes devoted to each activity by that divisor. The middle column shows the Hill-Huntington rounding points (RP) for the apportionment quotients, and the right column displays the Hill-Huntington tentative apportionment (TA). To increase the number of “seats” apportioned, the next trial divisor must be closer to 36 (but more than 36). We tried the divisor 36.2 and found that 99 “seats” were apportioned. Our third trial divisor was 36.1 (not shown in the table), which apportioned 101 “seats”—too many. The divisor we needed was therefore between 36.1 and 36.2. We set the divisor equal to 36.15, and the table shows that exactly 100 “seats” were apportioned. The right column of Table 14.19 displays the percentages that Professor Worktorule should put into her effort report.

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