表から，Aさんが1人で通販業務を行う場合，1週間の通販業務に要する作業時間は36時間である。このことから，1週間の通販業務の業務量に対して，Aさんが1時間でできる業務量は，その 1/36 である。
AさんとBさんが一緒に通販業務を行うこともある。この場合，1週間の通販業務の業務量に対して，AさんとBさんが1時間でできる業務量は，その 1/36 ＋ 1/45 である。
週の初めから A さんと B さんの 2 人が一緒に通販業務を行うとき, その週の作業は何時間で終わるか。
出典 平成21年度 春期 ITパスポート試験 問99
解説
問 99
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×Incorrectア
The work cannot be finished in 19 hours. Together the two complete exactly 1/36 ＋ 1/45 ＝ 1/20 of the workload per hour, so the time required is exactly 20 hours, with no fractional remainder. In 19 hours only 19/20 would be done, leaving 1/20 of the whole unfinished. This wrong choice is picked when the common denominator is worked out incorrectly, or when one estimates roughly that it should be a little less than 20.
○イ
正解The correct answer is イ (20 hours). This is a work-rate problem: let the total workload of one week of mail-order operations be 1. Mr. A takes 36 hours on his own, so the amount he completes in 1 hour is 1/36 of the whole. Mr. B takes 45 hours, so he completes 1/45 per hour. When the two work together, the amounts completed per hour are added, giving 1/36 ＋ 1/45, exactly as the preamble states. Putting them over a common denominator, the least common multiple of 36 and 45 is 180, so 1/36 ＝ 5/180 and 1/45 ＝ 4/180, for a total of 9/180 ＝ 1/20. That is, the two of them finish 1/20 of the whole in 1 hour. Finishing the whole amount 1 therefore takes 1 ÷ 1/20 ＝ 20 hours. The working days are the 5 days from Monday to Friday with 8 hours per day, so the week has 40 working hours; 20 hours completes the job in half a week, so the condition is satisfied as well. The same calculation can also be done as a product divided by a sum: 36×45 ÷ (36＋45) ＝ 1620 ÷ 81 ＝ 20.
×Incorrectウ
40 hours is obtained by averaging Mr. A's 36 hours and Mr. B's 45 hours as (36＋45) ÷ 2 ＝ 40.5 hours and then rounding down, so the underlying idea itself is wrong. Working times are not averaged; instead you add the amounts of work done per hour and then divide the total workload by that sum. Moreover, two people working together must always be faster than one person alone, yet 40 hours is longer than Mr. A's 36 hours by himself, which is clearly unreasonable.
×Incorrectエ
41 hours is likewise the average (36＋45) ÷ 2 ＝ 40.5 hours rounded up, and rests on the same mistake as ウ. Furthermore, the week contains only 5 days × 8 hours ＝ 40 working hours, so 41 hours cannot even satisfy the condition that the previous week's orders must all be shipped within the following week; it cannot hold as the answer.
ここがポイント
The basic form of a work-rate problem. Set the total amount of work to 1, obtain each person's amount of work per hour as 1 ÷ (time required), add these together, and finally divide 1 by that sum to get the time required for working together. In this question that is 1 ÷ (1/36 ＋ 1/45) ＝ 1 ÷ 1/20 ＝ 20 hours.
When the work is shared by several people, the time required is not the simple average of their individual times but is of the harmonic-mean type. For two people it can be computed as the product divided by the sum, 36×45 ÷ (36＋45) ＝ 20 hours, and the result must always be shorter than either person's time alone, which makes a good check.