問31
頻出度★★☆
A さんだと 10 日, B さんだと 15 日かかるプログラム開発の作業がある。これを A さんと B さんが一緒に作業した場合, 何日かかるか。ここで, 2 人で作業を行った場合もそれぞれの作業効率は変わらないものとする。
平成22年度 春期 ITパスポート試験 問31
テクノロジ系基礎理論 › 応用数学
This question's explanation and your answers so far.
解説
問 31
Incorrectア
5 days is simply A's 10 days divided by 2. That would be the answer if B worked at the same speed as A, which ignores the condition that B needs 15 days, that is, that B is slower than A. Checking with 5 days: A completes 5/10 = 0.5 and B completes 5/15 ≈ 0.33, for a total of only about 0.83, so the task is not finished.
- イ
正解 The standard approach is to set the whole task equal to 1. A finishes in 10 days, so A does 1/10 of the task per day; B finishes in 15 days, so B does 1/15 per day. Because the question states that each person's work efficiency remains unchanged even when the two work together, the daily amounts can simply be added: 1/10 + 1/15 = 3/30 + 2/30 = 5/30 = 1/6. Working through a task of size 1 at a pace of 1/6 per day takes 1 ÷ (1/6) = 6 days. As a check, in 6 days A completes 6/10 = 0.6 and B completes 6/15 = 0.4, which totals exactly 1.0, so the task is finished. Therefore イ, 6, is correct.
Incorrectウ
7.5 days is simply B's 15 days divided by 2. Dividing the number of days by 2 is wrong: what doubles when two people work together is the amount of work done per day (the speed), not the number of days. Checking with 7.5 days: A completes 7.5/10 = 0.75 and B completes 7.5/15 = 0.5, for a total of 1.25, which exceeds the total work of 1. In other words the task would already be finished before 7.5 days, so this value is too large.
Incorrectエ
12.5 days is the simple average of 10 days and 15 days, that is (10 + 15) / 2. Averaging the numbers of days has no meaning at all. Besides, when two people cooperate the result must always be shorter than the 10 days the faster person A would need alone, yet 12.5 days is longer than that, which is already plainly contradictory.
ここがポイント
- The basic form of a work-rate problem: set the whole task to 1, treat the reciprocal of each person's required days as the amount of work done per day (the work rate), and add those rates together. The combined rate 1/10 + 1/15 = 1/6, and its reciprocal is the required time, 6 days; in other words, take the reciprocal of the sum of the reciprocals. The most important point is that rates can be added while days cannot, so never average day counts together or divide them by 2. Note also that the harmonic mean of 10 and 15 is 12, which is a different thing from the 6 in this question, so do not confuse the two.
- A quick way to judge whether an answer is plausible: working together must take less than the 10 days the faster person A needs alone, and conversely, even if both worked as fast as A it would still take 5 days, so the answer must lie in the range 5 days < answer < 10 days. This range check rules out ア (5) and エ (12.5), but both イ (6) and ウ (7.5) survive it, so the final decision still has to come from computing 1/10 + 1/15.