正解The correct answer is ア, line ①. Let the workload of one week of mail-order operations be 1. Mr. A finishes it in 36 hours, so he processes 1/36 of it per hour, and Mr. B finishes it in 45 hours, so he processes 1/45 of it per hour. When the two work together, the amounts processed per hour are added: 1/36 + 1/45 = 5/180 + 4/180 = 9/180 = 1/20. In other words, together they reach 100 % in 20 hours. Stated as a progress rate (%), Mr. A advances about 2.8 % per hour and Mr. B about 2.2 % per hour, for a total of 5.0 % per hour, so 100 ÷ 5.0 = 20 hours. On the graph the slope of a line segment is the progress rate per hour, so the line for the two of them is steeper than both a and b, and the point where it reaches 100 % must lie to the left of a (the dotted line at 36 on the horizontal axis). The only rising line that reaches 100 % to the left of the dotted line at 36 is ①, so the answer is ①.
×Incorrectイ
② reaches 100 % between the dotted line at 36 and the dotted line at 45 on the horizontal axis, which means completion takes 36 to 45 hours. That is slower than when Mr. A works alone (36 hours), so it would mean the work becomes slower once Mr. B joins, which is a contradiction. A line whose slope is gentler than a cannot be the line for working together. Also, taking an average such as (36 + 45) ÷ 2 = 40.5 hours instead of working out the 20 hours makes it easy to pick the line at this position by mistake.
×Incorrectウ
③ reaches 100 % even further right than the dotted line at 45 on the horizontal axis; it has the gentlest slope, that is, it is the slowest of the rising lines in the figure. This would mean it is slower than when Mr. B works alone (45 hours), which cannot happen as the result of two people working together. This position corresponds to the mistake of adding the required times themselves, as in 36 + 45 = 81 hours, instead of adding 1/36 + 1/45.
×Incorrectエ
④ is a line that falls toward the lower right starting from 100 %, meaning the progress rate goes down as the working time increases, and it is already at 100 % when the working time is 0. The progress rate is a quantity that increases from 0 % as the work advances, so this line cannot represent progress at all (it is closer to a burndown line showing how the remaining workload changes). It has nothing to do with how fast the two people work together.
ここがポイント
Basics of work-rate problems: if the whole workload is taken as 1, a person who finishes it in t hours processes 1/t of it per hour. When several people work together, what you add is the amount processed per hour (the rate), not the required times. Since 1/36 + 1/45 = 1/20, the time required is 20 hours.
Reading the graph: in a straight-line graph of working time versus progress rate, the slope is the progress rate per hour, and the position on the horizontal axis where the line reaches 100 % is the time required. The line for working together is always steeper than each person's own line, and its 100 % point must lie to the left of both a and b — this qualitative judgement alone is enough to narrow the choices down to ①.