上限, 下限, 最大, and 最小: prescribed limits and data extrema
Compare prescribed upper and lower limits with the maximum and minimum of an explicitly stated data set, checking where each figure comes from.
A prescribed allowable range
上限, read じょうげん, is an upper limit; 下限, read かげん, is a lower limit. 一箱の個数は、下限が十個、上限が二十個です。 gives limits of ten and twenty items per box.
It states a prescribed range, not a finding that every box actually contained ten or twenty items.
Values found in the given data
For 三箱の個数は十二個、十五個、十八個です。, eighteen is the largest count and twelve the smallest. Here 最大, read さいだい, and 最小, read さいしょう, describe extrema in the data.
The observed maximum eighteen need not equal the prescribed upper limit twenty. Distinguish the restriction from the actual data set.
Do not fix the source from the word alone
この箱には最大で十冊の本が入る。 describes a capacity of at most ten books. 最大 is not restricted to a maximum already observed in data.
Check whether the sentence states a rule, capacity, or measurements. Equal numbers can serve different roles when their sources differ.
With an upper limit of twenty and observed counts of twelve, fifteen, and eighteen, is the data maximum twenty?
Show the answer and why
No. The maximum among these three boxes is eighteen. Twenty is the prescribed upper limit.