Changing useful work against a fixed input shows efficiency and waste as one checked budget. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
Thirty useful joules out of sixty gives one half
Input stays 60 joules. Useful work 30 joules leaves waste heat 30 joules, so efficiency is one half.
6030=21
Thirty-six useful joules gives three fifths
The same 60 joule input with useful work 36 joules leaves waste 24 joules and efficiency three fifths.
6036=53
Forty-five useful joules gives three fourths
Useful work 45 joules leaves only 15 joules of waste, so the efficiency rises to three fourths.