After 9 matches a batter's average score is 42 runs. How many runs must she make in the 10th match to take her average over the 10 matches to 45 runs?

She has $9\times 42=378$ runs and needs $10\times 45=450$, so the 10th match must bring 450 - 378 = 72 runs. The shortcut is new score = target + (old count × rise) $=45+9\times 3=72$: she must reach the new average AND make up 3 runs for each of the 9 earlier matches. 75 uses the new count of 10 for that repair instead of the 9 matches that actually fall short. 48 repairs only one match (45 + 3). 45 quotes the target itself and ignores the drag of the earlier scores altogether.