I'm stuck on this problem from the 2019 Grade 8 International Math Contest, which I'm using to help prepare my team for the Mathcounts competition:
Mrs Munch and Mrs. Crunch each bought the same number of identical cookie boxes from their local troop. Both women took either $2$ or $3$ days to finish each box of cookies. It took Mrs. Munch $43$ days to empty every last one of her cookie boxes; It took Mrs. Crunch $62$ days to empty all of hers. How many boxes of cookies did each woman buy?
I feel like I should be using a system of equations to solve this, but if that's correct, I can't figure out how to set it up.
Best Answer
I solved this immediatly by figuring out what the minimal number of boxes (finishing every box in 3 days) and the maximal number (finishing every box in 2 days) is.
Maximal:
Ms. Munch 21 boxes (eating 20 in 2 days and 1 in 3 days)
Ms. Crunch 31 boxes (eatin 31 in 2 days)
Minimal:
Ms. Munch 15 boxes (eating 13 in 3 days, and 2 in 2 days)
Ms. Crunch 21 boxes (eating 20 in 3 days and 1 in 2 days)
Which leads to one solution, that every person had 21 boxes.
Also this would lead to a general solving method, but of course it is here luck that it turns out this well.