June 2023 Problem of the Month

Window Problem
Submitted by Brijesh Dave

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Solution to the Problem:

They would save $100 if they purchased the windows together.

Extra credit: Amy should pay $553.33 and Brad should pay $646.67.

If Amy buys her windows alone, it would cost her $600 (six @$100 plus 1 free window).
If Brad buys his windows alone, it would cost him $700 ( seven @$100 plus 1 free window).

If they bought 15 windows together, it would cost $1200 (twelve @#100 plus three free windows).

They should split the $100 saving in the ratio of 7 to 8 (the number of windows each person bought).
So the savings should be $46.67 for Amy and $53.33 for Brad.

Many problem solvers said that Brad should pay $(8/15)*1200 = $640 and that Amy should pay $(7/15)*1200 = $560.   But then Brad has a savings of $60 on the $700 he originally was to pay which amounts to 8.57% while Amy saves $40 on her original cost of $600 which amounts to only 6.67%.

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Seth Cohen shared these comments about the extra credit:
"The extra credit is weird.   I feel like it's up to interpretation on how the price should be arranged.   On the one hand, you could argue that Brad should pay for 8/15 of the $1200 ($640) and Amy should pay 7/15 ($560).

But really, each of them gets a free window, and then the third window should be split evenly between them, since both contribute equally to "getting to" that third free window.   (In other words, if Amy wasn't there, Brad would only get 1 free window, and vice versa.   They both have equal stake in the third free window.) So in some sense, Brad pays for 8 - 1.5 = 6.5 windows, and Amy pays for 7 -1.5 = 5.5 windows.   So maybe Brad should pay 6.5/12 of the $1200 ($650) and Amy should pay 5.5/12 ($550).

OR, should the third free window not be split evenly?   Maybe Brad should get 8/15 of the free window, and Amy should get 7/15 of the free window.   In this case, Brad pays for 8 - 1 - (8/15) = 97/15 windows, and Amy pays for 7 - 1 - 7/15 = 83/15 windows.   Then Brad should pay (97/15)/12 of $1200 ($646.67) and Amy should pay (83/15)/12 ($553.33)."

I agree with Seth that the extra credit was up to interpretation.   But I believed that the $100 savings should be split in proportion to the number of windows purchased.

Colin Bowey also sent in several options and I liked the last one:
"Instead of splitting the $100 savings between themselves, they could donate the extra $100 savings to a charity or cause that they both support, they would be no worse off than if they hadn't purchased the windows together, but their supported charity would be better off."

Correctly solved by: