Question

A store purchased x packages of AAA batteries at $5 per package and y packages of AA batteries at $7 per package for a total cost of $135. If at least one package of each type of battery was purchased, what are the possibilities of the number of packages of AAA batteries and the number of packages of AA batteries?

Answer 1

(x,y) = (20, 5), (13, 10), (6,15)

Explanation:

Given that a store purchased x packages of AAA batteries at $5 per package and y packages of AA batteries at $7 per package for a total cost of $135.

Let x batteries of AAA and y batteries of AA be purchased.

Total cost = 5x+7y

Also since atleast one of each type to be purchased we must have

`x\geq 1\\y\geq 1`

5x+7y =135

have integral solutions for both x and y we have to find out

(This is because no of batteries can be only positive integers)

Compulsorily one AA and one AAA purchased

Thus remaining amount of 123 is utilized for purchase of remaining

i.e. 5x+7y = 123 can be considered

For numerical value of y, we must have 123-5x as a multiple of 7

This is possible only for 7y =98 when x = 5 or 7y = 63 when x =12 or 7y = 28 when x = 19.

So in total the possibilities are (including one compulsory)

(x,y) = (20, 5), (13, 10), (6,15)