Question

juile has $96 to spend on gifts for the holidays. she must buy at least 9 gifts. She plans to buy puzzles that cost $8 or $12. How many each puzzle can she buy? list at least three possible solutions?

Answer 1

The easiest way to start is by looking at the case where Julie buys puzzles that all have the same price. So, for example, if she buys all $8 puzzles then she can buy $96 divided $8 or 12 puzzles. Since she has to buy at least 9 gifts, and 12 is more than 9, this is one solution to the problem.

On the other hand, if she buys all $12 puzzles then she can buy only $96 divided by $12 or 8 puzzles. Since she has to buy 9 gifts, this is not a solution.

So she can't buy all $12 puzzles, but maybe she can buy a mix of $12 and $8 puzzles. One way to look at this is to start with the case where she buys 8 $12 puzzles, and start reducing the number of $12 puzzles, replacing each by an $8 puzzle instead. For each $12 puzzle that she replaces with an $8 puzzle she saves $4. So if she buys 7 $12 puzzles and 1 $8 puzzle she saves $4. If she buys 6 $12 puzzles and 2 $8 puzzles she saves $8 -- and now she has another to buy one more $8 puzzle, for a total of 3 $8 puzzles and 6 $12 puzzles, or 9 puzzles in all. To double-check, this will cost Julie 3 * 8$ + 6 * $12 = $24 + $72 = $96. So this is a second solution to the problem.

For a third solution, start with the case where she bought 12 $8 puzzles, and replace some of the $8 puzzles with $12 puzzles. I'll leave this one to you.

Answer 2

Thank you!