Harry is making gift bags for a holiday party using two types of candies. He has a total of 84 chocolate candies and 56 peppermint candies. Each gift bag he makes will be the same.
To make the gift bags, he needs to decide how many candies of each type should go into each bag. Let's assume he puts x chocolate candies and y peppermint candies in each bag.
Since he has a total of 84 chocolate candies, the number of gift bags he can make from these chocolates is 84/x.
Similarly, the number of gift bags he can make from the 56 peppermint candies is 56/y.
Since each gift bag will be the same, the number of gift bags made from chocolates should be equal to the number of gift bags made from peppermint candies.
Therefore, we have the equation:
84/x = 56/y
Cross-multiplying, we get:
84y = 56x
Simplifying, we can divide both sides by 28:
3y = 2x
This equation implies that the number of peppermint candies (y) must be divisible by 3, and the number of chocolate candies (x) must be divisible by 2. This is because the ratio of y to x is 3:2.
Therefore, possible combinations of x and y are:
x = 2, y = 3 (2 chocolate candies and 3 peppermint candies in each bag)
x = 4, y = 6 (4 chocolate candies and 6 peppermint candies in each bag)
x = 6, y = 9 (6 chocolate candies and 9 peppermint candies in each bag)
And so on.
This means that Harry can make gift bags with either 2 chocolate candies and 3 peppermint candies, 4 chocolate candies and 6 peppermint candies, or 6 chocolate candies and 9 peppermint candies in each bag (among other possible combinations).