Question

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.
If Harry uses all of his candy, he can make at most gift bags. Each gift bag will have chocolate candies and peppermint candies.

Answer 1

To determine how many gift bags Harry can make, we need to figure out how many of each type of candy he can use in each bag while ensuring that all gifts are the same.

Let \( x \) be the number of chocolate candies per bag and \( y \) be the number of peppermint candies per bag.

Given:
- Total chocolate candies: 84
- Total peppermint candies: 56

If he makes \( n \) gift bags, then:
\[
n \times x = 84 \quad \text{(1)}
\]
\[
n \times y = 56 \quad \text{(2)}
\]

From equation (1), we can express \( n \) in terms of \( x \):
\[
n = \frac{84}{x}
\]

From equation (2), we can express \( n \) in terms of \( y \):
\[
n = \frac{56}{y}
\]

Since both expressions represent \( n \), we can set them equal to each other:
\[
\frac{84}{x} = \frac{56}{y}
\]

Cross-multiplying gives:
\[
84y = 56x
\]

Dividing both sides by 28 yields:
\[
3y = 2x
\]
or
\[
x = \frac{3}{2}y
\]

Now, to find integer values of \( x \) and \( y \) that allow for \( n \) to remain an integer, \( y \) can take on integer values that keep \( x \) a whole number.

The simplest whole number to start with is \( y = 2 \):
\[
x = \frac{3}{2} \times 2 = 3
\]

Now substitute \( x = 3 \) and \( y = 2 \) back into equation (1) and (2), to find \( n \):
\[
n = \frac{84}{3} = 28
\]
\[
n = \frac{56}{2} = 28
\]

Both calculations confirm that he can make a maximum of 28 gift bags. Confirming the amounts, each gift bag will then have:
- 3 chocolate candies
- 2 peppermint candies

Thus, **Harry can make at most 28 gift bags, with each bag containing 3 chocolate candies and 2 peppermint candies.**