You have a bag with 30 fire balls and another bag with 42 jolly ranchers. For a party, you want to repackage the candy into smaller bags, and you want each bag to have the same number of fireballs and jolly ranchers as every other bag. How many bags will you be able to make?

Answer 1

To determine how many bags you can make with the same number of fireballs and jolly ranchers in each bag, you need to find the greatest common divisor (GCD) of the two quantities: 30 fireballs and 42 jolly ranchers.

1. Find the prime factorization of each number:

   • The prime factorization of 30 is \(2 \times 3 \times 5\).
   • The prime factorization of 42 is \(2 \times 3 \times 7\).

2. Identify the common prime factors: The common prime factors between 30 and 42 are 2 and 3.

3. Multiply the common prime factors: Multiply the common factors to find the GCD: \[ 2^1 \times 3^1 = 2 \times 3 = 6. \]

Thus, the greatest common divisor (GCD) of 30 and 42 is 6.

4. Find the number of bags: You can make 6 bags because that is the maximum number of bags that allows for the same quantity of fireballs and jolly ranchers in each bag.

To find out how many fireballs and jolly ranchers go into each bag:

• For fireballs: \[ \frac{30 \text{ fireballs}}{6 \text{ bags}} = 5 \text{ fireballs per bag}. \]
• For jolly ranchers: \[ \frac{42 \text{ jolly ranchers}}{6 \text{ bags}} = 7 \text{ jolly ranchers per bag}. \]

Therefore, you can make 6 bags containing 5 fireballs and 7 jolly ranchers each.