Final answer:
To make identical bouquets using all flowers, we need to find the greatest common divisor of the flower counts. Mr. Jones can create 16 bouquets with 2 roses, 4 carnations, and 3 peonies in each.
Step-by-step explanation:
The student's question is regarding the creation of bouquets using a certain number of roses, carnations, and peonies. To make all the bouquets exactly the same using all the flowers, we must find the greatest common divisor (GCD) of the quantities of each type of flower. Mr. Jones has 32 roses, 64 carnations, and 48 peonies.
Firstly, list the factors of each number:
Factors of 32: 1, 2, 4, 8, 16, 32
Factors of 64: 1, 2, 4, 8, 16, 32, 64
Factors of 48: 1, 2, 4, 6, 8, 12, 16, 24, 48
The GCD for 32, 64, and 48 is 16. Therefore, Mr. Jones can make 16 bouquets.
To determine how many of each flower is included in each individual bouquet:
Roses: 32 roses / 16 bouquets = 2 roses per bouquet
Carnations: 64 carnations / 16 bouquets = 4 carnations per bouquet
Peonies: 48 peonies / 16 bouquets = 3 peonies per bouquet