A florist has 36 roses, 27 tulips, and 18 carnations she must use to create bouquets. Each bouquet must have an equal number of each type of flower as the other bouquets. What is the largest number of bouquets she can make without having any flowers left over? How many roses will be in each bouquet? How many tulips will be in each bouquet? How many carnations will be in each bouquet?(4 points)
Bouquets
Roses in each bouquet
Tulips in each bouquet
Carnations in each bouquet
To find the largest number of bouquets the florist can make without having any flowers left over, we need to find the greatest common divisor (GCD) of the given numbers (36, 27, and 18). The GCD represents the maximum number of bouquets that can be made, with each bouquet having an equal number of each type of flower.
To find the GCD, we can factorize the given numbers:
36 = 2^2 * 3^2
27 = 3^3
18 = 2 * 3^2
From the factorizations, we can see that the common factors among the three numbers are 2 and 3. The highest power of 2 present in all three numbers is 2^1. The highest power of 3 present in all three numbers is 3^2.
Hence, the GCD of 36, 27, and 18 is 2^1 * 3^2 = 6.
Therefore, the florist can make a maximum of 6 bouquets without any flowers left over.
To calculate the number of roses, tulips, and carnations in each bouquet, we divide the respective quantities by the number of bouquets:
Roses in each bouquet = 36 / 6 = 6
Tulips in each bouquet = 27 / 6 = 4.5 (but since we cannot have fraction flowers, we round it down to the nearest whole number) = 4
Carnations in each bouquet = 18 / 6 = 3
Thus, each bouquet will have 6 roses, 4 tulips, and 3 carnations.
To find the largest number of bouquets the florist can make without any flowers left over, we need to find the greatest common divisor (GCD) of the given numbers: 36, 27, and 18.
The GCD is the largest number that divides each of the given numbers evenly.
Let's calculate the GCD of 36, 27, and 18 using the following steps:
Step 1: Find the GCD of 36 and 27.
- Divide 36 by 27: 36 ÷ 27 = 1 remainder 9
- Divide 27 by 9 (the remainder from the previous step): 27 ÷ 9 = 3
Step 2: Find the GCD of the result from Step 1 (which is 3) and 18.
- Divide 18 by 3: 18 ÷ 3 = 6
So, the GCD of 36, 27, and 18 is 6.
This means that the florist can make a maximum of 6 bouquets without any flowers left over.
To find out how many roses, tulips, and carnations will be in each bouquet, we divide the total number of each flower by the number of bouquets:
Roses in each bouquet = Total number of roses ÷ Number of bouquets
Tulips in each bouquet = Total number of tulips ÷ Number of bouquets
Carnations in each bouquet = Total number of carnations ÷ Number of bouquets
Let's calculate using the given numbers:
Roses in each bouquet = 36 ÷ 6 = 6
Tulips in each bouquet = 27 ÷ 6 = 4.5
Carnations in each bouquet = 18 ÷ 6 = 3
Since we cannot have a fraction of a flower in a bouquet, we need to round down the number of tulips in each bouquet. Therefore, each bouquet will have 6 roses, 4 tulips, and 3 carnations.
To summarize:
The largest number of bouquets the florist can make without having any flowers left over is 6.
Each bouquet will have:
- 6 roses
- 4 tulips
- 3 carnations