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 that can be made without any flowers left over, we need to find the greatest common divisor (GCD) of 36, 27, and 18.
The GCD of 36, 27, and 18 is 9.
Therefore, the largest number of bouquets that can be made is 9.
To determine the number of each type of flower in each bouquet, we divide the total number of each type of flower by the number of bouquets.
Roses in each bouquet = 36 / 9 = 4
Tulips in each bouquet = 27 / 9 = 3
Carnations in each bouquet = 18 / 9 = 2
Therefore, each bouquet will have 4 roses, 3 tulips, and 2 carnations.
how many bouquets though?
Apologies for the confusion.
The largest number of bouquets that can be made without any flowers left over is 9.
thankyou
You're welcome! If you have any more questions, feel free to ask.
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 number of roses, tulips, and carnations.
Step 1: Find the GCD of 36, 27, and 18.
The prime factorizations of the given numbers are:
36 = 2^2 x 3^2
27 = 3^3
18 = 2 x 3^2
The common factors are 2^1 and 3^2. Taking the minimum exponent of each common factor, we have:
GCD = 2^1 x 3^2 = 2 x 9 = 18.
So, the largest number of bouquets the florist can make without having any flowers left over is 18.
Step 2: Calculate the number of roses, tulips, and carnations in each bouquet.
To find out how many roses, tulips, and carnations will be in each bouquet, we divide the number of each flower by the number of bouquets.
Number of roses in each bouquet: 36 roses / 18 bouquets = 2 roses per bouquet.
Number of tulips in each bouquet: 27 tulips / 18 bouquets = 1.5 tulips per bouquet.
Number of carnations in each bouquet: 18 carnations / 18 bouquets = 1 carnation per bouquet.
However, we need whole numbers in each bouquet, so we'll round down the number of tulips to the nearest whole number.
Therefore, each bouquet will have:
- 2 roses
- 1 tulip
- 1 carnation.
To find the largest number of bouquets that can be made without any flowers left over, we need to find the greatest common divisor (GCD) of the number of roses, tulips, and carnations.
Step 1: Finding the GCD
The GCD of the given numbers can be calculated using the prime factorization method or the Euclidean algorithm. Let's use the Euclidean algorithm.
First, we find the GCD of the two smallest numbers: 27 (tulips) and 18 (carnations).
GCD(27, 18) = 9
Now, we find the GCD of the largest remaining number (36 roses) and the result from the previous step (9).
GCD(36, 9) = 9
Therefore, the GCD of 36, 27, and 18 is 9.
Step 2: Determining the number of bouquets
Since the GCD represents the common factor that each bouquet must have, the largest number of bouquets can be calculated by dividing the total number of each flower type by the GCD.
Number of bouquets = GCD(36, 27, 18) = 9
Step 3: Calculating the number of each flower type in each bouquet
To determine the number of each flower type in each bouquet, we divide the total number of each flower type by the number of bouquets.
Roses in each bouquet = 36 / 9 = 4
Tulips in each bouquet = 27 / 9 = 3
Carnations in each bouquet = 18 / 9 = 2
Therefore, the largest number of bouquets that can be made without any flowers left over is 9. Each bouquet will contain 4 roses, 3 tulips, and 2 carnations.