Question

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.
How many tulips will be in each bouquet?

Answer 1

To determine how many tulips will be in each bouquet, we first need to find the greatest common divisor (GCD) of the number of each type of flower available: 36 roses, 27 tulips, and 18 carnations. The GCD will give us the largest number of bouquets we can create, ensuring each bouquet has an equal number of each type of flower.

We can find the GCD by factoring each number:

- The prime factorization of 36 is:
\[
36 = 2^2 \times 3^2
\]

- The prime factorization of 27 is:
\[
27 = 3^3
\]

- The prime factorization of 18 is:
\[
18 = 2^1 \times 3^2
\]

Next, we take the lowest power of each prime factor common to all three numbers:

- For the prime factor 2, the lowest power among the factorizations is \(2^0 = 1\) (since 27 does not contain a factor of 2).
- For the prime factor 3, the lowest power is \(3^2 = 9\).

Thus, the GCD of 36, 27, and 18 is:
\[
GCD = 1 \times 9 = 9
\]

Now we can use this GCD to determine the number of bouquets we can create. Since we can make 9 bouquets:

1. For roses:
\[
\text{Roses per bouquet} = \frac{36}{9} = 4
\]

2. For tulips:
\[
\text{Tulips per bouquet} = \frac{27}{9} = 3
\]

3. For carnations:
\[
\text{Carnations per bouquet} = \frac{18}{9} = 2
\]

Thus, there will be 3 tulips in each bouquet. Therefore, the final answer is:
\[
\boxed{3}
\]