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 carnations will be in each bouquet?

Answer 1

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

First, we find the prime factorizations of the numbers:

1. **36 (roses)**:
\[
36 = 2^2 \times 3^2
\]

2. **27 (tulips)**:
\[
27 = 3^3
\]

3. **18 (carnations)**:
\[
18 = 2^1 \times 3^2
\]

Next, we identify the common prime factors and their lowest powers:

- The only common prime factor among the three factorizations is \(3\).
- In \(36\), the power of \(3\) is \(2\).
- In \(27\), the power of \(3\) is \(3\).
- In \(18\), the power of \(3\) is \(2\).

The lowest power of \(3\) among these numbers is \(2\). Therefore, the GCD is:
\[
GCD = 3^2 = 9
\]

Now that we have found the GCD of 36, 27, and 18 to be 9, we can find out how many flowers will be in each bouquet.

To find the number of carnations in each bouquet:
\[
\text{Number of carnations per bouquet} = \frac{\text{Total carnations}}{\text{Number of bouquets}} = \frac{18}{9} = 2
\]

Thus, the number of carnations in each bouquet is \(\boxed{2}\).