A floral shop receives a $780 order for roses and carnations. The order contains twice as many roses as carnations. The prices per dozen for the roses and carnations are $22 and $34, respectively. How many, in dozens, of each type of flower are in the order?

Let's solve this step by step.

### Step 1: Define the variables
- Let $ x $ be the number of dozens of carnations.
- Since there are twice as many roses as carnations, the number of dozens of roses would be $ 2x $.

### Step 2: Write the cost expressions
- The cost for the carnations is $ 34 \times x $ dollars.
- The cost for the roses is $ 22 \times 2x $ dollars.

### Step 3: Set up the equation
The total cost of the order is $780. Therefore, we can write the equation:
\[
34x + 22(2x) = 780
\]

### Step 4: Simplify and solve the equation
First, simplify the equation:
\[
34x + 44x = 780
\]
Combine like terms:
\[
78x = 780
\]
Solve for $ x $:
\[
x = \frac{780}{78} = 10
\]
So, $ x = 10 $ dozen carnations.

### Step 5: Calculate the number of dozens of roses
Since the number of dozens of roses is $ 2x $:
\[
2x = 2 \times 10 = 20
\]

### Final Answer:
- **Dozens of carnations**: 10
- **Dozens of roses**: 20

Would you like more details or have any questions? Here are 5 related questions:

1. What would happen if the prices of roses and carnations were swapped?
2. How would the number of dozens change if the order was $1000 instead of $780?
3. What if there were three times as many roses as carnations instead of twice as many?
4. How would the solution change if the price per dozen of carnations was $30?
5. How can we verify the solution using a different method?

**Tip:** When solving word problems, it's helpful to define variables clearly and translate the problem statement into mathematical equations step by step.

Learn how to solve a mathematical problem involving an order of roses and carnations at a floral shop. Understand the step-by-step solution process for determining the number of dozens of each type of flower based on given prices and total order cost.

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