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.