Billy's restaurant ordered 200 flowers for Mother's Day. They ordered carnations at $1.50 each, roses at $5.75 each, and daisies at $2.60 each. they ordered mostly carnations, and 20 fewer roses than daisies. the total order came to $589.50. How do i find out how many of each type of flower was ordered?

To find out how many carnations, roses, and daisies were ordered, let's define some variables and set up a system of equations:

### Step 1: Define the variables
Let:
- $ c $ be the number of carnations ordered.
- $ r $ be the number of roses ordered.
- $ d $ be the number of daisies ordered.

### Step 2: Write equations based on the problem

1. **Total number of flowers**:  
   Billy's restaurant ordered 200 flowers, so:
   \[
   c + r + d = 200
   \]

2. **Price equation**:  
   The total cost of the flowers is $589.50. The carnations cost $1.50 each, roses cost $5.75 each, and daisies cost $2.60 each. Therefore, we can write the total price equation as:
   \[
   1.50c + 5.75r + 2.60d = 589.50
   \]

3. **Relationship between roses and daisies**:  
   The problem says they ordered 20 fewer roses than daisies, so:
   \[
   r = d - 20
   \]

### Step 3: Substitute and solve the system of equations

We now have three equations:
1. $ c + r + d = 200 $
2. $ 1.50c + 5.75r + 2.60d = 589.50 $
3. $ r = d - 20 $

Start by substituting $ r = d - 20 $ into the first two equations.

#### Equation 1:
\[
c + (d - 20) + d = 200
\]
Simplifying:
\[
c + 2d - 20 = 200 \quad \text{or} \quad c + 2d = 220
\]
So we get:
\[
c = 220 - 2d \quad \text{(Equation 4)}
\]

#### Equation 2:
Substitute $ r = d - 20 $ into the price equation:
\[
1.50c + 5.75(d - 20) + 2.60d = 589.50
\]
Simplify this:
\[
1.50c + 5.75d - 115 + 2.60d = 589.50
\]
\[
1.50c + 8.35d = 704.50 \quad \text{(Equation 5)}
\]

### Step 4: Substitute $ c = 220 - 2d $ into Equation 5

Substitute Equation 4 into Equation 5:
\[
1.50(220 - 2d) + 8.35d = 704.50
\]
Simplifying:
\[
330 - 3d + 8.35d = 704.50
\]
\[
330 + 5.35d = 704.50
\]
\[
5.35d = 374.50
\]
\[
d = \frac{374.50}{5.35} \approx 70
\]

### Step 5: Find $ r $ and $ c $

Now that we know $ d \approx 70 $, we can find $ r $ and $ c $:
- $ r = d - 20 = 70 - 20 = 50 $
- $ c = 220 - 2d = 220 - 2(70) = 80 $

### Final Answer:
- Carnations: $ 80 $
- Roses: $ 50 $
- Daisies: $ 70 $

These are the number of each type of flower ordered.

Would you like a breakdown of any specific step?

Here are 5 related questions for practice:
1. What would be the total cost if the price of daisies increased by $1 each?
2. If the total number of flowers ordered was 250, how would the solution change?
3. What if the restaurant ordered 10 more roses than daisies instead of 20 fewer?
4. How would the system change if the cost of carnations was $2 each?
5. What if the total cost of the flowers was $700 instead of $589.50?

**Tip**: Solving systems of equations step-by-step, especially with substitution, helps to avoid mistakes in calculations.

Billy's restaurant ordered 200 flowers for Mother's Day. They ordered carnations at $1.50 each, roses at $5.75 each, and daisies at $2.60 each. They ordered mostly carnations, and 20 fewer roses than daisies. The total order came to $589.50. How do I find out how many of each type of flower was ordered?

This problem demonstrates how to solve a system of equations to determine how many carnations, roses, and daisies were ordered for Mother's Day. Learn the step-by-step solution, using algebraic methods like substitution, with real-world pricing and flower quantities. Suitable for students in Grades 9-11.

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