Caleb spends `$72.50` on 50 hamburgers for the marching band. If single burgers cost `$1.00` each and double burgers cost `$1.50` each, how many double burgers did the buy?

Answer

To solve the problem step by step, we will define the variables, set up the equations based on the information given, and then solve for the number of double burgers.

### Step 1: Define the Variables
Let:
- \( x \) = number of single hamburgers
- \( y \) = number of double hamburgers

### Step 2: Set Up the Equations
From the problem statement, we know:
1. The total number of hamburgers is 50:
\[ x + y = 50 \quad \text{(Equation 1)} \]

2. The total cost of the hamburgers is $72.50. The cost of single hamburgers is $1.00 each and the cost of double hamburgers is $1.50 each:
\[ 1 \cdot x + 1.5 \cdot y = 72.50 \quad \text{(Equation 2)} \]

### Step 3: Solve for One Variable
From Equation 1, we can express \( x \) in terms of \( y \):
\[ x = 50 - y \]

### Step 4: Substitute into the Cost Equation
Now, substitute \( x \) in Equation 2:
\[ 1(50 - y) + 1.5y = 72.50 \]

### Step 5: Simplify the Equation
Distributing the terms gives:
\[ 50 - y + 1.5y = 72.50 \]
Combine like terms:
\[ 50 + 0.5y = 72.50 \]

### Step 6: Isolate the Variable \( y \)
Subtract 50 from both sides:
\[ 0.5y = 72.50 - 50 \]
\[ 0.5y = 22.50 \]

### Step 7: Solve for \( y \)
Divide both sides by 0.5:
\[ y = \frac{22.50}{0.5} = 45 \]

### Step 8: Find \( x \)
Now that we have \( y \), we can find \( x \):
\[ x = 50 - y = 50 - 45 = 5 \]

### Conclusion
Caleb bought:
- **5 single hamburgers**
- **45 double hamburgers**

Thus, the number of double burgers Caleb bought is **45**.