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 he buy?

Answer

To solve the problem, we will set up a system of equations based on the information given.

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

### Step 2: Set Up the Equations
From the problem, we know:
1. The total number of burgers is 50:
\[ x + y = 50 \quad \text{(Equation 1)} \]
2. The total cost of the burgers is $72.50. The cost of a single burger is $1.00 and the cost of a double burger is $1.50:
\[ 1x + 1.5y = 72.50 \quad \text{(Equation 2)} \]

### Step 3: Solve the System of Equations
We can solve these equations simultaneously. First, we can express \( x \) in terms of \( y \) using Equation 1:
\[ x = 50 - y \]

Now, substitute \( x \) in Equation 2:
\[ 1(50 - y) + 1.5y = 72.50 \]

### Step 4: Simplify the Equation
Distributing the \( 1 \):
\[ 50 - y + 1.5y = 72.50 \]
Combine like terms:
\[ 50 + 0.5y = 72.50 \]

### Step 5: Isolate \( y \)
Subtract 50 from both sides:
\[ 0.5y = 72.50 - 50 \]
\[ 0.5y = 22.50 \]

### Step 6: Solve for \( y \)
Now, divide both sides by 0.5:
\[ y = \frac{22.50}{0.5} = 45 \]

### Step 7: Conclusion
Thus, Caleb bought \( y = 45 \) double burgers.

### Step 8: Find \( x \)
To find the number of single burgers:
\[ x = 50 - y = 50 - 45 = 5 \]

### Final Answer
Caleb bought 45 double burgers.

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