A manager purchased a total of 21 coffee mugs and key chains. Each coffee mug cost $8.50, and each key chain cost $2.75.

If the manager spent a total of $132.50, how many coffee mugs did the manager purchase?

Let the number of coffee mugs purchased be $ x $, and the number of key chains purchased be $ y $.

We are given two pieces of information:

1. The total number of items purchased is 21:
   \[
   x + y = 21
   \]

2. The total cost is $132.50, where each coffee mug costs $8.50 and each key chain costs $2.75:
   \[
   8.50x + 2.75y = 132.50
   \]

We can solve this system of equations using substitution or elimination. Let's use substitution.

### Step 1: Solve for $ y $ in terms of $ x $ from the first equation:
\[
y = 21 - x
\]

### Step 2: Substitute this into the second equation:
\[
8.50x + 2.75(21 - x) = 132.50
\]

### Step 3: Distribute the 2.75:
\[
8.50x + 2.75 \times 21 - 2.75x = 132.50
\]
\[
8.50x + 57.75 - 2.75x = 132.50
\]

### Step 4: Combine like terms:
\[
(8.50x - 2.75x) + 57.75 = 132.50
\]
\[
5.75x + 57.75 = 132.50
\]

### Step 5: Subtract 57.75 from both sides:
\[
5.75x = 132.50 - 57.75
\]
\[
5.75x = 74.75
\]

### Step 6: Solve for $ x $:
\[
x = \frac{74.75}{5.75}
\]
\[
x = 13
\]

Thus, the manager purchased **13 coffee mugs**.

### Step 7: Find the number of key chains:
\[
y = 21 - 13 = 8
\]

So, the manager purchased 13 coffee mugs and 8 key chains.

Would you like further details or clarification on any step?

Here are 5 related questions to expand on this problem:

1. What would be the cost if the manager bought 5 more coffee mugs?
2. How would the solution change if the price of key chains increased by $1?
3. What if the total number of items was 30 but with the same total cost?
4. How can this problem be solved using the elimination method instead?
5. How does solving a system of equations graphically differ from the algebraic method used here?

**Tip:** When dealing with systems of equations, substitution is a useful method when one equation is easily solvable for a variable.

A manager purchased a total of 21 coffee mugs and key chains. Each coffee mug cost $8.50, and each key chain cost $2.75. If the manager spent a total of $132.50, how many coffee mugs did the manager purchase?

This problem involves solving a system of linear equations to find how many coffee mugs and key chains were purchased. The problem uses algebraic concepts like substitution and linear equations with a step-by-step solution, suitable for Grades 8-10.

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