Let \( x \) be the number of coffee mugs purchased, and \( y \) be the number of key chains purchased. We can set up the following equations based on the given information:
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The total number of items: \[ x + y = 21 \]
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The total cost of the items: \[ 8.5x + 2.75y = 132.5 \]
We can solve this system of equations starting with the first equation. We can express \( y \) in terms of \( x \): \[ y = 21 - x \]
Now, substitute \( y \) into the second equation: \[ 8.5x + 2.75(21 - x) = 132.5 \]
Expanding the equation: \[ 8.5x + 57.75 - 2.75x = 132.5 \]
Combining like terms: \[ (8.5 - 2.75)x + 57.75 = 132.5 \] \[ 5.75x + 57.75 = 132.5 \]
Now, isolate \( x \): \[ 5.75x = 132.5 - 57.75 \] \[ 5.75x = 74.75 \] \[ x = \frac{74.75}{5.75} \] \[ x = 13 \]
Now that we have \( x \), we can find \( y \): \[ y = 21 - x = 21 - 13 = 8 \]
Thus, the manager purchased \( \boxed{13} \) coffee mugs.
To verify, we calculate the total cost with these values:
- Cost of coffee mugs: \( 13 \times 8.5 = 110.5 \)
- Cost of key chains: \( 8 \times 2.75 = 22 \)
Adding these: \[ 110.5 + 22 = 132.5 \]
Since the total cost matches, the solution is confirmed. The manager purchased \( \boxed{13} \) coffee mugs.
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