Let's denote the normal price of each shirt as \( p \). Since Brand Blast reduced the price by $5 for the bulk order, the price per shirt after the reduction becomes \( p - 5 \).
Principal Howard ordered 116 shirts, and he paid a total of $928. We can write the equation representing the total cost:
\[ 116(p - 5) = 928 \]
Next, we will simplify this equation:
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Distribute 116: \[ 116p - 580 = 928 \]
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Add 580 to both sides to isolate the term with \( p \): \[ 116p = 928 + 580 \] \[ 116p = 1508 \]
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Now, divide both sides by 116 to solve for \( p \): \[ p = \frac{1508}{116} \] \[ p = 13 \]
Therefore, the normal price for each shirt is \( \boxed{13} \).