Roy is starting a baking company and decides that he will sell cupcakes. He sells $n$ cupcakes for $(n + 20)(n + 15)$ cents. A man walks in and buys $\$10.50$ worth of cupcakes. Roy bakes cupcakes at a rate of $10$ cupcakes an hour. How many minutes will it take Roy to complete the order?
1. First, we need to determine the number of cupcakes \( n \) that the man buys for \$10.50. Since the price is given in cents, we convert \$10.50 to cents:
   \[
   10.50 \text{ dollars} = 1050 \text{ cents}
   \]

2. The price for \( n \) cupcakes is given by the expression \((n + 20)(n + 15)\). We set this equal to 1050 cents:
   \[
   (n + 20)(n + 15) = 1050
   \]

3. To find \( n \), we solve the quadratic equation. Notice that \( n + 20 \) and \( n + 15 \) are 5 units apart. We need to find two numbers that multiply to 1050 and are 5 units apart. Let's denote these numbers as \( a \) and \( b \) such that \( a = n + 20 \) and \( b = n + 15 \). Therefore:
   \[
   a \cdot b = 1050 \quad \text{and} \quad a - b = 5
   \]

4. Solving these equations, we can express \( a \) in terms of \( b \):
   \[
   a = b + 5
   \]

5. Substitute \( a = b + 5 \) into the product equation:
   \[
   (b + 5)b = 1050
   \]
   \[
   b^2 + 5b = 1050
   \]
   \[
   b^2 + 5b - 1050 = 0
   \]

6. Solve the quadratic equation using the quadratic formula \( b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \), where \( A = 1 \), \( B = 5 \), and \( C = -1050 \):
   \[
   b = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot (-1050)}}{2 \cdot 1}
   \]
   \[
   b = \frac{-5 \pm \sqrt{25 + 4200}}{2}
   \]
   \[
   b = \frac{-5 \pm \sqrt{4225}}{2}
   \]
   \[
   b = \frac{-5 \pm 65}{2}
   \]

7. This gives us two solutions for \( b \):
   \[
   b = \frac{60}{2} = 30 \quad \text{and} \quad b = \frac{-70}{2} = -35
   \]
   Since \( b \) must be positive, we have \( b = 30 \).

8. Now, we find \( a \):
   \[
   a = b + 5 = 30 + 5 = 35
   \]

9. Therefore, \( n + 15 = 30 \) and \( n + 20 = 35 \). Solving for \( n \):
   \[
   n + 15 = 30 \implies n = 15
   \]

10. Roy bakes cupcakes at a rate of 10 cupcakes per hour. To find the time it takes to bake 15 cupcakes:
    \[
    \text{Time} = \frac{15 \text{ cupcakes}}{10 \text{ cupcakes/hour}} = 1.5 \text{ hours}
    \]

11. Convert hours to minutes:
    \[
    1.5 \text{ hours} = 1.5 \times 60 \text{ minutes} = 90 \text{ minutes}
    \]

The final answer is \(\boxed{90 \text{ minutes}}\).