1. The problem states: The average of the numbers 1, 4, 10, P, and 16 is 9. 2. The formula for the average of $n$ numbers $x_1, x_2, ..., x_n$ is: $$\text{Average} = \frac{x_1 + x_2 + ... + x_n}{n}$$ 3. Here, the numbers are 1, 4, 10, P, and 16, so $n=5$ and the average is 9. 4. Substitute the values into the formula: $$9 = \frac{1 + 4 + 10 + P + 16}{5}$$ 5. Simplify the numerator: $$1 + 4 + 10 + 16 = 31$$ 6. So the equation becomes: $$9 = \frac{31 + P}{5}$$ 7. Multiply both sides by 5 to eliminate the denominator: $$9 \times 5 = 31 + P$$ $$45 = 31 + P$$ 8. Subtract 31 from both sides to solve for $P$: $$P = 45 - 31$$ $$P = 14$$ 9. Therefore, the value of $P$ is 14.