The sum of a non-zero number and twenty times its reciprocal is 9. What is the larger of the possible values of this number?

Introduction / Context:
This problem explores the relationship between a number and its reciprocal. You are told that the sum of a non zero number and twenty times its reciprocal equals 9. From this, you must find the larger of the possible values of the number. The question leads naturally to a quadratic equation and tests your ability to handle reciprocals and solve quadratics.

Given Data / Assumptions:

- Let the non zero number be x. - Its reciprocal is 1 / x. - The equation is x + 20 * (1 / x) = 9. - x is non zero, so division by x is valid. - We are specifically asked for the larger possible value of x.

Concept / Approach:
The main idea is to convert the given expression into a standard quadratic equation. Multiplying through by x eliminates the denominator and yields a polynomial equation. Factoring this quadratic gives two possible values for x. Because the problem asks for the larger value, we pick the bigger of the two real roots. This process combines algebraic manipulation with an understanding of reciprocals.

Step-by-Step Solution:
Step 1: Start from the equation x + 20 * (1 / x) = 9. Step 2: Write it as x + 20 / x = 9. Step 3: Multiply both sides of the equation by x to clear the denominator: x^2 + 20 = 9x. Step 4: Rearrange the equation into standard quadratic form: x^2 - 9x + 20 = 0. Step 5: Factor the quadratic: x^2 - 9x + 20 = (x - 4)(x - 5) = 0. Step 6: Set each factor to zero to find the roots: x - 4 = 0 or x - 5 = 0. Step 7: Thus, x = 4 or x = 5. Step 8: Both solutions are non zero and satisfy the original equation, as can be checked by substitution. Step 9: The larger of these two values is x = 5.

Verification / Alternative check:
Verify each root. For x = 4, compute x + 20 / x = 4 + 20 / 4 = 4 + 5 = 9. For x = 5, compute x + 20 / x = 5 + 20 / 5 = 5 + 4 = 9. Both satisfy the original equation. Since the question asks for the larger value, we select 5. This confirms that 5 is the correct answer.

Why Other Options Are Wrong:
Option A (-5) and option C (-3) give negative values that do not satisfy x + 20 / x = 9 when substituted. Option B (4) is a valid solution of the equation but it is the smaller of the two possible values, while the question explicitly asks for the larger one. Therefore, only 5 correctly answers what is asked.

Common Pitfalls:
Some learners try to guess x by mental arithmetic without forming the quadratic, which can be unreliable. Others may forget to multiply the entire equation by x, leading to an incorrect quadratic. A few may stop after finding only one root, ignoring the possibility of a second solution. Writing the equation in standard form, factoring carefully and then comparing the roots ensures a correct and complete solution.

Final Answer:
The larger possible value of the number is 5, which corresponds to option D.