The sum of a non zero number and nine times its reciprocal is 10. What is the value of the number?

Introduction / Context:
This question again deals with a number and its reciprocal, leading to a quadratic equation. It checks the ability to translate a word statement into algebra, manipulate equations involving reciprocals, and then correctly interpret the resulting solutions.

Given Data / Assumptions:
- Let the non zero number be x.
- The relationship given is x + 9 * (1 / x) = 10.
- x is not zero, so 1 / x is defined.

Concept / Approach:
We represent the sentence "the sum of a non zero number and nine times its reciprocal is 10" by an equation in x. Multiplying both sides by x removes the denominator and yields a quadratic equation. Solving this quadratic gives all possible values of x that satisfy the condition, and we check which of these values match the options.

Step-by-Step Solution:
Start from x + 9 * (1 / x) = 10. Multiply both sides by x: x^2 + 9 = 10x. Rearrange into standard quadratic form: x^2 − 10x + 9 = 0. Factorise: x^2 − 10x + 9 = (x − 1)(x − 9) = 0. Thus, possible solutions are x = 1 or x = 9.

Verification / Alternative Check:
Check x = 1: 1 + 9 * (1 / 1) = 1 + 9 = 10, so x = 1 satisfies the equation. Check x = 9: 9 + 9 * (1 / 9) = 9 + 1 = 10, so x = 9 also satisfies the equation. Both values are mathematically valid, but among the given options, only 9 appears, so 9 is the correct choice for this multiple choice question.

Why Other Options Are Wrong:
10 would give 10 + 9 * (1 / 10) = 10.9, not 10.
11 would give 11 + 9 / 11 which is greater than 11 and not equal to 10.
90 leads to 90 + 9 * (1 / 90) which is approximately 90.1, again not equal to 10.
8 yields 8 + 9 / 8 = 8 + 1.125 = 9.125, so it fails to satisfy the condition.

Common Pitfalls:
Students sometimes forget that quadratic equations can have two solutions and may stop after finding only one root. Another common mistake is incorrect multiplication when clearing denominators, which leads to a wrong quadratic. Finally, learners may not check which of the valid mathematical solutions actually appears among the options, which is crucial in multiple choice settings.

Final Answer:
The non zero number whose sum with nine times its reciprocal equals 10 is 9.