A non zero number is greater than seven times its reciprocal by 9.3. What is that number?

Introduction / Context:
This question involves an equation with a number and its reciprocal. It tests understanding of how to translate a verbal description into an algebraic equation and then solve a quadratic equation that arises from the relationship.

Given Data / Assumptions:
- Let the required non zero number be x.
- The phrase "greater than seven times its reciprocal by 9.3" means x − 7 * (1 / x) = 9.3.
- x is non zero so that its reciprocal 1 / x is well defined.

Concept / Approach:
We convert the English statement into an algebraic equation involving x and 1 / x. Multiplying both sides of the equation by x eliminates the denominator and gives a quadratic equation in x. Solving this quadratic equation will provide all possible values of x that satisfy the condition, and then we compare them with the options provided.

Step-by-Step Solution:
Let x be the unknown number. According to the statement, x − 7 * (1 / x) = 9.3. Multiply both sides by x to clear the denominator: x^2 − 7 = 9.3x. Rewrite as a standard quadratic: x^2 − 9.3x − 7 = 0. Solve this quadratic. The solutions are x = 10 and x = −0.7.

Verification / Alternative Check:
Check x = 10: 7 times the reciprocal is 7 * (1 / 10) = 0.7. The number minus this value is 10 − 0.7 = 9.3, which matches the condition. For x = −0.7, seven times the reciprocal is 7 * (1 / −0.7) = −10, and −0.7 minus (−10) equals 9.3 as well. Both values satisfy the equation, but only 10 appears among the answer options, so it is the required choice.

Why Other Options Are Wrong:
If x = 20, then 20 − 7 * (1 / 20) = 20 − 0.35 = 19.65, not 9.3.
If x = 5, then 5 − 7 * (1 / 5) = 5 − 1.4 = 3.6.
If x = 14, then 14 − 7 * (1 / 14) is approximately 13.5, again not 9.3.
Only x = 10 correctly satisfies the original relationship among the provided options.

Common Pitfalls:
Learners sometimes misread the phrase "greater than" and set 7 * (1 / x) − x = 9.3 instead of x − 7 * (1 / x) = 9.3. Another common error is to forget to multiply each term by x when clearing the denominator, which leads to an incorrect quadratic. Finally, after solving the quadratic, it is important to check which solution is presented in the options instead of assuming only one solution exists.

Final Answer:
The non zero number that is greater than seven times its reciprocal by 9.3 is 10.