The sum of a non zero number and ten times its reciprocal is 7. What is the value of the number?

Introduction / Context:
This problem tests your ability to translate a verbal statement involving a number and its reciprocal into an algebraic equation. Once the equation is set up, you must solve a quadratic to find possible values of the number. Questions of this type are common in algebra and aptitude exams, and they reinforce the relationship between a quantity and its reciprocal.

Given Data / Assumptions:

• The number is non zero.
• Let the number be x.
• The expression given is x plus ten times its reciprocal.
• x + 10/x = 7.

Concept / Approach:
The general method is to clear the denominator by multiplying both sides of the equation by x, which produces a quadratic equation. Then we factor the quadratic and identify the valid roots. Both roots are algebraically valid, but only the root that appears in the option list can be chosen as the correct multiple choice answer.

Step-by-Step Solution:
Step 1: Start with x + 10/x = 7.Step 2: Multiply both sides by x to remove the denominator: x^2 + 10 = 7x.Step 3: Rearrange to standard quadratic form: x^2 - 7x + 10 = 0.Step 4: Factor the quadratic: x^2 - 7x + 10 = (x - 5)(x - 2).Step 5: Set each factor equal to zero: x - 5 = 0 or x - 2 = 0, so x = 5 or x = 2.

Verification / Alternative check:
Check x = 5: 5 + 10/5 = 5 + 2 = 7, which satisfies the condition. Check x = 2: 2 + 10/2 = 2 + 5 = 7, which also satisfies the condition. However, in the given answer choices only 5 appears. Therefore, 5 is the correct option to select in this multiple choice setting.

Why Other Options Are Wrong:
Option 4 gives 4 + 10/4 = 4 + 2.5 = 6.5, not 7. Option 3 gives 3 + 10/3, which is greater than 6 and less than 7, so it does not match exactly. Option 6 yields 6 + 10/6, which is more than 7. Option 2 is algebraically correct but is not listed as a choice; in the context of this MCQ, the only matching available choice is 5.

Common Pitfalls:

• Forgetting to multiply every term by x when clearing the denominator.
• Making sign errors when rearranging the quadratic.
• Not checking which algebraic root actually appears among the given options.

Final Answer:
The required number, matching the given options, is 5.