The sum of a non-zero number and three times its reciprocal is 52/7. What is the larger of the possible values of this number?

Introduction / Context:
This problem tests your ability to set up and solve an equation that involves both a number and its reciprocal. Such questions are common in algebra and quantitative aptitude because they combine fractions, quadratic equations and careful interpretation of the worded condition. Here, the expression uses three times the reciprocal of the number, and the total is given as 52/7. We are specifically asked for the larger possible value of the number.

Given Data / Assumptions:

- Let the non-zero number be x. - Its reciprocal is 1 / x. - The sum x + 3 * (1 / x) is equal to 52 / 7. - x is real and non zero so that the reciprocal is defined. - We are asked to report the larger of the possible values of x.

Concept / Approach:
The key idea is to translate the verbal description into an algebraic equation and then clear the denominator by multiplying through by x. This produces a quadratic equation in standard form. Solving the quadratic using factoring or the quadratic formula yields two possible values for x. Finally, we compare these values and select the larger one as required by the question. Careful simplification of fractions is important to avoid errors.

Step-by-Step Solution:
Step 1: Write the equation from the statement: x + 3 * (1 / x) = 52 / 7. Step 2: Combine the terms: x + 3 / x = 52 / 7. Step 3: Multiply both sides by x to eliminate the denominator: x^2 + 3 = (52 / 7) * x. Step 4: Multiply both sides by 7 to clear the fraction: 7x^2 + 21 = 52x. Step 5: Bring all terms to one side: 7x^2 - 52x + 21 = 0. Step 6: This is a quadratic equation in standard form ax^2 + bx + c = 0. Step 7: Compute the discriminant: D = b^2 - 4ac = 52^2 - 4 * 7 * 21 = 2704 - 588 = 2116. Step 8: The square root of 2116 is 46, since 46 * 46 = 2116. Step 9: Use the quadratic formula: x = [52 ± 46] / (2 * 7) = [52 ± 46] / 14. Step 10: First root: x = (52 + 46) / 14 = 98 / 14 = 7. Step 11: Second root: x = (52 - 46) / 14 = 6 / 14 = 3 / 7. Step 12: The two possible values of x are 7 and 3 / 7. The larger of these is 7.

Verification / Alternative check:
Substitute x = 7 back into the original expression. The reciprocal is 1 / 7. Compute x + 3 * (1 / x) = 7 + 3 / 7 = (49 / 7) + (3 / 7) = 52 / 7, which matches the given value. For completeness, if x = 3 / 7, its reciprocal is 7 / 3, and x + 3 * (1 / x) again simplifies to 52 / 7. Both values satisfy the equation, so choosing the larger one, 7, is correct according to the question.

Why Other Options Are Wrong:
Values 6, 8 and 9 do not satisfy x + 3 / x = 52 / 7 when substituted. For example, with x = 8, we get 8 + 3 / 8, which is clearly far from 52 / 7. These values come from guessing rather than solving the equation and therefore cannot be correct answers.

Common Pitfalls:
Some learners forget to multiply the entire equation by x or 7 when clearing denominators, which leads to incomplete or incorrect equations. Others might mistakenly think there is only one solution and ignore the second root. It is important to remember that quadratic equations typically have two solutions and then use the wording of the problem to decide which one is required. Careful algebraic manipulation and verification by substitution help avoid such mistakes.

Final Answer:
The larger possible value of the number is 7, which matches option C.