Twice a certain fraction plus five times its reciprocal adds up to 7. What is the value of this fraction?

Introduction / Context:
This question is a classic algebraic equation involving a fraction and its reciprocal. You are told that twice the fraction plus five times its reciprocal sums to 7, and you must find the fraction. The problem tests your skill in setting up equations and solving quadratic equations.

Given Data / Assumptions:
Let the fraction be x, where x is a nonzero rational number.Twice the fraction is 2x.The reciprocal of the fraction is 1/x.Five times its reciprocal is 5/x.The equation given is 2x + 5/x = 7.

Concept / Approach:
Because the equation involves both x and 1/x, multiplying through by x will clear the denominator and convert the equation into a quadratic equation in x. Quadratic equations of the form ax^2 + bx + c = 0 can be solved using factorization or the quadratic formula. We then check which roots match the given options.

Step-by-Step Solution:
Step 1: Start from the equation 2x + 5/x = 7.Step 2: Multiply both sides by x (x ≠ 0) to eliminate the denominator: 2x^2 + 5 = 7x.Step 3: Rearrange the equation into standard quadratic form: 2x^2 − 7x + 5 = 0.Step 4: Compute the discriminant: D = b^2 − 4ac = (−7)^2 − 4 × 2 × 5 = 49 − 40 = 9.Step 5: Take the square root of the discriminant: √D = √9 = 3.Step 6: Solve for x using the quadratic formula x = [7 ± 3] / (2 × 2) = (7 ± 3) / 4.Step 7: The two solutions are x = (7 + 3) / 4 = 10 / 4 = 5/2 and x = (7 − 3) / 4 = 4 / 4 = 1.Step 8: Among the given options, only 5/2 appears; 1 is not listed as an option.

Verification / Alternative check:
Substitute x = 5/2 back into the original equation. Twice the fraction is 2 × 5/2 = 5. The reciprocal is 2/5, so five times the reciprocal is 5 × 2/5 = 2. Adding these gives 5 + 2 = 7, which satisfies the equation. This confirms that 5/2 is indeed a correct solution. Since 1 is not provided in the options, 5/2 is the only valid choice.

Why Other Options Are Wrong:
For x = 2/5, we get 2 × 2/5 + 5 × 5/2 = 4/5 + 25/2, which is far greater than 7. For x = 5/4 or 4/5, similar substitutions do not yield 7 as the total. Hence, none of these options satisfy the equation.

Common Pitfalls:
It is easy to make algebraic mistakes when clearing the denominator, such as forgetting to multiply every term by x. Another common error is mishandling the quadratic formula or miscomputing the discriminant. Carefully following each algebraic step and checking solutions in the original equation helps avoid these mistakes.

Final Answer:
The fraction that satisfies the condition is 5/2.