The sum of twice a fraction and three times its reciprocal is 29/3. What is the value of this fraction?

Introduction / Context:
This question again explores algebraic relationships between a fraction and its reciprocal. The expression given combines twice the fraction and three times its reciprocal to produce 29/3. Such problems test your skill in forming and solving quadratic equations based on verbal descriptions.

Given Data / Assumptions:

• Let the fraction be x, with x ≠ 0.
• Its reciprocal is 1/x.
• The given condition is 2x + 3 * (1/x) = 29/3.
• We must find the value of x from the options.

Concept / Approach:
We transform the verbal relation into an algebraic equation in x. To get rid of the denominator x, we multiply throughout by x. This produces a quadratic equation, which we solve using the quadratic formula. Then we compare the obtained roots to the answer choices and pick the appropriate one.

Step-by-Step Solution:
Step 1: Start from the equation 2x + 3/x = 29/3. Step 2: Multiply both sides by 3x to eliminate the denominator: 3x * 2x + 3x * (3/x) = 3x * 29/3. Step 3: Simplify: 6x^2 + 9 = 29x. Step 4: Rearrange to quadratic form: 6x^2 - 29x + 9 = 0. Step 5: Use the quadratic formula: x = [29 ± sqrt(29^2 - 4 * 6 * 9)] / (2 * 6). Step 6: Compute the discriminant: 29^2 = 841; 4 * 6 * 9 = 216; so discriminant = 841 - 216 = 625. Step 7: The square root of 625 is 25, so x = (29 ± 25) / 12. Step 8: The roots are x = (29 + 25)/12 = 54/12 = 9/2 and x = (29 - 25)/12 = 4/12 = 1/3.

Verification / Alternative Check:
Test x = 9/2. Then the reciprocal is 2/9. Compute 2x + 3/x = 2 * (9/2) + 3 * (2/9) = 9 + 6/9 = 9 + 2/3. Convert 9 to thirds: 9 = 27/3, so 27/3 + 2/3 = 29/3, matching the given value. The other root 1/3 also satisfies the equation but does not appear in the options provided. Therefore, 9/2 is the correct choice.

Why Other Options Are Wrong:
2/9, 5/4 and 4/5 do not satisfy the equation when substituted into 2x + 3/x. In each case, evaluating the expression will give a value different from 29/3, so these options must be rejected.

Common Pitfalls:
Learners sometimes mis-handle the multiplication by 3x and either forget a term or misplace coefficients. Another common issue is an error in computing the discriminant for the quadratic formula. Writing each intermediate step clearly helps reduce such mistakes.

Final Answer:
The fraction that satisfies the condition is 9/2.