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

Introduction / Context:
This algebraic question generalizes the earlier patterns by using twice a fraction and three times its reciprocal. It is designed to test your comfort with forming equations from verbal statements and solving quadratics.

Given Data / Assumptions:

• Let the fraction be x, with x not equal to 0.
• The reciprocal is 1/x.
• The condition given is 2x + 3 * (1/x) = 29/3.
• We must determine the value of x from the answer choices.

Concept / Approach:
We transform the condition into a quadratic equation by clearing denominators. After solving the quadratic, we test the resulting values against the options to identify the correct fraction.

Step-by-Step Solution:

Verification / Alternative check:
Check x = 9/2: 2 * (9/2) + 3 * (2/9) = 9 + 6/9 = 9 + 2/3 = 27/3 + 2/3 = 29/3. This matches the given condition. The other root x = 1/3 yields 2 * (1/3) + 3 * 3 = 2/3 + 9 = 29/3 as well, but 1/3 is not listed among the options.

Why Other Options Are Wrong:
Substituting 2/9, 5/4 or 4/5 into 2x + 3/x does not produce 29/3. Their resulting values are different, so they do not satisfy the equation.

Common Pitfalls:
Errors can occur when multiplying by the common denominator, especially if a term is accidentally omitted. Another pitfall is incorrect calculation of the discriminant, which leads to wrong roots. Finally, some students forget to check which algebraic solution appears in the multiple choice list.

Final Answer:
The fraction that satisfies the relationship and appears in the options is 9/2.