The sum of a positive fraction and four times its reciprocal is 13/3. If the fraction lies between 1 and 2, what is the exact value of the fraction?

Introduction / Context:
This question involves forming and solving a rational equation where a fraction and a multiple of its reciprocal sum to a given value. Because this leads to a quadratic equation in the fraction, we may obtain two algebraic solutions. The additional condition that the fraction lies between 1 and 2 is used to select the appropriate solution, which is a common technique in aptitude questions.

Given Data / Assumptions:

• Let the fraction be x (x > 0).
• x + 4(1/x) = 13/3.
• The fraction lies strictly between 1 and 2.
• x ≠ 0 since its reciprocal is involved.

Concept / Approach:
We treat x as an unknown and set up the equation x + 4/x = 13/3. Multiplying through by x clears the denominator and results in a quadratic equation. After solving this quadratic, we examine both roots and use the given range for x (between 1 and 2) to pick the correct value. This approach highlights the connection between rational equations and quadratic equations.

Step-by-Step Solution:
Start from x + 4/x = 13/3. Multiply both sides by x: x² + 4 = (13/3)x. Multiply everything by 3 to clear the denominator: 3x² + 12 = 13x. Rearrange to standard quadratic form: 3x² − 13x + 12 = 0. Solve the quadratic 3x² − 13x + 12 = 0. Compute the discriminant Δ = (−13)² − 4·3·12 = 169 − 144 = 25. √Δ = 5, so x = [13 ± 5] / (2·3). Thus x₁ = (13 + 5)/6 = 18/6 = 3, and x₂ = (13 − 5)/6 = 8/6 = 4/3. Given that the fraction lies between 1 and 2, we discard x = 3. So the valid fraction is x = 4/3.

Verification / Alternative check:
Substitute x = 4/3 into the original condition. Then 1/x = 3/4 and 4(1/x) = 3. So x + 4(1/x) = 4/3 + 3 = 4/3 + 9/3 = 13/3, which matches the given sum exactly. Also, 4/3 is approximately 1.333, which is indeed between 1 and 2, so it satisfies the range condition.

Why Other Options Are Wrong:
3/4 and 4/5 are less than 1 and violate the condition that the fraction lies between 1 and 2. 5/4 is between 1 and 2 but does not satisfy x + 4/x = 13/3 when substituted. 3/2 again fails the equation; its substitution gives a different sum. Only 4/3 both satisfies the rational equation and lies in the required interval.

Common Pitfalls:
A common error is to forget the range condition and accept both roots 3 and 4/3 as answers. Another mistake is miscalculating the discriminant or mishandling the algebra when clearing denominators. Always check each root in the original equation and apply any extra conditions given in the question.

Final Answer:
Therefore, the required fraction is 4/3.