The sum of a fraction and three times its reciprocal is 37/10. Given that the fraction is greater than 3/2, find the exact value of this fraction.

Introduction / Context:
This problem involves a rational equation where a fraction and a multiple of its reciprocal sum to a given value. The condition that the fraction is greater than 3/2 ensures a unique solution among the possible roots of the resulting quadratic equation.

Given Data / Assumptions:

• Let the fraction be x
• x + 3(1/x) = 37/10
• x is greater than 3/2
• x is non zero since its reciprocal is defined

Concept / Approach:
We treat x as an unknown and form an equation with rational terms. Multiplying through by x clears the denominator and yields a quadratic equation in x. Solving that quadratic provides two algebraic solutions, and the inequality condition x > 3/2 allows us to select the appropriate one.

Step-by-Step Solution:
Start from x + 3/x = 37/10 Multiply both sides by x (x ≠ 0) x² + 3 = (37/10)x Rearrange to standard form: 10x² + 30 = 37x So 10x² − 37x + 30 = 0 Solve the quadratic 10x² − 37x + 30 = 0 Discriminant Δ = 37² − 4·10·30 = 1369 − 1200 = 169 √Δ = 13 x = [37 ± 13] / (2·10) So x₁ = (37 + 13)/20 = 50/20 = 5/2 And x₂ = (37 − 13)/20 = 24/20 = 6/5 Given x > 3/2, we note 3/2 = 1.5, 6/5 = 1.2, and 5/2 = 2.5 Only x = 5/2 satisfies x > 3/2

Verification / Alternative check:
Substitute x = 5/2 into the original relation. Then 1/x = 2/5 and 3(1/x) = 6/5. Compute x + 3(1/x) = 5/2 + 6/5. With denominator 10 we have 25/10 + 12/10 = 37/10, which matches the given value exactly. Also x = 5/2 is indeed greater than 3/2 so it respects the condition.

Why Other Options Are Wrong:
2/5 and 4/5 are less than 1 and do not satisfy the original equation when substituted. 5/4 equals 1.25 and is still less than 3/2, so it violates the given condition. 3/2 matches the boundary but not the equation; substituting it does not yield 37/10. Only 5/2 both satisfies the equation and the inequality constraint x > 3/2.

Common Pitfalls:
Learners often forget that rational equations can yield two solutions and may omit checking given inequalities or restrictions. Another mistake is multiplying incorrectly when clearing denominators, leading to a wrong quadratic. Always verify both the algebraic solutions and any conditions stated in the question.

Final Answer:
Therefore, the fraction that satisfies the given condition is 5/2.