The sum of a fraction and ten times its reciprocal is 37/4. What is the fraction?

Introduction / Context:
This problem again uses a relationship between a fraction and its reciprocal, with a larger coefficient on the reciprocal term. It is an algebraic question that reinforces skills in forming and solving quadratic equations.

Given Data / Assumptions:

• Let the fraction be x, with x not equal to 0.
• The reciprocal is 1/x.
• The condition is x + 10 * (1/x) = 37/4.
• We need to find x among the given options.

Concept / Approach:
We translate the condition into an equation, clear denominators, and solve the quadratic equation that results. Finally, we see which algebraic solution appears in the answer choices.

Step-by-Step Solution:

Verification / Alternative check:
Check x = 5/4: 5/4 + 10 * (4/5) = 5/4 + 8 = 5/4 + 32/4 = 37/4, which matches the given sum. The value x = 8 also satisfies the equation, because 8 + 10 * (1/8) = 8 + 1.25 = 9.25 = 37/4. However, 8 is not listed among the options, whereas 5/4 is.

Why Other Options Are Wrong:
Substituting 4/5, 3/4 or 4/3 into x + 10/x does not produce 37/4. Their resulting sums are quite different, so these options do not satisfy the condition.

Common Pitfalls:
Mistakes can occur when clearing denominators and simplifying the quadratic. Some candidates may forget that there can be two valid algebraic solutions but only one matching the provided options. Always test candidate values back in the original equation.

Final Answer:
The fraction that fits the condition and appears in the choices is 5/4.