The sum of a fraction and three times its reciprocal is 19/4. What is the value of the fraction?

Introduction / Context:
This problem tests algebraic manipulation with fractions and reciprocals. You are given an equation involving an unknown fraction and its reciprocal, and you must solve for that fraction. Questions of this type commonly appear in aptitude tests to assess comfort with rational expressions and quadratic equations arising from them.

Given Data / Assumptions:
- Let the required fraction be x, where x is nonzero.
- The reciprocal of x is 1 / x.
- The sum of the fraction and three times its reciprocal is given as 19 / 4.
- Mathematically, x + 3 * (1 / x) = 19 / 4.
- We must find a value of x that satisfies this equation and matches one of the options.

Concept / Approach:
To solve equations involving a fraction and its reciprocal, we eliminate the denominator by multiplying through by x, which leads to a quadratic equation in x. Quadratic equations can then be solved using factoring or the quadratic formula. Finally, we match the solution to the options given. Since the problem is framed around fractions, the acceptable solution should be a rational value that makes sense in the context.

Step-by-Step Solution:
Step 1: Start from the equation x + 3 * (1 / x) = 19 / 4.Step 2: Multiply both sides of the equation by x to eliminate the denominator. This gives x^2 + 3 = (19 / 4) * x.Step 3: Multiply both sides by 4 to clear the remaining denominator: 4x^2 + 12 = 19x.Step 4: Rearrange the equation to standard quadratic form: 4x^2 - 19x + 12 = 0.Step 5: Solve this quadratic equation. Compute the discriminant D = 19^2 - 4 * 4 * 12 = 361 - 192 = 169.Step 6: The square root of 169 is 13, so the solutions are x = (19 ± 13) / (2 * 4) = (19 ± 13) / 8.Step 7: This gives two possible values: x = (19 + 13) / 8 = 32 / 8 = 4 and x = (19 - 13) / 8 = 6 / 8 = 3 / 4.Step 8: Both values satisfy the equation algebraically, but we must choose the one that appears among the answer options provided.Step 9: The fraction 4 is not listed, but 3 / 4 is given as one of the options.

Verification / Alternative check:
Check x = 3 / 4 in the original equation. The reciprocal of 3 / 4 is 4 / 3. Then x + 3 * (1 / x) becomes 3 / 4 + 3 * (4 / 3) = 3 / 4 + 4. The value 4 can be written as 16 / 4. Therefore, 3 / 4 + 16 / 4 = 19 / 4, which matches the given sum. This confirms that x = 3 / 4 is indeed a correct solution and is consistent with the options.

Why Other Options Are Wrong:
If x = 4 / 3, its reciprocal is 3 / 4, and x + 3 * (1 / x) becomes 4 / 3 + 3 * (3 / 4) = 4 / 3 + 9 / 4, which is not equal to 19 / 4. Similarly, using x = 5 / 4 or 4 / 5 or 2 / 3 in the equation will not produce 19 / 4 after simplification. Therefore, these options do not satisfy the condition given in the question.

Common Pitfalls:
Some learners forget to multiply by x to remove the denominator and instead try to treat x and 1 / x as independent. Others might solve the quadratic but discard the fractional solution or miscompute the discriminant. Carefully following the algebraic steps and checking each potential solution in the original equation ensures that you choose the correct fraction from the list of options.

Final Answer:
The required fraction is 3/4.