A certain real number is greater than 58 times its own reciprocal by 3/4. What is the value of this number?

Introduction / Context:
This question tests your understanding of reciprocal numbers and how to convert a word statement involving a number and its reciprocal into a quadratic equation. Problems of this type are very common in aptitude exams because they combine algebraic manipulation, equation formation, and the ability to select the correct solution from the given options. Here we are told that a number is greater than 58 times its reciprocal by a certain amount, and we must carefully translate that language into a mathematical equation and then solve it.

Given Data / Assumptions:

• Let the required real number be x.
• The reciprocal of the number is 1 / x, assuming x is not equal to 0.
• The number is greater than 58 times its reciprocal by 3/4.
• We assume standard real number arithmetic and look for values of x among the given options.

Concept / Approach:
The key idea is to translate the statement "a number is greater than 58 times its reciprocal by 3/4" into an algebraic equation. The phrase "greater by 3/4" means that if we subtract 58 times the reciprocal from the number, the result is 3/4. This leads to an equation of the form x - 58 * (1 / x) = 3/4. Once this equation is formed, we multiply through by a suitable factor to clear denominators and obtain a quadratic equation in x. Solving the quadratic gives two possible values, and we then match these with the given options to choose the correct answer.

Step-by-Step Solution:
Let the number be x and its reciprocal be 1 / x. According to the question, x is greater than 58 * (1 / x) by 3/4. So we write the equation: x - 58 / x = 3 / 4. Multiply both sides by 4x to remove denominators: 4x * x - 4x * (58 / x) = 3 * x. This simplifies to 4x^2 - 232 = 3x. Rearrange to standard quadratic form: 4x^2 - 3x - 232 = 0. Now solve using the quadratic formula: x = [3 ± sqrt(3^2 - 4 * 4 * (-232))] / (2 * 4). Compute the discriminant: 3^2 - 4 * 4 * (-232) = 9 + 3712 = 3721. The square root of 3721 is 61, so x = (3 ± 61) / 8. Thus x = (3 + 61) / 8 = 64 / 8 = 8, or x = (3 - 61) / 8 = -58 / 8 = -29 / 4. Among the options, only x = 8 is present, so the required number is 8.

Verification / Alternative Check:
We can verify the answer by substituting x = 8 back into the original relation. The reciprocal of 8 is 1 / 8. Then 58 times the reciprocal is 58 * (1 / 8) = 58 / 8 = 29 / 4. The difference x - 58 / x becomes 8 - 29 / 4. Write 8 as 32 / 4 to simplify the subtraction. Then 32 / 4 - 29 / 4 = 3 / 4, which matches exactly with the given condition. This confirms that x = 8 satisfies the statement. The other root, -29 / 4, is not listed among the options, so it cannot be the answer in this multiple choice setting.

Why Other Options Are Wrong:

• Option -8: If x = -8, then the reciprocal is -1 / 8 and x - 58 / x becomes -8 - 58 / (-8) = -8 + 29 / 4, which does not simplify to 3 / 4.
• Option 12: For x = 12, the reciprocal is 1 / 12 and x - 58 / x becomes 12 - 58 / 12, which does not equal 3 / 4.
• Option -12: For x = -12, the difference between x and 58 times its reciprocal is again not equal to 3 / 4.
• Option 4: With x = 4, the reciprocal is 1 / 4 and x - 58 / x becomes 4 - 58 / 4, clearly not 3 / 4.

Common Pitfalls:
A common mistake is to interpret the phrase "greater than 58 times its reciprocal by 3/4" incorrectly and write x = 58 / x * (3 / 4) or something similar. Another frequent error is to forget to multiply every term by the common denominator when clearing fractions, which leads to incorrect coefficients in the quadratic. Students also sometimes compute the discriminant incorrectly or make sign errors when applying the quadratic formula. Finally, many forget to check which root actually appears in the options, and they may choose a mathematically valid root that is not available as an answer choice.

Final Answer:
Therefore, the required number that is greater than 58 times its reciprocal by 3/4 is 8.