A number is greater than five times its reciprocal by 19 divided by 2. What is the value of the number that satisfies this condition?

Introduction / Context:
This problem involves a number and its reciprocal and asks you to build and solve a quadratic equation. The number is described as being greater than five times its reciprocal by 19 divided by 2. Problems of this type are common in aptitude tests to examine algebraic manipulation skills.

Given Data / Assumptions:
- Let the required number be x.
- The reciprocal of x is 1 / x, assuming x is not zero.
- The statement says that x is greater than 5 * (1 / x) by 19 / 2.
- We need to find the positive value of x that satisfies the relation.

Concept / Approach:
Translate the sentence into an equation. Saying that a number is greater than five times its reciprocal by 19 / 2 means x - 5 * (1 / x) = 19 / 2. Multiplying through by x clears the denominator and gives a quadratic equation in x. We then solve the quadratic using factorization or the quadratic formula and check which root matches the context.

Step-by-Step Solution:
Step 1: Let the number be x.Step 2: Set up the equation: x - 5 * (1 / x) = 19 / 2.Step 3: Multiply both sides by x to remove the denominator: x^2 - 5 = (19 / 2) * x.Step 4: Multiply both sides by 2 to clear the fraction: 2x^2 - 10 = 19x.Step 5: Rearrange to standard quadratic form: 2x^2 - 19x - 10 = 0.Step 6: Compute the discriminant: D = 19^2 - 4 * 2 * (-10) = 361 + 80 = 441 which is 21^2.Step 7: Solve for x: x = (19 ± 21) / (2 * 2) giving x = 40 / 4 = 10 or x = -2 / 4 = -0.5.Step 8: The context and options suggest a positive integer, so we take x = 10.

Verification / Alternative check:
Verify with x = 10. The reciprocal is 1 / 10. Then 5 times the reciprocal is 5 / 10 = 0.5. According to the condition, x - 5 * (1 / x) should equal 19 / 2. Compute 10 - 0.5 = 9.5 which is 19 / 2. So the condition holds exactly. The other root, -0.5, although algebraically valid, is not among the choices and is usually not intended in such aptitude settings.

Why Other Options Are Wrong:
Option 11 gives 11 - 5 * (1 / 11) which equals 11 - 5 / 11, not equal to 19 / 2. Option 9 leads to 9 - 5 / 9 which does not match 19 / 2. Option 8 produces 8 - 5 / 8, again different. Option 5 gives 5 - 1 which equals 4, far from 19 / 2. Only x = 10 satisfies the exact relation.

Common Pitfalls:
A common error is misinterpreting the phrase greater than five times its reciprocal by 19 / 2 and writing 5 / x - x = 19 / 2 instead of x - 5 / x = 19 / 2. Another typical mistake is to forget to multiply the entire equation by x or to make sign errors while rearranging the quadratic. Carefully handling fractions and algebraic steps avoids these issues.

Final Answer:
The number that is greater than five times its reciprocal by 19 divided by 2 is 10.