A positive number x satisfies: when x is increased by 10, the result equals 200 times its reciprocal (that is, x + 10 = 200 * (1 / x)). What is the value of x?
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A100
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B10
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C20
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D200
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E5
Answer
Correct Answer: 10
Explanation
Introduction / Context:This problem checks comfort with translating a verbal condition into an algebraic equation involving a reciprocal. Once the equation is set, solving a simple quadratic yields the positive value for the number. Such questions are common in aptitude tests and reinforce manipulation of fractions and quadratics.
Given Data / Assumptions:
- The number is positive and denoted by x.
- Condition: x + 10 = 200 * (1 / x).
- We seek the positive solution only, because the prompt specifies a positive number.
Concept / Approach:Clear the denominator to avoid fractions, then bring terms to one side and factor or use the quadratic formula. The reciprocal relation typically produces a quadratic in x. Choose the root that satisfies the positivity constraint.
Step-by-Step Solution:
Given x + 10 = 200 * (1 / x).Multiply both sides by x: x^2 + 10x = 200.Rearrange: x^2 + 10x − 200 = 0.Solve the quadratic: Discriminant D = 10^2 + 4 * 200 = 100 + 800 = 900, so sqrt(D) = 30.Roots: x = (−10 ± 30) / 2 → x = 10 or x = −20.Since x is positive, select x = 10.Verification / Alternative check:Check the condition: 10 + 10 = 20 and 200 * (1 / 10) = 20. Both sides match, confirming the result.
Why Other Options Are Wrong:Values 100, 20, 200, and 5 do not satisfy x + 10 = 200 / x when substituted.
Common Pitfalls:Forgetting to multiply through by x leads to errors with the reciprocal. Selecting the negative root violates the stated condition that the number is positive.
Final Answer:10