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A fraction f is such that 6 times the fraction is greater than 7 times its reciprocal by 11. If f is taken as a positive rational number, what is the exact value of the fraction?

Difficulty: Medium

Correct Answer: 7/3

Explanation:


Introduction / Context:
This is a classic reciprocal-based equation. You translate the English statement into an algebraic equation, clear fractions, and solve a quadratic. The phrase “greater than … by 11” means a difference of 11.


Given Data / Assumptions:

  • Let the fraction be f (assume f > 0).
  • 6 times the fraction is 11 more than 7 times its reciprocal.


Concept / Approach:
Convert to equation: 6f = 7*(1/f) + 11. Multiply both sides by f to remove the reciprocal, forming a quadratic equation in f, then solve and pick the positive value as required.


Step-by-Step Solution:

Form the equation: 6f = 7/f + 11. Multiply by f: 6f^2 = 7 + 11f. Bring all terms to one side: 6f^2 - 11f - 7 = 0. Compute discriminant: D = (-11)^2 - 4*6*(-7) = 121 + 168 = 289. Square root of D: sqrt(D) = 17. Solve: f = (11 ± 17)/(2*6) = (11 ± 17)/12. So f = 28/12 = 7/3 or f = -6/12 = -1/2. Since the problem states a positive fraction, choose f = 7/3.


Verification / Alternative check:
Check f = 7/3: 6f = 14. Reciprocal is 3/7, so 7*(1/f) = 3. Difference 14 - 3 = 11, which matches the statement.


Why Other Options Are Wrong:

5/3 and 5/4 do not satisfy 6f - 7/f = 11 when substituted. 4/5 and 3/7 are less than 1, making 7/f too large, so the difference cannot be 11.


Common Pitfalls:
Misreading “greater than … by 11” as multiplication instead of subtraction, or forgetting to multiply every term by f when clearing the denominator.


Final Answer:
f = 7/3

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