Find a fraction from an operation description A fraction f is multiplied by itself and then divided by its reciprocal. The result equals 18 26/27. Determine the fraction f in simplest form.

Introduction / Context:
This problem encodes an algebraic equation in words. Translating the description into symbolic form reveals an elegant simplification. It assesses comfort with reciprocals, mixed numbers, and cube roots of rational numbers.

Given Data / Assumptions:

• A fraction f is multiplied by itself (forming f^2).
• The result is divided by the reciprocal of the same fraction (1/f).
• The final value is 18 26/27, which is a mixed number.
• We must find f in lowest terms.

Concept / Approach:
Dividing by the reciprocal is equivalent to multiplying by the original fraction: f^2 ÷ (1/f) = f^2 * f = f^3. Hence f^3 equals the given mixed number converted to an improper fraction. Take the rational cube root to obtain f.

Step-by-Step Solution:
Convert the mixed number: 18 26/27 = (18*27 + 26)/27 = (486 + 26)/27 = 512/27.Set up the equation: f^3 = 512/27.Recognize 512 = 8^3 and 27 = 3^3.Therefore f = cube_root(512/27) = 8/3.Reduce if necessary (already simplest). So f = 8/3.

Verification / Alternative check:
Check: f^3 = (8/3)^3 = 512/27, which equals 18 26/27, matching the given result. The reverse operation (multiply by itself and divide by reciprocal) indeed returns f^3.

Why Other Options Are Wrong:
8/27 is far too small and (8/27)^3 = 512/19683; 11/3, 22/3, 32/3 all cube to values much larger than 512/27. Only 8/3 produces exactly 512/27.

Common Pitfalls:
Misreading “divided by the reciprocal” as dividing by the fraction itself (which would give f), or mishandling the mixed number to improper fraction conversion.

Final Answer:
8/3