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LCM of fractions (use LCM of numerators / GCD of denominators): Find the LCM of the fractions 1/3, 2/9, 5/6, and 4/27.

Difficulty: Easy

Correct Answer: 20/3

Explanation:


Introduction / Context:
The LCM of several fractions is computed by taking the LCM of their numerators and dividing by the GCD of their denominators. This rule is the fractional analogue of the integer LCM/GCD definitions and ensures each fraction divides the LCM exactly.


Given Data / Assumptions:

  • Fractions: 1/3, 2/9, 5/6, 4/27.
  • All are positive rational numbers.


Concept / Approach:
Use the formula: LCM of fractions = LCM(numerators) / GCD(denominators). Compute each component carefully by prime factorization or mental arithmetic with small numbers.


Step-by-Step Solution:

Numerators: 1, 2, 5, 4 ⇒ LCM = 20.Denominators: 3, 9, 6, 27 ⇒ GCD = 3.Therefore LCM = 20 / 3.


Verification / Alternative check:

Each given fraction divides 20/3: e.g., (20/3) ÷ (1/3) = 20, (20/3) ÷ (2/9) = 30, (20/3) ÷ (5/6) = 8, (20/3) ÷ (4/27) = 45; all integers.


Why Other Options Are Wrong:

  • 1/54, 10/27, 3/20 are either too small or inverted and will not be exact multiples of all the given fractions.


Common Pitfalls:

  • Confusing GCD and LCM roles for numerators and denominators.
  • Reducing prematurely and losing track of divisibility checks.


Final Answer:

20/3
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