Find the L.C.M. (least common multiple) of the fractions 2/7, 3/14, and 5/3. (Assume positive values.)

Introduction / Context:
For positive fractions, a standard rule is used to compute the least common multiple (LCM). Unlike integers, the LCM of fractions depends on both the numerators and denominators, combined through a specific relationship involving LCM and HCF (GCD).

Given Data / Assumptions:

• Fractions: 2/7, 3/14, 5/3.
• Assume all fractions are positive.

Concept / Approach:
The formula for the LCM of fractions a/b, c/d, e/f is: LCM = LCM(numerators) / HCF(denominators). We therefore find LCM of 2, 3, 5 and HCF (GCD) of 7, 14, 3.

Step-by-Step Solution:

Verification / Alternative check:
Each fraction should divide 30 (as a rational) exactly: 30 ÷ (2/7) = 105; 30 ÷ (3/14) = 140; 30 ÷ (5/3) = 18. All results are integers, validating 30 as a common multiple. Being constructed via the formula guarantees minimality.

Why Other Options Are Wrong:

• 35, 45, 25, 15: Do not satisfy the fraction LCM formula, and at least one of the fractions will not divide these cleanly in the rational sense.

Common Pitfalls:

• Using LCM(denominators) instead of HCF(denominators) in the fraction-LCM formula.
• Attempting integer-style LCM directly on fractions without the proper rule.

Final Answer:
30