Find the L.C.M. (least common multiple) of the fractions 2/7, 3/14, and 5/3. (Assume positive values.)
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A30
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B35
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C45
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D25
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E15
Answer
Correct Answer: 30
Explanation
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:
LCM of numerators 2, 3, 5 is 30 (since 2, 3, 5 are primes).HCF of denominators 7, 14, 3 is gcd(gcd(7, 14), 3) = gcd(7, 3) = 1.Therefore LCM of the fractions = 30 / 1 = 30.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