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Fractions & Rational Numbers

How to Find the LCM of Fractions

Wait—before you start, make sure you know which question your homework is asking! Most of the time, when people search for “LCM of fractions,” they just want a common denominator to add two fractions together (that is the Least Common Denominator). But if you genuinely need the smallest number that two whole fractions divide into cleanly, here is the exact formula.

The LCM of Fractions Formula

First, make sure every fraction is reduced to lowest terms. Then take the LCM of the top numbers (numerators) and divide it by the GCF of the bottom numbers (denominators):

LCM(a/b, c/d) = LCM(a, c) ÷ GCF(b, d)

Worked Example 1: Find the LCM of 2/3 and 4/9

  1. Check that both fractions are simplified: Both 2/3 and 4/9 are already in lowest terms.
  2. Find the LCM of the numerators (2 and 4): LCM(2, 4) = 4.
  3. Find the GCF of the denominators (3 and 9): GCF(3, 9) = 3.
  4. Put the numerator result over the denominator result: LCM(2/3, 4/9) = 4/3.
Check that it works: Does 4/3 divide by 2/3 to give a whole number? Yes: (4/3) ÷ (2/3) = 2. Does 4/3 divide by 4/9 to give a whole number? Yes: (4/3) ÷ (4/9) = 3.

Worked Example 2: Three Fractions (3/4, 5/6, and 15/8)

Suppose three pendulums swing with periods of 3/4 second, 5/6 second, and 15/8 seconds. When will all three be back at their starting point together?

  • Numerators (3, 5, 15): Both 3 and 5 divide evenly into 15, so LCM(3, 5, 15) = 15.
  • Denominators (4, 6, 8): The largest number that divides 4, 6, and 8 is GCF(4, 6, 8) = 2.
  • Result: 15/2 seconds (or 7.5 seconds). After 7.5 seconds, the three pendulums have completed 10, 9, and 4 full swings respectively.