ACT Math: Simplify Complex Fractions Using the LCM Method

A complex fraction is a fraction where the numerator or denominator (or both) contain fractions. Example: (1/2+1/3)/(1/4-1/6). Solution: Find the LCM of all denominators in the complex fraction. Here, LCM(2,3,4,6)=12. Multiply both numerator and denominator by 12: (12·1/2+12·1/3)/(12·1/4-12·1/6)=(6+4)/(3-2)=10/1=10. Multiplying by LCM eliminates all inner fractions, turning the complex fraction into a simple one.

Why this works: Multiplying by 12/12 doesn't change the fraction's value, but it scales out the small fractions. Instead of adding 1/2+1/3, you add 6+4 (whole numbers), which is much easier.

Mistake 1: Finding LCM of only the outer denominator. You must account for ALL denominators in the numerator and denominator. Mistake 2: Forgetting to multiply the entire numerator and denominator by the LCM. You must distribute; don't skip terms. Write out the multiplication step explicitly so you don't miss any terms.

Verification: After simplifying, plug in numbers to check. If the original complex fraction equals 10, your simplified answer should also equal 10.

Problem 1: (1/2)/(1/3). LCM=6. (6·1/2)/(6·1/3)=3/2. Problem 2: (1/x+1/2)/(1/x). LCM=2x. (2x·1/x+2x·1/2)/(2x·1/x)=(2+x)/2. Problem 3: (2-1/3)/(1+1/2). LCM=6. (6·2-6·1/3)/(6·1+6·1/2)=(12-2)/(6+3)=10/9. Complete all three; verify each by cross-multiplying or substituting values.

Challenge: Simplify (1/(x+1)+1/(x-1))/(1/(x²-1)). This requires factoring and algebraic manipulation after using LCM.

ACT Math includes complex fractions to test your fraction manipulation skills and ability to simplify nested operations. They appear 0-1 times per test but often scare students who don't know the method. Students who master the LCM method solve complex fraction questions in under two minutes, gaining confidence for harder algebra.

Spend 15 minutes this week solving 6-8 complex fraction problems using the LCM method. By test day, complex fractions will feel routine, and you'll move quickly through them.