How To Work With Fractions
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Vocabulary
Fractions feel harder than they are because the rules change depending on whether you’re adding, multiplying, or converting to a decimal. Adding 1/3 and 1/4 requires a common denominator; multiplying them doesn’t. Simplifying 24/36 requires finding a GCD; converting 1/7 to a decimal gives you an infinitely repeating pattern that confuses people into thinking they made an arithmetic error. Mixed numbers like 2 3/4 hide an extra step where you either convert to an improper fraction or keep them separate. This guide covers the vocabulary (proper, improper, mixed), the four operations with worked examples, simplification using the greatest common divisor, decimal conversion including repeating decimals, and the tricks that make fraction arithmetic feel mechanical instead of mysterious.
Simplifying with GCD
When the denominators share no common factors, the LCD is their product. When they share factors, find the LCD by multiplying each denominator’s unique prime factors at their highest power.
Adding and subtracting: you need a common denominator
Divide numerator by denominator. Some fractions terminate (3/4 = 0.75); others repeat (1/3 = 0.3333..., 1/7 = 0.142857142857...). A fraction terminates if and only if its simplified denominator’s prime factors are only 2 and 5. Everything else repeats.