How to Add and Simplify Fractions
Fractions follow four rules — one per operation — plus one finishing move: simplification. Master those five things and every fraction problem, from homework to halving a recipe, becomes mechanical.
Adding and subtracting: common denominators
You can only add pieces that are the same size. Halves and thirds aren't, so rewrite both fractions over a common denominator first:
- 1/2 + 1/3 → 3/6 + 2/6 = 5/6
- 3/4 − 1/6 → 9/12 − 2/12 = 7/12
The quick general formula multiplies the denominators: a/b + c/d = (ad + cb)/bd. It may produce a large denominator (that's fine — simplify at the end), and it's exactly what a calculator does internally.
The classic mistake is adding straight across: 1/2 + 1/3 is not 2/5. Adding numerators and denominators separately produces a number that isn't even between the two originals.
Multiplying: straight across (for real this time)
Multiplication is the easy one — multiply numerators together and denominators together:
- 3/4 × 2/5 = 6/20 = 3/10
Tip: cancel common factors before multiplying to keep numbers small. In 3/4 × 2/5, the 2 and 4 share a factor of 2, giving 3/2 × 1/5 = 3/10 directly.
Dividing: flip and multiply
Dividing by a fraction is multiplying by its reciprocal:
- 1/2 ÷ 1/4 = 1/2 × 4/1 = 4/2 = 2
The result makes sense when you read it as a question: “how many quarters fit in a half?” Two. If flip-and-multiply ever feels arbitrary, that phrasing is the intuition behind it.
Simplifying with the GCD
A fraction is fully simplified when the numerator and denominator share no factor except 1. The fastest route is dividing both by their greatest common divisor: for 12/18, the GCD is 6, so 12/18 = 2/3. Repeatedly halving works too, just slower. Test answers should almost always be given in simplified form — 6/20 and 3/10 are the same number, but only one usually earns the mark.
Improper fractions and mixed numbers
7/2 (improper) and 3 1/2 (mixed) are the same value in different clothes. Convert improper → mixed by dividing: 7 ÷ 2 = 3 remainder 1, so 3 1/2. Convert back with whole × denominator + numerator: 3 × 2 + 1 = 7 → 7/2. Do arithmetic in improper form; present results in whichever form the situation wants.
Where this shows up outside class
Recipes (3/4 cup halved is 3/8 cup), carpentry (5/8″ plus 3/16″ is 13/16″), fabric, medicine doses, time signatures — measurement systems built on halves, quarters and sixteenths keep fraction arithmetic alive long after school.
Check any answer instantly
The fraction calculator performs all four operations, always simplifies via GCD, and shows the improper, mixed and decimal forms together. For converting a fraction to a percentage, the percentage calculator finishes the job.