Adding & Subtracting Fractions

You cannot add 1/2 and 1/3 directly for the same reason you cannot add 2 apples and 3 oranges and call it 5 apples. The pieces are different sizes.

So rename both fractions with a common denominator. For 1/2 + 1/3 use sixths: 3/6 + 2/6 = 5/6. Now the pieces match and you just count them.

The denominator does not change when you add. You are counting pieces, not resizing them. 3/6 + 2/6 = 5/6, never 5/12.

The general form is:

a/b + c/d = (ad + cb)/(bd)

which is the common-denominator process compressed into one line. Using bd always works but is not always minimal; using LCM(b,d) keeps the numbers small and often skips a simplification step at the end.

Structurally, converting to a common denominator is applying the distributive property in reverse. Once both terms share the denominator you have (1/n)·a + (1/n)·c = (1/n)(a + c), so the addition happens purely in the numerators. That is the reason denominators do not add: they are a common factor being pulled out, not a quantity being combined.

The same procedure carries over unchanged to rational expressions in Algebra 2, where the denominators are polynomials and the LCM is found by factoring.