The rule
a/b + c/d = (ad + cb) ÷ bd, then simplify
Why it works
Adding fractions needs a common denominator because you can only add parts of the same size. Multiplying does not: numerators multiply, denominators multiply, and that is the whole rule. Dividing is multiplying by the reciprocal.
How to do it by hand
- For adding or subtracting, find the lowest common multiple of the denominators
- Rewrite both fractions with that denominator
- Operate on the numerators and keep the denominator
- Simplify by dividing both parts by their greatest common divisor
What is worth knowing
The rule everyone half-remembers is 'find a common denominator', and the part that gets forgotten is why: a half and a third cannot be added directly because they are different-sized pieces. Converting both to sixths makes them addable. Multiplication needs no such step, which is why it is the easier operation despite feeling harder — a fact that surprises students who expect difficulty to be consistent. The reciprocal rule for division follows from the same logic: dividing by 2/3 asks how many two-thirds fit, and that is the same as multiplying by 3/2.