Do arithmetic on two fractions and get the answer in lowest terms, as a mixed number and as a decimal. The converter works the other way too, turning a decimal into the simplest fraction that represents it rather than a technically-correct monster like 333333/1000000.
Put them over a common denominator first, then add the numerators. Multiplying the denominators together always works: 1/2 + 1/3 becomes 3/6 + 2/6 = 5/6. Using the lowest common denominator instead just saves reducing afterwards.
By using continued fractions, which find the simplest fraction within a tolerance rather than the literal decimal expansion. The same method gives 355/113 for pi - an approximation accurate to seven decimal places that has been known since the fifth century.
Dividing by a number is multiplying by its reciprocal, and the reciprocal of 2/3 is 3/2. Dividing by 1/2 gives twice as much because you are asking how many halves fit, and two do.