How fraction arithmetic works
Every mixed number is first converted to an improper fraction (whole × denominator + numerator), the operation is done on improper fractions, and the answer is reduced to lowest terms by dividing top and bottom by their greatest common divisor (GCD).
a/b + c/d = (ad + cb) ÷ bd · a/b × c/d = ac ÷ bd · a/b ÷ c/d = ad ÷ bc
Worked example: 3/4 + 5/6
The least common denominator of 4 and 6 is 12. Rewrite each fraction: 3/4 = 9/12 and 5/6 = 10/12. Add the numerators: 9/12 + 10/12 = 19/12. Since 19 and 12 share no common factor, that is already in lowest terms, and as a mixed number it is 1 7/12 (≈ 1.5833).
Multiplying and dividing
Multiplication is the easy one — multiply straight across: 3/4 × 5/6 = 15/24, which reduces by GCD 3 to 5/8. Division means multiplying by the reciprocal: 3/4 ÷ 5/6 = 3/4 × 6/5 = 18/20 = 9/10. You cannot divide by a fraction whose value is zero.
Mixed numbers
A whole part like 2 in "2 1/3" is folded in first: 2 1/3 = (2 × 3 + 1)/3 = 7/3. Negative mixed numbers follow the usual convention that the sign applies to the whole thing: −1 2/3 = −5/3. The calculator shows every one of these conversion steps under the result, so you can follow (or check homework against) the working line by line.
Why simplify?
19/12, 38/24 and 57/36 are all the same number, but lowest terms is the canonical way to write it — most textbooks and tests expect it, and it makes comparing fractions immediate.