fraction arithmetic
A fraction like 3/4 is a way of cutting a whole into equal parts and taking some of them. Fraction arithmetic is the collection of rules for combining such pieces. The key insight is that you can only add or subtract pieces of the same size, but you can multiply or divide pieces of any size directly.
To add or subtract, rewrite both fractions over a common denominator, then add or subtract the numerators: 1/2 + 1/3 = 3/6 + 2/6 = 5/6. To multiply, multiply numerators together and denominators together: (2/3)(4/5) = 8/15. To divide, multiply by the reciprocal of the second fraction: (2/3) ÷ (4/5) = (2/3)(5/4) = 10/12 = 5/6.
After any operation, reduce the result to lowest terms by dividing out common factors. A frequent error is adding denominators directly (writing 1/2 + 1/3 = 2/5), which is wrong because the halves and thirds are different-sized pieces. Multiplying does not require a common denominator — that requirement belongs only to addition and subtraction.
Compute 5/6 − 1/4. The least common denominator is 12: 5/6 = 10/12 and 1/4 = 3/12, so 10/12 − 3/12 = 7/12, already in lowest terms.
Convert to a common denominator, subtract numerators, then check it is in lowest terms.
Dividing by a fraction is the same as multiplying by its reciprocal, sometimes phrased “invert and multiply.” But the denominator can never be zero: 5/0 is undefined, because no number times zero gives 5.