Overview
Fractions describe division, so equivalent fractions preserve value when numerator and denominator are multiplied by the same nonzero quantity. Addition needs a common unit; multiplication combines factors directly. Dividing by a fraction means multiplying by its reciprocal because the product of a nonzero fraction and its reciprocal is 1.
Consider this example: Evaluate 5/6 − 3/4. Use twelfths: 5/6 = 10/12 and 3/4 = 9/12. Their difference is 1/12. Subtracting numerators and denominators would give 2/2, which ignores the unequal units.
The method also handles a different case: Evaluate (−7/10) ÷ (14/15). Multiply −7/10 by 15/14. Cancel 7 with 14 and 15 with 10 to obtain −3/4. The answer remains negative; multiplying the original fractions produces −49/75 instead.
For a further application, consider: A ribbon is 5 1/4 meters long. It is cut into equal pieces of length 3/8 meter with no material lost. How many pieces are obtained? Convert 5 1/4 to 21/4. Dividing by 3/8 gives (21/4)(8/3) = 14 pieces. Multiplication by the piece length would not count how many pieces fit.
Adding denominators treats unlike-sized pieces as though they were the same unit. Dividing numerator by numerator without accounting for denominators also loses scale. Estimate whether an answer should be positive, negative, larger or smaller before using a common denominator or reciprocal. Simplify factors before multiplication when possible.