Fraction Arithmetic Without Losing Exactness
Decimals are convenient, but homework and recipes often need exact rationals. This guide focuses on the habits that keep fraction work honest: common denominators, reciprocals, and GCD simplification—not a tour of every calculator button.
Addition Needs a Shared Unit
1/2 and 1/3 are different-sized pieces. Rewriting both with denominator 6 turns them into 3/6 and 2/6 so the numerators can add. Multiplication does not need that rewrite: multiply numerators and denominators, then simplify.
Division Is Multiply by the Reciprocal
a/b ÷ c/d means a/b × d/c, provided c ≠ 0. The reciprocal flips the second fraction. Students who memorize “keep-change-flip” without checking the zero numerator miss the only hard failure mode.
Improper vs Mixed
11/4 and 2 3/4 name the same rational. Improper form is better for further arithmetic; mixed form is better for measurement intuition. Convert before operating if your inputs start mixed.
Practice with two fractions
Run add/subtract/multiply/divide and inspect the simplified improper, mixed, and decimal views:
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