Comparing & Ordering Fractions

If the bottoms match, the bigger top wins: 5/8 > 3/8.

If the tops match, the *smaller* bottom wins: 3/4 > 3/7, because cutting something into 4 pieces gives bigger pieces than cutting it into 7.

Otherwise, cross multiply. To compare 3/5 and 5/8, multiply 3×8 = 24 and 5×5 = 25. The 25 came from the 5/8 side, so 5/8 is bigger. Converting both to decimals also works and is often faster on a calculator.

For positive b and d, a/b < c/d ⟺ ad < cb. The justification is multiplying both sides of the inequality by the positive quantity bd, which clears both denominators without flipping the inequality — and that positivity condition is exactly why the test must be stated carefully once negatives are allowed.

Two other comparisons are worth having as reflexes. Benchmarking against 1/2 or 1 sorts most pairs instantly: 7/13 > 1/2 because 7 is more than half of 13. And for fractions just below 1, the one with the larger denominator is closer to 1, since 8/9 is short by 1/9 while 5/6 is short by 1/6.