Comparing and ordering fractions

The digits in a fraction are not amounts, they are instructions. The bottom number says how many equal pieces the whole was cut into, and the top says how many you took. Cutting into 8 pieces makes each piece smaller than cutting into 3 pieces, so 5 small pieces can easily be less than 2 bigger ones. That is the whole reason 5/8 is smaller than 2/3, and it is why comparing tops and bottoms separately gives the wrong answer.

The reliable method is to give both fractions the same denominator, because then the pieces are the same size and the numerators can be compared directly. Rewrite 5/8 as 15/24 and 2/3 as 16/24. Now both are counted in twenty-fourths, and 15 is less than 16. The method is exactly the equivalence rule you already use when simplifying, run in the other direction.

Cross-multiplying is the same comparison done faster, and it is worth learning because it costs no writing. Multiply 5 by 3 and 2 by 8. The two products are 15 and 16, and the fraction on the side of the bigger product is the bigger fraction. This works because both products are the numerators of the same common denominator — 24 in this case — so you compared 15 and 16 without ever writing the denominator down.

Two quick checks save time when the fractions are far apart. Compare each to a half: anything with a top less than half the bottom is under a half, anything with a top more than half the bottom is over. And remember that a fraction is also a division, so a calculator check of 5 ÷ 8 against 2 ÷ 3 will settle any argument — the methods above are for when you want to be sure without one.