Ordering Fractions From Greatest to Least

To put fractions in order, rewrite them over a common denominator and compare the numerators. For 3/4, 2/3, 5/8 and 7/12, the least common denominator is 24, the smallest number that 4, 3, 8 and 12 all divide. Over 24 they are 18/24, 16/24, 15/24 and 14/24, so from greatest to least the order is 3/4, 2/3, 5/8, 7/12, and from least to greatest it is the reverse.

Converting to decimals is the other method: 0.75, 0.666..., 0.625 and 0.58333.... It is quick, but repeating decimals have to be rounded, and two fractions that differ only far past the decimal point can look equal after rounding. The common denominator is exact.

Two shortcuts save work. With equal numerators, the smaller denominator is the larger fraction, because its pieces are bigger: 3/5 > 3/8. Comparing each fraction with 1/2 also sorts many lists quickly: 4/9 is just under a half and 5/9 just over.

Mixed numbers sort by their whole part first, then by the fraction, so 2 1/3 comes after 1 7/8. Fractions, mixed numbers and decimals can be ordered together once they share a denominator: in the list 1/2, 0.5, 3/5 and 1 1/4, the denominator is 20, and 1/2 and 0.5 tie at 10/20.