There are two basic rules for ordering fractions.
- Bigger numerator, bigger fraction: A "seventh" or a "tenth" can be thought of as a slice of determined size. The more of them you have, the more the total. So 1/7 < 3/7 < 4/7 just as one orange is less than three oranges which is leas than four oranges.
- Bigger denominator, smaller fraction: A tenth represents a smaller slice than a seventh because the whole is divided more ways. So 2/10 <2/7, 3/10 < 3/7 , just as three nickels is less than three dimes, or three ounces is less than three pounds.
- These can also be combined into chains using "transitivity" and renaming:
1/3 = 3/9 < 3/7 < 3/5 < 4/5
- All negative fractions are less than all positive fractions, and among themselves they are ordered backwards:
-1/3 < -1/7 < 0 < 1/7 < 1/3
- You can compare any pair of fractions by cross-multiplying. To compare 2/3 and 5/7 compare 2x7 with 3x5. As 3x5 is bigger, the fraction with 5 in the _numerator_ is larger.
- There _are_ other rules. While fractions can be compared using very advanced algebra, the most advanced rule likely to be used at the school level says that if you add the numerators and add the denominators of two fractions, the result (which is NOT the sum!) is between them:
we know 2/3 < 7/10 < 5/7 because 7/10 = (2+5)/(3+7).