Fractions
Ordering Fractions – Smallest to Largest and Back
Ordering fractions means arranging a set of fractions from smallest to largest (ascending) or from largest to smallest (descending). The two most reliable methods are finding a common denominator and converting to decimals.
Ordering more than two fractions at once is a natural extension of comparing pairs, but it becomes far more efficient once every fraction shares a common denominator (or is converted to a decimal) — you sort once instead of comparing every pair separately. This exact skill shows up in sports league tables built from fractional win rates, in recipe scaling where several ingredient fractions must be checked against each other, and in probability, where outcomes are often ranked by likelihood expressed as fractions.
Method 1 – Common Denominator
- Find the LCM of all the denominators.
- Convert every fraction to an equivalent fraction with that denominator.
- Order by numerator.
- Rewrite in the original form.
LCM(2,3,8) = 24. Convert: 1/2 = 12/24, 2/3 = 16/24, 3/8 = 9/24.
Order of numerators: 9, 12, 16 → order: 3/8, 1/2, 2/3.
Method 2 – Decimal Conversion
5/6 ≈ 0.833, 3/4 = 0.75, 7/10 = 0.7, 2/3 ≈ 0.667.
Ascending: 0.667, 0.7, 0.75, 0.833 → 2/3, 7/10, 3/4, 5/6.
Mixed Sets – Including Improper Fractions and Mixed Numbers
Convert to decimals: 2.5, 2.333, 2.0, 1.75.
Ascending: 1¾, 8/4, 7/3, 2½
Ordering on a Number Line
Once fractions are converted to decimals, they slot naturally onto a number line. Mark each value as a point, read left to right for ascending, right to left for descending.
Key Takeaways
- Common denominator method: find LCM, convert, compare numerators.
- Decimal method: divide numerator by denominator, then order.
- Always check your final order by confirming each consecutive pair.
- Mixed numbers and improper fractions can be included by converting to decimals first.