Problem
Comparing sizes of fractions
Five fractions are given: 8/9, 11/12, 2/5, 4/16, 8/11. Keep each value as it is — no rounding. The answer must list all five exactly once. Put them in order from largest to smallest.
Reasoning · Grade 3-1 Comparing Fraction Sizes
Your answer
Watch out 4/16 looks big because of its denominator, but it reduces to 1/4 — the smallest of them.
How to solve
Strategy Organize Information in More Ways — Rather than forcing one common denominator for all five, we sort the fractions into useful groups: those very close to 1 (compare by the gap to 1), and the smaller ones (compare against the benchmark 1/2). Comparing within each group is easy, then we merge the groups into one ordered list.
1STEP 1
Simplify and benchmark each fraction
4/16 reduces to 1/4, and 1/2 splits them into two groups.
4/16 = 1/4, 8/11 ＞ 1/2 ＞ 2/5 ＞ 1/4
Comparing each fraction to the friendly benchmark 1/2 quickly splits the big ones from the small ones.
3.NF.A.3Organize Information In More Ways2STEP 2
Compare the two largest using the gap to 1
Short of 1 by 1/12 and 1/9, so 11/12 beats 8/9.
1-8/9=1/9, 1-11/12=1/12, 1/12 ＜ 1/9 → 11/12 ＞ 8/9
Whichever fraction has the smaller leftover piece to reach 1 is the larger fraction.
3.NF.A.3Organize Information In More WaysBetween two fractions just short of 1, the one with the smaller leftover piece is the larger fraction.
Why?
Each fraction is one whole with a single piece missing, and the piece missing is one of the equal pieces the whole was cut into.
🧱A fraction is equal shares of one wholeThe bottom number names how many equal pieces one whole was cut into and the top counts them, which is why same-sized pieces add just by counting.
Why?
Comparing the two leftovers settles which fraction is bigger, and those comparisons chain so the whole list can be ordered.
🧱Transitivity of orderIf the first is smaller than the second and the second smaller than the third, the first is smaller than the third, so pairwise comparisons chain into one full order.
3STEP 3
Place 8/11, then order the small group
8/11 is past 1/2 but not near 1, so it comes after 8/9.
2/5=0.4 ＞ 1/4=0.25
Quick decimal checks confirm the order of the smaller fractions.
3.NF.A.3Guess And Check4STEP 4
Merge into one ordered list
Below, 2/5 beats 1/4: 11/12, 8/9, 8/11, 2/5, 4/16.
11/12 ＞ 8/9 ＞ 8/11 ＞ 2/5 ＞ 4/16
Combining the near-1 group ahead of the smaller group gives the full ranking.
3.NF.A.3Organize Information In More WaysAnswer
11/12, 8/9, 8/11, 2/5, 4/16
Decimal values confirm the order: 11/12 ≈ 0.917, 8/9 ≈ 0.889, 8/11 ≈ 0.727, 2/5 = 0.4, 4/16 = 0.25, which is strictly decreasing as listed.
Takeaway
You don't always need a common denominator: compare fractions to friendly marks like 1/2 and 1, and the one with the smaller gap to 1 is bigger!
- Simplify and benchmark each fraction
- Compare the two largest using the gap to 1
- Place 8/11, then order the small group
- Merge into one ordered list