Problem
Compare and order fractions
Five numbers are given as fractions and mixed numbers. All lie between 1 and 2, so only the fractional parts matter. The forms are mixed and must be made comparable. List them from largest to smallest.
Reasoning · Grade 3-2 Size of Fractions
Your answer
Watch out Improper fractions look bigger, but all five sit between 1 and 2 — compare values, not forms.
How to solve
Strategy Organize Information in More Ways — The numbers are written in different forms (improper fractions and mixed numbers), so the key move is to re-organize them all into the same form. Turning each into a mixed number (whole part 1 plus a fractional part) lets us compare just the fractional parts, which we can size up by reasoning.
1STEP 1
Rewrite the improper fractions as mixed numbers
Converted, every number reads 1 and a fraction.
19/11 = 1 8/11 18/10 = 1 8/10
When everything starts with '1 and...', the whole-number 1 is the same for all, so only the fraction parts decide the order.
3.NF.A.3Organize Information In More WaysOnce every number is written as 1 and a bit, the shared whole number drops out and only the fraction parts decide the order.
Why?
Taking the same 1 away from every number slides them all down together, so the gaps between them are exactly as they were.
🧱Shift both by the same amount and the difference holdsSlide two points the same distance along the line and the gap between them never changes.
Why?
Comparing the fraction parts two at a time settles each pair, and those verdicts chain together into one full ordering.
🧱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.
2STEP 2
List the five fractional parts
The parts to compare are 8/11, 1/4, 13/15, 8/10, 2/5.
8/11, 1/4, 13/15, 8/10, 2/5
Stripping off the shared whole number leaves a cleaner, smaller comparison.
4.NF.A.2Organize Information In More Ways3STEP 3
Compare the parts using benchmarks
Above 1/2 the order is 13/15, 8/10, 8/11.
13/15 ＞ 8/10 ＞ 8/11 ＞ 2/5 ＞ 1/4
Checking each part against 1/2 (and against 1) sorts them fast without finding one giant common denominator.
4.NF.A.2Guess And Check4STEP 4
Put the whole numbers back
Restoring the 1 gives 1 13/15, 18/10, 19/11, 1 2/5, 1 1/4.
1 13/15 ＞ 18/10 ＞ 19/11 ＞ 1 2/5 ＞ 1 1/4
The fraction parts already decided the order, so we just rename them back to how the problem wrote them.
4.NF.A.2Organize Information In More WaysAnswer
1 13/15, 18/10, 19/11, 1 2/5, 1 1/4
As decimals the five numbers are about 1.87, 1.80, 1.73, 1.40, 1.25, which decrease in exactly the listed order, confirming greatest-to-least. All sit between 1 and 2 as expected.
Takeaway
Write every number the same way first, then compare just the leftover fraction parts against an easy landmark like one-half!
- Rewrite the improper fractions as mixed numbers
- List the five fractional parts
- Compare the parts using benchmarks
- Put the whole numbers back