Problem
Same numerator: smaller denominator is larger
Order the three fractions 144/143, 353/352, and 279/278 from greatest to least.
Fractions
Your answer
Watch out It's easy to think a bigger denominator means a bigger fraction (like comparing whole numbers), which flips the order to 353/352, 279/278, 144/143 — but for same-numerator fractions, the smaller denominator makes the bigger piece.
How to solve
Strategy Organize Information in More Ways — Rewrite each fraction as 1 plus a unit fraction. Then the comparison becomes comparing unit fractions with the same numerator (1), where the smaller denominator means the larger value.
1STEP 1
Split each fraction into 1 plus a small part
Each fraction's top is one more than its bottom, so it's 1 plus a unit fraction: 144/143=1+1/143, 279/278=1+1/278, 353/352=1+1/352.
144/143 = 1 + 1/143, 279/278 = 1 + 1/278, 353/352 = 1 + 1/352
Each fraction is just a little over 1, so what matters is which 'little extra piece' is biggest.
3.NF.A.3Organize Information In More Ways2STEP 2
Compare the unit fractions by their denominators
Fractions with the same numerator grow as the denominator shrinks, so 1/143 is greatest, 1/278 next, 1/352 least.
143 is less than 278, which is less than 352 → 1/143 is greater than 1/278, which is greater than 1/352
Cutting a whole into fewer pieces makes each piece bigger, so 1/143 is the biggest extra piece.
3.NF.A.3Look For A PatternWhen two unit fractions share the numerator 1, the one with the smaller denominator is the larger fraction, so 1/143 is bigger than 1/278.
Why?
Each of these is one equal piece of the very same whole, where 1/143 cuts that whole into 143 pieces and 1/278 cuts it into 278 pieces.
Why?
Cutting the same whole into fewer equal pieces leaves each piece holding more, and 143 pieces is fewer than 278.
Why?
The equal pieces together make up exactly one whole with nothing left over, so when fewer of them must fill that same whole, each single piece has to be bigger.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
3STEP 3
Write the fractions in order
The biggest extra piece makes the biggest fraction. So 144/143 is greatest, 279/278 is next, and 353/352 is least.
144/143 greater than 279/278 greater than 353/352
Order the wholes-plus-pieces by the size of the extra piece.
3.NF.A.3Look For A PatternAnswer
144/143, 279/278, 353/352
All three are a tiny bit more than 1. 144/143 is about 1.007, 279/278 is about 1.0036, and 353/352 is about 1.0028, which matches the order greatest to least.
Takeaway
This only needs the Grade 3 idea that same-numerator fractions get bigger as the denominator gets smaller!
- Split each fraction into 1 plus a small part
- Compare the unit fractions by their denominators
- Write the fractions in order
Where next?
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