Same numerator: smaller denominator is larger

3.NF.A.3 · take · grade 3

Archetype: Compare Fractions and Decimals by Structure · step in a 7-type progression

▶ Practice — 12 problems

Write the three fractions in order from greatest to least.

$\frac{144}{143}, \quad \frac{353}{352}, \quad \frac{279}{278}$

Show solution

Understand

Order the three fractions 144/143, 353/352, and 279/278 from greatest to least.

Givens

The three fractions are 144/143, 353/352, and 279/278.
Each fraction is slightly bigger than 1 because the numerator is one more than the denominator.

Unknowns

The order of the three fractions from greatest to least.

Constraints

All three fractions are greater than 1.

Plan

#15 Organize Information in More Ways · also uses: #5 Look for a Pattern

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.

Execute

#15 Organize Information in More Ways 3.NF.A.3

In each fraction the top is one more than the bottom, so each equals 1 plus a unit fraction: 144/143 = 1 + 1/143, 279/278 = 1 + 1/278, and 353/352 = 1 + 1/352.

\dfrac{144}{143}=1+\dfrac{1}{143},\quad \dfrac{279}{278}=1+\dfrac{1}{278},\quad \dfrac{353}{352}=1+\dfrac{1}{352}

Each fraction is just a little over 1, so what matters is which 'little extra piece' is biggest.

#5 Look for a Pattern 3.NF.A.3

The extra pieces 1/143, 1/278, 1/352 all have numerator 1. When fractions share the same numerator, the one with the smaller denominator is larger. Since 143 < 278 < 352, we get 1/143 > 1/278 > 1/352.

143 < 278 < 352 \;\Rightarrow\; \dfrac{1}{143} > \dfrac{1}{278} > \dfrac{1}{352}

Cutting a whole into fewer pieces makes each piece bigger, so 1/143 is the biggest extra piece.

#5 Look for a Pattern 3.NF.A.3

The biggest extra piece makes the biggest fraction. So 144/143 is greatest, 279/278 is next, and 353/352 is least.

\dfrac{144}{143} > \dfrac{279}{278} > \dfrac{353}{352}

Order the wholes-plus-pieces by the size of the extra piece.

Answer: 144/143, 279/278, 353/352

Review

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.

Guess and check with cross-multiplication (tool 6): 144 x 278 = 40032 vs 279 x 143 = 39897, so 144/143 > 279/278, confirming the same order by direct comparison.

Standards · min grade 3

3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Comparing fractions by rewriting them as 1 plus a unit fraction and using denominator size.

💡 This only needs the Grade 3 idea that same-numerator fractions get bigger as the denominator gets smaller!