← Same numerator: smaller denominator is larger · Compare Fractions and Decimals by Structure

Same numerator: smaller denominator is larger · 12 practice problems

3.NF.A.3

Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: 3/2, 4/3, 6/5

Write the three fractions in order from greatest to least.

$\frac{3}{2}, \quad \frac{6}{5}, \quad \frac{4}{3}$

Show solution

1 · Understandwhat's really being asked

Order the three fractions 3/2, 6/5, 4/3 from greatest to least.

Givens

The three fractions are 3/2, 6/5, 4/3.
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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Split each fraction into 1 plus a small part

#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: 3/2 = 1 + 1/2, 6/5 = 1 + 1/5, 4/3 = 1 + 1/3.

\dfrac{3}{2}=1+\dfrac{1}{2},\quad \dfrac{6}{5}=1+\dfrac{1}{5},\quad \dfrac{4}{3}=1+\dfrac{1}{3}

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

2Compare the unit fractions by their denominators

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

The extra pieces 1/2, 1/5, 1/3 all have numerator 1. When fractions share the same numerator, the one with the smaller denominator is larger. Since 2 < 3 < 5, we get 1/2 > 1/3 > 1/5.

2 < 3 < 5 \;\Rightarrow\; \dfrac{1}{2} > \dfrac{1}{3} > \dfrac{1}{5}

Cutting a whole into fewer pieces makes each piece bigger, so 1/2 is the biggest extra piece.

3Write the fractions in order

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

The biggest extra piece makes the biggest fraction. So 3/2 is greatest, 4/3 is next, and 6/5 is least.

\dfrac{3}{2} > \dfrac{4}{3} > \dfrac{6}{5}

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

Answer: 3/2, 4/3, 6/5

4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. 3/2 is about 1.5000, 4/3 is about 1.3333, 6/5 is about 1.2000, which matches the order greatest to least.

Another way: Guess and check with cross-multiplication (tool 6): for the top two, 3 x 3 = 9 vs 4 x 2 = 8, so 3/2 > 4/3, confirming the same order by direct comparison.

Standardsmin 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.

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

Variant 2 easy answer: 6/5, 7/6, 9/8

Write the three fractions in order from greatest to least.

$\frac{6}{5}, \quad \frac{9}{8}, \quad \frac{7}{6}$

Show solution

1 · Understandwhat's really being asked

Order the three fractions 6/5, 9/8, 7/6 from greatest to least.

Givens

The three fractions are 6/5, 9/8, 7/6.
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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Split each fraction into 1 plus a small part

#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: 6/5 = 1 + 1/5, 9/8 = 1 + 1/8, 7/6 = 1 + 1/6.

\dfrac{6}{5}=1+\dfrac{1}{5},\quad \dfrac{9}{8}=1+\dfrac{1}{8},\quad \dfrac{7}{6}=1+\dfrac{1}{6}

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

2Compare the unit fractions by their denominators

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

The extra pieces 1/5, 1/8, 1/6 all have numerator 1. When fractions share the same numerator, the one with the smaller denominator is larger. Since 5 < 6 < 8, we get 1/5 > 1/6 > 1/8.

5 < 6 < 8 \;\Rightarrow\; \dfrac{1}{5} > \dfrac{1}{6} > \dfrac{1}{8}

Cutting a whole into fewer pieces makes each piece bigger, so 1/5 is the biggest extra piece.

3Write the fractions in order

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

The biggest extra piece makes the biggest fraction. So 6/5 is greatest, 7/6 is next, and 9/8 is least.

\dfrac{6}{5} > \dfrac{7}{6} > \dfrac{9}{8}

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

Answer: 6/5, 7/6, 9/8

4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. 6/5 is about 1.2000, 7/6 is about 1.1667, 9/8 is about 1.1250, which matches the order greatest to least.

Another way: Guess and check with cross-multiplication (tool 6): for the top two, 6 x 6 = 36 vs 7 x 5 = 35, so 6/5 > 7/6, confirming the same order by direct comparison.

Standardsmin 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.

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

Variant 3 easy answer: 5/4, 8/7, 10/9

Write the three fractions in order from greatest to least.

$\frac{10}{9}, \quad \frac{5}{4}, \quad \frac{8}{7}$

Show solution

1 · Understandwhat's really being asked

Order the three fractions 10/9, 5/4, 8/7 from greatest to least.

Givens

The three fractions are 10/9, 5/4, 8/7.
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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Split each fraction into 1 plus a small part

#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: 10/9 = 1 + 1/9, 5/4 = 1 + 1/4, 8/7 = 1 + 1/7.

\dfrac{10}{9}=1+\dfrac{1}{9},\quad \dfrac{5}{4}=1+\dfrac{1}{4},\quad \dfrac{8}{7}=1+\dfrac{1}{7}

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

2Compare the unit fractions by their denominators

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

The extra pieces 1/9, 1/4, 1/7 all have numerator 1. When fractions share the same numerator, the one with the smaller denominator is larger. Since 4 < 7 < 9, we get 1/4 > 1/7 > 1/9.

4 < 7 < 9 \;\Rightarrow\; \dfrac{1}{4} > \dfrac{1}{7} > \dfrac{1}{9}

Cutting a whole into fewer pieces makes each piece bigger, so 1/4 is the biggest extra piece.

3Write the fractions in order

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

The biggest extra piece makes the biggest fraction. So 5/4 is greatest, 8/7 is next, and 10/9 is least.

\dfrac{5}{4} > \dfrac{8}{7} > \dfrac{10}{9}

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

Answer: 5/4, 8/7, 10/9

4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. 5/4 is about 1.2500, 8/7 is about 1.1429, 10/9 is about 1.1111, which matches the order greatest to least.

Another way: Guess and check with cross-multiplication (tool 6): for the top two, 5 x 7 = 35 vs 8 x 4 = 32, so 5/4 > 8/7, confirming the same order by direct comparison.

Standardsmin 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.

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

Variant 4 easy answer: 5/4, 7/6, 10/9

Write the three fractions in order from greatest to least.

$\frac{7}{6}, \quad \frac{10}{9}, \quad \frac{5}{4}$

Show solution

1 · Understandwhat's really being asked

Order the three fractions 7/6, 10/9, 5/4 from greatest to least.

Givens

The three fractions are 7/6, 10/9, 5/4.
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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Split each fraction into 1 plus a small part

#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: 7/6 = 1 + 1/6, 10/9 = 1 + 1/9, 5/4 = 1 + 1/4.

\dfrac{7}{6}=1+\dfrac{1}{6},\quad \dfrac{10}{9}=1+\dfrac{1}{9},\quad \dfrac{5}{4}=1+\dfrac{1}{4}

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

2Compare the unit fractions by their denominators

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

The extra pieces 1/6, 1/9, 1/4 all have numerator 1. When fractions share the same numerator, the one with the smaller denominator is larger. Since 4 < 6 < 9, we get 1/4 > 1/6 > 1/9.

4 < 6 < 9 \;\Rightarrow\; \dfrac{1}{4} > \dfrac{1}{6} > \dfrac{1}{9}

Cutting a whole into fewer pieces makes each piece bigger, so 1/4 is the biggest extra piece.

3Write the fractions in order

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

The biggest extra piece makes the biggest fraction. So 5/4 is greatest, 7/6 is next, and 10/9 is least.

\dfrac{5}{4} > \dfrac{7}{6} > \dfrac{10}{9}

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

Answer: 5/4, 7/6, 10/9

4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. 5/4 is about 1.2500, 7/6 is about 1.1667, 10/9 is about 1.1111, which matches the order greatest to least.

Another way: Guess and check with cross-multiplication (tool 6): for the top two, 5 x 6 = 30 vs 7 x 4 = 28, so 5/4 > 7/6, confirming the same order by direct comparison.

Standardsmin 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.

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

Variant 5 medium answer: 4/3, 8/7, 12/11

Write the three fractions in order from greatest to least.

$\frac{4}{3}, \quad \frac{12}{11}, \quad \frac{8}{7}$

Show solution

1 · Understandwhat's really being asked

Order the three fractions 4/3, 12/11, 8/7 from greatest to least.

Givens

The three fractions are 4/3, 12/11, 8/7.
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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Split each fraction into 1 plus a small part

#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: 4/3 = 1 + 1/3, 12/11 = 1 + 1/11, 8/7 = 1 + 1/7.

\dfrac{4}{3}=1+\dfrac{1}{3},\quad \dfrac{12}{11}=1+\dfrac{1}{11},\quad \dfrac{8}{7}=1+\dfrac{1}{7}

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

2Compare the unit fractions by their denominators

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

The extra pieces 1/3, 1/11, 1/7 all have numerator 1. When fractions share the same numerator, the one with the smaller denominator is larger. Since 3 < 7 < 11, we get 1/3 > 1/7 > 1/11.

3 < 7 < 11 \;\Rightarrow\; \dfrac{1}{3} > \dfrac{1}{7} > \dfrac{1}{11}

Cutting a whole into fewer pieces makes each piece bigger, so 1/3 is the biggest extra piece.

3Write the fractions in order

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

The biggest extra piece makes the biggest fraction. So 4/3 is greatest, 8/7 is next, and 12/11 is least.

\dfrac{4}{3} > \dfrac{8}{7} > \dfrac{12}{11}

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

Answer: 4/3, 8/7, 12/11

4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. 4/3 is about 1.3333, 8/7 is about 1.1429, 12/11 is about 1.0909, which matches the order greatest to least.

Another way: Guess and check with cross-multiplication (tool 6): for the top two, 4 x 7 = 28 vs 8 x 3 = 24, so 4/3 > 8/7, confirming the same order by direct comparison.

Standardsmin 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.

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

Variant 6 medium answer: 13/12, 16/15, 21/20

Write the three fractions in order from greatest to least.

$\frac{13}{12}, \quad \frac{21}{20}, \quad \frac{16}{15}$

Show solution

1 · Understandwhat's really being asked

Order the three fractions 13/12, 21/20, 16/15 from greatest to least.

Givens

The three fractions are 13/12, 21/20, 16/15.
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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Split each fraction into 1 plus a small part

#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: 13/12 = 1 + 1/12, 21/20 = 1 + 1/20, 16/15 = 1 + 1/15.

\dfrac{13}{12}=1+\dfrac{1}{12},\quad \dfrac{21}{20}=1+\dfrac{1}{20},\quad \dfrac{16}{15}=1+\dfrac{1}{15}

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

2Compare the unit fractions by their denominators

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

The extra pieces 1/12, 1/20, 1/15 all have numerator 1. When fractions share the same numerator, the one with the smaller denominator is larger. Since 12 < 15 < 20, we get 1/12 > 1/15 > 1/20.

12 < 15 < 20 \;\Rightarrow\; \dfrac{1}{12} > \dfrac{1}{15} > \dfrac{1}{20}

Cutting a whole into fewer pieces makes each piece bigger, so 1/12 is the biggest extra piece.

3Write the fractions in order

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

The biggest extra piece makes the biggest fraction. So 13/12 is greatest, 16/15 is next, and 21/20 is least.

\dfrac{13}{12} > \dfrac{16}{15} > \dfrac{21}{20}

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

Answer: 13/12, 16/15, 21/20

4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. 13/12 is about 1.0833, 16/15 is about 1.0667, 21/20 is about 1.0500, which matches the order greatest to least.

Another way: Guess and check with cross-multiplication (tool 6): for the top two, 13 x 15 = 195 vs 16 x 12 = 192, so 13/12 > 16/15, confirming the same order by direct comparison.

Standardsmin 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.

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

Variant 7 medium answer: 18/17, 20/19, 24/23

Write the three fractions in order from greatest to least.

$\frac{18}{17}, \quad \frac{24}{23}, \quad \frac{20}{19}$

Show solution

1 · Understandwhat's really being asked

Order the three fractions 18/17, 24/23, 20/19 from greatest to least.

Givens

The three fractions are 18/17, 24/23, 20/19.
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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Split each fraction into 1 plus a small part

#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: 18/17 = 1 + 1/17, 24/23 = 1 + 1/23, 20/19 = 1 + 1/19.

\dfrac{18}{17}=1+\dfrac{1}{17},\quad \dfrac{24}{23}=1+\dfrac{1}{23},\quad \dfrac{20}{19}=1+\dfrac{1}{19}

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

2Compare the unit fractions by their denominators

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

The extra pieces 1/17, 1/23, 1/19 all have numerator 1. When fractions share the same numerator, the one with the smaller denominator is larger. Since 17 < 19 < 23, we get 1/17 > 1/19 > 1/23.

17 < 19 < 23 \;\Rightarrow\; \dfrac{1}{17} > \dfrac{1}{19} > \dfrac{1}{23}

Cutting a whole into fewer pieces makes each piece bigger, so 1/17 is the biggest extra piece.

3Write the fractions in order

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

The biggest extra piece makes the biggest fraction. So 18/17 is greatest, 20/19 is next, and 24/23 is least.

\dfrac{18}{17} > \dfrac{20}{19} > \dfrac{24}{23}

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

Answer: 18/17, 20/19, 24/23

4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. 18/17 is about 1.0588, 20/19 is about 1.0526, 24/23 is about 1.0435, which matches the order greatest to least.

Another way: Guess and check with cross-multiplication (tool 6): for the top two, 18 x 19 = 342 vs 20 x 17 = 340, so 18/17 > 20/19, confirming the same order by direct comparison.

Standardsmin 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.

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

Variant 8 medium answer: 15/14, 22/21, 36/35

Write the three fractions in order from greatest to least.

$\frac{22}{21}, \quad \frac{15}{14}, \quad \frac{36}{35}$

Show solution

1 · Understandwhat's really being asked

Order the three fractions 22/21, 15/14, 36/35 from greatest to least.

Givens

The three fractions are 22/21, 15/14, 36/35.
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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Split each fraction into 1 plus a small part

#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: 22/21 = 1 + 1/21, 15/14 = 1 + 1/14, 36/35 = 1 + 1/35.

\dfrac{22}{21}=1+\dfrac{1}{21},\quad \dfrac{15}{14}=1+\dfrac{1}{14},\quad \dfrac{36}{35}=1+\dfrac{1}{35}

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

2Compare the unit fractions by their denominators

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

The extra pieces 1/21, 1/14, 1/35 all have numerator 1. When fractions share the same numerator, the one with the smaller denominator is larger. Since 14 < 21 < 35, we get 1/14 > 1/21 > 1/35.

14 < 21 < 35 \;\Rightarrow\; \dfrac{1}{14} > \dfrac{1}{21} > \dfrac{1}{35}

Cutting a whole into fewer pieces makes each piece bigger, so 1/14 is the biggest extra piece.

3Write the fractions in order

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

The biggest extra piece makes the biggest fraction. So 15/14 is greatest, 22/21 is next, and 36/35 is least.

\dfrac{15}{14} > \dfrac{22}{21} > \dfrac{36}{35}

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

Answer: 15/14, 22/21, 36/35

4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. 15/14 is about 1.0714, 22/21 is about 1.0476, 36/35 is about 1.0286, which matches the order greatest to least.

Another way: Guess and check with cross-multiplication (tool 6): for the top two, 15 x 21 = 315 vs 22 x 14 = 308, so 15/14 > 22/21, confirming the same order by direct comparison.

Standardsmin 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.

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

Variant 9 hard answer: 26/25, 41/40, 51/50

Write the three fractions in order from greatest to least.

$\frac{51}{50}, \quad \frac{26}{25}, \quad \frac{41}{40}$

Show solution

1 · Understandwhat's really being asked

Order the three fractions 51/50, 26/25, 41/40 from greatest to least.

Givens

The three fractions are 51/50, 26/25, 41/40.
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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Split each fraction into 1 plus a small part

#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: 51/50 = 1 + 1/50, 26/25 = 1 + 1/25, 41/40 = 1 + 1/40.

\dfrac{51}{50}=1+\dfrac{1}{50},\quad \dfrac{26}{25}=1+\dfrac{1}{25},\quad \dfrac{41}{40}=1+\dfrac{1}{40}

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

2Compare the unit fractions by their denominators

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

The extra pieces 1/50, 1/25, 1/40 all have numerator 1. When fractions share the same numerator, the one with the smaller denominator is larger. Since 25 < 40 < 50, we get 1/25 > 1/40 > 1/50.

25 < 40 < 50 \;\Rightarrow\; \dfrac{1}{25} > \dfrac{1}{40} > \dfrac{1}{50}

Cutting a whole into fewer pieces makes each piece bigger, so 1/25 is the biggest extra piece.

3Write the fractions in order

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

The biggest extra piece makes the biggest fraction. So 26/25 is greatest, 41/40 is next, and 51/50 is least.

\dfrac{26}{25} > \dfrac{41}{40} > \dfrac{51}{50}

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

Answer: 26/25, 41/40, 51/50

4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. 26/25 is about 1.0400, 41/40 is about 1.0250, 51/50 is about 1.0200, which matches the order greatest to least.

Another way: Guess and check with cross-multiplication (tool 6): for the top two, 26 x 40 = 1040 vs 41 x 25 = 1025, so 26/25 > 41/40, confirming the same order by direct comparison.

Standardsmin 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.

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

Variant 10 hard answer: 45/44, 67/66, 89/88

Write the three fractions in order from greatest to least.

$\frac{89}{88}, \quad \frac{45}{44}, \quad \frac{67}{66}$

Show solution

1 · Understandwhat's really being asked

Order the three fractions 89/88, 45/44, 67/66 from greatest to least.

Givens

The three fractions are 89/88, 45/44, 67/66.
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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Split each fraction into 1 plus a small part

#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: 89/88 = 1 + 1/88, 45/44 = 1 + 1/44, 67/66 = 1 + 1/66.

\dfrac{89}{88}=1+\dfrac{1}{88},\quad \dfrac{45}{44}=1+\dfrac{1}{44},\quad \dfrac{67}{66}=1+\dfrac{1}{66}

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

2Compare the unit fractions by their denominators

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

The extra pieces 1/88, 1/44, 1/66 all have numerator 1. When fractions share the same numerator, the one with the smaller denominator is larger. Since 44 < 66 < 88, we get 1/44 > 1/66 > 1/88.

44 < 66 < 88 \;\Rightarrow\; \dfrac{1}{44} > \dfrac{1}{66} > \dfrac{1}{88}

Cutting a whole into fewer pieces makes each piece bigger, so 1/44 is the biggest extra piece.

3Write the fractions in order

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

The biggest extra piece makes the biggest fraction. So 45/44 is greatest, 67/66 is next, and 89/88 is least.

\dfrac{45}{44} > \dfrac{67}{66} > \dfrac{89}{88}

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

Answer: 45/44, 67/66, 89/88

4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. 45/44 is about 1.0227, 67/66 is about 1.0152, 89/88 is about 1.0114, which matches the order greatest to least.

Another way: Guess and check with cross-multiplication (tool 6): for the top two, 45 x 66 = 2970 vs 67 x 44 = 2948, so 45/44 > 67/66, confirming the same order by direct comparison.

Standardsmin 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.

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

Variant 11 hard answer: 61/60, 76/75, 101/100

Write the three fractions in order from greatest to least.

$\frac{101}{100}, \quad \frac{76}{75}, \quad \frac{61}{60}$

Show solution

1 · Understandwhat's really being asked

Order the three fractions 101/100, 76/75, 61/60 from greatest to least.

Givens

The three fractions are 101/100, 76/75, 61/60.
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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Split each fraction into 1 plus a small part

#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: 101/100 = 1 + 1/100, 76/75 = 1 + 1/75, 61/60 = 1 + 1/60.

\dfrac{101}{100}=1+\dfrac{1}{100},\quad \dfrac{76}{75}=1+\dfrac{1}{75},\quad \dfrac{61}{60}=1+\dfrac{1}{60}

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

2Compare the unit fractions by their denominators

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

The extra pieces 1/100, 1/75, 1/60 all have numerator 1. When fractions share the same numerator, the one with the smaller denominator is larger. Since 60 < 75 < 100, we get 1/60 > 1/75 > 1/100.

60 < 75 < 100 \;\Rightarrow\; \dfrac{1}{60} > \dfrac{1}{75} > \dfrac{1}{100}

Cutting a whole into fewer pieces makes each piece bigger, so 1/60 is the biggest extra piece.

3Write the fractions in order

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

The biggest extra piece makes the biggest fraction. So 61/60 is greatest, 76/75 is next, and 101/100 is least.

\dfrac{61}{60} > \dfrac{76}{75} > \dfrac{101}{100}

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

Answer: 61/60, 76/75, 101/100

4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. 61/60 is about 1.0167, 76/75 is about 1.0133, 101/100 is about 1.0100, which matches the order greatest to least.

Another way: Guess and check with cross-multiplication (tool 6): for the top two, 61 x 75 = 4575 vs 76 x 60 = 4560, so 61/60 > 76/75, confirming the same order by direct comparison.

Standardsmin 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.

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

Variant 12 hard answer: 144/143, 279/278, 353/352

Write the three fractions in order from greatest to least.

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

Show solution

1 · Understandwhat's really being asked

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

Givens

The three fractions are 144/143, 353/352, 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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Split each fraction into 1 plus a small part

#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, 353/352 = 1 + 1/352, 279/278 = 1 + 1/278.

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

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

2Compare the unit fractions by their denominators

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

The extra pieces 1/143, 1/352, 1/278 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.

3Write the fractions in order

#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

4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. 144/143 is about 1.0070, 279/278 is about 1.0036, 353/352 is about 1.0028, which matches the order greatest to least.

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

Standardsmin 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.

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