← 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: 32\frac{3}{2}, 43\frac{4}{3}, 65\frac{6}{5}

Write the three fractions in order from greatest to least.

32,65,43\frac{3}{2}, \quad \frac{6}{5}, \quad \frac{4}{3}

Show solution
1 · Understandwhat's really being asked

Order the three fractions 32\frac{3}{2}, 65\frac{6}{5}, 43\frac{4}{3} from greatest to least.

Givens
  • The three fractions are 32\frac{3}{2}, 65\frac{6}{5}, 43\frac{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: 32\frac{3}{2} = 1 + 12\frac{1}{2}, 65\frac{6}{5} = 1 + 15\frac{1}{5}, 43\frac{4}{3} = 1 + 13\frac{1}{3}.
32=1+12,65=1+15,43=1+13\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 12\frac{1}{2}, 15\frac{1}{5}, 13\frac{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 12\frac{1}{2} > 13\frac{1}{3} > 15\frac{1}{5}.
2<3<5  ⇒  12>13>152 < 3 < 5 \;\Rightarrow\; \dfrac{1}{2} > \dfrac{1}{3} > \dfrac{1}{5}
Cutting a whole into fewer pieces makes each piece bigger, so 12\frac{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 32\frac{3}{2} is greatest, 43\frac{4}{3} is next, and 65\frac{6}{5} is least.
32>43>65\dfrac{3}{2} > \dfrac{4}{3} > \dfrac{6}{5}
Order the wholes-plus-pieces by the size of the extra piece.
Answer: 32\frac{3}{2}, 43\frac{4}{3}, 65\frac{6}{5}
4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. 32\frac{3}{2} is about 1.5000, 43\frac{4}{3} is about 1.3333, 65\frac{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 32\frac{3}{2} > 43\frac{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: 65\frac{6}{5}, 76\frac{7}{6}, 98\frac{9}{8}

Write the three fractions in order from greatest to least.

65,98,76\frac{6}{5}, \quad \frac{9}{8}, \quad \frac{7}{6}

Show solution
1 · Understandwhat's really being asked

Order the three fractions 65\frac{6}{5}, 98\frac{9}{8}, 76\frac{7}{6} from greatest to least.

Givens
  • The three fractions are 65\frac{6}{5}, 98\frac{9}{8}, 76\frac{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: 65=1+15\frac{6}{5} = 1 + \frac{1}{5}, 98=1+18\frac{9}{8} = 1 + \frac{1}{8}, 76=1+16\frac{7}{6} = 1 + \frac{1}{6}.
65=1+15,98=1+18,76=1+16\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 15\frac{1}{5}, 18\frac{1}{8}, 16\frac{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 15>16>18\frac{1}{5} > \frac{1}{6} > \frac{1}{8}.
5<6<8  ⇒  15>16>185 < 6 < 8 \;\Rightarrow\; \dfrac{1}{5} > \dfrac{1}{6} > \dfrac{1}{8}
Cutting a whole into fewer pieces makes each piece bigger, so 15\frac{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 65\frac{6}{5} is greatest, 76\frac{7}{6} is next, and 98\frac{9}{8} is least.
65>76>98\dfrac{6}{5} > \dfrac{7}{6} > \dfrac{9}{8}
Order the wholes-plus-pieces by the size of the extra piece.
Answer: 65\frac{6}{5}, 76\frac{7}{6}, 98\frac{9}{8}
4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. 65\frac{6}{5} is about 1.2000, 76\frac{7}{6} is about 1.1667, 98\frac{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 65>76\frac{6}{5} > \frac{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: 54\frac{5}{4}, 87\frac{8}{7}, 109\frac{10}{9}

Write the three fractions in order from greatest to least.

109,54,87\frac{10}{9}, \quad \frac{5}{4}, \quad \frac{8}{7}

Show solution
1 · Understandwhat's really being asked

Order the three fractions 109\frac{10}{9}, 54\frac{5}{4}, 87\frac{8}{7} from greatest to least.

Givens
  • The three fractions are 109\frac{10}{9}, 54\frac{5}{4}, 87\frac{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: 109\frac{10}{9} = 1 + 19\frac{1}{9}, 54\frac{5}{4} = 1 + 14\frac{1}{4}, 87\frac{8}{7} = 1 + 17\frac{1}{7}.
109=1+19,54=1+14,87=1+17\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 19\frac{1}{9}, 14\frac{1}{4}, 17\frac{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 14\frac{1}{4} > 17\frac{1}{7} > 19\frac{1}{9}.
4<7<9  ⇒  14>17>194 < 7 < 9 \;\Rightarrow\; \dfrac{1}{4} > \dfrac{1}{7} > \dfrac{1}{9}
Cutting a whole into fewer pieces makes each piece bigger, so 14\frac{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 54\frac{5}{4} is greatest, 87\frac{8}{7} is next, and 109\frac{10}{9} is least.
54>87>109\dfrac{5}{4} > \dfrac{8}{7} > \dfrac{10}{9}
Order the wholes-plus-pieces by the size of the extra piece.
Answer: 54\frac{5}{4}, 87\frac{8}{7}, 109\frac{10}{9}
4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. 54\frac{5}{4} is about 1.2500, 87\frac{8}{7} is about 1.1429, 109\frac{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 54\frac{5}{4} > 87\frac{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: 54\frac{5}{4}, 76\frac{7}{6}, 109\frac{10}{9}

Write the three fractions in order from greatest to least.

76,109,54\frac{7}{6}, \quad \frac{10}{9}, \quad \frac{5}{4}

Show solution
1 · Understandwhat's really being asked

Order the three fractions 76\frac{7}{6}, 109\frac{10}{9}, 54\frac{5}{4} from greatest to least.

Givens
  • The three fractions are 76\frac{7}{6}, 109\frac{10}{9}, 54\frac{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: 76=1+16\frac{7}{6} = 1 + \frac{1}{6}, 109=1+19\frac{10}{9} = 1 + \frac{1}{9}, 54=1+14\frac{5}{4} = 1 + \frac{1}{4}.
76=1+16,109=1+19,54=1+14\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 16\frac{1}{6}, 19\frac{1}{9}, 14\frac{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 14>16>19\frac{1}{4} > \frac{1}{6} > \frac{1}{9}.
4<6<9  ⇒  14>16>194 < 6 < 9 \;\Rightarrow\; \dfrac{1}{4} > \dfrac{1}{6} > \dfrac{1}{9}
Cutting a whole into fewer pieces makes each piece bigger, so 14\frac{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 54\frac{5}{4} is greatest, 76\frac{7}{6} is next, and 109\frac{10}{9} is least.
54>76>109\dfrac{5}{4} > \dfrac{7}{6} > \dfrac{10}{9}
Order the wholes-plus-pieces by the size of the extra piece.
Answer: 54\frac{5}{4}, 76\frac{7}{6}, 109\frac{10}{9}
4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. 54\frac{5}{4} is about 1.2500, 76\frac{7}{6} is about 1.1667, 109\frac{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 54>76\frac{5}{4} > \frac{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: 43\frac{4}{3}, 87\frac{8}{7}, 1211\frac{12}{11}

Write the three fractions in order from greatest to least.

43,1211,87\frac{4}{3}, \quad \frac{12}{11}, \quad \frac{8}{7}

Show solution
1 · Understandwhat's really being asked

Order the three fractions 43\frac{4}{3}, 1211\frac{12}{11}, 87\frac{8}{7} from greatest to least.

Givens
  • The three fractions are 43\frac{4}{3}, 1211\frac{12}{11}, 87\frac{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: 43=1+13\frac{4}{3} = 1 + \frac{1}{3}, 1211=1+111\frac{12}{11} = 1 + \frac{1}{11}, 87=1+17\frac{8}{7} = 1 + \frac{1}{7}.
43=1+13,1211=1+111,87=1+17\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 13\frac{1}{3}, 111\frac{1}{11}, 17\frac{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 13>17>111\frac{1}{3} > \frac{1}{7} > \frac{1}{11}.
3<7<11  ⇒  13>17>1113 < 7 < 11 \;\Rightarrow\; \dfrac{1}{3} > \dfrac{1}{7} > \dfrac{1}{11}
Cutting a whole into fewer pieces makes each piece bigger, so 13\frac{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 43\frac{4}{3} is greatest, 87\frac{8}{7} is next, and 1211\frac{12}{11} is least.
43>87>1211\dfrac{4}{3} > \dfrac{8}{7} > \dfrac{12}{11}
Order the wholes-plus-pieces by the size of the extra piece.
Answer: 43\frac{4}{3}, 87\frac{8}{7}, 1211\frac{12}{11}
4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. 43\frac{4}{3} is about 1.3333, 87\frac{8}{7} is about 1.1429, 1211\frac{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 43>87\frac{4}{3} > \frac{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: 1312\frac{13}{12}, 1615\frac{16}{15}, 2120\frac{21}{20}

Write the three fractions in order from greatest to least.

1312,2120,1615\frac{13}{12}, \quad \frac{21}{20}, \quad \frac{16}{15}

Show solution
1 · Understandwhat's really being asked

Order the three fractions 1312\frac{13}{12}, 2120\frac{21}{20}, 1615\frac{16}{15} from greatest to least.

Givens
  • The three fractions are 1312\frac{13}{12}, 2120\frac{21}{20}, 1615\frac{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: 1312\frac{13}{12} = 1 + 112\frac{1}{12}, 2120\frac{21}{20} = 1 + 120\frac{1}{20}, 1615\frac{16}{15} = 1 + 115\frac{1}{15}.
1312=1+112,2120=1+120,1615=1+115\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 112\frac{1}{12}, 120\frac{1}{20}, 115\frac{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 112\frac{1}{12} > 115\frac{1}{15} > 120\frac{1}{20}.
12<15<20  ⇒  112>115>12012 < 15 < 20 \;\Rightarrow\; \dfrac{1}{12} > \dfrac{1}{15} > \dfrac{1}{20}
Cutting a whole into fewer pieces makes each piece bigger, so 112\frac{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 1312\frac{13}{12} is greatest, 1615\frac{16}{15} is next, and 2120\frac{21}{20} is least.
1312>1615>2120\dfrac{13}{12} > \dfrac{16}{15} > \dfrac{21}{20}
Order the wholes-plus-pieces by the size of the extra piece.
Answer: 1312\frac{13}{12}, 1615\frac{16}{15}, 2120\frac{21}{20}
4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. 1312\frac{13}{12} is about 1.0833, 1615\frac{16}{15} is about 1.0667, 2120\frac{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 1312\frac{13}{12} > 1615\frac{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: 1817\frac{18}{17}, 2019\frac{20}{19}, 2423\frac{24}{23}

Write the three fractions in order from greatest to least.

1817,2423,2019\frac{18}{17}, \quad \frac{24}{23}, \quad \frac{20}{19}

Show solution
1 · Understandwhat's really being asked

Order the three fractions 1817\frac{18}{17}, 2423\frac{24}{23}, 2019\frac{20}{19} from greatest to least.

Givens
  • The three fractions are 1817\frac{18}{17}, 2423\frac{24}{23}, 2019\frac{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: 1817\frac{18}{17} = 1 + 117\frac{1}{17}, 2423\frac{24}{23} = 1 + 123\frac{1}{23}, 2019\frac{20}{19} = 1 + 119\frac{1}{19}.
1817=1+117,2423=1+123,2019=1+119\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 117\frac{1}{17}, 123\frac{1}{23}, 119\frac{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 117\frac{1}{17} > 119\frac{1}{19} > 123\frac{1}{23}.
17<19<23  ⇒  117>119>12317 < 19 < 23 \;\Rightarrow\; \dfrac{1}{17} > \dfrac{1}{19} > \dfrac{1}{23}
Cutting a whole into fewer pieces makes each piece bigger, so 117\frac{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 1817\frac{18}{17} is greatest, 2019\frac{20}{19} is next, and 2423\frac{24}{23} is least.
1817>2019>2423\dfrac{18}{17} > \dfrac{20}{19} > \dfrac{24}{23}
Order the wholes-plus-pieces by the size of the extra piece.
Answer: 1817\frac{18}{17}, 2019\frac{20}{19}, 2423\frac{24}{23}
4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. 1817\frac{18}{17} is about 1.0588, 2019\frac{20}{19} is about 1.0526, 2423\frac{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 1817\frac{18}{17} > 2019\frac{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: 1514\frac{15}{14}, 2221\frac{22}{21}, 3635\frac{36}{35}

Write the three fractions in order from greatest to least.

2221,1514,3635\frac{22}{21}, \quad \frac{15}{14}, \quad \frac{36}{35}

Show solution
1 · Understandwhat's really being asked

Order the three fractions 2221\frac{22}{21}, 1514\frac{15}{14}, 3635\frac{36}{35} from greatest to least.

Givens
  • The three fractions are 2221\frac{22}{21}, 1514\frac{15}{14}, 3635\frac{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: 2221\frac{22}{21} = 1 + 121\frac{1}{21}, 1514\frac{15}{14} = 1 + 114\frac{1}{14}, 3635\frac{36}{35} = 1 + 135\frac{1}{35}.
2221=1+121,1514=1+114,3635=1+135\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 121\frac{1}{21}, 114\frac{1}{14}, 135\frac{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 114\frac{1}{14} > 121\frac{1}{21} > 135\frac{1}{35}.
14<21<35  ⇒  114>121>13514 < 21 < 35 \;\Rightarrow\; \dfrac{1}{14} > \dfrac{1}{21} > \dfrac{1}{35}
Cutting a whole into fewer pieces makes each piece bigger, so 114\frac{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 1514\frac{15}{14} is greatest, 2221\frac{22}{21} is next, and 3635\frac{36}{35} is least.
1514>2221>3635\dfrac{15}{14} > \dfrac{22}{21} > \dfrac{36}{35}
Order the wholes-plus-pieces by the size of the extra piece.
Answer: 1514\frac{15}{14}, 2221\frac{22}{21}, 3635\frac{36}{35}
4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. 1514\frac{15}{14} is about 1.0714, 2221\frac{22}{21} is about 1.0476, 3635\frac{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 1514\frac{15}{14} > 2221\frac{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: 2625\frac{26}{25}, 4140\frac{41}{40}, 5150\frac{51}{50}

Write the three fractions in order from greatest to least.

5150,2625,4140\frac{51}{50}, \quad \frac{26}{25}, \quad \frac{41}{40}

Show solution
1 · Understandwhat's really being asked

Order the three fractions 5150\frac{51}{50}, 2625\frac{26}{25}, 4140\frac{41}{40} from greatest to least.

Givens
  • The three fractions are 5150\frac{51}{50}, 2625\frac{26}{25}, 4140\frac{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: 5150\frac{51}{50} = 1 + 150\frac{1}{50}, 2625\frac{26}{25} = 1 + 125\frac{1}{25}, 4140\frac{41}{40} = 1 + 140\frac{1}{40}.
5150=1+150,2625=1+125,4140=1+140\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 150\frac{1}{50}, 125\frac{1}{25}, 140\frac{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 125\frac{1}{25} > 140\frac{1}{40} > 150\frac{1}{50}.
25<40<50  ⇒  125>140>15025 < 40 < 50 \;\Rightarrow\; \dfrac{1}{25} > \dfrac{1}{40} > \dfrac{1}{50}
Cutting a whole into fewer pieces makes each piece bigger, so 125\frac{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 2625\frac{26}{25} is greatest, 4140\frac{41}{40} is next, and 5150\frac{51}{50} is least.
2625>4140>5150\dfrac{26}{25} > \dfrac{41}{40} > \dfrac{51}{50}
Order the wholes-plus-pieces by the size of the extra piece.
Answer: 2625\frac{26}{25}, 4140\frac{41}{40}, 5150\frac{51}{50}
4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. 2625\frac{26}{25} is about 1.0400, 4140\frac{41}{40} is about 1.0250, 5150\frac{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 2625\frac{26}{25} > 4140\frac{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: 4544\frac{45}{44}, 6766\frac{67}{66}, 8988\frac{89}{88}

Write the three fractions in order from greatest to least.

8988,4544,6766\frac{89}{88}, \quad \frac{45}{44}, \quad \frac{67}{66}

Show solution
1 · Understandwhat's really being asked

Order the three fractions 8988\frac{89}{88}, 4544\frac{45}{44}, 6766\frac{67}{66} from greatest to least.

Givens
  • The three fractions are 8988\frac{89}{88}, 4544\frac{45}{44}, 6766\frac{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: 8988\frac{89}{88} = 1 + 188\frac{1}{88}, 4544\frac{45}{44} = 1 + 144\frac{1}{44}, 6766\frac{67}{66} = 1 + 166\frac{1}{66}.
8988=1+188,4544=1+144,6766=1+166\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 188\frac{1}{88}, 144\frac{1}{44}, 166\frac{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 144\frac{1}{44} > 166\frac{1}{66} > 188\frac{1}{88}.
44<66<88  ⇒  144>166>18844 < 66 < 88 \;\Rightarrow\; \dfrac{1}{44} > \dfrac{1}{66} > \dfrac{1}{88}
Cutting a whole into fewer pieces makes each piece bigger, so 144\frac{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 4544\frac{45}{44} is greatest, 6766\frac{67}{66} is next, and 8988\frac{89}{88} is least.
4544>6766>8988\dfrac{45}{44} > \dfrac{67}{66} > \dfrac{89}{88}
Order the wholes-plus-pieces by the size of the extra piece.
Answer: 4544\frac{45}{44}, 6766\frac{67}{66}, 8988\frac{89}{88}
4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. 4544\frac{45}{44} is about 1.0227, 6766\frac{67}{66} is about 1.0152, 8988\frac{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 4544\frac{45}{44} > 6766\frac{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: 6160\frac{61}{60}, 7675\frac{76}{75}, 101100\frac{101}{100}

Write the three fractions in order from greatest to least.

101100,7675,6160\frac{101}{100}, \quad \frac{76}{75}, \quad \frac{61}{60}

Show solution
1 · Understandwhat's really being asked

Order the three fractions 101100\frac{101}{100}, 7675\frac{76}{75}, 6160\frac{61}{60} from greatest to least.

Givens
  • The three fractions are 101100\frac{101}{100}, 7675\frac{76}{75}, 6160\frac{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: 101100\frac{101}{100} = 1 + 1100\frac{1}{100}, 7675\frac{76}{75} = 1 + 175\frac{1}{75}, 6160\frac{61}{60} = 1 + 160\frac{1}{60}.
101100=1+1100,7675=1+175,6160=1+160\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 1100\frac{1}{100}, 175\frac{1}{75}, 160\frac{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 160\frac{1}{60} > 175\frac{1}{75} > 1100\frac{1}{100}.
60<75<100  ⇒  160>175>110060 < 75 < 100 \;\Rightarrow\; \dfrac{1}{60} > \dfrac{1}{75} > \dfrac{1}{100}
Cutting a whole into fewer pieces makes each piece bigger, so 160\frac{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 6160\frac{61}{60} is greatest, 7675\frac{76}{75} is next, and 101100\frac{101}{100} is least.
6160>7675>101100\dfrac{61}{60} > \dfrac{76}{75} > \dfrac{101}{100}
Order the wholes-plus-pieces by the size of the extra piece.
Answer: 6160\frac{61}{60}, 7675\frac{76}{75}, 101100\frac{101}{100}
4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. 6160\frac{61}{60} is about 1.0167, 7675\frac{76}{75} is about 1.0133, 101100\frac{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 6160\frac{61}{60} > 7675\frac{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: 144143\frac{144}{143}, 279278\frac{279}{278}, 353352\frac{353}{352}

Write the three fractions in order from greatest to least.

144143,353352,279278\frac{144}{143}, \quad \frac{353}{352}, \quad \frac{279}{278}

Show solution
1 · Understandwhat's really being asked

Order the three fractions 144143\frac{144}{143}, 353352\frac{353}{352}, 279278\frac{279}{278} from greatest to least.

Givens
  • The three fractions are 144143\frac{144}{143}, 353352\frac{353}{352}, 279278\frac{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: 144143\frac{144}{143} = 1 + 1143\frac{1}{143}, 353352\frac{353}{352} = 1 + 1352\frac{1}{352}, 279278\frac{279}{278} = 1 + 1278\frac{1}{278}.
144143=1+1143,353352=1+1352,279278=1+1278\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 1143\frac{1}{143}, 1352\frac{1}{352}, 1278\frac{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 1143\frac{1}{143} > 1278\frac{1}{278} > 1352\frac{1}{352}.
143<278<352  ⇒  1143>1278>1352143 < 278 < 352 \;\Rightarrow\; \dfrac{1}{143} > \dfrac{1}{278} > \dfrac{1}{352}
Cutting a whole into fewer pieces makes each piece bigger, so 1143\frac{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 144143\frac{144}{143} is greatest, 279278\frac{279}{278} is next, and 353352\frac{353}{352} is least.
144143>279278>353352\dfrac{144}{143} > \dfrac{279}{278} > \dfrac{353}{352}
Order the wholes-plus-pieces by the size of the extra piece.
Answer: 144143\frac{144}{143}, 279278\frac{279}{278}, 353352\frac{353}{352}
4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. 144143\frac{144}{143} is about 1.0070, 279278\frac{279}{278} is about 1.0036, 353352\frac{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 144143\frac{144}{143} > 279278\frac{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!