Practice · Same numerator: smaller denominator is larger · Sensim Math

Variant 1 easy

Write the three fractions in order from greatest to least.

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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

Givens

The three fractions are $\frac{3}{2}$ , $\frac{6}{5}$ , $\frac{4}{3}$ .
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:

\frac{3}{2}

= 1 +

\frac{1}{2}

,

\frac{6}{5}

= 1 +

\frac{1}{5}

,

\frac{4}{3}

= 1 +

\frac{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

\frac{1}{2}

,

\frac{1}{5}

,

\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

\frac{1}{2}

>

\frac{1}{3}

>

\frac{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

\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

\frac{3}{2}

is greatest,

\frac{4}{3}

is next, and

\frac{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:

\frac{3}{2}

,

\frac{4}{3}

,

\frac{6}{5}

4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. $\frac{3}{2}$ is about 1.5000, $\frac{4}{3}$ is about 1.3333, $\frac{6}{5}$ 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 $\frac{3}{2}$ > $\frac{4}{3}$ , 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

Write the three fractions in order from greatest to least.

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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

Givens

The three fractions are $\frac{6}{5}$ , $\frac{9}{8}$ , $\frac{7}{6}$ .
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:

\frac{6}{5} = 1 + \frac{1}{5}

,

\frac{9}{8} = 1 + \frac{1}{8}

,

\frac{7}{6} = 1 + \frac{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

\frac{1}{5}

,

\frac{1}{8}

,

\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

\frac{1}{5} > \frac{1}{6} > \frac{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

\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

\frac{6}{5}

is greatest,

\frac{7}{6}

is next, and

\frac{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:

\frac{6}{5}

,

\frac{7}{6}

,

\frac{9}{8}

4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. $\frac{6}{5}$ is about 1.2000, $\frac{7}{6}$ is about 1.1667, $\frac{9}{8}$ 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 $\frac{6}{5} > \frac{7}{6}$ , 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

Write the three fractions in order from greatest to least.

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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

Givens

The three fractions are $\frac{10}{9}$ , $\frac{5}{4}$ , $\frac{8}{7}$ .
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:

\frac{10}{9}

= 1 +

\frac{1}{9}

,

\frac{5}{4}

= 1 +

\frac{1}{4}

,

\frac{8}{7}

= 1 +

\frac{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

\frac{1}{9}

,

\frac{1}{4}

,

\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

\frac{1}{4}

>

\frac{1}{7}

>

\frac{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

\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

\frac{5}{4}

is greatest,

\frac{8}{7}

is next, and

\frac{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:

\frac{5}{4}

,

\frac{8}{7}

,

\frac{10}{9}

4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. $\frac{5}{4}$ is about 1.2500, $\frac{8}{7}$ is about 1.1429, $\frac{10}{9}$ 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 $\frac{5}{4}$ > $\frac{8}{7}$ , 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

Write the three fractions in order from greatest to least.

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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

Givens

The three fractions are $\frac{7}{6}$ , $\frac{10}{9}$ , $\frac{5}{4}$ .
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:

\frac{7}{6} = 1 + \frac{1}{6}

,

\frac{10}{9} = 1 + \frac{1}{9}

,

\frac{5}{4} = 1 + \frac{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

\frac{1}{6}

,

\frac{1}{9}

,

\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

\frac{1}{4} > \frac{1}{6} > \frac{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

\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

\frac{5}{4}

is greatest,

\frac{7}{6}

is next, and

\frac{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:

\frac{5}{4}

,

\frac{7}{6}

,

\frac{10}{9}

4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. $\frac{5}{4}$ is about 1.2500, $\frac{7}{6}$ is about 1.1667, $\frac{10}{9}$ 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 $\frac{5}{4} > \frac{7}{6}$ , 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

Write the three fractions in order from greatest to least.

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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

Givens

The three fractions are $\frac{4}{3}$ , $\frac{12}{11}$ , $\frac{8}{7}$ .
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:

\frac{4}{3} = 1 + \frac{1}{3}

,

\frac{12}{11} = 1 + \frac{1}{11}

,

\frac{8}{7} = 1 + \frac{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

\frac{1}{3}

,

\frac{1}{11}

,

\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

\frac{1}{3} > \frac{1}{7} > \frac{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

\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

\frac{4}{3}

is greatest,

\frac{8}{7}

is next, and

\frac{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:

\frac{4}{3}

,

\frac{8}{7}

,

\frac{12}{11}

4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. $\frac{4}{3}$ is about 1.3333, $\frac{8}{7}$ is about 1.1429, $\frac{12}{11}$ 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 $\frac{4}{3} > \frac{8}{7}$ , 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

Write the three fractions in order from greatest to least.

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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

Givens

The three fractions are $\frac{13}{12}$ , $\frac{21}{20}$ , $\frac{16}{15}$ .
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:

\frac{13}{12}

= 1 +

\frac{1}{12}

,

\frac{21}{20}

= 1 +

\frac{1}{20}

,

\frac{16}{15}

= 1 +

\frac{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

\frac{1}{12}

,

\frac{1}{20}

,

\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

\frac{1}{12}

>

\frac{1}{15}

>

\frac{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

\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

\frac{13}{12}

is greatest,

\frac{16}{15}

is next, and

\frac{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:

\frac{13}{12}

,

\frac{16}{15}

,

\frac{21}{20}

4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. $\frac{13}{12}$ is about 1.0833, $\frac{16}{15}$ is about 1.0667, $\frac{21}{20}$ 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 $\frac{13}{12}$ > $\frac{16}{15}$ , 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

Write the three fractions in order from greatest to least.

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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

Givens

The three fractions are $\frac{18}{17}$ , $\frac{24}{23}$ , $\frac{20}{19}$ .
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:

\frac{18}{17}

= 1 +

\frac{1}{17}

,

\frac{24}{23}

= 1 +

\frac{1}{23}

,

\frac{20}{19}

= 1 +

\frac{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

\frac{1}{17}

,

\frac{1}{23}

,

\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

\frac{1}{17}

>

\frac{1}{19}

>

\frac{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

\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

\frac{18}{17}

is greatest,

\frac{20}{19}

is next, and

\frac{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:

\frac{18}{17}

,

\frac{20}{19}

,

\frac{24}{23}

4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. $\frac{18}{17}$ is about 1.0588, $\frac{20}{19}$ is about 1.0526, $\frac{24}{23}$ 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 $\frac{18}{17}$ > $\frac{20}{19}$ , 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

Write the three fractions in order from greatest to least.

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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

Givens

The three fractions are $\frac{22}{21}$ , $\frac{15}{14}$ , $\frac{36}{35}$ .
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:

\frac{22}{21}

= 1 +

\frac{1}{21}

,

\frac{15}{14}

= 1 +

\frac{1}{14}

,

\frac{36}{35}

= 1 +

\frac{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

\frac{1}{21}

,

\frac{1}{14}

,

\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

\frac{1}{14}

>

\frac{1}{21}

>

\frac{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

\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

\frac{15}{14}

is greatest,

\frac{22}{21}

is next, and

\frac{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:

\frac{15}{14}

,

\frac{22}{21}

,

\frac{36}{35}

4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. $\frac{15}{14}$ is about 1.0714, $\frac{22}{21}$ is about 1.0476, $\frac{36}{35}$ 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 $\frac{15}{14}$ > $\frac{22}{21}$ , 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

Write the three fractions in order from greatest to least.

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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

Givens

The three fractions are $\frac{51}{50}$ , $\frac{26}{25}$ , $\frac{41}{40}$ .
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:

\frac{51}{50}

= 1 +

\frac{1}{50}

,

\frac{26}{25}

= 1 +

\frac{1}{25}

,

\frac{41}{40}

= 1 +

\frac{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

\frac{1}{50}

,

\frac{1}{25}

,

\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

\frac{1}{25}

>

\frac{1}{40}

>

\frac{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

\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

\frac{26}{25}

is greatest,

\frac{41}{40}

is next, and

\frac{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:

\frac{26}{25}

,

\frac{41}{40}

,

\frac{51}{50}

4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. $\frac{26}{25}$ is about 1.0400, $\frac{41}{40}$ is about 1.0250, $\frac{51}{50}$ 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 $\frac{26}{25}$ > $\frac{41}{40}$ , 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

Write the three fractions in order from greatest to least.

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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

Givens

The three fractions are $\frac{89}{88}$ , $\frac{45}{44}$ , $\frac{67}{66}$ .
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:

\frac{89}{88}

= 1 +

\frac{1}{88}

,

\frac{45}{44}

= 1 +

\frac{1}{44}

,

\frac{67}{66}

= 1 +

\frac{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

\frac{1}{88}

,

\frac{1}{44}

,

\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

\frac{1}{44}

>

\frac{1}{66}

>

\frac{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

\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

\frac{45}{44}

is greatest,

\frac{67}{66}

is next, and

\frac{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:

\frac{45}{44}

,

\frac{67}{66}

,

\frac{89}{88}

4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. $\frac{45}{44}$ is about 1.0227, $\frac{67}{66}$ is about 1.0152, $\frac{89}{88}$ 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 $\frac{45}{44}$ > $\frac{67}{66}$ , 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

Write the three fractions in order from greatest to least.

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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

Givens

The three fractions are $\frac{101}{100}$ , $\frac{76}{75}$ , $\frac{61}{60}$ .
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:

\frac{101}{100}

= 1 +

\frac{1}{100}

,

\frac{76}{75}

= 1 +

\frac{1}{75}

,

\frac{61}{60}

= 1 +

\frac{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

\frac{1}{100}

,

\frac{1}{75}

,

\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

\frac{1}{60}

>

\frac{1}{75}

>

\frac{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

\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

\frac{61}{60}

is greatest,

\frac{76}{75}

is next, and

\frac{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:

\frac{61}{60}

,

\frac{76}{75}

,

\frac{101}{100}

4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. $\frac{61}{60}$ is about 1.0167, $\frac{76}{75}$ is about 1.0133, $\frac{101}{100}$ 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 $\frac{61}{60}$ > $\frac{76}{75}$ , 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

Write the three fractions in order from greatest to least.

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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

Givens

The three fractions are $\frac{144}{143}$ , $\frac{353}{352}$ , $\frac{279}{278}$ .
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:

\frac{144}{143}

= 1 +

\frac{1}{143}

,

\frac{353}{352}

= 1 +

\frac{1}{352}

,

\frac{279}{278}

= 1 +

\frac{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

\frac{1}{143}

,

\frac{1}{352}

,

\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

\frac{1}{143}

>

\frac{1}{278}

>

\frac{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

\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

\frac{144}{143}

is greatest,

\frac{279}{278}

is next, and

\frac{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:

\frac{144}{143}

,

\frac{279}{278}

,

\frac{353}{352}

4 · Reviewdoes it hold up?

All three are a tiny bit more than 1. $\frac{144}{143}$ is about 1.0070, $\frac{279}{278}$ is about 1.0036, $\frac{353}{352}$ 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 $\frac{144}{143}$ > $\frac{279}{278}$ , 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!

Same numerator: smaller denominator is larger · 12 practice problems

Generated variants — 12

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