← 3-1 · Compare fractions sharing numerator or denominator · Compare Fractions and Decimals by Structure

Compare fractions sharing numerator or denominator · 8 practice problems

3.NF.A.3

Generated variants — 8

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

Variant 1 answer: 2 numbers

Among the numbers from 1 to 9, how many numbers can go in the $\bigstar$ ?

$\dfrac{1}{3} < \dfrac{1}{\bigstar}$

Show solution

Understand

I need to count how many of the digits 1 through 9 can be placed in the star so that the fraction 1/3 is less than 1/star.

Givens

The inequality is 1/3 < 1/(star).
The star must be a whole number from 1 to 9.

Unknowns

How many values of the star make the inequality true.

Constraints

Both fractions are unit fractions (numerator 1).
For unit fractions, a smaller denominator means a larger fraction.

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List

Both fractions have numerator 1, so the comparison depends only on the denominators: the unit fraction with the smaller denominator is larger. That pattern instantly tells me 1/star > 1/3 exactly when star < 3, and I can list those digits to count them.

Execute

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

Both fractions split one whole into equal parts. The more parts you cut a whole into, the smaller each part. So with numerator 1, the fraction is larger when the denominator is smaller.

\dfrac{1}{3} < \dfrac{1}{\bigstar} \iff \bigstar < 3

Sharing a pizza among fewer people gives each person a bigger slice, so smaller denominator = bigger unit fraction.

#2 Make a Systematic List 3.NF.A.3

We need star < 3, with star chosen from 1 through 9. The digits less than 3 are 1, 2.

\bigstar \in \{1, 2\}

Only denominators smaller than 3 make a slice bigger than 1/3; 3 itself ties and larger ones give smaller slices.

#2 Make a Systematic List 3.NF.A.3

Count the digits in the list 1, 2.

2 \text{ numbers}

There are exactly 2 whole numbers from 1 to 2, so 2 choices work.

Answer: 2 numbers

Review

Spot-check the boundaries: 1/2 > 1/3 (true, 2 counts) and 1/3 < 1/3 is false (3 excluded). Only 1-2 qualify, giving 2 - consistent with the answer.

Guess and check (tool 6): test each star from 1 to 9 in 1/3 < 1/star; it holds for 1, 2 and fails for the rest, again giving 2 values.

Standards · min grade 3

3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Comparing unit fractions by their denominators to determine which stars satisfy 1/3 < 1/star.

💡 For 1-over-something fractions, fewer parts means bigger pieces - so just count the denominators smaller than 3!

Variant 2 answer: 8 numbers

Among the numbers from 1 to 9, how many numbers can go in the $\bigstar$ ?

$\dfrac{1}{9} < \dfrac{1}{\bigstar}$

Show solution

Understand

I need to count how many of the digits 1 through 9 can be placed in the star so that the fraction 1/9 is less than 1/star.

Givens

The inequality is 1/9 < 1/(star).
The star must be a whole number from 1 to 9.

Unknowns

How many values of the star make the inequality true.

Constraints

Both fractions are unit fractions (numerator 1).
For unit fractions, a smaller denominator means a larger fraction.

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List

Both fractions have numerator 1, so the comparison depends only on the denominators: the unit fraction with the smaller denominator is larger. That pattern instantly tells me 1/star > 1/9 exactly when star < 9, and I can list those digits to count them.

Execute

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

Both fractions split one whole into equal parts. The more parts you cut a whole into, the smaller each part. So with numerator 1, the fraction is larger when the denominator is smaller.

\dfrac{1}{9} < \dfrac{1}{\bigstar} \iff \bigstar < 9

Sharing a pizza among fewer people gives each person a bigger slice, so smaller denominator = bigger unit fraction.

#2 Make a Systematic List 3.NF.A.3

We need star < 9, with star chosen from 1 through 9. The digits less than 9 are 1, 2, 3, 4, 5, 6, 7, 8.

\bigstar \in \{1, 2, 3, 4, 5, 6, 7, 8\}

Only denominators smaller than 9 make a slice bigger than 1/9; 9 itself ties and larger ones give smaller slices.

#2 Make a Systematic List 3.NF.A.3

Count the digits in the list 1, 2, 3, 4, 5, 6, 7, 8.

8 \text{ numbers}

There are exactly 8 whole numbers from 1 to 8, so 8 choices work.

Answer: 8 numbers

Review

Spot-check the boundaries: 1/8 > 1/9 (true, 8 counts) and 1/9 < 1/9 is false (9 excluded). Only 1-8 qualify, giving 8 - consistent with the answer.

Guess and check (tool 6): test each star from 1 to 9 in 1/9 < 1/star; it holds for 1, 2, 3, 4, 5, 6, 7, 8 and fails for the rest, again giving 8 values.

Standards · min grade 3

3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Comparing unit fractions by their denominators to determine which stars satisfy 1/9 < 1/star.

💡 For 1-over-something fractions, fewer parts means bigger pieces - so just count the denominators smaller than 9!

Variant 3 answer: 5 numbers

Among the numbers from 1 to 9, how many numbers can go in the $\bigstar$ ?

$\dfrac{1}{6} < \dfrac{1}{\bigstar}$

Show solution

Understand

I need to count how many of the digits 1 through 9 can be placed in the star so that the fraction 1/6 is less than 1/star.

Givens

The inequality is 1/6 < 1/(star).
The star must be a whole number from 1 to 9.

Unknowns

How many values of the star make the inequality true.

Constraints

Both fractions are unit fractions (numerator 1).
For unit fractions, a smaller denominator means a larger fraction.

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List

Both fractions have numerator 1, so the comparison depends only on the denominators: the unit fraction with the smaller denominator is larger. That pattern instantly tells me 1/star > 1/6 exactly when star < 6, and I can list those digits to count them.

Execute

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

Both fractions split one whole into equal parts. The more parts you cut a whole into, the smaller each part. So with numerator 1, the fraction is larger when the denominator is smaller.

\dfrac{1}{6} < \dfrac{1}{\bigstar} \iff \bigstar < 6

Sharing a pizza among fewer people gives each person a bigger slice, so smaller denominator = bigger unit fraction.

#2 Make a Systematic List 3.NF.A.3

We need star < 6, with star chosen from 1 through 9. The digits less than 6 are 1, 2, 3, 4, 5.

\bigstar \in \{1, 2, 3, 4, 5\}

Only denominators smaller than 6 make a slice bigger than 1/6; 6 itself ties and larger ones give smaller slices.

#2 Make a Systematic List 3.NF.A.3

Count the digits in the list 1, 2, 3, 4, 5.

5 \text{ numbers}

There are exactly 5 whole numbers from 1 to 5, so 5 choices work.

Answer: 5 numbers

Review

Spot-check the boundaries: 1/5 > 1/6 (true, 5 counts) and 1/6 < 1/6 is false (6 excluded). Only 1-5 qualify, giving 5 - consistent with the answer.

Guess and check (tool 6): test each star from 1 to 9 in 1/6 < 1/star; it holds for 1, 2, 3, 4, 5 and fails for the rest, again giving 5 values.

Standards · min grade 3

3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Comparing unit fractions by their denominators to determine which stars satisfy 1/6 < 1/star.

💡 For 1-over-something fractions, fewer parts means bigger pieces - so just count the denominators smaller than 6!

Variant 4 answer: 7 numbers

Among the numbers from 1 to 9, how many numbers can go in the $\bigstar$ ?

$\dfrac{1}{8} < \dfrac{1}{\bigstar}$

Show solution

Understand

I need to count how many of the digits 1 through 9 can be placed in the star so that the fraction 1/8 is less than 1/star.

Givens

The inequality is 1/8 < 1/(star).
The star must be a whole number from 1 to 9.

Unknowns

How many values of the star make the inequality true.

Constraints

Both fractions are unit fractions (numerator 1).
For unit fractions, a smaller denominator means a larger fraction.

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List

Both fractions have numerator 1, so the comparison depends only on the denominators: the unit fraction with the smaller denominator is larger. That pattern instantly tells me 1/star > 1/8 exactly when star < 8, and I can list those digits to count them.

Execute

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

Both fractions split one whole into equal parts. The more parts you cut a whole into, the smaller each part. So with numerator 1, the fraction is larger when the denominator is smaller.

\dfrac{1}{8} < \dfrac{1}{\bigstar} \iff \bigstar < 8

Sharing a pizza among fewer people gives each person a bigger slice, so smaller denominator = bigger unit fraction.

#2 Make a Systematic List 3.NF.A.3

We need star < 8, with star chosen from 1 through 9. The digits less than 8 are 1, 2, 3, 4, 5, 6, 7.

\bigstar \in \{1, 2, 3, 4, 5, 6, 7\}

Only denominators smaller than 8 make a slice bigger than 1/8; 8 itself ties and larger ones give smaller slices.

#2 Make a Systematic List 3.NF.A.3

Count the digits in the list 1, 2, 3, 4, 5, 6, 7.

7 \text{ numbers}

There are exactly 7 whole numbers from 1 to 7, so 7 choices work.

Answer: 7 numbers

Review

Spot-check the boundaries: 1/7 > 1/8 (true, 7 counts) and 1/8 < 1/8 is false (8 excluded). Only 1-7 qualify, giving 7 - consistent with the answer.

Guess and check (tool 6): test each star from 1 to 9 in 1/8 < 1/star; it holds for 1, 2, 3, 4, 5, 6, 7 and fails for the rest, again giving 7 values.

Standards · min grade 3

3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Comparing unit fractions by their denominators to determine which stars satisfy 1/8 < 1/star.

💡 For 1-over-something fractions, fewer parts means bigger pieces - so just count the denominators smaller than 8!

Variant 5 answer: 1 numbers

Among the numbers from 1 to 9, how many numbers can go in the $\bigstar$ ?

$\dfrac{1}{2} < \dfrac{1}{\bigstar}$

Show solution

Understand

I need to count how many of the digits 1 through 9 can be placed in the star so that the fraction 1/2 is less than 1/star.

Givens

The inequality is 1/2 < 1/(star).
The star must be a whole number from 1 to 9.

Unknowns

How many values of the star make the inequality true.

Constraints

Both fractions are unit fractions (numerator 1).
For unit fractions, a smaller denominator means a larger fraction.

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List

Both fractions have numerator 1, so the comparison depends only on the denominators: the unit fraction with the smaller denominator is larger. That pattern instantly tells me 1/star > 1/2 exactly when star < 2, and I can list those digits to count them.

Execute

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

Both fractions split one whole into equal parts. The more parts you cut a whole into, the smaller each part. So with numerator 1, the fraction is larger when the denominator is smaller.

\dfrac{1}{2} < \dfrac{1}{\bigstar} \iff \bigstar < 2

Sharing a pizza among fewer people gives each person a bigger slice, so smaller denominator = bigger unit fraction.

#2 Make a Systematic List 3.NF.A.3

We need star < 2, with star chosen from 1 through 9. The digits less than 2 are 1.

\bigstar \in \{1\}

Only denominators smaller than 2 make a slice bigger than 1/2; 2 itself ties and larger ones give smaller slices.

#2 Make a Systematic List 3.NF.A.3

Count the digits in the list 1.

1 \text{ numbers}

There are exactly 1 whole numbers from 1 to 1, so 1 choices work.

Answer: 1 numbers

Review

Spot-check the boundaries: 1/1 > 1/2 (true, 1 counts) and 1/2 < 1/2 is false (2 excluded). Only 1-1 qualify, giving 1 - consistent with the answer.

Guess and check (tool 6): test each star from 1 to 9 in 1/2 < 1/star; it holds for 1 and fails for the rest, again giving 1 values.

Standards · min grade 3

3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Comparing unit fractions by their denominators to determine which stars satisfy 1/2 < 1/star.

💡 For 1-over-something fractions, fewer parts means bigger pieces - so just count the denominators smaller than 2!

Variant 6 answer: 3 numbers

Among the numbers from 1 to 9, how many numbers can go in the $\bigstar$ ?

$\dfrac{1}{4} < \dfrac{1}{\bigstar}$

Show solution

Understand

I need to count how many of the digits 1 through 9 can be placed in the star so that the fraction 1/4 is less than 1/star.

Givens

The inequality is 1/4 < 1/(star).
The star must be a whole number from 1 to 9.

Unknowns

How many values of the star make the inequality true.

Constraints

Both fractions are unit fractions (numerator 1).
For unit fractions, a smaller denominator means a larger fraction.

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List

Both fractions have numerator 1, so the comparison depends only on the denominators: the unit fraction with the smaller denominator is larger. That pattern instantly tells me 1/star > 1/4 exactly when star < 4, and I can list those digits to count them.

Execute

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

Both fractions split one whole into equal parts. The more parts you cut a whole into, the smaller each part. So with numerator 1, the fraction is larger when the denominator is smaller.

\dfrac{1}{4} < \dfrac{1}{\bigstar} \iff \bigstar < 4

Sharing a pizza among fewer people gives each person a bigger slice, so smaller denominator = bigger unit fraction.

#2 Make a Systematic List 3.NF.A.3

We need star < 4, with star chosen from 1 through 9. The digits less than 4 are 1, 2, 3.

\bigstar \in \{1, 2, 3\}

Only denominators smaller than 4 make a slice bigger than 1/4; 4 itself ties and larger ones give smaller slices.

#2 Make a Systematic List 3.NF.A.3

Count the digits in the list 1, 2, 3.

3 \text{ numbers}

There are exactly 3 whole numbers from 1 to 3, so 3 choices work.

Answer: 3 numbers

Review

Spot-check the boundaries: 1/3 > 1/4 (true, 3 counts) and 1/4 < 1/4 is false (4 excluded). Only 1-3 qualify, giving 3 - consistent with the answer.

Guess and check (tool 6): test each star from 1 to 9 in 1/4 < 1/star; it holds for 1, 2, 3 and fails for the rest, again giving 3 values.

Standards · min grade 3

3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Comparing unit fractions by their denominators to determine which stars satisfy 1/4 < 1/star.

💡 For 1-over-something fractions, fewer parts means bigger pieces - so just count the denominators smaller than 4!

Variant 7 answer: 6 numbers

Among the numbers from 1 to 9, how many numbers can go in the $\bigstar$ ?

$\dfrac{1}{7} < \dfrac{1}{\bigstar}$

Show solution

Understand

I need to count how many of the digits 1 through 9 can be placed in the star so that the fraction 1/7 is less than 1/star.

Givens

The inequality is 1/7 < 1/(star).
The star must be a whole number from 1 to 9.

Unknowns

How many values of the star make the inequality true.

Constraints

Both fractions are unit fractions (numerator 1).
For unit fractions, a smaller denominator means a larger fraction.

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List

Both fractions have numerator 1, so the comparison depends only on the denominators: the unit fraction with the smaller denominator is larger. That pattern instantly tells me 1/star > 1/7 exactly when star < 7, and I can list those digits to count them.

Execute

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

Both fractions split one whole into equal parts. The more parts you cut a whole into, the smaller each part. So with numerator 1, the fraction is larger when the denominator is smaller.

\dfrac{1}{7} < \dfrac{1}{\bigstar} \iff \bigstar < 7

Sharing a pizza among fewer people gives each person a bigger slice, so smaller denominator = bigger unit fraction.

#2 Make a Systematic List 3.NF.A.3

We need star < 7, with star chosen from 1 through 9. The digits less than 7 are 1, 2, 3, 4, 5, 6.

\bigstar \in \{1, 2, 3, 4, 5, 6\}

Only denominators smaller than 7 make a slice bigger than 1/7; 7 itself ties and larger ones give smaller slices.

#2 Make a Systematic List 3.NF.A.3

Count the digits in the list 1, 2, 3, 4, 5, 6.

6 \text{ numbers}

There are exactly 6 whole numbers from 1 to 6, so 6 choices work.

Answer: 6 numbers

Review

Spot-check the boundaries: 1/6 > 1/7 (true, 6 counts) and 1/7 < 1/7 is false (7 excluded). Only 1-6 qualify, giving 6 - consistent with the answer.

Guess and check (tool 6): test each star from 1 to 9 in 1/7 < 1/star; it holds for 1, 2, 3, 4, 5, 6 and fails for the rest, again giving 6 values.

Standards · min grade 3

3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Comparing unit fractions by their denominators to determine which stars satisfy 1/7 < 1/star.

💡 For 1-over-something fractions, fewer parts means bigger pieces - so just count the denominators smaller than 7!

Variant 8 answer: 4 numbers

Among the numbers from 1 to 9, how many numbers can go in the $\bigstar$ ?

$\dfrac{1}{5} < \dfrac{1}{\bigstar}$

Show solution

Understand

I need to count how many of the digits 1 through 9 can be placed in the star so that the fraction 1/5 is less than 1/star.

Givens

The inequality is 1/5 < 1/(star).
The star must be a whole number from 1 to 9.

Unknowns

How many values of the star make the inequality true.

Constraints

Both fractions are unit fractions (numerator 1).
For unit fractions, a smaller denominator means a larger fraction.

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List

Both fractions have numerator 1, so the comparison depends only on the denominators: the unit fraction with the smaller denominator is larger. That pattern instantly tells me 1/star > 1/5 exactly when star < 5, and I can list those digits to count them.

Execute

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

Both fractions split one whole into equal parts. The more parts you cut a whole into, the smaller each part. So with numerator 1, the fraction is larger when the denominator is smaller.

\dfrac{1}{5} < \dfrac{1}{\bigstar} \iff \bigstar < 5

Sharing a pizza among fewer people gives each person a bigger slice, so smaller denominator = bigger unit fraction.

#2 Make a Systematic List 3.NF.A.3

We need star < 5, with star chosen from 1 through 9. The digits less than 5 are 1, 2, 3, 4.

\bigstar \in \{1, 2, 3, 4\}

Only denominators smaller than 5 make a slice bigger than 1/5; 5 itself ties and larger ones give smaller slices.

#2 Make a Systematic List 3.NF.A.3

Count the digits in the list 1, 2, 3, 4.

4 \text{ numbers}

There are exactly 4 whole numbers from 1 to 4, so 4 choices work.

Answer: 4 numbers

Review

Spot-check the boundaries: 1/4 > 1/5 (true, 4 counts) and 1/5 < 1/5 is false (5 excluded). Only 1-4 qualify, giving 4 - consistent with the answer.

Guess and check (tool 6): test each star from 1 to 9 in 1/5 < 1/star; it holds for 1, 2, 3, 4 and fails for the rest, again giving 4 values.

Standards · min grade 3

3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Comparing unit fractions by their denominators to determine which stars satisfy 1/5 < 1/star.

💡 For 1-over-something fractions, fewer parts means bigger pieces - so just count the denominators smaller than 5!