Practice · Apply conditions step by step, then count · Sensim Math

Variant 1 easy

How many fractions satisfy all of the following conditions?

It is equal in size to $\frac{2}{5}$ .
Its denominator is greater than 15 and less than 45.
Its numerator is greater than 5 and less than 20.

Your answer

Show solution

1 · Understandwhat's really being asked

We want fractions worth exactly 2 over 5, whose denominator lies between 15 and 45 and whose numerator lies between 5 and 20. All three conditions must hold at once.

Givens

The value must equal $\frac{2}{5}$ .
The denominator is greater than 15 and less than 45.
The numerator is greater than 5 and less than 20.

Unknowns

How many fractions meet every condition.

Constraints

All four bounds are strict, so none of the endpoints counts.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Every fraction equal to $\frac{2}{5}$ is that fraction scaled by some whole number. So the whole question is about which scale factors survive both windows -- one list, then a filter.

3 · Execute4 carry out the plan

1Write the form of every equal fraction

#2 Make a Systematic List 4.NF.A.1

Multiplying top and bottom by the same whole number keeps the value, so every candidate looks like this.

\frac{2 \times k}{5 \times k}

One unknown, k, instead of two.

2Use the denominator condition to limit k

#2 Make a Systematic List 4.OA.B.4

The denominator 5k has to sit strictly between 15 and 45.

15 < 5k < 45 \Rightarrow k \in \{4, 5, 6, 7, 8\}

5 scale factors survive the first window.

3Check the numerator condition for those k

#6 Guess and Check 4.NBT.B.5

For each surviving k the numerator 2k already lands strictly between 5 and 20, so none is lost.

k \in \{4, 5, 6, 7, 8\}

5 scale factors pass both windows.

4Count the fractions that pass everything

#2 Make a Systematic List 4.NF.A.1

Each surviving k gives exactly one fraction.

\frac{8}{20}, \frac{10}{25}, \frac{12}{30}, \frac{14}{35}, \frac{16}{40}

5 fractions in all.

Answer: 5 fractions

4 · Reviewdoes it hold up?

Check the ends of the list: the smallest is $\frac{8}{20}$ and the largest $\frac{16}{40}$ , and both sit inside every window.

Another way: Listing every denominator from 16 to 44 and testing which are multiples of 5 finds the same k values, more slowly.

Standardsmin grade 4

4.NF.A.1 Explain why a fraction is equivalent to another fraction — Writing every fraction equal to $\frac{2}{5}$ as a scaling of it.
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Finding the multiples of the denominator inside its window.
4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Checking each candidate numerator against its own window.

💡Takeaway. Every fraction with the same value is the simplest one scaled up, so conditions on both parts turn into conditions on one scale.

Variant 2 easy

How many fractions satisfy all of the following conditions?

It is equal in size to $\frac{4}{7}$ .
Its denominator is greater than 20 and less than 55.
Its numerator is greater than 10 and less than 35.

Your answer

Show solution

1 · Understandwhat's really being asked

We want fractions worth exactly 4 over 7, whose denominator lies between 20 and 55 and whose numerator lies between 10 and 35. All three conditions must hold at once.

Givens

The value must equal $\frac{4}{7}$ .
The denominator is greater than 20 and less than 55.
The numerator is greater than 10 and less than 35.

Unknowns

How many fractions meet every condition.

Constraints

All four bounds are strict, so none of the endpoints counts.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Every fraction equal to $\frac{4}{7}$ is that fraction scaled by some whole number. So the whole question is about which scale factors survive both windows -- one list, then a filter.

3 · Execute4 carry out the plan

1Write the form of every equal fraction

#2 Make a Systematic List 4.NF.A.1

Multiplying top and bottom by the same whole number keeps the value, so every candidate looks like this.

\frac{4 \times k}{7 \times k}

One unknown, k, instead of two.

2Use the denominator condition to limit k

#2 Make a Systematic List 4.OA.B.4

The denominator 7k has to sit strictly between 20 and 55.

20 < 7k < 55 \Rightarrow k \in \{3, 4, 5, 6, 7\}

5 scale factors survive the first window.

3Check the numerator condition for those k

#6 Guess and Check 4.NBT.B.5

For each surviving k the numerator 4k already lands strictly between 10 and 35, so none is lost.

k \in \{3, 4, 5, 6, 7\}

5 scale factors pass both windows.

4Count the fractions that pass everything

#2 Make a Systematic List 4.NF.A.1

Each surviving k gives exactly one fraction.

\frac{12}{21}, \frac{16}{28}, \frac{20}{35}, \frac{24}{42}, \frac{28}{49}

5 fractions in all.

Answer: 5 fractions

4 · Reviewdoes it hold up?

Check the ends of the list: the smallest is $\frac{12}{21}$ and the largest $\frac{28}{49}$ , and both sit inside every window.

Another way: Listing every denominator from 21 to 54 and testing which are multiples of 7 finds the same k values, more slowly.

Standardsmin grade 4

4.NF.A.1 Explain why a fraction is equivalent to another fraction — Writing every fraction equal to $\frac{4}{7}$ as a scaling of it.
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Finding the multiples of the denominator inside its window.
4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Checking each candidate numerator against its own window.

💡Takeaway. Every fraction with the same value is the simplest one scaled up, so conditions on both parts turn into conditions on one scale.

Variant 3 easy

How many fractions satisfy all of the following conditions?

It is equal in size to $\frac{3}{7}$ .
Its denominator is greater than 20 and less than 60.
Its numerator is greater than 8 and less than 30.

Your answer

Show solution

1 · Understandwhat's really being asked

We want fractions worth exactly 3 over 7, whose denominator lies between 20 and 60 and whose numerator lies between 8 and 30. All three conditions must hold at once.

Givens

The value must equal $\frac{3}{7}$ .
The denominator is greater than 20 and less than 60.
The numerator is greater than 8 and less than 30.

Unknowns

How many fractions meet every condition.

Constraints

All four bounds are strict, so none of the endpoints counts.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Every fraction equal to $\frac{3}{7}$ is that fraction scaled by some whole number. So the whole question is about which scale factors survive both windows -- one list, then a filter.

3 · Execute4 carry out the plan

1Write the form of every equal fraction

#2 Make a Systematic List 4.NF.A.1

Multiplying top and bottom by the same whole number keeps the value, so every candidate looks like this.

\frac{3 \times k}{7 \times k}

One unknown, k, instead of two.

2Use the denominator condition to limit k

#2 Make a Systematic List 4.OA.B.4

The denominator 7k has to sit strictly between 20 and 60.

20 < 7k < 60 \Rightarrow k \in \{3, 4, 5, 6, 7, 8\}

6 scale factors survive the first window.

3Check the numerator condition for those k

#6 Guess and Check 4.NBT.B.5

For each surviving k the numerator 3k already lands strictly between 8 and 30, so none is lost.

k \in \{3, 4, 5, 6, 7, 8\}

6 scale factors pass both windows.

4Count the fractions that pass everything

#2 Make a Systematic List 4.NF.A.1

Each surviving k gives exactly one fraction.

\frac{9}{21}, \frac{12}{28}, \frac{15}{35}, \frac{18}{42}, \frac{21}{49}, \frac{24}{56}

6 fractions in all.

Answer: 6 fractions

4 · Reviewdoes it hold up?

Check the ends of the list: the smallest is $\frac{9}{21}$ and the largest $\frac{24}{56}$ , and both sit inside every window.

Another way: Listing every denominator from 21 to 59 and testing which are multiples of 7 finds the same k values, more slowly.

Standardsmin grade 4

4.NF.A.1 Explain why a fraction is equivalent to another fraction — Writing every fraction equal to $\frac{3}{7}$ as a scaling of it.
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Finding the multiples of the denominator inside its window.
4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Checking each candidate numerator against its own window.

💡Takeaway. Every fraction with the same value is the simplest one scaled up, so conditions on both parts turn into conditions on one scale.

Variant 4 easy

How many fractions satisfy all of the following conditions?

It is equal in size to $\frac{4}{9}$ .
Its denominator is greater than 25 and less than 70.
Its numerator is greater than 10 and less than 35.

Your answer

Show solution

1 · Understandwhat's really being asked

We want fractions worth exactly 4 over 9, whose denominator lies between 25 and 70 and whose numerator lies between 10 and 35. All three conditions must hold at once.

Givens

The value must equal $\frac{4}{9}$ .
The denominator is greater than 25 and less than 70.
The numerator is greater than 10 and less than 35.

Unknowns

How many fractions meet every condition.

Constraints

All four bounds are strict, so none of the endpoints counts.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Every fraction equal to $\frac{4}{9}$ is that fraction scaled by some whole number. So the whole question is about which scale factors survive both windows -- one list, then a filter.

3 · Execute4 carry out the plan

1Write the form of every equal fraction

#2 Make a Systematic List 4.NF.A.1

Multiplying top and bottom by the same whole number keeps the value, so every candidate looks like this.

\frac{4 \times k}{9 \times k}

One unknown, k, instead of two.

2Use the denominator condition to limit k

#2 Make a Systematic List 4.OA.B.4

The denominator 9k has to sit strictly between 25 and 70.

25 < 9k < 70 \Rightarrow k \in \{3, 4, 5, 6, 7\}

5 scale factors survive the first window.

3Check the numerator condition for those k

#6 Guess and Check 4.NBT.B.5

For each surviving k the numerator 4k already lands strictly between 10 and 35, so none is lost.

k \in \{3, 4, 5, 6, 7\}

5 scale factors pass both windows.

4Count the fractions that pass everything

#2 Make a Systematic List 4.NF.A.1

Each surviving k gives exactly one fraction.

\frac{12}{27}, \frac{16}{36}, \frac{20}{45}, \frac{24}{54}, \frac{28}{63}

5 fractions in all.

Answer: 5 fractions

4 · Reviewdoes it hold up?

Check the ends of the list: the smallest is $\frac{12}{27}$ and the largest $\frac{28}{63}$ , and both sit inside every window.

Another way: Listing every denominator from 26 to 69 and testing which are multiples of 9 finds the same k values, more slowly.

Standardsmin grade 4

4.NF.A.1 Explain why a fraction is equivalent to another fraction — Writing every fraction equal to $\frac{4}{9}$ as a scaling of it.
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Finding the multiples of the denominator inside its window.
4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Checking each candidate numerator against its own window.

💡Takeaway. Every fraction with the same value is the simplest one scaled up, so conditions on both parts turn into conditions on one scale.

Variant 5 medium

How many fractions satisfy all of the following conditions?

It is equal in size to $\frac{5}{8}$ .
Its denominator is greater than 30 and less than 70.
Its numerator is greater than 18 and less than 50.

Your answer

Show solution

1 · Understandwhat's really being asked

We want fractions worth exactly 5 over 8, whose denominator lies between 30 and 70 and whose numerator lies between 18 and 50. All three conditions must hold at once.

Givens

The value must equal $\frac{5}{8}$ .
The denominator is greater than 30 and less than 70.
The numerator is greater than 18 and less than 50.

Unknowns

How many fractions meet every condition.

Constraints

All four bounds are strict, so none of the endpoints counts.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Every fraction equal to $\frac{5}{8}$ is that fraction scaled by some whole number. So the whole question is about which scale factors survive both windows -- one list, then a filter.

3 · Execute4 carry out the plan

1Write the form of every equal fraction

#2 Make a Systematic List 4.NF.A.1

Multiplying top and bottom by the same whole number keeps the value, so every candidate looks like this.

\frac{5 \times k}{8 \times k}

One unknown, k, instead of two.

2Use the denominator condition to limit k

#2 Make a Systematic List 4.OA.B.4

The denominator 8k has to sit strictly between 30 and 70.

30 < 8k < 70 \Rightarrow k \in \{4, 5, 6, 7, 8\}

5 scale factors survive the first window.

3Check the numerator condition for those k

#6 Guess and Check 4.NBT.B.5

For each surviving k the numerator 5k already lands strictly between 18 and 50, so none is lost.

k \in \{4, 5, 6, 7, 8\}

5 scale factors pass both windows.

4Count the fractions that pass everything

#2 Make a Systematic List 4.NF.A.1

Each surviving k gives exactly one fraction.

\frac{20}{32}, \frac{25}{40}, \frac{30}{48}, \frac{35}{56}, \frac{40}{64}

5 fractions in all.

Answer: 5 fractions

4 · Reviewdoes it hold up?

Check the ends of the list: the smallest is $\frac{20}{32}$ and the largest $\frac{40}{64}$ , and both sit inside every window.

Another way: Listing every denominator from 31 to 69 and testing which are multiples of 8 finds the same k values, more slowly.

Standardsmin grade 4

4.NF.A.1 Explain why a fraction is equivalent to another fraction — Writing every fraction equal to $\frac{5}{8}$ as a scaling of it.
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Finding the multiples of the denominator inside its window.
4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Checking each candidate numerator against its own window.

💡Takeaway. Every fraction with the same value is the simplest one scaled up, so conditions on both parts turn into conditions on one scale.

Variant 6 medium

How many fractions satisfy all of the following conditions?

It is equal in size to $\frac{2}{9}$ .
Its denominator is greater than 30 and less than 80.
Its numerator is greater than 6 and less than 20.

Your answer

Show solution

1 · Understandwhat's really being asked

We want fractions worth exactly 2 over 9, whose denominator lies between 30 and 80 and whose numerator lies between 6 and 20. All three conditions must hold at once.

Givens

The value must equal $\frac{2}{9}$ .
The denominator is greater than 30 and less than 80.
The numerator is greater than 6 and less than 20.

Unknowns

How many fractions meet every condition.

Constraints

All four bounds are strict, so none of the endpoints counts.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Every fraction equal to $\frac{2}{9}$ is that fraction scaled by some whole number. So the whole question is about which scale factors survive both windows -- one list, then a filter.

3 · Execute4 carry out the plan

1Write the form of every equal fraction

#2 Make a Systematic List 4.NF.A.1

Multiplying top and bottom by the same whole number keeps the value, so every candidate looks like this.

\frac{2 \times k}{9 \times k}

One unknown, k, instead of two.

2Use the denominator condition to limit k

#2 Make a Systematic List 4.OA.B.4

The denominator 9k has to sit strictly between 30 and 80.

30 < 9k < 80 \Rightarrow k \in \{4, 5, 6, 7, 8\}

5 scale factors survive the first window.

3Check the numerator condition for those k

#6 Guess and Check 4.NBT.B.5

For each surviving k the numerator 2k already lands strictly between 6 and 20, so none is lost.

k \in \{4, 5, 6, 7, 8\}

5 scale factors pass both windows.

4Count the fractions that pass everything

#2 Make a Systematic List 4.NF.A.1

Each surviving k gives exactly one fraction.

\frac{8}{36}, \frac{10}{45}, \frac{12}{54}, \frac{14}{63}, \frac{16}{72}

5 fractions in all.

Answer: 5 fractions

4 · Reviewdoes it hold up?

Check the ends of the list: the smallest is $\frac{8}{36}$ and the largest $\frac{16}{72}$ , and both sit inside every window.

Another way: Listing every denominator from 31 to 79 and testing which are multiples of 9 finds the same k values, more slowly.

Standardsmin grade 4

4.NF.A.1 Explain why a fraction is equivalent to another fraction — Writing every fraction equal to $\frac{2}{9}$ as a scaling of it.
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Finding the multiples of the denominator inside its window.
4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Checking each candidate numerator against its own window.

💡Takeaway. Every fraction with the same value is the simplest one scaled up, so conditions on both parts turn into conditions on one scale.

Variant 7 medium

How many fractions satisfy all of the following conditions?

It is equal in size to $\frac{5}{11}$ .
Its denominator is greater than 40 and less than 90.
Its numerator is greater than 15 and less than 45.

Your answer

Show solution

1 · Understandwhat's really being asked

We want fractions worth exactly 5 over 11, whose denominator lies between 40 and 90 and whose numerator lies between 15 and 45. All three conditions must hold at once.

Givens

The value must equal $\frac{5}{11}$ .
The denominator is greater than 40 and less than 90.
The numerator is greater than 15 and less than 45.

Unknowns

How many fractions meet every condition.

Constraints

All four bounds are strict, so none of the endpoints counts.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Every fraction equal to $\frac{5}{11}$ is that fraction scaled by some whole number. So the whole question is about which scale factors survive both windows -- one list, then a filter.

3 · Execute4 carry out the plan

1Write the form of every equal fraction

#2 Make a Systematic List 4.NF.A.1

Multiplying top and bottom by the same whole number keeps the value, so every candidate looks like this.

\frac{5 \times k}{11 \times k}

One unknown, k, instead of two.

2Use the denominator condition to limit k

#2 Make a Systematic List 4.OA.B.4

The denominator 11k has to sit strictly between 40 and 90.

40 < 11k < 90 \Rightarrow k \in \{4, 5, 6, 7, 8\}

5 scale factors survive the first window.

3Check the numerator condition for those k

#6 Guess and Check 4.NBT.B.5

For each surviving k the numerator 5k already lands strictly between 15 and 45, so none is lost.

k \in \{4, 5, 6, 7, 8\}

5 scale factors pass both windows.

4Count the fractions that pass everything

#2 Make a Systematic List 4.NF.A.1

Each surviving k gives exactly one fraction.

\frac{20}{44}, \frac{25}{55}, \frac{30}{66}, \frac{35}{77}, \frac{40}{88}

5 fractions in all.

Answer: 5 fractions

4 · Reviewdoes it hold up?

Check the ends of the list: the smallest is $\frac{20}{44}$ and the largest $\frac{40}{88}$ , and both sit inside every window.

Another way: Listing every denominator from 41 to 89 and testing which are multiples of 11 finds the same k values, more slowly.

Standardsmin grade 4

4.NF.A.1 Explain why a fraction is equivalent to another fraction — Writing every fraction equal to $\frac{5}{11}$ as a scaling of it.
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Finding the multiples of the denominator inside its window.
4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Checking each candidate numerator against its own window.

💡Takeaway. Every fraction with the same value is the simplest one scaled up, so conditions on both parts turn into conditions on one scale.

Variant 8 hard

How many fractions satisfy all of the following conditions?

It is equal in size to $\frac{3}{11}$ .
Its denominator is greater than 40 and less than 100.
Its numerator is greater than 10 and less than 30.

Your answer

Show solution

1 · Understandwhat's really being asked

We want fractions worth exactly 3 over 11, whose denominator lies between 40 and 100 and whose numerator lies between 10 and 30. All three conditions must hold at once.

Givens

The value must equal $\frac{3}{11}$ .
The denominator is greater than 40 and less than 100.
The numerator is greater than 10 and less than 30.

Unknowns

How many fractions meet every condition.

Constraints

All four bounds are strict, so none of the endpoints counts.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Every fraction equal to $\frac{3}{11}$ is that fraction scaled by some whole number. So the whole question is about which scale factors survive both windows -- one list, then a filter.

3 · Execute4 carry out the plan

1Write the form of every equal fraction

#2 Make a Systematic List 4.NF.A.1

Multiplying top and bottom by the same whole number keeps the value, so every candidate looks like this.

\frac{3 \times k}{11 \times k}

One unknown, k, instead of two.

2Use the denominator condition to limit k

#2 Make a Systematic List 4.OA.B.4

The denominator 11k has to sit strictly between 40 and 100.

40 < 11k < 100 \Rightarrow k \in \{4, 5, 6, 7, 8, 9\}

6 scale factors survive the first window.

3Check the numerator condition for those k

#6 Guess and Check 4.NBT.B.5

For each surviving k the numerator 3k already lands strictly between 10 and 30, so none is lost.

k \in \{4, 5, 6, 7, 8, 9\}

6 scale factors pass both windows.

4Count the fractions that pass everything

#2 Make a Systematic List 4.NF.A.1

Each surviving k gives exactly one fraction.

\frac{12}{44}, \frac{15}{55}, \frac{18}{66}, \frac{21}{77}, \frac{24}{88}, \frac{27}{99}

6 fractions in all.

Answer: 6 fractions

4 · Reviewdoes it hold up?

Check the ends of the list: the smallest is $\frac{12}{44}$ and the largest $\frac{27}{99}$ , and both sit inside every window.

Another way: Listing every denominator from 41 to 99 and testing which are multiples of 11 finds the same k values, more slowly.

Standardsmin grade 4

4.NF.A.1 Explain why a fraction is equivalent to another fraction — Writing every fraction equal to $\frac{3}{11}$ as a scaling of it.
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Finding the multiples of the denominator inside its window.
4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Checking each candidate numerator against its own window.

💡Takeaway. Every fraction with the same value is the simplest one scaled up, so conditions on both parts turn into conditions on one scale.

Variant 9 medium

How many fractions satisfy all of the following conditions?

It is equal in size to $\frac{6}{13}$ .
Its denominator is greater than 50 and less than 100.
Its numerator is greater than 20 and less than 80.

Your answer

Show solution

1 · Understandwhat's really being asked

We want fractions worth exactly 6 over 13, whose denominator lies between 50 and 100 and whose numerator lies between 20 and 80. All three conditions must hold at once.

Givens

The value must equal $\frac{6}{13}$ .
The denominator is greater than 50 and less than 100.
The numerator is greater than 20 and less than 80.

Unknowns

How many fractions meet every condition.

Constraints

All four bounds are strict, so none of the endpoints counts.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Every fraction equal to $\frac{6}{13}$ is that fraction scaled by some whole number. So the whole question is about which scale factors survive both windows -- one list, then a filter.

3 · Execute4 carry out the plan

1Write the form of every equal fraction

#2 Make a Systematic List 4.NF.A.1

Multiplying top and bottom by the same whole number keeps the value, so every candidate looks like this.

\frac{6 \times k}{13 \times k}

One unknown, k, instead of two.

2Use the denominator condition to limit k

#2 Make a Systematic List 4.OA.B.4

The denominator 13k has to sit strictly between 50 and 100.

50 < 13k < 100 \Rightarrow k \in \{4, 5, 6, 7\}

4 scale factors survive the first window.

3Check the numerator condition for those k

#6 Guess and Check 4.NBT.B.5

For each surviving k the numerator 6k already lands strictly between 20 and 80, so none is lost.

k \in \{4, 5, 6, 7\}

4 scale factors pass both windows.

4Count the fractions that pass everything

#2 Make a Systematic List 4.NF.A.1

Each surviving k gives exactly one fraction.

\frac{24}{52}, \frac{30}{65}, \frac{36}{78}, \frac{42}{91}

4 fractions in all.

Answer: 4 fractions

4 · Reviewdoes it hold up?

Check the ends of the list: the smallest is $\frac{24}{52}$ and the largest $\frac{42}{91}$ , and both sit inside every window.

Another way: Listing every denominator from 51 to 99 and testing which are multiples of 13 finds the same k values, more slowly.

Standardsmin grade 4

4.NF.A.1 Explain why a fraction is equivalent to another fraction — Writing every fraction equal to $\frac{6}{13}$ as a scaling of it.
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Finding the multiples of the denominator inside its window.
4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Checking each candidate numerator against its own window.

💡Takeaway. Every fraction with the same value is the simplest one scaled up, so conditions on both parts turn into conditions on one scale.

Variant 10 hard

How many fractions satisfy all of the following conditions?

It is equal in size to $\frac{9}{14}$ .
Its denominator is greater than 40 and less than 100.
Its numerator is greater than 25 and less than 70.

Your answer

Show solution

1 · Understandwhat's really being asked

We want fractions worth exactly 9 over 14, whose denominator lies between 40 and 100 and whose numerator lies between 25 and 70. All three conditions must hold at once.

Givens

The value must equal $\frac{9}{14}$ .
The denominator is greater than 40 and less than 100.
The numerator is greater than 25 and less than 70.

Unknowns

How many fractions meet every condition.

Constraints

All four bounds are strict, so none of the endpoints counts.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Every fraction equal to $\frac{9}{14}$ is that fraction scaled by some whole number. So the whole question is about which scale factors survive both windows -- one list, then a filter.

3 · Execute4 carry out the plan

1Write the form of every equal fraction

#2 Make a Systematic List 4.NF.A.1

Multiplying top and bottom by the same whole number keeps the value, so every candidate looks like this.

\frac{9 \times k}{14 \times k}

One unknown, k, instead of two.

2Use the denominator condition to limit k

#2 Make a Systematic List 4.OA.B.4

The denominator 14k has to sit strictly between 40 and 100.

40 < 14k < 100 \Rightarrow k \in \{3, 4, 5, 6, 7\}

5 scale factors survive the first window.

3Check the numerator condition for those k

#6 Guess and Check 4.NBT.B.5

For each surviving k the numerator 9k already lands strictly between 25 and 70, so none is lost.

k \in \{3, 4, 5, 6, 7\}

5 scale factors pass both windows.

4Count the fractions that pass everything

#2 Make a Systematic List 4.NF.A.1

Each surviving k gives exactly one fraction.

\frac{27}{42}, \frac{36}{56}, \frac{45}{70}, \frac{54}{84}, \frac{63}{98}

5 fractions in all.

Answer: 5 fractions

4 · Reviewdoes it hold up?

Check the ends of the list: the smallest is $\frac{27}{42}$ and the largest $\frac{63}{98}$ , and both sit inside every window.

Another way: Listing every denominator from 41 to 99 and testing which are multiples of 14 finds the same k values, more slowly.

Standardsmin grade 4

4.NF.A.1 Explain why a fraction is equivalent to another fraction — Writing every fraction equal to $\frac{9}{14}$ as a scaling of it.
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Finding the multiples of the denominator inside its window.
4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Checking each candidate numerator against its own window.

💡Takeaway. Every fraction with the same value is the simplest one scaled up, so conditions on both parts turn into conditions on one scale.

Variant 11 hard

How many fractions satisfy all of the following conditions?

It is equal in size to $\frac{7}{12}$ .
Its denominator is greater than 50 and less than 110.
Its numerator is greater than 25 and less than 70.

Your answer

Show solution

1 · Understandwhat's really being asked

We want fractions worth exactly 7 over 12, whose denominator lies between 50 and 110 and whose numerator lies between 25 and 70. All three conditions must hold at once.

Givens

The value must equal $\frac{7}{12}$ .
The denominator is greater than 50 and less than 110.
The numerator is greater than 25 and less than 70.

Unknowns

How many fractions meet every condition.

Constraints

All four bounds are strict, so none of the endpoints counts.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Every fraction equal to $\frac{7}{12}$ is that fraction scaled by some whole number. So the whole question is about which scale factors survive both windows -- one list, then a filter.

3 · Execute4 carry out the plan

1Write the form of every equal fraction

#2 Make a Systematic List 4.NF.A.1

Multiplying top and bottom by the same whole number keeps the value, so every candidate looks like this.

\frac{7 \times k}{12 \times k}

One unknown, k, instead of two.

2Use the denominator condition to limit k

#2 Make a Systematic List 4.OA.B.4

The denominator 12k has to sit strictly between 50 and 110.

50 < 12k < 110 \Rightarrow k \in \{5, 6, 7, 8, 9\}

5 scale factors survive the first window.

3Check the numerator condition for those k

#6 Guess and Check 4.NBT.B.5

For each surviving k the numerator 7k already lands strictly between 25 and 70, so none is lost.

k \in \{5, 6, 7, 8, 9\}

5 scale factors pass both windows.

4Count the fractions that pass everything

#2 Make a Systematic List 4.NF.A.1

Each surviving k gives exactly one fraction.

\frac{35}{60}, \frac{42}{72}, \frac{49}{84}, \frac{56}{96}, \frac{63}{108}

5 fractions in all.

Answer: 5 fractions

4 · Reviewdoes it hold up?

Check the ends of the list: the smallest is $\frac{35}{60}$ and the largest $\frac{63}{108}$ , and both sit inside every window.

Another way: Listing every denominator from 51 to 109 and testing which are multiples of 12 finds the same k values, more slowly.

Standardsmin grade 4

4.NF.A.1 Explain why a fraction is equivalent to another fraction — Writing every fraction equal to $\frac{7}{12}$ as a scaling of it.
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Finding the multiples of the denominator inside its window.
4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Checking each candidate numerator against its own window.

💡Takeaway. Every fraction with the same value is the simplest one scaled up, so conditions on both parts turn into conditions on one scale.

Variant 12 hard

How many fractions satisfy all of the following conditions?

It is equal in size to $\frac{7}{15}$ .
Its denominator is greater than 45 and less than 120.
Its numerator is greater than 20 and less than 60.

Your answer

Show solution

1 · Understandwhat's really being asked

We want fractions worth exactly 7 over 15, whose denominator lies between 45 and 120 and whose numerator lies between 20 and 60. All three conditions must hold at once.

Givens

The value must equal $\frac{7}{15}$ .
The denominator is greater than 45 and less than 120.
The numerator is greater than 20 and less than 60.

Unknowns

How many fractions meet every condition.

Constraints

All four bounds are strict, so none of the endpoints counts.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Every fraction equal to $\frac{7}{15}$ is that fraction scaled by some whole number. So the whole question is about which scale factors survive both windows -- one list, then a filter.

3 · Execute4 carry out the plan

1Write the form of every equal fraction

#2 Make a Systematic List 4.NF.A.1

Multiplying top and bottom by the same whole number keeps the value, so every candidate looks like this.

\frac{7 \times k}{15 \times k}

One unknown, k, instead of two.

2Use the denominator condition to limit k

#2 Make a Systematic List 4.OA.B.4

The denominator 15k has to sit strictly between 45 and 120.

45 < 15k < 120 \Rightarrow k \in \{4, 5, 6, 7\}

4 scale factors survive the first window.

3Check the numerator condition for those k

#6 Guess and Check 4.NBT.B.5

For each surviving k the numerator 7k already lands strictly between 20 and 60, so none is lost.

k \in \{4, 5, 6, 7\}

4 scale factors pass both windows.

4Count the fractions that pass everything

#2 Make a Systematic List 4.NF.A.1

Each surviving k gives exactly one fraction.

\frac{28}{60}, \frac{35}{75}, \frac{42}{90}, \frac{49}{105}

4 fractions in all.

Answer: 4 fractions

4 · Reviewdoes it hold up?

Check the ends of the list: the smallest is $\frac{28}{60}$ and the largest $\frac{49}{105}$ , and both sit inside every window.

Another way: Listing every denominator from 46 to 119 and testing which are multiples of 15 finds the same k values, more slowly.

Standardsmin grade 4

4.NF.A.1 Explain why a fraction is equivalent to another fraction — Writing every fraction equal to $\frac{7}{15}$ as a scaling of it.
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Finding the multiples of the denominator inside its window.
4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Checking each candidate numerator against its own window.

💡Takeaway. Every fraction with the same value is the simplest one scaled up, so conditions on both parts turn into conditions on one scale.

Apply conditions step by step, then count · 12 practice problems

Generated variants — 12

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