← Apply conditions step by step, then count · Equivalent Fractions by Scaling

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

4.NF.A.14.OA.B.44.NBT.B.5

Generated variants — 12

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

Variant 1 easy answer: 5 fractions

How many fractions satisfy all of the following conditions?

  • It is equal in size to 25\frac{2}{5}.
  • Its denominator is greater than 15 and less than 45.
  • Its numerator is greater than 5 and less than 20.
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 25\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 25\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.
2×k5×k\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⇒k∈{4,5,6,7,8}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∈{4,5,6,7,8}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.
820,1025,1230,1435,1640\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 820\frac{8}{20} and the largest 1640\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 answer: 5 fractions

How many fractions satisfy all of the following conditions?

  • It is equal in size to 47\frac{4}{7}.
  • Its denominator is greater than 20 and less than 55.
  • Its numerator is greater than 10 and less than 35.
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 47\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 47\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.
4×k7×k\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⇒k∈{3,4,5,6,7}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∈{3,4,5,6,7}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.
1221,1628,2035,2442,2849\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 1221\frac{12}{21} and the largest 2849\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 answer: 6 fractions

How many fractions satisfy all of the following conditions?

  • It is equal in size to 37\frac{3}{7}.
  • Its denominator is greater than 20 and less than 60.
  • Its numerator is greater than 8 and less than 30.
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 37\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 37\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.
3×k7×k\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⇒k∈{3,4,5,6,7,8}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∈{3,4,5,6,7,8}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.
921,1228,1535,1842,2149,2456\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 921\frac{9}{21} and the largest 2456\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 answer: 5 fractions

How many fractions satisfy all of the following conditions?

  • It is equal in size to 49\frac{4}{9}.
  • Its denominator is greater than 25 and less than 70.
  • Its numerator is greater than 10 and less than 35.
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 49\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 49\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.
4×k9×k\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⇒k∈{3,4,5,6,7}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∈{3,4,5,6,7}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.
1227,1636,2045,2454,2863\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 1227\frac{12}{27} and the largest 2863\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 answer: 5 fractions

How many fractions satisfy all of the following conditions?

  • It is equal in size to 58\frac{5}{8}.
  • Its denominator is greater than 30 and less than 70.
  • Its numerator is greater than 18 and less than 50.
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 58\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 58\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.
5×k8×k\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⇒k∈{4,5,6,7,8}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∈{4,5,6,7,8}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.
2032,2540,3048,3556,4064\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 2032\frac{20}{32} and the largest 4064\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 answer: 5 fractions

How many fractions satisfy all of the following conditions?

  • It is equal in size to 29\frac{2}{9}.
  • Its denominator is greater than 30 and less than 80.
  • Its numerator is greater than 6 and less than 20.
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 29\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 29\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.
2×k9×k\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⇒k∈{4,5,6,7,8}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∈{4,5,6,7,8}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.
836,1045,1254,1463,1672\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 836\frac{8}{36} and the largest 1672\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 answer: 5 fractions

How many fractions satisfy all of the following conditions?

  • It is equal in size to 511\frac{5}{11}.
  • Its denominator is greater than 40 and less than 90.
  • Its numerator is greater than 15 and less than 45.
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 511\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 511\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.
5×k11×k\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⇒k∈{4,5,6,7,8}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∈{4,5,6,7,8}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.
2044,2555,3066,3577,4088\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 2044\frac{20}{44} and the largest 4088\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 answer: 6 fractions

How many fractions satisfy all of the following conditions?

  • It is equal in size to 311\frac{3}{11}.
  • Its denominator is greater than 40 and less than 100.
  • Its numerator is greater than 10 and less than 30.
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 311\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 311\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.
3×k11×k\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⇒k∈{4,5,6,7,8,9}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∈{4,5,6,7,8,9}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.
1244,1555,1866,2177,2488,2799\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 1244\frac{12}{44} and the largest 2799\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 answer: 4 fractions

How many fractions satisfy all of the following conditions?

  • It is equal in size to 613\frac{6}{13}.
  • Its denominator is greater than 50 and less than 100.
  • Its numerator is greater than 20 and less than 80.
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 613\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 613\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.
6×k13×k\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⇒k∈{4,5,6,7}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∈{4,5,6,7}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.
2452,3065,3678,4291\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 2452\frac{24}{52} and the largest 4291\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 answer: 5 fractions

How many fractions satisfy all of the following conditions?

  • It is equal in size to 914\frac{9}{14}.
  • Its denominator is greater than 40 and less than 100.
  • Its numerator is greater than 25 and less than 70.
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 914\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 914\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.
9×k14×k\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⇒k∈{3,4,5,6,7}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∈{3,4,5,6,7}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.
2742,3656,4570,5484,6398\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 2742\frac{27}{42} and the largest 6398\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 answer: 5 fractions

How many fractions satisfy all of the following conditions?

  • It is equal in size to 712\frac{7}{12}.
  • Its denominator is greater than 50 and less than 110.
  • Its numerator is greater than 25 and less than 70.
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 712\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 712\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.
7×k12×k\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⇒k∈{5,6,7,8,9}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∈{5,6,7,8,9}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.
3560,4272,4984,5696,63108\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 3560\frac{35}{60} and the largest 63108\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 answer: 4 fractions

How many fractions satisfy all of the following conditions?

  • It is equal in size to 715\frac{7}{15}.
  • Its denominator is greater than 45 and less than 120.
  • Its numerator is greater than 20 and less than 60.
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 715\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 715\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.
7×k15×k\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⇒k∈{4,5,6,7}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∈{4,5,6,7}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.
2860,3575,4290,49105\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 2860\frac{28}{60} and the largest 49105\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.