Practice · Scaling numerator and denominator keeps value · Sensim Math

Variant 1 easy

Find the fraction whose denominator minus its numerator is 15 and which, when written in lowest terms, equals $\dfrac{4}{7}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A fraction reduces to 4 over 7. In that unreduced fraction the denominator is 15 more than the numerator. We need the fraction itself.

Givens

In lowest terms the fraction is $\frac{4}{7}$ .
Denominator minus numerator is 15.

Unknowns

The fraction before it was reduced.

Constraints

Reducing means both parts were divided by the same number, so the fraction we want is $\frac{4}{7}$ scaled up.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

Write out the fractions equal to $\frac{4}{7}$ and watch what the gap between bottom and top does. It grows in steps, and finding which step reaches 15 pins the fraction down.

3 · Execute4 carry out the plan

1List equivalent fractions

#5 Look for a Pattern 4.NF.A.1

Multiplying top and bottom by the same number does not change a fraction's value, so these are all equal to

\frac{4}{7}

.

\frac{4}{7} = \frac{8}{14} = \frac{12}{21} = \cdots

The list runs

\frac{4}{7}

,

\frac{8}{14}

,

\frac{12}{21}

,

\frac{16}{28}

, and so on.

2Track the gap between bottom and top

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

Subtract top from bottom in each one: the gaps are 3, 6, 9, 12. They go up by 3 every time.

7 - 4 = 3,\quad 14 - 8 = 6,\quad 21 - 12 = 9

Scaling by a number scales the gap by that same number.

3Find the multiplier that gives a gap of 15

#6 Guess and Check 4.NF.A.1

We need 3 multiplied by something to reach 15, so divide.

15 \div 3 = 5

The fraction is

\frac{4}{7}

scaled by 5.

4Build the fraction

#6 Guess and Check 4.NF.A.1

Multiply both the 4 and the 7 by 5.

\frac{4 \times 5}{7 \times 5} = \frac{20}{35}

The fraction is

\frac{20}{35}

.

Answer:

\frac{20}{35}

4 · Reviewdoes it hold up?

Check both conditions: 35 - 20 = 15, and dividing both by 5 gives back $\frac{4}{7}$ .

Another way: Since the gap is 3 for every 1 of scale, 15 of gap means 5 lots of $\frac{4}{7}$ 's parts: 4 x 5 over 7 x 5.

Standardsmin grade 4

4.NF.A.1 Explain why a fraction is equivalent to another fraction and generate equivalents — Generating fractions equivalent to $\frac{4}{7}$ and scaling by 5 to build the answer.
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning about size — Reasoning that the gap between the parts scales with them.

💡Takeaway. Scaling a fraction scales the gap between its two numbers by exactly the same amount, so the gap tells you the scale.

Variant 2 easy

Find the fraction whose denominator minus its numerator is 18 and which, when written in lowest terms, equals $\dfrac{2}{5}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A fraction reduces to 2 over 5. In that unreduced fraction the denominator is 18 more than the numerator. We need the fraction itself.

Givens

In lowest terms the fraction is $\frac{2}{5}$ .
Denominator minus numerator is 18.

Unknowns

The fraction before it was reduced.

Constraints

Reducing means both parts were divided by the same number, so the fraction we want is $\frac{2}{5}$ scaled up.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

Write out the fractions equal to $\frac{2}{5}$ and watch what the gap between bottom and top does. It grows in steps, and finding which step reaches 18 pins the fraction down.

3 · Execute4 carry out the plan

1List equivalent fractions

#5 Look for a Pattern 4.NF.A.1

Multiplying top and bottom by the same number does not change a fraction's value, so these are all equal to

\frac{2}{5}

.

\frac{2}{5} = \frac{4}{10} = \frac{6}{15} = \cdots

The list runs

\frac{2}{5}

,

\frac{4}{10}

,

\frac{6}{15}

,

\frac{8}{20}

, and so on.

2Track the gap between bottom and top

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

Subtract top from bottom in each one: the gaps are 3, 6, 9, 12. They go up by 3 every time.

5 - 2 = 3,\quad 10 - 4 = 6,\quad 15 - 6 = 9

Scaling by a number scales the gap by that same number.

3Find the multiplier that gives a gap of 18

#6 Guess and Check 4.NF.A.1

We need 3 multiplied by something to reach 18, so divide.

18 \div 3 = 6

The fraction is

\frac{2}{5}

scaled by 6.

4Build the fraction

#6 Guess and Check 4.NF.A.1

Multiply both the 2 and the 5 by 6.

\frac{2 \times 6}{5 \times 6} = \frac{12}{30}

The fraction is

\frac{12}{30}

.

Answer:

\frac{12}{30}

4 · Reviewdoes it hold up?

Check both conditions: 30 - 12 = 18, and dividing both by 6 gives back $\frac{2}{5}$ .

Another way: Since the gap is 3 for every 1 of scale, 18 of gap means 6 lots of $\frac{2}{5}$ 's parts: 2 x 6 over 5 x 6.

Standardsmin grade 4

4.NF.A.1 Explain why a fraction is equivalent to another fraction and generate equivalents — Generating fractions equivalent to $\frac{2}{5}$ and scaling by 6 to build the answer.
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning about size — Reasoning that the gap between the parts scales with them.

💡Takeaway. Scaling a fraction scales the gap between its two numbers by exactly the same amount, so the gap tells you the scale.

Variant 3 easy

Find the fraction whose denominator minus its numerator is 20 and which, when written in lowest terms, equals $\dfrac{3}{8}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A fraction reduces to 3 over 8. In that unreduced fraction the denominator is 20 more than the numerator. We need the fraction itself.

Givens

In lowest terms the fraction is $\frac{3}{8}$ .
Denominator minus numerator is 20.

Unknowns

The fraction before it was reduced.

Constraints

Reducing means both parts were divided by the same number, so the fraction we want is $\frac{3}{8}$ scaled up.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

Write out the fractions equal to $\frac{3}{8}$ and watch what the gap between bottom and top does. It grows in steps, and finding which step reaches 20 pins the fraction down.

3 · Execute4 carry out the plan

1List equivalent fractions

#5 Look for a Pattern 4.NF.A.1

Multiplying top and bottom by the same number does not change a fraction's value, so these are all equal to

\frac{3}{8}

.

\frac{3}{8} = \frac{6}{16} = \frac{9}{24} = \cdots

The list runs

\frac{3}{8}

,

\frac{6}{16}

,

\frac{9}{24}

,

\frac{12}{32}

, and so on.

2Track the gap between bottom and top

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

Subtract top from bottom in each one: the gaps are 5, 10, 15, 20. They go up by 5 every time.

8 - 3 = 5,\quad 16 - 6 = 10,\quad 24 - 9 = 15

Scaling by a number scales the gap by that same number.

3Find the multiplier that gives a gap of 20

#6 Guess and Check 4.NF.A.1

We need 5 multiplied by something to reach 20, so divide.

20 \div 5 = 4

The fraction is

\frac{3}{8}

scaled by 4.

4Build the fraction

#6 Guess and Check 4.NF.A.1

Multiply both the 3 and the 8 by 4.

\frac{3 \times 4}{8 \times 4} = \frac{12}{32}

The fraction is

\frac{12}{32}

.

Answer:

\frac{12}{32}

4 · Reviewdoes it hold up?

Check both conditions: 32 - 12 = 20, and dividing both by 4 gives back $\frac{3}{8}$ .

Another way: Since the gap is 5 for every 1 of scale, 20 of gap means 4 lots of $\frac{3}{8}$ 's parts: 3 x 4 over 8 x 4.

Standardsmin grade 4

4.NF.A.1 Explain why a fraction is equivalent to another fraction and generate equivalents — Generating fractions equivalent to $\frac{3}{8}$ and scaling by 4 to build the answer.
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning about size — Reasoning that the gap between the parts scales with them.

💡Takeaway. Scaling a fraction scales the gap between its two numbers by exactly the same amount, so the gap tells you the scale.

Variant 4 easy

Find the fraction whose denominator minus its numerator is 21 and which, when written in lowest terms, equals $\dfrac{2}{9}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A fraction reduces to 2 over 9. In that unreduced fraction the denominator is 21 more than the numerator. We need the fraction itself.

Givens

In lowest terms the fraction is $\frac{2}{9}$ .
Denominator minus numerator is 21.

Unknowns

The fraction before it was reduced.

Constraints

Reducing means both parts were divided by the same number, so the fraction we want is $\frac{2}{9}$ scaled up.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

Write out the fractions equal to $\frac{2}{9}$ and watch what the gap between bottom and top does. It grows in steps, and finding which step reaches 21 pins the fraction down.

3 · Execute4 carry out the plan

1List equivalent fractions

#5 Look for a Pattern 4.NF.A.1

Multiplying top and bottom by the same number does not change a fraction's value, so these are all equal to

\frac{2}{9}

.

\frac{2}{9} = \frac{4}{18} = \frac{6}{27} = \cdots

The list runs

\frac{2}{9}

,

\frac{4}{18}

,

\frac{6}{27}

,

\frac{8}{36}

, and so on.

2Track the gap between bottom and top

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

Subtract top from bottom in each one: the gaps are 7, 14, 21, 28. They go up by 7 every time.

9 - 2 = 7,\quad 18 - 4 = 14,\quad 27 - 6 = 21

Scaling by a number scales the gap by that same number.

3Find the multiplier that gives a gap of 21

#6 Guess and Check 4.NF.A.1

We need 7 multiplied by something to reach 21, so divide.

21 \div 7 = 3

The fraction is

\frac{2}{9}

scaled by 3.

4Build the fraction

#6 Guess and Check 4.NF.A.1

Multiply both the 2 and the 9 by 3.

\frac{2 \times 3}{9 \times 3} = \frac{6}{27}

The fraction is

\frac{6}{27}

.

Answer:

\frac{6}{27}

4 · Reviewdoes it hold up?

Check both conditions: 27 - 6 = 21, and dividing both by 3 gives back $\frac{2}{9}$ .

Another way: Since the gap is 7 for every 1 of scale, 21 of gap means 3 lots of $\frac{2}{9}$ 's parts: 2 x 3 over 9 x 3.

Standardsmin grade 4

4.NF.A.1 Explain why a fraction is equivalent to another fraction and generate equivalents — Generating fractions equivalent to $\frac{2}{9}$ and scaling by 3 to build the answer.
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning about size — Reasoning that the gap between the parts scales with them.

💡Takeaway. Scaling a fraction scales the gap between its two numbers by exactly the same amount, so the gap tells you the scale.

Variant 5 medium

Find the fraction whose denominator minus its numerator is 24 and which, when written in lowest terms, equals $\dfrac{3}{7}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A fraction reduces to 3 over 7. In that unreduced fraction the denominator is 24 more than the numerator. We need the fraction itself.

Givens

In lowest terms the fraction is $\frac{3}{7}$ .
Denominator minus numerator is 24.

Unknowns

The fraction before it was reduced.

Constraints

Reducing means both parts were divided by the same number, so the fraction we want is $\frac{3}{7}$ scaled up.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

Write out the fractions equal to $\frac{3}{7}$ and watch what the gap between bottom and top does. It grows in steps, and finding which step reaches 24 pins the fraction down.

3 · Execute4 carry out the plan

1List equivalent fractions

#5 Look for a Pattern 4.NF.A.1

Multiplying top and bottom by the same number does not change a fraction's value, so these are all equal to

\frac{3}{7}

.

\frac{3}{7} = \frac{6}{14} = \frac{9}{21} = \cdots

The list runs

\frac{3}{7}

,

\frac{6}{14}

,

\frac{9}{21}

,

\frac{12}{28}

, and so on.

2Track the gap between bottom and top

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

Subtract top from bottom in each one: the gaps are 4, 8, 12, 16. They go up by 4 every time.

7 - 3 = 4,\quad 14 - 6 = 8,\quad 21 - 9 = 12

Scaling by a number scales the gap by that same number.

3Find the multiplier that gives a gap of 24

#6 Guess and Check 4.NF.A.1

We need 4 multiplied by something to reach 24, so divide.

24 \div 4 = 6

The fraction is

\frac{3}{7}

scaled by 6.

4Build the fraction

#6 Guess and Check 4.NF.A.1

Multiply both the 3 and the 7 by 6.

\frac{3 \times 6}{7 \times 6} = \frac{18}{42}

The fraction is

\frac{18}{42}

.

Answer:

\frac{18}{42}

4 · Reviewdoes it hold up?

Check both conditions: 42 - 18 = 24, and dividing both by 6 gives back $\frac{3}{7}$ .

Another way: Since the gap is 4 for every 1 of scale, 24 of gap means 6 lots of $\frac{3}{7}$ 's parts: 3 x 6 over 7 x 6.

Standardsmin grade 4

4.NF.A.1 Explain why a fraction is equivalent to another fraction and generate equivalents — Generating fractions equivalent to $\frac{3}{7}$ and scaling by 6 to build the answer.
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning about size — Reasoning that the gap between the parts scales with them.

💡Takeaway. Scaling a fraction scales the gap between its two numbers by exactly the same amount, so the gap tells you the scale.

Variant 6 medium

Find the fraction whose denominator minus its numerator is 24 and which, when written in lowest terms, equals $\dfrac{7}{10}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A fraction reduces to 7 over 10. In that unreduced fraction the denominator is 24 more than the numerator. We need the fraction itself.

Givens

In lowest terms the fraction is $\frac{7}{10}$ .
Denominator minus numerator is 24.

Unknowns

The fraction before it was reduced.

Constraints

Reducing means both parts were divided by the same number, so the fraction we want is $\frac{7}{10}$ scaled up.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

Write out the fractions equal to $\frac{7}{10}$ and watch what the gap between bottom and top does. It grows in steps, and finding which step reaches 24 pins the fraction down.

3 · Execute4 carry out the plan

1List equivalent fractions

#5 Look for a Pattern 4.NF.A.1

Multiplying top and bottom by the same number does not change a fraction's value, so these are all equal to

\frac{7}{10}

.

\frac{7}{10} = \frac{14}{20} = \frac{21}{30} = \cdots

The list runs

\frac{7}{10}

,

\frac{14}{20}

,

\frac{21}{30}

,

\frac{28}{40}

, and so on.

2Track the gap between bottom and top

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

Subtract top from bottom in each one: the gaps are 3, 6, 9, 12. They go up by 3 every time.

10 - 7 = 3,\quad 20 - 14 = 6,\quad 30 - 21 = 9

Scaling by a number scales the gap by that same number.

3Find the multiplier that gives a gap of 24

#6 Guess and Check 4.NF.A.1

We need 3 multiplied by something to reach 24, so divide.

24 \div 3 = 8

The fraction is

\frac{7}{10}

scaled by 8.

4Build the fraction

#6 Guess and Check 4.NF.A.1

Multiply both the 7 and the 10 by 8.

\frac{7 \times 8}{10 \times 8} = \frac{56}{80}

The fraction is

\frac{56}{80}

.

Answer:

\frac{56}{80}

4 · Reviewdoes it hold up?

Check both conditions: 80 - 56 = 24, and dividing both by 8 gives back $\frac{7}{10}$ .

Another way: Since the gap is 3 for every 1 of scale, 24 of gap means 8 lots of $\frac{7}{10}$ 's parts: 7 x 8 over 10 x 8.

Standardsmin grade 4

4.NF.A.1 Explain why a fraction is equivalent to another fraction and generate equivalents — Generating fractions equivalent to $\frac{7}{10}$ and scaling by 8 to build the answer.
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning about size — Reasoning that the gap between the parts scales with them.

💡Takeaway. Scaling a fraction scales the gap between its two numbers by exactly the same amount, so the gap tells you the scale.

Variant 7 medium

Find the fraction whose denominator minus its numerator is 25 and which, when written in lowest terms, equals $\dfrac{4}{9}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A fraction reduces to 4 over 9. In that unreduced fraction the denominator is 25 more than the numerator. We need the fraction itself.

Givens

In lowest terms the fraction is $\frac{4}{9}$ .
Denominator minus numerator is 25.

Unknowns

The fraction before it was reduced.

Constraints

Reducing means both parts were divided by the same number, so the fraction we want is $\frac{4}{9}$ scaled up.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

Write out the fractions equal to $\frac{4}{9}$ and watch what the gap between bottom and top does. It grows in steps, and finding which step reaches 25 pins the fraction down.

3 · Execute4 carry out the plan

1List equivalent fractions

#5 Look for a Pattern 4.NF.A.1

Multiplying top and bottom by the same number does not change a fraction's value, so these are all equal to

\frac{4}{9}

.

\frac{4}{9} = \frac{8}{18} = \frac{12}{27} = \cdots

The list runs

\frac{4}{9}

,

\frac{8}{18}

,

\frac{12}{27}

,

\frac{16}{36}

, and so on.

2Track the gap between bottom and top

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

Subtract top from bottom in each one: the gaps are 5, 10, 15, 20. They go up by 5 every time.

9 - 4 = 5,\quad 18 - 8 = 10,\quad 27 - 12 = 15

Scaling by a number scales the gap by that same number.

3Find the multiplier that gives a gap of 25

#6 Guess and Check 4.NF.A.1

We need 5 multiplied by something to reach 25, so divide.

25 \div 5 = 5

The fraction is

\frac{4}{9}

scaled by 5.

4Build the fraction

#6 Guess and Check 4.NF.A.1

Multiply both the 4 and the 9 by 5.

\frac{4 \times 5}{9 \times 5} = \frac{20}{45}

The fraction is

\frac{20}{45}

.

Answer:

\frac{20}{45}

4 · Reviewdoes it hold up?

Check both conditions: 45 - 20 = 25, and dividing both by 5 gives back $\frac{4}{9}$ .

Another way: Since the gap is 5 for every 1 of scale, 25 of gap means 5 lots of $\frac{4}{9}$ 's parts: 4 x 5 over 9 x 5.

Standardsmin grade 4

4.NF.A.1 Explain why a fraction is equivalent to another fraction and generate equivalents — Generating fractions equivalent to $\frac{4}{9}$ and scaling by 5 to build the answer.
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning about size — Reasoning that the gap between the parts scales with them.

💡Takeaway. Scaling a fraction scales the gap between its two numbers by exactly the same amount, so the gap tells you the scale.

Variant 8 medium

Find the fraction whose denominator minus its numerator is 27 and which, when written in lowest terms, equals $\dfrac{1}{4}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A fraction reduces to 1 over 4. In that unreduced fraction the denominator is 27 more than the numerator. We need the fraction itself.

Givens

In lowest terms the fraction is $\frac{1}{4}$ .
Denominator minus numerator is 27.

Unknowns

The fraction before it was reduced.

Constraints

Reducing means both parts were divided by the same number, so the fraction we want is $\frac{1}{4}$ scaled up.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

Write out the fractions equal to $\frac{1}{4}$ and watch what the gap between bottom and top does. It grows in steps, and finding which step reaches 27 pins the fraction down.

3 · Execute4 carry out the plan

1List equivalent fractions

#5 Look for a Pattern 4.NF.A.1

Multiplying top and bottom by the same number does not change a fraction's value, so these are all equal to

\frac{1}{4}

.

\frac{1}{4} = \frac{2}{8} = \frac{3}{12} = \cdots

The list runs

\frac{1}{4}

,

\frac{2}{8}

,

\frac{3}{12}

,

\frac{4}{16}

, and so on.

2Track the gap between bottom and top

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

Subtract top from bottom in each one: the gaps are 3, 6, 9, 12. They go up by 3 every time.

4 - 1 = 3,\quad 8 - 2 = 6,\quad 12 - 3 = 9

Scaling by a number scales the gap by that same number.

3Find the multiplier that gives a gap of 27

#6 Guess and Check 4.NF.A.1

We need 3 multiplied by something to reach 27, so divide.

27 \div 3 = 9

The fraction is

\frac{1}{4}

scaled by 9.

4Build the fraction

#6 Guess and Check 4.NF.A.1

Multiply both the 1 and the 4 by 9.

\frac{1 \times 9}{4 \times 9} = \frac{9}{36}

The fraction is

\frac{9}{36}

.

Answer:

\frac{9}{36}

4 · Reviewdoes it hold up?

Check both conditions: 36 - 9 = 27, and dividing both by 9 gives back $\frac{1}{4}$ .

Another way: Since the gap is 3 for every 1 of scale, 27 of gap means 9 lots of $\frac{1}{4}$ 's parts: 1 x 9 over 4 x 9.

Standardsmin grade 4

4.NF.A.1 Explain why a fraction is equivalent to another fraction and generate equivalents — Generating fractions equivalent to $\frac{1}{4}$ and scaling by 9 to build the answer.
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning about size — Reasoning that the gap between the parts scales with them.

💡Takeaway. Scaling a fraction scales the gap between its two numbers by exactly the same amount, so the gap tells you the scale.

Variant 9 hard

Find the fraction whose denominator minus its numerator is 28 and which, when written in lowest terms, equals $\dfrac{5}{12}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A fraction reduces to 5 over 12. In that unreduced fraction the denominator is 28 more than the numerator. We need the fraction itself.

Givens

In lowest terms the fraction is $\frac{5}{12}$ .
Denominator minus numerator is 28.

Unknowns

The fraction before it was reduced.

Constraints

Reducing means both parts were divided by the same number, so the fraction we want is $\frac{5}{12}$ scaled up.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

Write out the fractions equal to $\frac{5}{12}$ and watch what the gap between bottom and top does. It grows in steps, and finding which step reaches 28 pins the fraction down.

3 · Execute4 carry out the plan

1List equivalent fractions

#5 Look for a Pattern 4.NF.A.1

Multiplying top and bottom by the same number does not change a fraction's value, so these are all equal to

\frac{5}{12}

.

\frac{5}{12} = \frac{10}{24} = \frac{15}{36} = \cdots

The list runs

\frac{5}{12}

,

\frac{10}{24}

,

\frac{15}{36}

,

\frac{20}{48}

, and so on.

2Track the gap between bottom and top

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

Subtract top from bottom in each one: the gaps are 7, 14, 21, 28. They go up by 7 every time.

12 - 5 = 7,\quad 24 - 10 = 14,\quad 36 - 15 = 21

Scaling by a number scales the gap by that same number.

3Find the multiplier that gives a gap of 28

#6 Guess and Check 4.NF.A.1

We need 7 multiplied by something to reach 28, so divide.

28 \div 7 = 4

The fraction is

\frac{5}{12}

scaled by 4.

4Build the fraction

#6 Guess and Check 4.NF.A.1

Multiply both the 5 and the 12 by 4.

\frac{5 \times 4}{12 \times 4} = \frac{20}{48}

The fraction is

\frac{20}{48}

.

Answer:

\frac{20}{48}

4 · Reviewdoes it hold up?

Check both conditions: 48 - 20 = 28, and dividing both by 4 gives back $\frac{5}{12}$ .

Another way: Since the gap is 7 for every 1 of scale, 28 of gap means 4 lots of $\frac{5}{12}$ 's parts: 5 x 4 over 12 x 4.

Standardsmin grade 4

4.NF.A.1 Explain why a fraction is equivalent to another fraction and generate equivalents — Generating fractions equivalent to $\frac{5}{12}$ and scaling by 4 to build the answer.
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning about size — Reasoning that the gap between the parts scales with them.

💡Takeaway. Scaling a fraction scales the gap between its two numbers by exactly the same amount, so the gap tells you the scale.

Variant 10 hard

Find the fraction whose denominator minus its numerator is 28 and which, when written in lowest terms, equals $\dfrac{5}{9}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A fraction reduces to 5 over 9. In that unreduced fraction the denominator is 28 more than the numerator. We need the fraction itself.

Givens

In lowest terms the fraction is $\frac{5}{9}$ .
Denominator minus numerator is 28.

Unknowns

The fraction before it was reduced.

Constraints

Reducing means both parts were divided by the same number, so the fraction we want is $\frac{5}{9}$ scaled up.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

Write out the fractions equal to $\frac{5}{9}$ and watch what the gap between bottom and top does. It grows in steps, and finding which step reaches 28 pins the fraction down.

3 · Execute4 carry out the plan

1List equivalent fractions

#5 Look for a Pattern 4.NF.A.1

Multiplying top and bottom by the same number does not change a fraction's value, so these are all equal to

\frac{5}{9}

.

\frac{5}{9} = \frac{10}{18} = \frac{15}{27} = \cdots

The list runs

\frac{5}{9}

,

\frac{10}{18}

,

\frac{15}{27}

,

\frac{20}{36}

, and so on.

2Track the gap between bottom and top

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

Subtract top from bottom in each one: the gaps are 4, 8, 12, 16. They go up by 4 every time.

9 - 5 = 4,\quad 18 - 10 = 8,\quad 27 - 15 = 12

Scaling by a number scales the gap by that same number.

3Find the multiplier that gives a gap of 28

#6 Guess and Check 4.NF.A.1

We need 4 multiplied by something to reach 28, so divide.

28 \div 4 = 7

The fraction is

\frac{5}{9}

scaled by 7.

4Build the fraction

#6 Guess and Check 4.NF.A.1

Multiply both the 5 and the 9 by 7.

\frac{5 \times 7}{9 \times 7} = \frac{35}{63}

The fraction is

\frac{35}{63}

.

Answer:

\frac{35}{63}

4 · Reviewdoes it hold up?

Check both conditions: 63 - 35 = 28, and dividing both by 7 gives back $\frac{5}{9}$ .

Another way: Since the gap is 4 for every 1 of scale, 28 of gap means 7 lots of $\frac{5}{9}$ 's parts: 5 x 7 over 9 x 7.

Standardsmin grade 4

4.NF.A.1 Explain why a fraction is equivalent to another fraction and generate equivalents — Generating fractions equivalent to $\frac{5}{9}$ and scaling by 7 to build the answer.
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning about size — Reasoning that the gap between the parts scales with them.

💡Takeaway. Scaling a fraction scales the gap between its two numbers by exactly the same amount, so the gap tells you the scale.

Variant 11 hard

Find the fraction whose denominator minus its numerator is 33 and which, when written in lowest terms, equals $\dfrac{5}{8}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A fraction reduces to 5 over 8. In that unreduced fraction the denominator is 33 more than the numerator. We need the fraction itself.

Givens

In lowest terms the fraction is $\frac{5}{8}$ .
Denominator minus numerator is 33.

Unknowns

The fraction before it was reduced.

Constraints

Reducing means both parts were divided by the same number, so the fraction we want is $\frac{5}{8}$ scaled up.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

Write out the fractions equal to $\frac{5}{8}$ and watch what the gap between bottom and top does. It grows in steps, and finding which step reaches 33 pins the fraction down.

3 · Execute4 carry out the plan

1List equivalent fractions

#5 Look for a Pattern 4.NF.A.1

Multiplying top and bottom by the same number does not change a fraction's value, so these are all equal to

\frac{5}{8}

.

\frac{5}{8} = \frac{10}{16} = \frac{15}{24} = \cdots

The list runs

\frac{5}{8}

,

\frac{10}{16}

,

\frac{15}{24}

,

\frac{20}{32}

, and so on.

2Track the gap between bottom and top

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

Subtract top from bottom in each one: the gaps are 3, 6, 9, 12. They go up by 3 every time.

8 - 5 = 3,\quad 16 - 10 = 6,\quad 24 - 15 = 9

Scaling by a number scales the gap by that same number.

3Find the multiplier that gives a gap of 33

#6 Guess and Check 4.NF.A.1

We need 3 multiplied by something to reach 33, so divide.

33 \div 3 = 11

The fraction is

\frac{5}{8}

scaled by 11.

4Build the fraction

#6 Guess and Check 4.NF.A.1

Multiply both the 5 and the 8 by 11.

\frac{5 \times 11}{8 \times 11} = \frac{55}{88}

The fraction is

\frac{55}{88}

.

Answer:

\frac{55}{88}

4 · Reviewdoes it hold up?

Check both conditions: 88 - 55 = 33, and dividing both by 11 gives back $\frac{5}{8}$ .

Another way: Since the gap is 3 for every 1 of scale, 33 of gap means 11 lots of $\frac{5}{8}$ 's parts: 5 x 11 over 8 x 11.

Standardsmin grade 4

4.NF.A.1 Explain why a fraction is equivalent to another fraction and generate equivalents — Generating fractions equivalent to $\frac{5}{8}$ and scaling by 11 to build the answer.
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning about size — Reasoning that the gap between the parts scales with them.

💡Takeaway. Scaling a fraction scales the gap between its two numbers by exactly the same amount, so the gap tells you the scale.

Variant 12 hard

Find the fraction whose denominator minus its numerator is 35 and which, when written in lowest terms, equals $\dfrac{6}{11}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A fraction reduces to 6 over 11. In that unreduced fraction the denominator is 35 more than the numerator. We need the fraction itself.

Givens

In lowest terms the fraction is $\frac{6}{11}$ .
Denominator minus numerator is 35.

Unknowns

The fraction before it was reduced.

Constraints

Reducing means both parts were divided by the same number, so the fraction we want is $\frac{6}{11}$ scaled up.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

Write out the fractions equal to $\frac{6}{11}$ and watch what the gap between bottom and top does. It grows in steps, and finding which step reaches 35 pins the fraction down.

3 · Execute4 carry out the plan

1List equivalent fractions

#5 Look for a Pattern 4.NF.A.1

Multiplying top and bottom by the same number does not change a fraction's value, so these are all equal to

\frac{6}{11}

.

\frac{6}{11} = \frac{12}{22} = \frac{18}{33} = \cdots

The list runs

\frac{6}{11}

,

\frac{12}{22}

,

\frac{18}{33}

,

\frac{24}{44}

, and so on.

2Track the gap between bottom and top

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

Subtract top from bottom in each one: the gaps are 5, 10, 15, 20. They go up by 5 every time.

11 - 6 = 5,\quad 22 - 12 = 10,\quad 33 - 18 = 15

Scaling by a number scales the gap by that same number.

3Find the multiplier that gives a gap of 35

#6 Guess and Check 4.NF.A.1

We need 5 multiplied by something to reach 35, so divide.

35 \div 5 = 7

The fraction is

\frac{6}{11}

scaled by 7.

4Build the fraction

#6 Guess and Check 4.NF.A.1

Multiply both the 6 and the 11 by 7.

\frac{6 \times 7}{11 \times 7} = \frac{42}{77}

The fraction is

\frac{42}{77}

.

Answer:

\frac{42}{77}

4 · Reviewdoes it hold up?

Check both conditions: 77 - 42 = 35, and dividing both by 7 gives back $\frac{6}{11}$ .

Another way: Since the gap is 5 for every 1 of scale, 35 of gap means 7 lots of $\frac{6}{11}$ 's parts: 6 x 7 over 11 x 7.

Standardsmin grade 4

4.NF.A.1 Explain why a fraction is equivalent to another fraction and generate equivalents — Generating fractions equivalent to $\frac{6}{11}$ and scaling by 7 to build the answer.
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning about size — Reasoning that the gap between the parts scales with them.

💡Takeaway. Scaling a fraction scales the gap between its two numbers by exactly the same amount, so the gap tells you the scale.

Scaling numerator and denominator keeps value · 12 practice problems

Generated variants — 12

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