← Scaling numerator and denominator keeps value · Equivalent Fractions by Scaling

Scaling numerator and denominator keeps value · 12 practice problems

4.NF.A.13.NF.A.3

Generated variants — 12

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

Variant 1 easy answer: 2035\frac{20}{35}

Find the fraction whose denominator minus its numerator is 15 and which, when written in lowest terms, equals 47\dfrac{4}{7}.

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 47\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 47\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 47\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 47\frac{4}{7}.
47=814=1221=⋯\frac{4}{7} = \frac{8}{14} = \frac{12}{21} = \cdots
The list runs 47\frac{4}{7}, 814\frac{8}{14}, 1221\frac{12}{21}, 1628\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=3,14−8=6,21−12=97 - 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÷3=515 \div 3 = 5
The fraction is 47\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.
4×57×5=2035\frac{4 \times 5}{7 \times 5} = \frac{20}{35}
The fraction is 2035\frac{20}{35}.
Answer: 2035\frac{20}{35}
4 · Reviewdoes it hold up?

Check both conditions: 35 - 20 = 15, and dividing both by 5 gives back 47\frac{4}{7}.

Another way: Since the gap is 3 for every 1 of scale, 15 of gap means 5 lots of 47\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 answer: 1230\frac{12}{30}

Find the fraction whose denominator minus its numerator is 18 and which, when written in lowest terms, equals 25\dfrac{2}{5}.

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 25\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 25\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 25\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 25\frac{2}{5}.
25=410=615=⋯\frac{2}{5} = \frac{4}{10} = \frac{6}{15} = \cdots
The list runs 25\frac{2}{5}, 410\frac{4}{10}, 615\frac{6}{15}, 820\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=3,10−4=6,15−6=95 - 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÷3=618 \div 3 = 6
The fraction is 25\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.
2×65×6=1230\frac{2 \times 6}{5 \times 6} = \frac{12}{30}
The fraction is 1230\frac{12}{30}.
Answer: 1230\frac{12}{30}
4 · Reviewdoes it hold up?

Check both conditions: 30 - 12 = 18, and dividing both by 6 gives back 25\frac{2}{5}.

Another way: Since the gap is 3 for every 1 of scale, 18 of gap means 6 lots of 25\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 answer: 1232\frac{12}{32}

Find the fraction whose denominator minus its numerator is 20 and which, when written in lowest terms, equals 38\dfrac{3}{8}.

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 38\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 38\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 38\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 38\frac{3}{8}.
38=616=924=⋯\frac{3}{8} = \frac{6}{16} = \frac{9}{24} = \cdots
The list runs 38\frac{3}{8}, 616\frac{6}{16}, 924\frac{9}{24}, 1232\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=5,16−6=10,24−9=158 - 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÷5=420 \div 5 = 4
The fraction is 38\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.
3×48×4=1232\frac{3 \times 4}{8 \times 4} = \frac{12}{32}
The fraction is 1232\frac{12}{32}.
Answer: 1232\frac{12}{32}
4 · Reviewdoes it hold up?

Check both conditions: 32 - 12 = 20, and dividing both by 4 gives back 38\frac{3}{8}.

Another way: Since the gap is 5 for every 1 of scale, 20 of gap means 4 lots of 38\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 answer: 627\frac{6}{27}

Find the fraction whose denominator minus its numerator is 21 and which, when written in lowest terms, equals 29\dfrac{2}{9}.

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 29\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 29\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 29\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 29\frac{2}{9}.
29=418=627=⋯\frac{2}{9} = \frac{4}{18} = \frac{6}{27} = \cdots
The list runs 29\frac{2}{9}, 418\frac{4}{18}, 627\frac{6}{27}, 836\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=7,18−4=14,27−6=219 - 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÷7=321 \div 7 = 3
The fraction is 29\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.
2×39×3=627\frac{2 \times 3}{9 \times 3} = \frac{6}{27}
The fraction is 627\frac{6}{27}.
Answer: 627\frac{6}{27}
4 · Reviewdoes it hold up?

Check both conditions: 27 - 6 = 21, and dividing both by 3 gives back 29\frac{2}{9}.

Another way: Since the gap is 7 for every 1 of scale, 21 of gap means 3 lots of 29\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 answer: 1842\frac{18}{42}

Find the fraction whose denominator minus its numerator is 24 and which, when written in lowest terms, equals 37\dfrac{3}{7}.

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 37\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 37\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 37\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 37\frac{3}{7}.
37=614=921=⋯\frac{3}{7} = \frac{6}{14} = \frac{9}{21} = \cdots
The list runs 37\frac{3}{7}, 614\frac{6}{14}, 921\frac{9}{21}, 1228\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=4,14−6=8,21−9=127 - 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÷4=624 \div 4 = 6
The fraction is 37\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.
3×67×6=1842\frac{3 \times 6}{7 \times 6} = \frac{18}{42}
The fraction is 1842\frac{18}{42}.
Answer: 1842\frac{18}{42}
4 · Reviewdoes it hold up?

Check both conditions: 42 - 18 = 24, and dividing both by 6 gives back 37\frac{3}{7}.

Another way: Since the gap is 4 for every 1 of scale, 24 of gap means 6 lots of 37\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 answer: 5680\frac{56}{80}

Find the fraction whose denominator minus its numerator is 24 and which, when written in lowest terms, equals 710\dfrac{7}{10}.

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 710\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 710\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 710\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 710\frac{7}{10}.
710=1420=2130=⋯\frac{7}{10} = \frac{14}{20} = \frac{21}{30} = \cdots
The list runs 710\frac{7}{10}, 1420\frac{14}{20}, 2130\frac{21}{30}, 2840\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=3,20−14=6,30−21=910 - 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÷3=824 \div 3 = 8
The fraction is 710\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.
7×810×8=5680\frac{7 \times 8}{10 \times 8} = \frac{56}{80}
The fraction is 5680\frac{56}{80}.
Answer: 5680\frac{56}{80}
4 · Reviewdoes it hold up?

Check both conditions: 80 - 56 = 24, and dividing both by 8 gives back 710\frac{7}{10}.

Another way: Since the gap is 3 for every 1 of scale, 24 of gap means 8 lots of 710\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 answer: 2045\frac{20}{45}

Find the fraction whose denominator minus its numerator is 25 and which, when written in lowest terms, equals 49\dfrac{4}{9}.

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 49\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 49\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 49\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 49\frac{4}{9}.
49=818=1227=⋯\frac{4}{9} = \frac{8}{18} = \frac{12}{27} = \cdots
The list runs 49\frac{4}{9}, 818\frac{8}{18}, 1227\frac{12}{27}, 1636\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=5,18−8=10,27−12=159 - 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÷5=525 \div 5 = 5
The fraction is 49\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.
4×59×5=2045\frac{4 \times 5}{9 \times 5} = \frac{20}{45}
The fraction is 2045\frac{20}{45}.
Answer: 2045\frac{20}{45}
4 · Reviewdoes it hold up?

Check both conditions: 45 - 20 = 25, and dividing both by 5 gives back 49\frac{4}{9}.

Another way: Since the gap is 5 for every 1 of scale, 25 of gap means 5 lots of 49\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 answer: 936\frac{9}{36}

Find the fraction whose denominator minus its numerator is 27 and which, when written in lowest terms, equals 14\dfrac{1}{4}.

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 14\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 14\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 14\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 14\frac{1}{4}.
14=28=312=⋯\frac{1}{4} = \frac{2}{8} = \frac{3}{12} = \cdots
The list runs 14\frac{1}{4}, 28\frac{2}{8}, 312\frac{3}{12}, 416\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=3,8−2=6,12−3=94 - 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÷3=927 \div 3 = 9
The fraction is 14\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.
1×94×9=936\frac{1 \times 9}{4 \times 9} = \frac{9}{36}
The fraction is 936\frac{9}{36}.
Answer: 936\frac{9}{36}
4 · Reviewdoes it hold up?

Check both conditions: 36 - 9 = 27, and dividing both by 9 gives back 14\frac{1}{4}.

Another way: Since the gap is 3 for every 1 of scale, 27 of gap means 9 lots of 14\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 answer: 2048\frac{20}{48}

Find the fraction whose denominator minus its numerator is 28 and which, when written in lowest terms, equals 512\dfrac{5}{12}.

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 512\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 512\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 512\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 512\frac{5}{12}.
512=1024=1536=⋯\frac{5}{12} = \frac{10}{24} = \frac{15}{36} = \cdots
The list runs 512\frac{5}{12}, 1024\frac{10}{24}, 1536\frac{15}{36}, 2048\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=7,24−10=14,36−15=2112 - 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÷7=428 \div 7 = 4
The fraction is 512\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.
5×412×4=2048\frac{5 \times 4}{12 \times 4} = \frac{20}{48}
The fraction is 2048\frac{20}{48}.
Answer: 2048\frac{20}{48}
4 · Reviewdoes it hold up?

Check both conditions: 48 - 20 = 28, and dividing both by 4 gives back 512\frac{5}{12}.

Another way: Since the gap is 7 for every 1 of scale, 28 of gap means 4 lots of 512\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 answer: 3563\frac{35}{63}

Find the fraction whose denominator minus its numerator is 28 and which, when written in lowest terms, equals 59\dfrac{5}{9}.

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 59\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 59\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 59\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 59\frac{5}{9}.
59=1018=1527=⋯\frac{5}{9} = \frac{10}{18} = \frac{15}{27} = \cdots
The list runs 59\frac{5}{9}, 1018\frac{10}{18}, 1527\frac{15}{27}, 2036\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=4,18−10=8,27−15=129 - 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÷4=728 \div 4 = 7
The fraction is 59\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.
5×79×7=3563\frac{5 \times 7}{9 \times 7} = \frac{35}{63}
The fraction is 3563\frac{35}{63}.
Answer: 3563\frac{35}{63}
4 · Reviewdoes it hold up?

Check both conditions: 63 - 35 = 28, and dividing both by 7 gives back 59\frac{5}{9}.

Another way: Since the gap is 4 for every 1 of scale, 28 of gap means 7 lots of 59\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 answer: 5588\frac{55}{88}

Find the fraction whose denominator minus its numerator is 33 and which, when written in lowest terms, equals 58\dfrac{5}{8}.

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 58\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 58\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 58\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 58\frac{5}{8}.
58=1016=1524=⋯\frac{5}{8} = \frac{10}{16} = \frac{15}{24} = \cdots
The list runs 58\frac{5}{8}, 1016\frac{10}{16}, 1524\frac{15}{24}, 2032\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=3,16−10=6,24−15=98 - 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÷3=1133 \div 3 = 11
The fraction is 58\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.
5×118×11=5588\frac{5 \times 11}{8 \times 11} = \frac{55}{88}
The fraction is 5588\frac{55}{88}.
Answer: 5588\frac{55}{88}
4 · Reviewdoes it hold up?

Check both conditions: 88 - 55 = 33, and dividing both by 11 gives back 58\frac{5}{8}.

Another way: Since the gap is 3 for every 1 of scale, 33 of gap means 11 lots of 58\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 answer: 4277\frac{42}{77}

Find the fraction whose denominator minus its numerator is 35 and which, when written in lowest terms, equals 611\dfrac{6}{11}.

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 611\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 611\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 611\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 611\frac{6}{11}.
611=1222=1833=⋯\frac{6}{11} = \frac{12}{22} = \frac{18}{33} = \cdots
The list runs 611\frac{6}{11}, 1222\frac{12}{22}, 1833\frac{18}{33}, 2444\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=5,22−12=10,33−18=1511 - 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÷5=735 \div 5 = 7
The fraction is 611\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.
6×711×7=4277\frac{6 \times 7}{11 \times 7} = \frac{42}{77}
The fraction is 4277\frac{42}{77}.
Answer: 4277\frac{42}{77}
4 · Reviewdoes it hold up?

Check both conditions: 77 - 42 = 35, and dividing both by 7 gives back 611\frac{6}{11}.

Another way: Since the gap is 5 for every 1 of scale, 35 of gap means 7 lots of 611\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.