← Reverse the steps to recover the original fraction · Work Backwards to Recover a Start Value

Reverse the steps to recover the original fraction · 12 practice problems

4.NF.A.14.OA.A.3

Generated variants — 12

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

Variant 1 easy answer: 1637\frac{16}{37}

Start with a certain fraction. Subtract 7 from its denominator, add 2 to its numerator, and then reduce the result to lowest terms by dividing by 6, giving 35\frac{3}{5}. Find the original fraction.

Show solution
1 · Understandwhat's really being asked

A fraction had 7 taken off its denominator and 2 added to its numerator. Dividing both parts of the result by 6 gave 3 over 5. We need the fraction it started as.

Givens
  • The denominator lost 7.
  • The numerator gained 2.
  • Dividing both parts by 6 gave 35\frac{3}{5}.
Unknowns
  • The original fraction.
Constraints
  • The three changes happened in that order, so undoing them runs the other way.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

The reducing came last, so it is undone first. Then the numerator and denominator can each be walked back on their own.

3 · Execute3 carry out the plan

1Undo the reducing (multiply back by 6)

#11 Work Backwards 4.NF.A.1
Reducing divided both parts by 6, so multiply both back by 6 to see the fraction as it was just before.
3×65×6=1830\frac{3 \times 6}{5 \times 6} = \frac{18}{30}
Before reducing it was 1830\frac{18}{30}.

2Undo adding 2 to the numerator (subtract 2)

#11 Work Backwards 4.OA.A.3
The numerator had gained 2, so take it off again.
18−2=1618 - 2 = 16
The original numerator was 16.

3Undo subtracting 7 from the denominator (add 7)

#11 Work Backwards 4.OA.A.3
The denominator had lost 7, so put it back.
30+7=3730 + 7 = 37
The original fraction was 1637\frac{16}{37}.
Answer: 1637\frac{16}{37}
4 · Reviewdoes it hold up?

Run it forwards: 37 - 7 = 30, 16 + 2 = 18, and dividing 1830\frac{18}{30} by 6 gives 35\frac{3}{5}. It matches.

Another way: Guess and check would mean trying fractions until one survives all three steps; undoing gets there in three moves with nothing to search.

Standardsmin grade 4
  • 4.NF.A.1 Explain why a fraction is equivalent to another fraction — Multiplying both parts by 6 to undo the reducing.
  • 4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Undoing the change to each part of the fraction in turn.
💡Takeaway. Undo the LAST thing first. Reducing happened at the end, so multiplying back is where the trip home starts.
Variant 2 easy answer: 2132\frac{21}{32}

Start with a certain fraction. Subtract 4 from its denominator, add 3 to its numerator, and then reduce the result to lowest terms by dividing by 4, giving 67\frac{6}{7}. Find the original fraction.

Show solution
1 · Understandwhat's really being asked

A fraction had 4 taken off its denominator and 3 added to its numerator. Dividing both parts of the result by 4 gave 6 over 7. We need the fraction it started as.

Givens
  • The denominator lost 4.
  • The numerator gained 3.
  • Dividing both parts by 4 gave 67\frac{6}{7}.
Unknowns
  • The original fraction.
Constraints
  • The three changes happened in that order, so undoing them runs the other way.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

The reducing came last, so it is undone first. Then the numerator and denominator can each be walked back on their own.

3 · Execute3 carry out the plan

1Undo the reducing (multiply back by 4)

#11 Work Backwards 4.NF.A.1
Reducing divided both parts by 4, so multiply both back by 4 to see the fraction as it was just before.
6×47×4=2428\frac{6 \times 4}{7 \times 4} = \frac{24}{28}
Before reducing it was 2428\frac{24}{28}.

2Undo adding 3 to the numerator (subtract 3)

#11 Work Backwards 4.OA.A.3
The numerator had gained 3, so take it off again.
24−3=2124 - 3 = 21
The original numerator was 21.

3Undo subtracting 4 from the denominator (add 4)

#11 Work Backwards 4.OA.A.3
The denominator had lost 4, so put it back.
28+4=3228 + 4 = 32
The original fraction was 2132\frac{21}{32}.
Answer: 2132\frac{21}{32}
4 · Reviewdoes it hold up?

Run it forwards: 32 - 4 = 28, 21 + 3 = 24, and dividing 2428\frac{24}{28} by 4 gives 67\frac{6}{7}. It matches.

Another way: Guess and check would mean trying fractions until one survives all three steps; undoing gets there in three moves with nothing to search.

Standardsmin grade 4
  • 4.NF.A.1 Explain why a fraction is equivalent to another fraction — Multiplying both parts by 4 to undo the reducing.
  • 4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Undoing the change to each part of the fraction in turn.
💡Takeaway. Undo the LAST thing first. Reducing happened at the end, so multiplying back is where the trip home starts.
Variant 3 easy answer: 1229\frac{12}{29}

Start with a certain fraction. Subtract 5 from its denominator, add 4 to its numerator, and then reduce the result to lowest terms by dividing by 8, giving 23\frac{2}{3}. Find the original fraction.

Show solution
1 · Understandwhat's really being asked

A fraction had 5 taken off its denominator and 4 added to its numerator. Dividing both parts of the result by 8 gave 2 over 3. We need the fraction it started as.

Givens
  • The denominator lost 5.
  • The numerator gained 4.
  • Dividing both parts by 8 gave 23\frac{2}{3}.
Unknowns
  • The original fraction.
Constraints
  • The three changes happened in that order, so undoing them runs the other way.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

The reducing came last, so it is undone first. Then the numerator and denominator can each be walked back on their own.

3 · Execute3 carry out the plan

1Undo the reducing (multiply back by 8)

#11 Work Backwards 4.NF.A.1
Reducing divided both parts by 8, so multiply both back by 8 to see the fraction as it was just before.
2×83×8=1624\frac{2 \times 8}{3 \times 8} = \frac{16}{24}
Before reducing it was 1624\frac{16}{24}.

2Undo adding 4 to the numerator (subtract 4)

#11 Work Backwards 4.OA.A.3
The numerator had gained 4, so take it off again.
16−4=1216 - 4 = 12
The original numerator was 12.

3Undo subtracting 5 from the denominator (add 5)

#11 Work Backwards 4.OA.A.3
The denominator had lost 5, so put it back.
24+5=2924 + 5 = 29
The original fraction was 1229\frac{12}{29}.
Answer: 1229\frac{12}{29}
4 · Reviewdoes it hold up?

Run it forwards: 29 - 5 = 24, 12 + 4 = 16, and dividing 1624\frac{16}{24} by 8 gives 23\frac{2}{3}. It matches.

Another way: Guess and check would mean trying fractions until one survives all three steps; undoing gets there in three moves with nothing to search.

Standardsmin grade 4
  • 4.NF.A.1 Explain why a fraction is equivalent to another fraction — Multiplying both parts by 8 to undo the reducing.
  • 4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Undoing the change to each part of the fraction in turn.
💡Takeaway. Undo the LAST thing first. Reducing happened at the end, so multiplying back is where the trip home starts.
Variant 4 easy answer: 2744\frac{27}{44}

Start with a certain fraction. Subtract 2 from its denominator, add 8 to its numerator, and then reduce the result to lowest terms by dividing by 7, giving 56\frac{5}{6}. Find the original fraction.

Show solution
1 · Understandwhat's really being asked

A fraction had 2 taken off its denominator and 8 added to its numerator. Dividing both parts of the result by 7 gave 5 over 6. We need the fraction it started as.

Givens
  • The denominator lost 2.
  • The numerator gained 8.
  • Dividing both parts by 7 gave 56\frac{5}{6}.
Unknowns
  • The original fraction.
Constraints
  • The three changes happened in that order, so undoing them runs the other way.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

The reducing came last, so it is undone first. Then the numerator and denominator can each be walked back on their own.

3 · Execute3 carry out the plan

1Undo the reducing (multiply back by 7)

#11 Work Backwards 4.NF.A.1
Reducing divided both parts by 7, so multiply both back by 7 to see the fraction as it was just before.
5×76×7=3542\frac{5 \times 7}{6 \times 7} = \frac{35}{42}
Before reducing it was 3542\frac{35}{42}.

2Undo adding 8 to the numerator (subtract 8)

#11 Work Backwards 4.OA.A.3
The numerator had gained 8, so take it off again.
35−8=2735 - 8 = 27
The original numerator was 27.

3Undo subtracting 2 from the denominator (add 2)

#11 Work Backwards 4.OA.A.3
The denominator had lost 2, so put it back.
42+2=4442 + 2 = 44
The original fraction was 2744\frac{27}{44}.
Answer: 2744\frac{27}{44}
4 · Reviewdoes it hold up?

Run it forwards: 44 - 2 = 42, 27 + 8 = 35, and dividing 3542\frac{35}{42} by 7 gives 56\frac{5}{6}. It matches.

Another way: Guess and check would mean trying fractions until one survives all three steps; undoing gets there in three moves with nothing to search.

Standardsmin grade 4
  • 4.NF.A.1 Explain why a fraction is equivalent to another fraction — Multiplying both parts by 7 to undo the reducing.
  • 4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Undoing the change to each part of the fraction in turn.
💡Takeaway. Undo the LAST thing first. Reducing happened at the end, so multiplying back is where the trip home starts.
Variant 5 medium answer: 1448\frac{14}{48}

Start with a certain fraction. Subtract 3 from its denominator, add 6 to its numerator, and then reduce the result to lowest terms by dividing by 5, giving 49\frac{4}{9}. Find the original fraction.

Show solution
1 · Understandwhat's really being asked

A fraction had 3 taken off its denominator and 6 added to its numerator. Dividing both parts of the result by 5 gave 4 over 9. We need the fraction it started as.

Givens
  • The denominator lost 3.
  • The numerator gained 6.
  • Dividing both parts by 5 gave 49\frac{4}{9}.
Unknowns
  • The original fraction.
Constraints
  • The three changes happened in that order, so undoing them runs the other way.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

The reducing came last, so it is undone first. Then the numerator and denominator can each be walked back on their own.

3 · Execute3 carry out the plan

1Undo the reducing (multiply back by 5)

#11 Work Backwards 4.NF.A.1
Reducing divided both parts by 5, so multiply both back by 5 to see the fraction as it was just before.
4×59×5=2045\frac{4 \times 5}{9 \times 5} = \frac{20}{45}
Before reducing it was 2045\frac{20}{45}.

2Undo adding 6 to the numerator (subtract 6)

#11 Work Backwards 4.OA.A.3
The numerator had gained 6, so take it off again.
20−6=1420 - 6 = 14
The original numerator was 14.

3Undo subtracting 3 from the denominator (add 3)

#11 Work Backwards 4.OA.A.3
The denominator had lost 3, so put it back.
45+3=4845 + 3 = 48
The original fraction was 1448\frac{14}{48}.
Answer: 1448\frac{14}{48}
4 · Reviewdoes it hold up?

Run it forwards: 48 - 3 = 45, 14 + 6 = 20, and dividing 2045\frac{20}{45} by 5 gives 49\frac{4}{9}. It matches.

Another way: Guess and check would mean trying fractions until one survives all three steps; undoing gets there in three moves with nothing to search.

Standardsmin grade 4
  • 4.NF.A.1 Explain why a fraction is equivalent to another fraction — Multiplying both parts by 5 to undo the reducing.
  • 4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Undoing the change to each part of the fraction in turn.
💡Takeaway. Undo the LAST thing first. Reducing happened at the end, so multiplying back is where the trip home starts.
Variant 6 medium answer: 2267\frac{22}{67}

Start with a certain fraction. Subtract 4 from its denominator, add 5 to its numerator, and then reduce the result to lowest terms by dividing by 9, giving 37\frac{3}{7}. Find the original fraction.

Show solution
1 · Understandwhat's really being asked

A fraction had 4 taken off its denominator and 5 added to its numerator. Dividing both parts of the result by 9 gave 3 over 7. We need the fraction it started as.

Givens
  • The denominator lost 4.
  • The numerator gained 5.
  • Dividing both parts by 9 gave 37\frac{3}{7}.
Unknowns
  • The original fraction.
Constraints
  • The three changes happened in that order, so undoing them runs the other way.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

The reducing came last, so it is undone first. Then the numerator and denominator can each be walked back on their own.

3 · Execute3 carry out the plan

1Undo the reducing (multiply back by 9)

#11 Work Backwards 4.NF.A.1
Reducing divided both parts by 9, so multiply both back by 9 to see the fraction as it was just before.
3×97×9=2763\frac{3 \times 9}{7 \times 9} = \frac{27}{63}
Before reducing it was 2763\frac{27}{63}.

2Undo adding 5 to the numerator (subtract 5)

#11 Work Backwards 4.OA.A.3
The numerator had gained 5, so take it off again.
27−5=2227 - 5 = 22
The original numerator was 22.

3Undo subtracting 4 from the denominator (add 4)

#11 Work Backwards 4.OA.A.3
The denominator had lost 4, so put it back.
63+4=6763 + 4 = 67
The original fraction was 2267\frac{22}{67}.
Answer: 2267\frac{22}{67}
4 · Reviewdoes it hold up?

Run it forwards: 67 - 4 = 63, 22 + 5 = 27, and dividing 2763\frac{27}{63} by 9 gives 37\frac{3}{7}. It matches.

Another way: Guess and check would mean trying fractions until one survives all three steps; undoing gets there in three moves with nothing to search.

Standardsmin grade 4
  • 4.NF.A.1 Explain why a fraction is equivalent to another fraction — Multiplying both parts by 9 to undo the reducing.
  • 4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Undoing the change to each part of the fraction in turn.
💡Takeaway. Undo the LAST thing first. Reducing happened at the end, so multiplying back is where the trip home starts.
Variant 7 medium answer: 1633\frac{16}{33}

Start with a certain fraction. Subtract 9 from its denominator, add 5 to its numerator, and then reduce the result to lowest terms by dividing by 3, giving 78\frac{7}{8}. Find the original fraction.

Show solution
1 · Understandwhat's really being asked

A fraction had 9 taken off its denominator and 5 added to its numerator. Dividing both parts of the result by 3 gave 7 over 8. We need the fraction it started as.

Givens
  • The denominator lost 9.
  • The numerator gained 5.
  • Dividing both parts by 3 gave 78\frac{7}{8}.
Unknowns
  • The original fraction.
Constraints
  • The three changes happened in that order, so undoing them runs the other way.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

The reducing came last, so it is undone first. Then the numerator and denominator can each be walked back on their own.

3 · Execute3 carry out the plan

1Undo the reducing (multiply back by 3)

#11 Work Backwards 4.NF.A.1
Reducing divided both parts by 3, so multiply both back by 3 to see the fraction as it was just before.
7×38×3=2124\frac{7 \times 3}{8 \times 3} = \frac{21}{24}
Before reducing it was 2124\frac{21}{24}.

2Undo adding 5 to the numerator (subtract 5)

#11 Work Backwards 4.OA.A.3
The numerator had gained 5, so take it off again.
21−5=1621 - 5 = 16
The original numerator was 16.

3Undo subtracting 9 from the denominator (add 9)

#11 Work Backwards 4.OA.A.3
The denominator had lost 9, so put it back.
24+9=3324 + 9 = 33
The original fraction was 1633\frac{16}{33}.
Answer: 1633\frac{16}{33}
4 · Reviewdoes it hold up?

Run it forwards: 33 - 9 = 24, 16 + 5 = 21, and dividing 2124\frac{21}{24} by 3 gives 78\frac{7}{8}. It matches.

Another way: Guess and check would mean trying fractions until one survives all three steps; undoing gets there in three moves with nothing to search.

Standardsmin grade 4
  • 4.NF.A.1 Explain why a fraction is equivalent to another fraction — Multiplying both parts by 3 to undo the reducing.
  • 4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Undoing the change to each part of the fraction in turn.
💡Takeaway. Undo the LAST thing first. Reducing happened at the end, so multiplying back is where the trip home starts.
Variant 8 medium answer: 4187\frac{41}{87}

Start with a certain fraction. Subtract 7 from its denominator, add 9 to its numerator, and then reduce the result to lowest terms by dividing by 10, giving 58\frac{5}{8}. Find the original fraction.

Show solution
1 · Understandwhat's really being asked

A fraction had 7 taken off its denominator and 9 added to its numerator. Dividing both parts of the result by 10 gave 5 over 8. We need the fraction it started as.

Givens
  • The denominator lost 7.
  • The numerator gained 9.
  • Dividing both parts by 10 gave 58\frac{5}{8}.
Unknowns
  • The original fraction.
Constraints
  • The three changes happened in that order, so undoing them runs the other way.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

The reducing came last, so it is undone first. Then the numerator and denominator can each be walked back on their own.

3 · Execute3 carry out the plan

1Undo the reducing (multiply back by 10)

#11 Work Backwards 4.NF.A.1
Reducing divided both parts by 10, so multiply both back by 10 to see the fraction as it was just before.
5×108×10=5080\frac{5 \times 10}{8 \times 10} = \frac{50}{80}
Before reducing it was 5080\frac{50}{80}.

2Undo adding 9 to the numerator (subtract 9)

#11 Work Backwards 4.OA.A.3
The numerator had gained 9, so take it off again.
50−9=4150 - 9 = 41
The original numerator was 41.

3Undo subtracting 7 from the denominator (add 7)

#11 Work Backwards 4.OA.A.3
The denominator had lost 7, so put it back.
80+7=8780 + 7 = 87
The original fraction was 4187\frac{41}{87}.
Answer: 4187\frac{41}{87}
4 · Reviewdoes it hold up?

Run it forwards: 87 - 7 = 80, 41 + 9 = 50, and dividing 5080\frac{50}{80} by 10 gives 58\frac{5}{8}. It matches.

Another way: Guess and check would mean trying fractions until one survives all three steps; undoing gets there in three moves with nothing to search.

Standardsmin grade 4
  • 4.NF.A.1 Explain why a fraction is equivalent to another fraction — Multiplying both parts by 10 to undo the reducing.
  • 4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Undoing the change to each part of the fraction in turn.
💡Takeaway. Undo the LAST thing first. Reducing happened at the end, so multiplying back is where the trip home starts.
Variant 9 hard answer: 928\frac{9}{28}

Start with a certain fraction. Subtract 6 from its denominator, add 7 to its numerator, and then reduce the result to lowest terms by dividing by 2, giving 811\frac{8}{11}. Find the original fraction.

Show solution
1 · Understandwhat's really being asked

A fraction had 6 taken off its denominator and 7 added to its numerator. Dividing both parts of the result by 2 gave 8 over 11. We need the fraction it started as.

Givens
  • The denominator lost 6.
  • The numerator gained 7.
  • Dividing both parts by 2 gave 811\frac{8}{11}.
Unknowns
  • The original fraction.
Constraints
  • The three changes happened in that order, so undoing them runs the other way.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

The reducing came last, so it is undone first. Then the numerator and denominator can each be walked back on their own.

3 · Execute3 carry out the plan

1Undo the reducing (multiply back by 2)

#11 Work Backwards 4.NF.A.1
Reducing divided both parts by 2, so multiply both back by 2 to see the fraction as it was just before.
8×211×2=1622\frac{8 \times 2}{11 \times 2} = \frac{16}{22}
Before reducing it was 1622\frac{16}{22}.

2Undo adding 7 to the numerator (subtract 7)

#11 Work Backwards 4.OA.A.3
The numerator had gained 7, so take it off again.
16−7=916 - 7 = 9
The original numerator was 9.

3Undo subtracting 6 from the denominator (add 6)

#11 Work Backwards 4.OA.A.3
The denominator had lost 6, so put it back.
22+6=2822 + 6 = 28
The original fraction was 928\frac{9}{28}.
Answer: 928\frac{9}{28}
4 · Reviewdoes it hold up?

Run it forwards: 28 - 6 = 22, 9 + 7 = 16, and dividing 1622\frac{16}{22} by 2 gives 811\frac{8}{11}. It matches.

Another way: Guess and check would mean trying fractions until one survives all three steps; undoing gets there in three moves with nothing to search.

Standardsmin grade 4
  • 4.NF.A.1 Explain why a fraction is equivalent to another fraction — Multiplying both parts by 2 to undo the reducing.
  • 4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Undoing the change to each part of the fraction in turn.
💡Takeaway. Undo the LAST thing first. Reducing happened at the end, so multiplying back is where the trip home starts.
Variant 10 hard answer: 2548\frac{25}{48}

Start with a certain fraction. Subtract 8 from its denominator, add 11 to its numerator, and then reduce the result to lowest terms by dividing by 4, giving 910\frac{9}{10}. Find the original fraction.

Show solution
1 · Understandwhat's really being asked

A fraction had 8 taken off its denominator and 11 added to its numerator. Dividing both parts of the result by 4 gave 9 over 10. We need the fraction it started as.

Givens
  • The denominator lost 8.
  • The numerator gained 11.
  • Dividing both parts by 4 gave 910\frac{9}{10}.
Unknowns
  • The original fraction.
Constraints
  • The three changes happened in that order, so undoing them runs the other way.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

The reducing came last, so it is undone first. Then the numerator and denominator can each be walked back on their own.

3 · Execute3 carry out the plan

1Undo the reducing (multiply back by 4)

#11 Work Backwards 4.NF.A.1
Reducing divided both parts by 4, so multiply both back by 4 to see the fraction as it was just before.
9×410×4=3640\frac{9 \times 4}{10 \times 4} = \frac{36}{40}
Before reducing it was 3640\frac{36}{40}.

2Undo adding 11 to the numerator (subtract 11)

#11 Work Backwards 4.OA.A.3
The numerator had gained 11, so take it off again.
36−11=2536 - 11 = 25
The original numerator was 25.

3Undo subtracting 8 from the denominator (add 8)

#11 Work Backwards 4.OA.A.3
The denominator had lost 8, so put it back.
40+8=4840 + 8 = 48
The original fraction was 2548\frac{25}{48}.
Answer: 2548\frac{25}{48}
4 · Reviewdoes it hold up?

Run it forwards: 48 - 8 = 40, 25 + 11 = 36, and dividing 3640\frac{36}{40} by 4 gives 910\frac{9}{10}. It matches.

Another way: Guess and check would mean trying fractions until one survives all three steps; undoing gets there in three moves with nothing to search.

Standardsmin grade 4
  • 4.NF.A.1 Explain why a fraction is equivalent to another fraction — Multiplying both parts by 4 to undo the reducing.
  • 4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Undoing the change to each part of the fraction in turn.
💡Takeaway. Undo the LAST thing first. Reducing happened at the end, so multiplying back is where the trip home starts.
Variant 11 hard answer: 4570\frac{45}{70}

Start with a certain fraction. Subtract 10 from its denominator, add 3 to its numerator, and then reduce the result to lowest terms by dividing by 12, giving 45\frac{4}{5}. Find the original fraction.

Show solution
1 · Understandwhat's really being asked

A fraction had 10 taken off its denominator and 3 added to its numerator. Dividing both parts of the result by 12 gave 4 over 5. We need the fraction it started as.

Givens
  • The denominator lost 10.
  • The numerator gained 3.
  • Dividing both parts by 12 gave 45\frac{4}{5}.
Unknowns
  • The original fraction.
Constraints
  • The three changes happened in that order, so undoing them runs the other way.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

The reducing came last, so it is undone first. Then the numerator and denominator can each be walked back on their own.

3 · Execute3 carry out the plan

1Undo the reducing (multiply back by 12)

#11 Work Backwards 4.NF.A.1
Reducing divided both parts by 12, so multiply both back by 12 to see the fraction as it was just before.
4×125×12=4860\frac{4 \times 12}{5 \times 12} = \frac{48}{60}
Before reducing it was 4860\frac{48}{60}.

2Undo adding 3 to the numerator (subtract 3)

#11 Work Backwards 4.OA.A.3
The numerator had gained 3, so take it off again.
48−3=4548 - 3 = 45
The original numerator was 45.

3Undo subtracting 10 from the denominator (add 10)

#11 Work Backwards 4.OA.A.3
The denominator had lost 10, so put it back.
60+10=7060 + 10 = 70
The original fraction was 4570\frac{45}{70}.
Answer: 4570\frac{45}{70}
4 · Reviewdoes it hold up?

Run it forwards: 70 - 10 = 60, 45 + 3 = 48, and dividing 4860\frac{48}{60} by 12 gives 45\frac{4}{5}. It matches.

Another way: Guess and check would mean trying fractions until one survives all three steps; undoing gets there in three moves with nothing to search.

Standardsmin grade 4
  • 4.NF.A.1 Explain why a fraction is equivalent to another fraction — Multiplying both parts by 12 to undo the reducing.
  • 4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Undoing the change to each part of the fraction in turn.
💡Takeaway. Undo the LAST thing first. Reducing happened at the end, so multiplying back is where the trip home starts.
Variant 12 hard answer: 2959\frac{29}{59}

Start with a certain fraction. Subtract 5 from its denominator, add 13 to its numerator, and then reduce the result to lowest terms by dividing by 6, giving 79\frac{7}{9}. Find the original fraction.

Show solution
1 · Understandwhat's really being asked

A fraction had 5 taken off its denominator and 13 added to its numerator. Dividing both parts of the result by 6 gave 7 over 9. We need the fraction it started as.

Givens
  • The denominator lost 5.
  • The numerator gained 13.
  • Dividing both parts by 6 gave 79\frac{7}{9}.
Unknowns
  • The original fraction.
Constraints
  • The three changes happened in that order, so undoing them runs the other way.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

The reducing came last, so it is undone first. Then the numerator and denominator can each be walked back on their own.

3 · Execute3 carry out the plan

1Undo the reducing (multiply back by 6)

#11 Work Backwards 4.NF.A.1
Reducing divided both parts by 6, so multiply both back by 6 to see the fraction as it was just before.
7×69×6=4254\frac{7 \times 6}{9 \times 6} = \frac{42}{54}
Before reducing it was 4254\frac{42}{54}.

2Undo adding 13 to the numerator (subtract 13)

#11 Work Backwards 4.OA.A.3
The numerator had gained 13, so take it off again.
42−13=2942 - 13 = 29
The original numerator was 29.

3Undo subtracting 5 from the denominator (add 5)

#11 Work Backwards 4.OA.A.3
The denominator had lost 5, so put it back.
54+5=5954 + 5 = 59
The original fraction was 2959\frac{29}{59}.
Answer: 2959\frac{29}{59}
4 · Reviewdoes it hold up?

Run it forwards: 59 - 5 = 54, 29 + 13 = 42, and dividing 4254\frac{42}{54} by 6 gives 79\frac{7}{9}. It matches.

Another way: Guess and check would mean trying fractions until one survives all three steps; undoing gets there in three moves with nothing to search.

Standardsmin grade 4
  • 4.NF.A.1 Explain why a fraction is equivalent to another fraction — Multiplying both parts by 6 to undo the reducing.
  • 4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Undoing the change to each part of the fraction in turn.
💡Takeaway. Undo the LAST thing first. Reducing happened at the end, so multiplying back is where the trip home starts.