← 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

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Generated variants — 12

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

Variant 1 easy

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 $\frac{3}{5}$ . Find the original fraction.

Your answer

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 $\frac{3}{5}$ .

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.

\frac{3 \times 6}{5 \times 6} = \frac{18}{30}

Before reducing it was

\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 = 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 = 37

The original fraction was

\frac{16}{37}

Answer:

\frac{16}{37}

4 · Reviewdoes it hold up?

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

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 $\frac{6}{7}$ . Find the original fraction.

Your answer

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 $\frac{6}{7}$ .

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.

\frac{6 \times 4}{7 \times 4} = \frac{24}{28}

Before reducing it was

\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 = 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 = 32

The original fraction was

\frac{21}{32}

Answer:

\frac{21}{32}

4 · Reviewdoes it hold up?

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

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 $\frac{2}{3}$ . Find the original fraction.

Your answer

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 $\frac{2}{3}$ .

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.

\frac{2 \times 8}{3 \times 8} = \frac{16}{24}

Before reducing it was

\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 = 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 = 29

The original fraction was

\frac{12}{29}

Answer:

\frac{12}{29}

4 · Reviewdoes it hold up?

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

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 $\frac{5}{6}$ . Find the original fraction.

Your answer

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 $\frac{5}{6}$ .

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.

\frac{5 \times 7}{6 \times 7} = \frac{35}{42}

Before reducing it was

\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 = 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 = 44

The original fraction was

\frac{27}{44}

Answer:

\frac{27}{44}

4 · Reviewdoes it hold up?

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

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 $\frac{4}{9}$ . Find the original fraction.

Your answer

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 $\frac{4}{9}$ .

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.

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

Before reducing it was

\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 = 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 = 48

The original fraction was

\frac{14}{48}

Answer:

\frac{14}{48}

4 · Reviewdoes it hold up?

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

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 $\frac{3}{7}$ . Find the original fraction.

Your answer

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 $\frac{3}{7}$ .

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.

\frac{3 \times 9}{7 \times 9} = \frac{27}{63}

Before reducing it was

\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 = 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 = 67

The original fraction was

\frac{22}{67}

Answer:

\frac{22}{67}

4 · Reviewdoes it hold up?

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

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 $\frac{7}{8}$ . Find the original fraction.

Your answer

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 $\frac{7}{8}$ .

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.

\frac{7 \times 3}{8 \times 3} = \frac{21}{24}

Before reducing it was

\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 = 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 = 33

The original fraction was

\frac{16}{33}

Answer:

\frac{16}{33}

4 · Reviewdoes it hold up?

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

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 $\frac{5}{8}$ . Find the original fraction.

Your answer

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 $\frac{5}{8}$ .

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.

\frac{5 \times 10}{8 \times 10} = \frac{50}{80}

Before reducing it was

\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 = 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 = 87

The original fraction was

\frac{41}{87}

Answer:

\frac{41}{87}

4 · Reviewdoes it hold up?

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

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 $\frac{8}{11}$ . Find the original fraction.

Your answer

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 $\frac{8}{11}$ .

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.

\frac{8 \times 2}{11 \times 2} = \frac{16}{22}

Before reducing it was

\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 = 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 = 28

The original fraction was

\frac{9}{28}

Answer:

\frac{9}{28}

4 · Reviewdoes it hold up?

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

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 $\frac{9}{10}$ . Find the original fraction.

Your answer

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 $\frac{9}{10}$ .

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.

\frac{9 \times 4}{10 \times 4} = \frac{36}{40}

Before reducing it was

\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 = 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 = 48

The original fraction was

\frac{25}{48}

Answer:

\frac{25}{48}

4 · Reviewdoes it hold up?

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

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 $\frac{4}{5}$ . Find the original fraction.

Your answer

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 $\frac{4}{5}$ .

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.

\frac{4 \times 12}{5 \times 12} = \frac{48}{60}

Before reducing it was

\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 = 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 = 70

The original fraction was

\frac{45}{70}

Answer:

\frac{45}{70}

4 · Reviewdoes it hold up?

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

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 $\frac{7}{9}$ . Find the original fraction.

Your answer

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 $\frac{7}{9}$ .

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.

\frac{7 \times 6}{9 \times 6} = \frac{42}{54}

Before reducing it was

\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 = 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 = 59

The original fraction was

\frac{29}{59}

Answer:

\frac{29}{59}

4 · Reviewdoes it hold up?

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

Where next?

← Back to the problem: Reverse the steps to recover the original fraction