← Dividing by a fraction undoes taking that fraction of something · Part-Whole Fraction Reasoning

Dividing by a fraction undoes taking that fraction of something · 12 practice problems

6.NS.A.16.RP.A.36.EE.B.7

Generated variants — 12

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

Variant 1 easy answer: 3500 won

Suho saved $\frac{2}{7}$ of his allowance. If he saved $1400$ won, how much of his allowance is left after saving?

Show solution

1 · Understandwhat's really being asked

Saving 2/7 of an allowance came to 1400 won. We want what is left after that saving.

Givens

The saving is 2/7 of the allowance.
The saving is 1400 won.

Unknowns

How much of the allowance is left.

Constraints

Only the saved part is given; the allowance itself is not.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#13 Convert to Algebra

Taking a fraction of the allowance produced the saving, so dividing by that fraction takes it back. Then read the question again -- it asks for the remainder, not the allowance.

3 · Execute3 carry out the plan

1Find one part

#7 Identify Subproblems 6.NS.A.1

1400 won is 2 of the 7 equal parts, so one part is that divided by 2.

1400 \div 2 = 700

Each 7th is 700 won.

2Build the whole allowance

#11 Work Backwards 6.EE.B.7

There are 7 of those parts in the whole.

700 \times 7 = 4900

The allowance is 4900 won.

3Take the saving off

#13 Convert to Algebra 6.RP.A.3

The question wants what is left, not the allowance itself.

4900 - 1400 = 3500

3500 won is left.

Answer: 3500 won

4 · Reviewdoes it hold up?

2/7 of 4900 won is 1400 won, the amount saved -- and what is left, 3500 won, is the other 5/7.

Another way: The leftover is 5 of the 7 parts, so 700 times 5 gives 3500 won without ever naming the whole allowance.

Standardsmin grade 6

6.NS.A.1 Divide fractions by fractions — Dividing by the fraction to recover the whole.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Reading the saving as a share of the allowance.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the allowance from one known part.

💡Takeaway. Find what one part is worth and the whole is easy. Then check whether the question wanted the whole or what is left of it.

Variant 2 easy answer: 2500 won

Suho saved $\frac{3}{8}$ of his allowance. If he saved $1500$ won, how much of his allowance is left after saving?

Show solution

1 · Understandwhat's really being asked

Saving 3/8 of an allowance came to 1500 won. We want what is left after that saving.

Givens

The saving is 3/8 of the allowance.
The saving is 1500 won.

Unknowns

How much of the allowance is left.

Constraints

Only the saved part is given; the allowance itself is not.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#13 Convert to Algebra

Taking a fraction of the allowance produced the saving, so dividing by that fraction takes it back. Then read the question again -- it asks for the remainder, not the allowance.

3 · Execute3 carry out the plan

1Find one part

#7 Identify Subproblems 6.NS.A.1

1500 won is 3 of the 8 equal parts, so one part is that divided by 3.

1500 \div 3 = 500

Each 8th is 500 won.

2Build the whole allowance

#11 Work Backwards 6.EE.B.7

There are 8 of those parts in the whole.

500 \times 8 = 4000

The allowance is 4000 won.

3Take the saving off

#13 Convert to Algebra 6.RP.A.3

The question wants what is left, not the allowance itself.

4000 - 1500 = 2500

2500 won is left.

Answer: 2500 won

4 · Reviewdoes it hold up?

3/8 of 4000 won is 1500 won, the amount saved -- and what is left, 2500 won, is the other 5/8.

Another way: The leftover is 5 of the 8 parts, so 500 times 5 gives 2500 won without ever naming the whole allowance.

Standardsmin grade 6

6.NS.A.1 Divide fractions by fractions — Dividing by the fraction to recover the whole.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Reading the saving as a share of the allowance.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the allowance from one known part.

💡Takeaway. Find what one part is worth and the whole is easy. Then check whether the question wanted the whole or what is left of it.

Variant 3 easy answer: 4200 won

Suho saved $\frac{3}{10}$ of his allowance. If he saved $1800$ won, how much of his allowance is left after saving?

Show solution

1 · Understandwhat's really being asked

Saving 3/10 of an allowance came to 1800 won. We want what is left after that saving.

Givens

The saving is 3/10 of the allowance.
The saving is 1800 won.

Unknowns

How much of the allowance is left.

Constraints

Only the saved part is given; the allowance itself is not.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#13 Convert to Algebra

Taking a fraction of the allowance produced the saving, so dividing by that fraction takes it back. Then read the question again -- it asks for the remainder, not the allowance.

3 · Execute3 carry out the plan

1Find one part

#7 Identify Subproblems 6.NS.A.1

1800 won is 3 of the 10 equal parts, so one part is that divided by 3.

1800 \div 3 = 600

Each 10th is 600 won.

2Build the whole allowance

#11 Work Backwards 6.EE.B.7

There are 10 of those parts in the whole.

600 \times 10 = 6000

The allowance is 6000 won.

3Take the saving off

#13 Convert to Algebra 6.RP.A.3

The question wants what is left, not the allowance itself.

6000 - 1800 = 4200

4200 won is left.

Answer: 4200 won

4 · Reviewdoes it hold up?

3/10 of 6000 won is 1800 won, the amount saved -- and what is left, 4200 won, is the other 7/10.

Another way: The leftover is 7 of the 10 parts, so 600 times 7 gives 4200 won without ever naming the whole allowance.

Standardsmin grade 6

6.NS.A.1 Divide fractions by fractions — Dividing by the fraction to recover the whole.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Reading the saving as a share of the allowance.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the allowance from one known part.

💡Takeaway. Find what one part is worth and the whole is easy. Then check whether the question wanted the whole or what is left of it.

Variant 4 easy answer: 3000 won

Suho saved $\frac{2}{5}$ of his allowance. If he saved $2000$ won, how much of his allowance is left after saving?

Show solution

1 · Understandwhat's really being asked

Saving 2/5 of an allowance came to 2000 won. We want what is left after that saving.

Givens

The saving is 2/5 of the allowance.
The saving is 2000 won.

Unknowns

How much of the allowance is left.

Constraints

Only the saved part is given; the allowance itself is not.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#13 Convert to Algebra

Taking a fraction of the allowance produced the saving, so dividing by that fraction takes it back. Then read the question again -- it asks for the remainder, not the allowance.

3 · Execute3 carry out the plan

1Find one part

#7 Identify Subproblems 6.NS.A.1

2000 won is 2 of the 5 equal parts, so one part is that divided by 2.

2000 \div 2 = 1000

Each 5th is 1000 won.

2Build the whole allowance

#11 Work Backwards 6.EE.B.7

There are 5 of those parts in the whole.

1000 \times 5 = 5000

The allowance is 5000 won.

3Take the saving off

#13 Convert to Algebra 6.RP.A.3

The question wants what is left, not the allowance itself.

5000 - 2000 = 3000

3000 won is left.

Answer: 3000 won

4 · Reviewdoes it hold up?

2/5 of 5000 won is 2000 won, the amount saved -- and what is left, 3000 won, is the other 3/5.

Another way: The leftover is 3 of the 5 parts, so 1000 times 3 gives 3000 won without ever naming the whole allowance.

Standardsmin grade 6

6.NS.A.1 Divide fractions by fractions — Dividing by the fraction to recover the whole.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Reading the saving as a share of the allowance.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the allowance from one known part.

💡Takeaway. Find what one part is worth and the whole is easy. Then check whether the question wanted the whole or what is left of it.

Variant 5 medium answer: 2800 won

Suho saved $\frac{3}{7}$ of his allowance. If he saved $2100$ won, how much of his allowance is left after saving?

Show solution

1 · Understandwhat's really being asked

Saving 3/7 of an allowance came to 2100 won. We want what is left after that saving.

Givens

The saving is 3/7 of the allowance.
The saving is 2100 won.

Unknowns

How much of the allowance is left.

Constraints

Only the saved part is given; the allowance itself is not.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#13 Convert to Algebra

Taking a fraction of the allowance produced the saving, so dividing by that fraction takes it back. Then read the question again -- it asks for the remainder, not the allowance.

3 · Execute3 carry out the plan

1Find one part

#7 Identify Subproblems 6.NS.A.1

2100 won is 3 of the 7 equal parts, so one part is that divided by 3.

2100 \div 3 = 700

Each 7th is 700 won.

2Build the whole allowance

#11 Work Backwards 6.EE.B.7

There are 7 of those parts in the whole.

700 \times 7 = 4900

The allowance is 4900 won.

3Take the saving off

#13 Convert to Algebra 6.RP.A.3

The question wants what is left, not the allowance itself.

4900 - 2100 = 2800

2800 won is left.

Answer: 2800 won

4 · Reviewdoes it hold up?

3/7 of 4900 won is 2100 won, the amount saved -- and what is left, 2800 won, is the other 4/7.

Another way: The leftover is 4 of the 7 parts, so 700 times 4 gives 2800 won without ever naming the whole allowance.

Standardsmin grade 6

6.NS.A.1 Divide fractions by fractions — Dividing by the fraction to recover the whole.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Reading the saving as a share of the allowance.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the allowance from one known part.

💡Takeaway. Find what one part is worth and the whole is easy. Then check whether the question wanted the whole or what is left of it.

Variant 6 medium answer: 4200 won

Suho saved $\frac{4}{11}$ of his allowance. If he saved $2400$ won, how much of his allowance is left after saving?

Show solution

1 · Understandwhat's really being asked

Saving 4/11 of an allowance came to 2400 won. We want what is left after that saving.

Givens

The saving is 4/11 of the allowance.
The saving is 2400 won.

Unknowns

How much of the allowance is left.

Constraints

Only the saved part is given; the allowance itself is not.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#13 Convert to Algebra

Taking a fraction of the allowance produced the saving, so dividing by that fraction takes it back. Then read the question again -- it asks for the remainder, not the allowance.

3 · Execute3 carry out the plan

1Find one part

#7 Identify Subproblems 6.NS.A.1

2400 won is 4 of the 11 equal parts, so one part is that divided by 4.

2400 \div 4 = 600

Each 11th is 600 won.

2Build the whole allowance

#11 Work Backwards 6.EE.B.7

There are 11 of those parts in the whole.

600 \times 11 = 6600

The allowance is 6600 won.

3Take the saving off

#13 Convert to Algebra 6.RP.A.3

The question wants what is left, not the allowance itself.

6600 - 2400 = 4200

4200 won is left.

Answer: 4200 won

4 · Reviewdoes it hold up?

4/11 of 6600 won is 2400 won, the amount saved -- and what is left, 4200 won, is the other 7/11.

Another way: The leftover is 7 of the 11 parts, so 600 times 7 gives 4200 won without ever naming the whole allowance.

Standardsmin grade 6

6.NS.A.1 Divide fractions by fractions — Dividing by the fraction to recover the whole.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Reading the saving as a share of the allowance.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the allowance from one known part.

💡Takeaway. Find what one part is worth and the whole is easy. Then check whether the question wanted the whole or what is left of it.

Variant 7 medium answer: 3500 won

Suho saved $\frac{5}{12}$ of his allowance. If he saved $2500$ won, how much of his allowance is left after saving?

Show solution

1 · Understandwhat's really being asked

Saving 5/12 of an allowance came to 2500 won. We want what is left after that saving.

Givens

The saving is 5/12 of the allowance.
The saving is 2500 won.

Unknowns

How much of the allowance is left.

Constraints

Only the saved part is given; the allowance itself is not.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#13 Convert to Algebra

Taking a fraction of the allowance produced the saving, so dividing by that fraction takes it back. Then read the question again -- it asks for the remainder, not the allowance.

3 · Execute3 carry out the plan

1Find one part

#7 Identify Subproblems 6.NS.A.1

2500 won is 5 of the 12 equal parts, so one part is that divided by 5.

2500 \div 5 = 500

Each 12th is 500 won.

2Build the whole allowance

#11 Work Backwards 6.EE.B.7

There are 12 of those parts in the whole.

500 \times 12 = 6000

The allowance is 6000 won.

3Take the saving off

#13 Convert to Algebra 6.RP.A.3

The question wants what is left, not the allowance itself.

6000 - 2500 = 3500

3500 won is left.

Answer: 3500 won

4 · Reviewdoes it hold up?

5/12 of 6000 won is 2500 won, the amount saved -- and what is left, 3500 won, is the other 7/12.

Another way: The leftover is 7 of the 12 parts, so 500 times 7 gives 3500 won without ever naming the whole allowance.

Standardsmin grade 6

6.NS.A.1 Divide fractions by fractions — Dividing by the fraction to recover the whole.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Reading the saving as a share of the allowance.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the allowance from one known part.

💡Takeaway. Find what one part is worth and the whole is easy. Then check whether the question wanted the whole or what is left of it.

Variant 8 medium answer: 3200 won

Suho saved $\frac{7}{15}$ of his allowance. If he saved $2800$ won, how much of his allowance is left after saving?

Show solution

1 · Understandwhat's really being asked

Saving 7/15 of an allowance came to 2800 won. We want what is left after that saving.

Givens

The saving is 7/15 of the allowance.
The saving is 2800 won.

Unknowns

How much of the allowance is left.

Constraints

Only the saved part is given; the allowance itself is not.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#13 Convert to Algebra

Taking a fraction of the allowance produced the saving, so dividing by that fraction takes it back. Then read the question again -- it asks for the remainder, not the allowance.

3 · Execute3 carry out the plan

1Find one part

#7 Identify Subproblems 6.NS.A.1

2800 won is 7 of the 15 equal parts, so one part is that divided by 7.

2800 \div 7 = 400

Each 15th is 400 won.

2Build the whole allowance

#11 Work Backwards 6.EE.B.7

There are 15 of those parts in the whole.

400 \times 15 = 6000

The allowance is 6000 won.

3Take the saving off

#13 Convert to Algebra 6.RP.A.3

The question wants what is left, not the allowance itself.

6000 - 2800 = 3200

3200 won is left.

Answer: 3200 won

4 · Reviewdoes it hold up?

7/15 of 6000 won is 2800 won, the amount saved -- and what is left, 3200 won, is the other 8/15.

Another way: The leftover is 8 of the 15 parts, so 400 times 8 gives 3200 won without ever naming the whole allowance.

Standardsmin grade 6

6.NS.A.1 Divide fractions by fractions — Dividing by the fraction to recover the whole.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Reading the saving as a share of the allowance.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the allowance from one known part.

💡Takeaway. Find what one part is worth and the whole is easy. Then check whether the question wanted the whole or what is left of it.

Variant 9 hard answer: 4000 won

Suho saved $\frac{4}{9}$ of his allowance. If he saved $3200$ won, how much of his allowance is left after saving?

Show solution

1 · Understandwhat's really being asked

Saving 4/9 of an allowance came to 3200 won. We want what is left after that saving.

Givens

The saving is 4/9 of the allowance.
The saving is 3200 won.

Unknowns

How much of the allowance is left.

Constraints

Only the saved part is given; the allowance itself is not.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#13 Convert to Algebra

Taking a fraction of the allowance produced the saving, so dividing by that fraction takes it back. Then read the question again -- it asks for the remainder, not the allowance.

3 · Execute3 carry out the plan

1Find one part

#7 Identify Subproblems 6.NS.A.1

3200 won is 4 of the 9 equal parts, so one part is that divided by 4.

3200 \div 4 = 800

Each 9th is 800 won.

2Build the whole allowance

#11 Work Backwards 6.EE.B.7

There are 9 of those parts in the whole.

800 \times 9 = 7200

The allowance is 7200 won.

3Take the saving off

#13 Convert to Algebra 6.RP.A.3

The question wants what is left, not the allowance itself.

7200 - 3200 = 4000

4000 won is left.

Answer: 4000 won

4 · Reviewdoes it hold up?

4/9 of 7200 won is 3200 won, the amount saved -- and what is left, 4000 won, is the other 5/9.

Another way: The leftover is 5 of the 9 parts, so 800 times 5 gives 4000 won without ever naming the whole allowance.

Standardsmin grade 6

6.NS.A.1 Divide fractions by fractions — Dividing by the fraction to recover the whole.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Reading the saving as a share of the allowance.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the allowance from one known part.

💡Takeaway. Find what one part is worth and the whole is easy. Then check whether the question wanted the whole or what is left of it.

Variant 10 hard answer: 2500 won

Suho saved $\frac{7}{12}$ of his allowance. If he saved $3500$ won, how much of his allowance is left after saving?

Show solution

1 · Understandwhat's really being asked

Saving 7/12 of an allowance came to 3500 won. We want what is left after that saving.

Givens

The saving is 7/12 of the allowance.
The saving is 3500 won.

Unknowns

How much of the allowance is left.

Constraints

Only the saved part is given; the allowance itself is not.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#13 Convert to Algebra

Taking a fraction of the allowance produced the saving, so dividing by that fraction takes it back. Then read the question again -- it asks for the remainder, not the allowance.

3 · Execute3 carry out the plan

1Find one part

#7 Identify Subproblems 6.NS.A.1

3500 won is 7 of the 12 equal parts, so one part is that divided by 7.

3500 \div 7 = 500

Each 12th is 500 won.

2Build the whole allowance

#11 Work Backwards 6.EE.B.7

There are 12 of those parts in the whole.

500 \times 12 = 6000

The allowance is 6000 won.

3Take the saving off

#13 Convert to Algebra 6.RP.A.3

The question wants what is left, not the allowance itself.

6000 - 3500 = 2500

2500 won is left.

Answer: 2500 won

4 · Reviewdoes it hold up?

7/12 of 6000 won is 3500 won, the amount saved -- and what is left, 2500 won, is the other 5/12.

Another way: The leftover is 5 of the 12 parts, so 500 times 5 gives 2500 won without ever naming the whole allowance.

Standardsmin grade 6

6.NS.A.1 Divide fractions by fractions — Dividing by the fraction to recover the whole.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Reading the saving as a share of the allowance.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the allowance from one known part.

💡Takeaway. Find what one part is worth and the whole is easy. Then check whether the question wanted the whole or what is left of it.

Variant 11 hard answer: 2400 won

Suho saved $\frac{5}{8}$ of his allowance. If he saved $4000$ won, how much of his allowance is left after saving?

Show solution

1 · Understandwhat's really being asked

Saving 5/8 of an allowance came to 4000 won. We want what is left after that saving.

Givens

The saving is 5/8 of the allowance.
The saving is 4000 won.

Unknowns

How much of the allowance is left.

Constraints

Only the saved part is given; the allowance itself is not.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#13 Convert to Algebra

Taking a fraction of the allowance produced the saving, so dividing by that fraction takes it back. Then read the question again -- it asks for the remainder, not the allowance.

3 · Execute3 carry out the plan

1Find one part

#7 Identify Subproblems 6.NS.A.1

4000 won is 5 of the 8 equal parts, so one part is that divided by 5.

4000 \div 5 = 800

Each 8th is 800 won.

2Build the whole allowance

#11 Work Backwards 6.EE.B.7

There are 8 of those parts in the whole.

800 \times 8 = 6400

The allowance is 6400 won.

3Take the saving off

#13 Convert to Algebra 6.RP.A.3

The question wants what is left, not the allowance itself.

6400 - 4000 = 2400

2400 won is left.

Answer: 2400 won

4 · Reviewdoes it hold up?

5/8 of 6400 won is 4000 won, the amount saved -- and what is left, 2400 won, is the other 3/8.

Another way: The leftover is 3 of the 8 parts, so 800 times 3 gives 2400 won without ever naming the whole allowance.

Standardsmin grade 6

6.NS.A.1 Divide fractions by fractions — Dividing by the fraction to recover the whole.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Reading the saving as a share of the allowance.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the allowance from one known part.

💡Takeaway. Find what one part is worth and the whole is easy. Then check whether the question wanted the whole or what is left of it.

Variant 12 hard answer: 4000 won

Suho saved $\frac{5}{9}$ of his allowance. If he saved $5000$ won, how much of his allowance is left after saving?

Show solution

1 · Understandwhat's really being asked

Saving 5/9 of an allowance came to 5000 won. We want what is left after that saving.

Givens

The saving is 5/9 of the allowance.
The saving is 5000 won.

Unknowns

How much of the allowance is left.

Constraints

Only the saved part is given; the allowance itself is not.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#13 Convert to Algebra

Taking a fraction of the allowance produced the saving, so dividing by that fraction takes it back. Then read the question again -- it asks for the remainder, not the allowance.

3 · Execute3 carry out the plan

1Find one part

#7 Identify Subproblems 6.NS.A.1

5000 won is 5 of the 9 equal parts, so one part is that divided by 5.

5000 \div 5 = 1000

Each 9th is 1000 won.

2Build the whole allowance

#11 Work Backwards 6.EE.B.7

There are 9 of those parts in the whole.

1000 \times 9 = 9000

The allowance is 9000 won.

3Take the saving off

#13 Convert to Algebra 6.RP.A.3

The question wants what is left, not the allowance itself.

9000 - 5000 = 4000

4000 won is left.

Answer: 4000 won

4 · Reviewdoes it hold up?

5/9 of 9000 won is 5000 won, the amount saved -- and what is left, 4000 won, is the other 4/9.

Another way: The leftover is 4 of the 9 parts, so 1000 times 4 gives 4000 won without ever naming the whole allowance.

Standardsmin grade 6

6.NS.A.1 Divide fractions by fractions — Dividing by the fraction to recover the whole.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Reading the saving as a share of the allowance.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the allowance from one known part.

💡Takeaway. Find what one part is worth and the whole is easy. Then check whether the question wanted the whole or what is left of it.