← The whole is one; find the remaining fraction · Part-Whole Fraction Reasoning

The whole is one; find the remaining fraction · 12 practice problems

3.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: 2/7

Sea spent $\dfrac{2}{7}$ of the money she had on snacks, then spent $\dfrac{3}{5}$ of what was left on school supplies. Write, as a fraction, what part of the money she started with is left.

Show solution

1 · Understandwhat's really being asked

Sea spends 2/7 of her money on snacks, then spends 3/5 of what is left on school supplies. Write, as a fraction of the money she started with, how much is left.

Givens

The starting amount of money counts as the whole, 1
She spends 2/7 of the whole on snacks
Then she spends 3/5 of what remains on school supplies

Unknowns

The fraction of the original money that is left at the end

Constraints

The whole is 1
The 3/5 is taken from the remainder after snacks, not from the original whole

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Solve it in stages: first the money left after snacks (whole minus 2/7), then the fraction of that remainder still left after spending 3/5 of it. A bar model makes the 'fraction of what is left' step clear.

3 · Execute3 carry out the plan

1Money left after snacks

#7 Identify Subproblems 3.NF.A.3

The whole is 1 = 7/7. Spending 2/7 leaves 7/7 - 2/7 = 5/7 of the original money.

1 - \dfrac{2}{7} = \dfrac{7}{7} - \dfrac{2}{7} = \dfrac{5}{7}

Since the whole is 1, what is left is simply 1 minus the part spent.

2Fraction of the remainder still left

#1 Draw a Diagram 3.NF.A.1

She spends 3/5 of what was left, so 5/5 - 3/5 = 2/5 of the remainder stays. The 2/5 applies to the 5/7 she had after snacks.

1 - \dfrac{3}{5} = \dfrac{2}{5} \text{ of the } \dfrac{5}{7} \text{ left}

Spending 3 of every 5 equal parts of the remainder leaves 2 of those 5 parts.

3Combine to a fraction of the whole

#7 Identify Subproblems 3.NF.A.1

Take 2/5 of 5/7. Split the 5/7 into 5 equal parts of 1/7 each; 2 of them are left, which is 2/7 of the original money.

\dfrac{2}{5} \text{ of } \dfrac{5}{7} = \dfrac{2 \times 5}{5 \times 7} = \dfrac{2}{7}

5 7ths split into 5 equal pieces gives pieces of 1/7, and keeping 2 of them is 2/7.

Answer: 2/7

4 · Reviewdoes it hold up?

After snacks 5/7 remains; she then spends a bit more, so the final amount must be less than 5/7. The answer 2/7 is less than 5/7 and still positive, which fits. Check: spent 2/7 on snacks plus 3/7 on supplies (3/5 of 5/7) = 5/7 spent, leaving 2/7.

Another way: Work it as parts of 7ths (tool 15): the 5/7 left is 5 7ths; spending 3/5 of those 5 7ths means spending exactly 3 7ths, so 5/7 - 3/7 = 2/7 remains.

Standardsmin grade 3

3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Taking 2/5 of the 5/7 remainder as 2 equal parts of 1/7
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Computing 1 - 2/7 = 5/7 with the whole written as 7/7

💡Takeaway. The whole is just 1, so subtract each part spent in turn to find the fraction that is left!

Variant 2 easy answer: 3/8

Sea spent $\dfrac{1}{8}$ of the money she had on snacks, then spent $\dfrac{4}{7}$ of what was left on school supplies. Write, as a fraction, what part of the money she started with is left.

Show solution

1 · Understandwhat's really being asked

Sea spends 1/8 of her money on snacks, then spends 4/7 of what is left on school supplies. Write, as a fraction of the money she started with, how much is left.

Givens

The starting amount of money counts as the whole, 1
She spends 1/8 of the whole on snacks
Then she spends 4/7 of what remains on school supplies

Unknowns

The fraction of the original money that is left at the end

Constraints

The whole is 1
The 4/7 is taken from the remainder after snacks, not from the original whole

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Solve it in stages: first the money left after snacks (whole minus 1/8), then the fraction of that remainder still left after spending 4/7 of it. A bar model makes the 'fraction of what is left' step clear.

3 · Execute3 carry out the plan

1Money left after snacks

#7 Identify Subproblems 3.NF.A.3

The whole is 1 = 8/8. Spending 1/8 leaves 8/8 - 1/8 = 7/8 of the original money.

1 - \dfrac{1}{8} = \dfrac{8}{8} - \dfrac{1}{8} = \dfrac{7}{8}

Since the whole is 1, what is left is simply 1 minus the part spent.

2Fraction of the remainder still left

#1 Draw a Diagram 3.NF.A.1

She spends 4/7 of what was left, so 7/7 - 4/7 = 3/7 of the remainder stays. The 3/7 applies to the 7/8 she had after snacks.

1 - \dfrac{4}{7} = \dfrac{3}{7} \text{ of the } \dfrac{7}{8} \text{ left}

Spending 4 of every 7 equal parts of the remainder leaves 3 of those 7 parts.

3Combine to a fraction of the whole

#7 Identify Subproblems 3.NF.A.1

Take 3/7 of 7/8. Split the 7/8 into 7 equal parts of 1/8 each; 3 of them are left, which is 3/8 of the original money.

\dfrac{3}{7} \text{ of } \dfrac{7}{8} = \dfrac{3 \times 7}{7 \times 8} = \dfrac{3}{8}

7 8ths split into 7 equal pieces gives pieces of 1/8, and keeping 3 of them is 3/8.

Answer: 3/8

4 · Reviewdoes it hold up?

After snacks 7/8 remains; she then spends a bit more, so the final amount must be less than 7/8. The answer 3/8 is less than 7/8 and still positive, which fits. Check: spent 1/8 on snacks plus 4/8 on supplies (4/7 of 7/8) = 5/8 spent, leaving 3/8.

Another way: Work it as parts of 8ths (tool 15): the 7/8 left is 7 8ths; spending 4/7 of those 7 8ths means spending exactly 4 8ths, so 7/8 - 4/8 = 3/8 remains.

Standardsmin grade 3

3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Taking 3/7 of the 7/8 remainder as 3 equal parts of 1/8
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Computing 1 - 1/8 = 7/8 with the whole written as 8/8

💡Takeaway. The whole is just 1, so subtract each part spent in turn to find the fraction that is left!

Variant 3 easy answer: 4/9

Sea spent $\dfrac{2}{9}$ of the money she had on snacks, then spent $\dfrac{3}{7}$ of what was left on school supplies. Write, as a fraction, what part of the money she started with is left.

Show solution

1 · Understandwhat's really being asked

Sea spends 2/9 of her money on snacks, then spends 3/7 of what is left on school supplies. Write, as a fraction of the money she started with, how much is left.

Givens

The starting amount of money counts as the whole, 1
She spends 2/9 of the whole on snacks
Then she spends 3/7 of what remains on school supplies

Unknowns

The fraction of the original money that is left at the end

Constraints

The whole is 1
The 3/7 is taken from the remainder after snacks, not from the original whole

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Solve it in stages: first the money left after snacks (whole minus 2/9), then the fraction of that remainder still left after spending 3/7 of it. A bar model makes the 'fraction of what is left' step clear.

3 · Execute3 carry out the plan

1Money left after snacks

#7 Identify Subproblems 3.NF.A.3

The whole is 1 = 9/9. Spending 2/9 leaves 9/9 - 2/9 = 7/9 of the original money.

1 - \dfrac{2}{9} = \dfrac{9}{9} - \dfrac{2}{9} = \dfrac{7}{9}

Since the whole is 1, what is left is simply 1 minus the part spent.

2Fraction of the remainder still left

#1 Draw a Diagram 3.NF.A.1

She spends 3/7 of what was left, so 7/7 - 3/7 = 4/7 of the remainder stays. The 4/7 applies to the 7/9 she had after snacks.

1 - \dfrac{3}{7} = \dfrac{4}{7} \text{ of the } \dfrac{7}{9} \text{ left}

Spending 3 of every 7 equal parts of the remainder leaves 4 of those 7 parts.

3Combine to a fraction of the whole

#7 Identify Subproblems 3.NF.A.1

Take 4/7 of 7/9. Split the 7/9 into 7 equal parts of 1/9 each; 4 of them are left, which is 4/9 of the original money.

\dfrac{4}{7} \text{ of } \dfrac{7}{9} = \dfrac{4 \times 7}{7 \times 9} = \dfrac{4}{9}

7 9ths split into 7 equal pieces gives pieces of 1/9, and keeping 4 of them is 4/9.

Answer: 4/9

4 · Reviewdoes it hold up?

After snacks 7/9 remains; she then spends a bit more, so the final amount must be less than 7/9. The answer 4/9 is less than 7/9 and still positive, which fits. Check: spent 2/9 on snacks plus 3/9 on supplies (3/7 of 7/9) = 5/9 spent, leaving 4/9.

Another way: Work it as parts of 9ths (tool 15): the 7/9 left is 7 9ths; spending 3/7 of those 7 9ths means spending exactly 3 9ths, so 7/9 - 3/9 = 4/9 remains.

Standardsmin grade 3

3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Taking 4/7 of the 7/9 remainder as 4 equal parts of 1/9
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Computing 1 - 2/9 = 7/9 with the whole written as 9/9

💡Takeaway. The whole is just 1, so subtract each part spent in turn to find the fraction that is left!

Variant 4 medium answer: 4/10

Sea spent $\dfrac{4}{10}$ of the money she had on snacks, then spent $\dfrac{2}{6}$ of what was left on school supplies. Write, as a fraction, what part of the money she started with is left.

Show solution

1 · Understandwhat's really being asked

Sea spends 4/10 of her money on snacks, then spends 2/6 of what is left on school supplies. Write, as a fraction of the money she started with, how much is left.

Givens

The starting amount of money counts as the whole, 1
She spends 4/10 of the whole on snacks
Then she spends 2/6 of what remains on school supplies

Unknowns

The fraction of the original money that is left at the end

Constraints

The whole is 1
The 2/6 is taken from the remainder after snacks, not from the original whole

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Solve it in stages: first the money left after snacks (whole minus 4/10), then the fraction of that remainder still left after spending 2/6 of it. A bar model makes the 'fraction of what is left' step clear.

3 · Execute3 carry out the plan

1Money left after snacks

#7 Identify Subproblems 3.NF.A.3

The whole is 1 = 10/10. Spending 4/10 leaves 10/10 - 4/10 = 6/10 of the original money.

1 - \dfrac{4}{10} = \dfrac{10}{10} - \dfrac{4}{10} = \dfrac{6}{10}

Since the whole is 1, what is left is simply 1 minus the part spent.

2Fraction of the remainder still left

#1 Draw a Diagram 3.NF.A.1

She spends 2/6 of what was left, so 6/6 - 2/6 = 4/6 of the remainder stays. The 4/6 applies to the 6/10 she had after snacks.

1 - \dfrac{2}{6} = \dfrac{4}{6} \text{ of the } \dfrac{6}{10} \text{ left}

Spending 2 of every 6 equal parts of the remainder leaves 4 of those 6 parts.

3Combine to a fraction of the whole

#7 Identify Subproblems 3.NF.A.1

Take 4/6 of 6/10. Split the 6/10 into 6 equal parts of 1/10 each; 4 of them are left, which is 4/10 of the original money.

\dfrac{4}{6} \text{ of } \dfrac{6}{10} = \dfrac{4 \times 6}{6 \times 10} = \dfrac{4}{10}

6 10ths split into 6 equal pieces gives pieces of 1/10, and keeping 4 of them is 4/10.

Answer: 4/10

4 · Reviewdoes it hold up?

After snacks 6/10 remains; she then spends a bit more, so the final amount must be less than 6/10. The answer 4/10 is less than 6/10 and still positive, which fits. Check: spent 4/10 on snacks plus 2/10 on supplies (2/6 of 6/10) = 6/10 spent, leaving 4/10.

Another way: Work it as parts of 10ths (tool 15): the 6/10 left is 6 10ths; spending 2/6 of those 6 10ths means spending exactly 2 10ths, so 6/10 - 2/10 = 4/10 remains.

Standardsmin grade 3

3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Taking 4/6 of the 6/10 remainder as 4 equal parts of 1/10
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Computing 1 - 4/10 = 6/10 with the whole written as 10/10

💡Takeaway. The whole is just 1, so subtract each part spent in turn to find the fraction that is left!

Variant 5 easy answer: 3/10

Sea spent $\dfrac{3}{10}$ of the money she had on snacks, then spent $\dfrac{4}{7}$ of what was left on school supplies. Write, as a fraction, what part of the money she started with is left.

Show solution

1 · Understandwhat's really being asked

Sea spends 3/10 of her money on snacks, then spends 4/7 of what is left on school supplies. Write, as a fraction of the money she started with, how much is left.

Givens

The starting amount of money counts as the whole, 1
She spends 3/10 of the whole on snacks
Then she spends 4/7 of what remains on school supplies

Unknowns

The fraction of the original money that is left at the end

Constraints

The whole is 1
The 4/7 is taken from the remainder after snacks, not from the original whole

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Solve it in stages: first the money left after snacks (whole minus 3/10), then the fraction of that remainder still left after spending 4/7 of it. A bar model makes the 'fraction of what is left' step clear.

3 · Execute3 carry out the plan

1Money left after snacks

#7 Identify Subproblems 3.NF.A.3

The whole is 1 = 10/10. Spending 3/10 leaves 10/10 - 3/10 = 7/10 of the original money.

1 - \dfrac{3}{10} = \dfrac{10}{10} - \dfrac{3}{10} = \dfrac{7}{10}

Since the whole is 1, what is left is simply 1 minus the part spent.

2Fraction of the remainder still left

#1 Draw a Diagram 3.NF.A.1

She spends 4/7 of what was left, so 7/7 - 4/7 = 3/7 of the remainder stays. The 3/7 applies to the 7/10 she had after snacks.

1 - \dfrac{4}{7} = \dfrac{3}{7} \text{ of the } \dfrac{7}{10} \text{ left}

Spending 4 of every 7 equal parts of the remainder leaves 3 of those 7 parts.

3Combine to a fraction of the whole

#7 Identify Subproblems 3.NF.A.1

Take 3/7 of 7/10. Split the 7/10 into 7 equal parts of 1/10 each; 3 of them are left, which is 3/10 of the original money.

\dfrac{3}{7} \text{ of } \dfrac{7}{10} = \dfrac{3 \times 7}{7 \times 10} = \dfrac{3}{10}

7 10ths split into 7 equal pieces gives pieces of 1/10, and keeping 3 of them is 3/10.

Answer: 3/10

4 · Reviewdoes it hold up?

After snacks 7/10 remains; she then spends a bit more, so the final amount must be less than 7/10. The answer 3/10 is less than 7/10 and still positive, which fits. Check: spent 3/10 on snacks plus 4/10 on supplies (4/7 of 7/10) = 7/10 spent, leaving 3/10.

Another way: Work it as parts of 10ths (tool 15): the 7/10 left is 7 10ths; spending 4/7 of those 7 10ths means spending exactly 4 10ths, so 7/10 - 4/10 = 3/10 remains.

Standardsmin grade 3

3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Taking 3/7 of the 7/10 remainder as 3 equal parts of 1/10
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Computing 1 - 3/10 = 7/10 with the whole written as 10/10

💡Takeaway. The whole is just 1, so subtract each part spent in turn to find the fraction that is left!

Variant 6 medium answer: 2/11

Sea spent $\dfrac{4}{11}$ of the money she had on snacks, then spent $\dfrac{5}{7}$ of what was left on school supplies. Write, as a fraction, what part of the money she started with is left.

Show solution

1 · Understandwhat's really being asked

Sea spends 4/11 of her money on snacks, then spends 5/7 of what is left on school supplies. Write, as a fraction of the money she started with, how much is left.

Givens

The starting amount of money counts as the whole, 1
She spends 4/11 of the whole on snacks
Then she spends 5/7 of what remains on school supplies

Unknowns

The fraction of the original money that is left at the end

Constraints

The whole is 1
The 5/7 is taken from the remainder after snacks, not from the original whole

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Solve it in stages: first the money left after snacks (whole minus 4/11), then the fraction of that remainder still left after spending 5/7 of it. A bar model makes the 'fraction of what is left' step clear.

3 · Execute3 carry out the plan

1Money left after snacks

#7 Identify Subproblems 3.NF.A.3

The whole is 1 = 11/11. Spending 4/11 leaves 11/11 - 4/11 = 7/11 of the original money.

1 - \dfrac{4}{11} = \dfrac{11}{11} - \dfrac{4}{11} = \dfrac{7}{11}

Since the whole is 1, what is left is simply 1 minus the part spent.

2Fraction of the remainder still left

#1 Draw a Diagram 3.NF.A.1

She spends 5/7 of what was left, so 7/7 - 5/7 = 2/7 of the remainder stays. The 2/7 applies to the 7/11 she had after snacks.

1 - \dfrac{5}{7} = \dfrac{2}{7} \text{ of the } \dfrac{7}{11} \text{ left}

Spending 5 of every 7 equal parts of the remainder leaves 2 of those 7 parts.

3Combine to a fraction of the whole

#7 Identify Subproblems 3.NF.A.1

Take 2/7 of 7/11. Split the 7/11 into 7 equal parts of 1/11 each; 2 of them are left, which is 2/11 of the original money.

\dfrac{2}{7} \text{ of } \dfrac{7}{11} = \dfrac{2 \times 7}{7 \times 11} = \dfrac{2}{11}

7 11ths split into 7 equal pieces gives pieces of 1/11, and keeping 2 of them is 2/11.

Answer: 2/11

4 · Reviewdoes it hold up?

After snacks 7/11 remains; she then spends a bit more, so the final amount must be less than 7/11. The answer 2/11 is less than 7/11 and still positive, which fits. Check: spent 4/11 on snacks plus 5/11 on supplies (5/7 of 7/11) = 9/11 spent, leaving 2/11.

Another way: Work it as parts of 11ths (tool 15): the 7/11 left is 7 11ths; spending 5/7 of those 7 11ths means spending exactly 5 11ths, so 7/11 - 5/11 = 2/11 remains.

Standardsmin grade 3

3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Taking 2/7 of the 7/11 remainder as 2 equal parts of 1/11
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Computing 1 - 4/11 = 7/11 with the whole written as 11/11

💡Takeaway. The whole is just 1, so subtract each part spent in turn to find the fraction that is left!

Variant 7 medium answer: 5/12

Sea spent $\dfrac{5}{12}$ of the money she had on snacks, then spent $\dfrac{2}{7}$ of what was left on school supplies. Write, as a fraction, what part of the money she started with is left.

Show solution

1 · Understandwhat's really being asked

Sea spends 5/12 of her money on snacks, then spends 2/7 of what is left on school supplies. Write, as a fraction of the money she started with, how much is left.

Givens

The starting amount of money counts as the whole, 1
She spends 5/12 of the whole on snacks
Then she spends 2/7 of what remains on school supplies

Unknowns

The fraction of the original money that is left at the end

Constraints

The whole is 1
The 2/7 is taken from the remainder after snacks, not from the original whole

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Solve it in stages: first the money left after snacks (whole minus 5/12), then the fraction of that remainder still left after spending 2/7 of it. A bar model makes the 'fraction of what is left' step clear.

3 · Execute3 carry out the plan

1Money left after snacks

#7 Identify Subproblems 3.NF.A.3

The whole is 1 = 12/12. Spending 5/12 leaves 12/12 - 5/12 = 7/12 of the original money.

1 - \dfrac{5}{12} = \dfrac{12}{12} - \dfrac{5}{12} = \dfrac{7}{12}

Since the whole is 1, what is left is simply 1 minus the part spent.

2Fraction of the remainder still left

#1 Draw a Diagram 3.NF.A.1

She spends 2/7 of what was left, so 7/7 - 2/7 = 5/7 of the remainder stays. The 5/7 applies to the 7/12 she had after snacks.

1 - \dfrac{2}{7} = \dfrac{5}{7} \text{ of the } \dfrac{7}{12} \text{ left}

Spending 2 of every 7 equal parts of the remainder leaves 5 of those 7 parts.

3Combine to a fraction of the whole

#7 Identify Subproblems 3.NF.A.1

Take 5/7 of 7/12. Split the 7/12 into 7 equal parts of 1/12 each; 5 of them are left, which is 5/12 of the original money.

\dfrac{5}{7} \text{ of } \dfrac{7}{12} = \dfrac{5 \times 7}{7 \times 12} = \dfrac{5}{12}

7 12ths split into 7 equal pieces gives pieces of 1/12, and keeping 5 of them is 5/12.

Answer: 5/12

4 · Reviewdoes it hold up?

After snacks 7/12 remains; she then spends a bit more, so the final amount must be less than 7/12. The answer 5/12 is less than 7/12 and still positive, which fits. Check: spent 5/12 on snacks plus 2/12 on supplies (2/7 of 7/12) = 7/12 spent, leaving 5/12.

Another way: Work it as parts of 12ths (tool 15): the 7/12 left is 7 12ths; spending 2/7 of those 7 12ths means spending exactly 2 12ths, so 7/12 - 2/12 = 5/12 remains.

Standardsmin grade 3

3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Taking 5/7 of the 7/12 remainder as 5 equal parts of 1/12
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Computing 1 - 5/12 = 7/12 with the whole written as 12/12

💡Takeaway. The whole is just 1, so subtract each part spent in turn to find the fraction that is left!

Variant 8 medium answer: 5/13

Sea spent $\dfrac{6}{13}$ of the money she had on snacks, then spent $\dfrac{2}{7}$ of what was left on school supplies. Write, as a fraction, what part of the money she started with is left.

Show solution

1 · Understandwhat's really being asked

Sea spends 6/13 of her money on snacks, then spends 2/7 of what is left on school supplies. Write, as a fraction of the money she started with, how much is left.

Givens

The starting amount of money counts as the whole, 1
She spends 6/13 of the whole on snacks
Then she spends 2/7 of what remains on school supplies

Unknowns

The fraction of the original money that is left at the end

Constraints

The whole is 1
The 2/7 is taken from the remainder after snacks, not from the original whole

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Solve it in stages: first the money left after snacks (whole minus 6/13), then the fraction of that remainder still left after spending 2/7 of it. A bar model makes the 'fraction of what is left' step clear.

3 · Execute3 carry out the plan

1Money left after snacks

#7 Identify Subproblems 3.NF.A.3

The whole is 1 = 13/13. Spending 6/13 leaves 13/13 - 6/13 = 7/13 of the original money.

1 - \dfrac{6}{13} = \dfrac{13}{13} - \dfrac{6}{13} = \dfrac{7}{13}

Since the whole is 1, what is left is simply 1 minus the part spent.

2Fraction of the remainder still left

#1 Draw a Diagram 3.NF.A.1

She spends 2/7 of what was left, so 7/7 - 2/7 = 5/7 of the remainder stays. The 5/7 applies to the 7/13 she had after snacks.

1 - \dfrac{2}{7} = \dfrac{5}{7} \text{ of the } \dfrac{7}{13} \text{ left}

Spending 2 of every 7 equal parts of the remainder leaves 5 of those 7 parts.

3Combine to a fraction of the whole

#7 Identify Subproblems 3.NF.A.1

Take 5/7 of 7/13. Split the 7/13 into 7 equal parts of 1/13 each; 5 of them are left, which is 5/13 of the original money.

\dfrac{5}{7} \text{ of } \dfrac{7}{13} = \dfrac{5 \times 7}{7 \times 13} = \dfrac{5}{13}

7 13ths split into 7 equal pieces gives pieces of 1/13, and keeping 5 of them is 5/13.

Answer: 5/13

4 · Reviewdoes it hold up?

After snacks 7/13 remains; she then spends a bit more, so the final amount must be less than 7/13. The answer 5/13 is less than 7/13 and still positive, which fits. Check: spent 6/13 on snacks plus 2/13 on supplies (2/7 of 7/13) = 8/13 spent, leaving 5/13.

Another way: Work it as parts of 13ths (tool 15): the 7/13 left is 7 13ths; spending 2/7 of those 7 13ths means spending exactly 2 13ths, so 7/13 - 2/13 = 5/13 remains.

Standardsmin grade 3

3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Taking 5/7 of the 7/13 remainder as 5 equal parts of 1/13
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Computing 1 - 6/13 = 7/13 with the whole written as 13/13

💡Takeaway. The whole is just 1, so subtract each part spent in turn to find the fraction that is left!

Variant 9 hard answer: 4/14

Sea spent $\dfrac{3}{14}$ of the money she had on snacks, then spent $\dfrac{7}{11}$ of what was left on school supplies. Write, as a fraction, what part of the money she started with is left.

Show solution

1 · Understandwhat's really being asked

Sea spends 3/14 of her money on snacks, then spends 7/11 of what is left on school supplies. Write, as a fraction of the money she started with, how much is left.

Givens

The starting amount of money counts as the whole, 1
She spends 3/14 of the whole on snacks
Then she spends 7/11 of what remains on school supplies

Unknowns

The fraction of the original money that is left at the end

Constraints

The whole is 1
The 7/11 is taken from the remainder after snacks, not from the original whole

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Solve it in stages: first the money left after snacks (whole minus 3/14), then the fraction of that remainder still left after spending 7/11 of it. A bar model makes the 'fraction of what is left' step clear.

3 · Execute3 carry out the plan

1Money left after snacks

#7 Identify Subproblems 3.NF.A.3

The whole is 1 = 14/14. Spending 3/14 leaves 14/14 - 3/14 = 11/14 of the original money.

1 - \dfrac{3}{14} = \dfrac{14}{14} - \dfrac{3}{14} = \dfrac{11}{14}

Since the whole is 1, what is left is simply 1 minus the part spent.

2Fraction of the remainder still left

#1 Draw a Diagram 3.NF.A.1

She spends 7/11 of what was left, so 11/11 - 7/11 = 4/11 of the remainder stays. The 4/11 applies to the 11/14 she had after snacks.

1 - \dfrac{7}{11} = \dfrac{4}{11} \text{ of the } \dfrac{11}{14} \text{ left}

Spending 7 of every 11 equal parts of the remainder leaves 4 of those 11 parts.

3Combine to a fraction of the whole

#7 Identify Subproblems 3.NF.A.1

Take 4/11 of 11/14. Split the 11/14 into 11 equal parts of 1/14 each; 4 of them are left, which is 4/14 of the original money.

\dfrac{4}{11} \text{ of } \dfrac{11}{14} = \dfrac{4 \times 11}{11 \times 14} = \dfrac{4}{14}

11 14ths split into 11 equal pieces gives pieces of 1/14, and keeping 4 of them is 4/14.

Answer: 4/14

4 · Reviewdoes it hold up?

After snacks 11/14 remains; she then spends a bit more, so the final amount must be less than 11/14. The answer 4/14 is less than 11/14 and still positive, which fits. Check: spent 3/14 on snacks plus 7/14 on supplies (7/11 of 11/14) = 10/14 spent, leaving 4/14.

Another way: Work it as parts of 14ths (tool 15): the 11/14 left is 11 14ths; spending 7/11 of those 11 14ths means spending exactly 7 14ths, so 11/14 - 7/14 = 4/14 remains.

Standardsmin grade 3

3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Taking 4/11 of the 11/14 remainder as 4 equal parts of 1/14
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Computing 1 - 3/14 = 11/14 with the whole written as 14/14

💡Takeaway. The whole is just 1, so subtract each part spent in turn to find the fraction that is left!

Variant 10 hard answer: 2/15

Sea spent $\dfrac{7}{15}$ of the money she had on snacks, then spent $\dfrac{6}{8}$ of what was left on school supplies. Write, as a fraction, what part of the money she started with is left.

Show solution

1 · Understandwhat's really being asked

Sea spends 7/15 of her money on snacks, then spends 6/8 of what is left on school supplies. Write, as a fraction of the money she started with, how much is left.

Givens

The starting amount of money counts as the whole, 1
She spends 7/15 of the whole on snacks
Then she spends 6/8 of what remains on school supplies

Unknowns

The fraction of the original money that is left at the end

Constraints

The whole is 1
The 6/8 is taken from the remainder after snacks, not from the original whole

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Solve it in stages: first the money left after snacks (whole minus 7/15), then the fraction of that remainder still left after spending 6/8 of it. A bar model makes the 'fraction of what is left' step clear.

3 · Execute3 carry out the plan

1Money left after snacks

#7 Identify Subproblems 3.NF.A.3

The whole is 1 = 15/15. Spending 7/15 leaves 15/15 - 7/15 = 8/15 of the original money.

1 - \dfrac{7}{15} = \dfrac{15}{15} - \dfrac{7}{15} = \dfrac{8}{15}

Since the whole is 1, what is left is simply 1 minus the part spent.

2Fraction of the remainder still left

#1 Draw a Diagram 3.NF.A.1

She spends 6/8 of what was left, so 8/8 - 6/8 = 2/8 of the remainder stays. The 2/8 applies to the 8/15 she had after snacks.

1 - \dfrac{6}{8} = \dfrac{2}{8} \text{ of the } \dfrac{8}{15} \text{ left}

Spending 6 of every 8 equal parts of the remainder leaves 2 of those 8 parts.

3Combine to a fraction of the whole

#7 Identify Subproblems 3.NF.A.1

Take 2/8 of 8/15. Split the 8/15 into 8 equal parts of 1/15 each; 2 of them are left, which is 2/15 of the original money.

\dfrac{2}{8} \text{ of } \dfrac{8}{15} = \dfrac{2 \times 8}{8 \times 15} = \dfrac{2}{15}

8 15ths split into 8 equal pieces gives pieces of 1/15, and keeping 2 of them is 2/15.

Answer: 2/15

4 · Reviewdoes it hold up?

After snacks 8/15 remains; she then spends a bit more, so the final amount must be less than 8/15. The answer 2/15 is less than 8/15 and still positive, which fits. Check: spent 7/15 on snacks plus 6/15 on supplies (6/8 of 8/15) = 13/15 spent, leaving 2/15.

Another way: Work it as parts of 15ths (tool 15): the 8/15 left is 8 15ths; spending 6/8 of those 8 15ths means spending exactly 6 15ths, so 8/15 - 6/15 = 2/15 remains.

Standardsmin grade 3

3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Taking 2/8 of the 8/15 remainder as 2 equal parts of 1/15
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Computing 1 - 7/15 = 8/15 with the whole written as 15/15

💡Takeaway. The whole is just 1, so subtract each part spent in turn to find the fraction that is left!

Variant 11 hard answer: 3/16

Sea spent $\dfrac{5}{16}$ of the money she had on snacks, then spent $\dfrac{8}{11}$ of what was left on school supplies. Write, as a fraction, what part of the money she started with is left.

Show solution

1 · Understandwhat's really being asked

Sea spends 5/16 of her money on snacks, then spends 8/11 of what is left on school supplies. Write, as a fraction of the money she started with, how much is left.

Givens

The starting amount of money counts as the whole, 1
She spends 5/16 of the whole on snacks
Then she spends 8/11 of what remains on school supplies

Unknowns

The fraction of the original money that is left at the end

Constraints

The whole is 1
The 8/11 is taken from the remainder after snacks, not from the original whole

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Solve it in stages: first the money left after snacks (whole minus 5/16), then the fraction of that remainder still left after spending 8/11 of it. A bar model makes the 'fraction of what is left' step clear.

3 · Execute3 carry out the plan

1Money left after snacks

#7 Identify Subproblems 3.NF.A.3

The whole is 1 = 16/16. Spending 5/16 leaves 16/16 - 5/16 = 11/16 of the original money.

1 - \dfrac{5}{16} = \dfrac{16}{16} - \dfrac{5}{16} = \dfrac{11}{16}

Since the whole is 1, what is left is simply 1 minus the part spent.

2Fraction of the remainder still left

#1 Draw a Diagram 3.NF.A.1

She spends 8/11 of what was left, so 11/11 - 8/11 = 3/11 of the remainder stays. The 3/11 applies to the 11/16 she had after snacks.

1 - \dfrac{8}{11} = \dfrac{3}{11} \text{ of the } \dfrac{11}{16} \text{ left}

Spending 8 of every 11 equal parts of the remainder leaves 3 of those 11 parts.

3Combine to a fraction of the whole

#7 Identify Subproblems 3.NF.A.1

Take 3/11 of 11/16. Split the 11/16 into 11 equal parts of 1/16 each; 3 of them are left, which is 3/16 of the original money.

\dfrac{3}{11} \text{ of } \dfrac{11}{16} = \dfrac{3 \times 11}{11 \times 16} = \dfrac{3}{16}

11 16ths split into 11 equal pieces gives pieces of 1/16, and keeping 3 of them is 3/16.

Answer: 3/16

4 · Reviewdoes it hold up?

After snacks 11/16 remains; she then spends a bit more, so the final amount must be less than 11/16. The answer 3/16 is less than 11/16 and still positive, which fits. Check: spent 5/16 on snacks plus 8/16 on supplies (8/11 of 11/16) = 13/16 spent, leaving 3/16.

Another way: Work it as parts of 16ths (tool 15): the 11/16 left is 11 16ths; spending 8/11 of those 11 16ths means spending exactly 8 16ths, so 11/16 - 8/16 = 3/16 remains.

Standardsmin grade 3

3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Taking 3/11 of the 11/16 remainder as 3 equal parts of 1/16
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Computing 1 - 5/16 = 11/16 with the whole written as 16/16

💡Takeaway. The whole is just 1, so subtract each part spent in turn to find the fraction that is left!

Variant 12 hard answer: 7/20

Sea spent $\dfrac{8}{20}$ of the money she had on snacks, then spent $\dfrac{5}{12}$ of what was left on school supplies. Write, as a fraction, what part of the money she started with is left.

Show solution

1 · Understandwhat's really being asked

Sea spends 8/20 of her money on snacks, then spends 5/12 of what is left on school supplies. Write, as a fraction of the money she started with, how much is left.

Givens

The starting amount of money counts as the whole, 1
She spends 8/20 of the whole on snacks
Then she spends 5/12 of what remains on school supplies

Unknowns

The fraction of the original money that is left at the end

Constraints

The whole is 1
The 5/12 is taken from the remainder after snacks, not from the original whole

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Solve it in stages: first the money left after snacks (whole minus 8/20), then the fraction of that remainder still left after spending 5/12 of it. A bar model makes the 'fraction of what is left' step clear.

3 · Execute3 carry out the plan

1Money left after snacks

#7 Identify Subproblems 3.NF.A.3

The whole is 1 = 20/20. Spending 8/20 leaves 20/20 - 8/20 = 12/20 of the original money.

1 - \dfrac{8}{20} = \dfrac{20}{20} - \dfrac{8}{20} = \dfrac{12}{20}

Since the whole is 1, what is left is simply 1 minus the part spent.

2Fraction of the remainder still left

#1 Draw a Diagram 3.NF.A.1

She spends 5/12 of what was left, so 12/12 - 5/12 = 7/12 of the remainder stays. The 7/12 applies to the 12/20 she had after snacks.

1 - \dfrac{5}{12} = \dfrac{7}{12} \text{ of the } \dfrac{12}{20} \text{ left}

Spending 5 of every 12 equal parts of the remainder leaves 7 of those 12 parts.

3Combine to a fraction of the whole

#7 Identify Subproblems 3.NF.A.1

Take 7/12 of 12/20. Split the 12/20 into 12 equal parts of 1/20 each; 7 of them are left, which is 7/20 of the original money.

\dfrac{7}{12} \text{ of } \dfrac{12}{20} = \dfrac{7 \times 12}{12 \times 20} = \dfrac{7}{20}

12 20ths split into 12 equal pieces gives pieces of 1/20, and keeping 7 of them is 7/20.

Answer: 7/20

4 · Reviewdoes it hold up?

After snacks 12/20 remains; she then spends a bit more, so the final amount must be less than 12/20. The answer 7/20 is less than 12/20 and still positive, which fits. Check: spent 8/20 on snacks plus 5/20 on supplies (5/12 of 12/20) = 13/20 spent, leaving 7/20.

Another way: Work it as parts of 20ths (tool 15): the 12/20 left is 12 20ths; spending 5/12 of those 12 20ths means spending exactly 5 20ths, so 12/20 - 5/20 = 7/20 remains.

Standardsmin grade 3

3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Taking 7/12 of the 12/20 remainder as 7 equal parts of 1/20
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Computing 1 - 8/20 = 12/20 with the whole written as 20/20

💡Takeaway. The whole is just 1, so subtract each part spent in turn to find the fraction that is left!