Practice · The whole is one; find the remaining fraction · Sensim Math

Variant 1 easy answer: $\frac{2}{7}$

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 $\frac{2}{7}$ of her money on snacks, then spends $\frac{3}{5}$ 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 $\frac{2}{7}$ of the whole on snacks
Then she spends $\frac{3}{5}$ 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 $\frac{3}{5}$ 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 $\frac{2}{7}$ ), then the fraction of that remainder still left after spending $\frac{3}{5}$ 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 =

\frac{7}{7}

. Spending

\frac{2}{7}

leaves

\frac{7}{7}

-

\frac{2}{7}

=

\frac{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

\frac{3}{5}

of what was left, so

\frac{5}{5}

-

\frac{3}{5}

=

\frac{2}{5}

of the remainder stays. The

\frac{2}{5}

applies to the

\frac{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

\frac{2}{5}

of

\frac{5}{7}

. Split the

\frac{5}{7}

into 5 equal parts of

\frac{1}{7}

each; 2 of them are left, which is

\frac{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

\frac{1}{7}

, and keeping 2 of them is

\frac{2}{7}

.

Answer:

\frac{2}{7}

4 · Reviewdoes it hold up?

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

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

Standardsmin grade 3

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

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

Variant 2 easy answer: $\frac{3}{8}$

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 $\frac{1}{8}$ of her money on snacks, then spends $\frac{4}{7}$ 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 $\frac{1}{8}$ of the whole on snacks
Then she spends $\frac{4}{7}$ 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 $\frac{4}{7}$ 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 $\frac{1}{8}$ ), then the fraction of that remainder still left after spending $\frac{4}{7}$ 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 =

\frac{8}{8}

. Spending

\frac{1}{8}

leaves

\frac{8}{8}

-

\frac{1}{8}

=

\frac{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

\frac{4}{7}

of what was left, so

\frac{7}{7}

-

\frac{4}{7}

=

\frac{3}{7}

of the remainder stays. The

\frac{3}{7}

applies to the

\frac{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

\frac{3}{7}

of

\frac{7}{8}

. Split the

\frac{7}{8}

into 7 equal parts of

\frac{1}{8}

each; 3 of them are left, which is

\frac{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

\frac{1}{8}

, and keeping 3 of them is

\frac{3}{8}

.

Answer:

\frac{3}{8}

4 · Reviewdoes it hold up?

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

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

Standardsmin grade 3

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

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

Variant 3 easy answer: $\frac{4}{9}$

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 $\frac{2}{9}$ of her money on snacks, then spends $\frac{3}{7}$ 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 $\frac{2}{9}$ of the whole on snacks
Then she spends $\frac{3}{7}$ 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 $\frac{3}{7}$ 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 $\frac{2}{9}$ ), then the fraction of that remainder still left after spending $\frac{3}{7}$ 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 =

\frac{9}{9}

. Spending

\frac{2}{9}

leaves

\frac{9}{9}

-

\frac{2}{9}

=

\frac{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

\frac{3}{7}

of what was left, so

\frac{7}{7}

-

\frac{3}{7}

=

\frac{4}{7}

of the remainder stays. The

\frac{4}{7}

applies to the

\frac{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

\frac{4}{7}

of

\frac{7}{9}

. Split the

\frac{7}{9}

into 7 equal parts of

\frac{1}{9}

each; 4 of them are left, which is

\frac{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

\frac{1}{9}

, and keeping 4 of them is

\frac{4}{9}

.

Answer:

\frac{4}{9}

4 · Reviewdoes it hold up?

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

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

Standardsmin grade 3

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

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

Variant 4 medium answer: $\frac{4}{10}$

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 $\frac{4}{10}$ of her money on snacks, then spends $\frac{2}{6}$ 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 $\frac{4}{10}$ of the whole on snacks
Then she spends $\frac{2}{6}$ 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 $\frac{2}{6}$ 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 $\frac{4}{10}$ ), then the fraction of that remainder still left after spending $\frac{2}{6}$ 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 =

\frac{10}{10}

. Spending

\frac{4}{10}

leaves

\frac{10}{10}

-

\frac{4}{10}

=

\frac{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

\frac{2}{6}

of what was left, so

\frac{6}{6}

-

\frac{2}{6}

=

\frac{4}{6}

of the remainder stays. The

\frac{4}{6}

applies to the

\frac{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

\frac{4}{6}

of

\frac{6}{10}

. Split the

\frac{6}{10}

into 6 equal parts of

\frac{1}{10}

each; 4 of them are left, which is

\frac{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

\frac{1}{10}

, and keeping 4 of them is

\frac{4}{10}

.

Answer:

\frac{4}{10}

4 · Reviewdoes it hold up?

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

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

Standardsmin grade 3

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

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

Variant 5 easy answer: $\frac{3}{10}$

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 $\frac{3}{10}$ of her money on snacks, then spends $\frac{4}{7}$ 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 $\frac{3}{10}$ of the whole on snacks
Then she spends $\frac{4}{7}$ 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 $\frac{4}{7}$ 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 $\frac{3}{10}$ ), then the fraction of that remainder still left after spending $\frac{4}{7}$ 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 =

\frac{10}{10}

. Spending

\frac{3}{10}

leaves

\frac{10}{10}

-

\frac{3}{10}

=

\frac{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

\frac{4}{7}

of what was left, so

\frac{7}{7}

-

\frac{4}{7}

=

\frac{3}{7}

of the remainder stays. The

\frac{3}{7}

applies to the

\frac{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

\frac{3}{7}

of

\frac{7}{10}

. Split the

\frac{7}{10}

into 7 equal parts of

\frac{1}{10}

each; 3 of them are left, which is

\frac{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

\frac{1}{10}

, and keeping 3 of them is

\frac{3}{10}

.

Answer:

\frac{3}{10}

4 · Reviewdoes it hold up?

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

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

Standardsmin grade 3

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

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

Variant 6 medium answer: $\frac{2}{11}$

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 $\frac{4}{11}$ of her money on snacks, then spends $\frac{5}{7}$ 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 $\frac{4}{11}$ of the whole on snacks
Then she spends $\frac{5}{7}$ 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 $\frac{5}{7}$ 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 $\frac{4}{11}$ ), then the fraction of that remainder still left after spending $\frac{5}{7}$ 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 =

\frac{11}{11}

. Spending

\frac{4}{11}

leaves

\frac{11}{11}

-

\frac{4}{11}

=

\frac{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

\frac{5}{7}

of what was left, so

\frac{7}{7}

-

\frac{5}{7}

=

\frac{2}{7}

of the remainder stays. The

\frac{2}{7}

applies to the

\frac{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

\frac{2}{7}

of

\frac{7}{11}

. Split the

\frac{7}{11}

into 7 equal parts of

\frac{1}{11}

each; 2 of them are left, which is

\frac{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

\frac{1}{11}

, and keeping 2 of them is

\frac{2}{11}

.

Answer:

\frac{2}{11}

4 · Reviewdoes it hold up?

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

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

Standardsmin grade 3

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

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

Variant 7 medium answer: $\frac{5}{12}$

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 $\frac{5}{12}$ of her money on snacks, then spends $\frac{2}{7}$ 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 $\frac{5}{12}$ of the whole on snacks
Then she spends $\frac{2}{7}$ 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 $\frac{2}{7}$ 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 $\frac{5}{12}$ ), then the fraction of that remainder still left after spending $\frac{2}{7}$ 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 =

\frac{12}{12}

. Spending

\frac{5}{12}

leaves

\frac{12}{12}

-

\frac{5}{12}

=

\frac{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

\frac{2}{7}

of what was left, so

\frac{7}{7}

-

\frac{2}{7}

=

\frac{5}{7}

of the remainder stays. The

\frac{5}{7}

applies to the

\frac{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

\frac{5}{7}

of

\frac{7}{12}

. Split the

\frac{7}{12}

into 7 equal parts of

\frac{1}{12}

each; 5 of them are left, which is

\frac{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

\frac{1}{12}

, and keeping 5 of them is

\frac{5}{12}

.

Answer:

\frac{5}{12}

4 · Reviewdoes it hold up?

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

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

Standardsmin grade 3

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

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

Variant 8 medium answer: $\frac{5}{13}$

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 $\frac{6}{13}$ of her money on snacks, then spends $\frac{2}{7}$ 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 $\frac{6}{13}$ of the whole on snacks
Then she spends $\frac{2}{7}$ 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 $\frac{2}{7}$ 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 $\frac{6}{13}$ ), then the fraction of that remainder still left after spending $\frac{2}{7}$ 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 =

\frac{13}{13}

. Spending

\frac{6}{13}

leaves

\frac{13}{13}

-

\frac{6}{13}

=

\frac{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

\frac{2}{7}

of what was left, so

\frac{7}{7}

-

\frac{2}{7}

=

\frac{5}{7}

of the remainder stays. The

\frac{5}{7}

applies to the

\frac{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

\frac{5}{7}

of

\frac{7}{13}

. Split the

\frac{7}{13}

into 7 equal parts of

\frac{1}{13}

each; 5 of them are left, which is

\frac{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

\frac{1}{13}

, and keeping 5 of them is

\frac{5}{13}

.

Answer:

\frac{5}{13}

4 · Reviewdoes it hold up?

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

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

Standardsmin grade 3

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

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

Variant 9 hard answer: $\frac{4}{14}$

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 $\frac{3}{14}$ of her money on snacks, then spends $\frac{7}{11}$ 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 $\frac{3}{14}$ of the whole on snacks
Then she spends $\frac{7}{11}$ 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 $\frac{7}{11}$ 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 $\frac{3}{14}$ ), then the fraction of that remainder still left after spending $\frac{7}{11}$ 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 =

\frac{14}{14}

. Spending

\frac{3}{14}

leaves

\frac{14}{14}

-

\frac{3}{14}

=

\frac{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

\frac{7}{11}

of what was left, so

\frac{11}{11}

-

\frac{7}{11}

=

\frac{4}{11}

of the remainder stays. The

\frac{4}{11}

applies to the

\frac{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

\frac{4}{11}

of

\frac{11}{14}

. Split the

\frac{11}{14}

into 11 equal parts of

\frac{1}{14}

each; 4 of them are left, which is

\frac{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

\frac{1}{14}

, and keeping 4 of them is

\frac{4}{14}

.

Answer:

\frac{4}{14}

4 · Reviewdoes it hold up?

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

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

Standardsmin grade 3

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

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

Variant 10 hard answer: $\frac{2}{15}$

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 $\frac{7}{15}$ of her money on snacks, then spends $\frac{6}{8}$ 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 $\frac{7}{15}$ of the whole on snacks
Then she spends $\frac{6}{8}$ 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 $\frac{6}{8}$ 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 $\frac{7}{15}$ ), then the fraction of that remainder still left after spending $\frac{6}{8}$ 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 =

\frac{15}{15}

. Spending

\frac{7}{15}

leaves

\frac{15}{15}

-

\frac{7}{15}

=

\frac{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

\frac{6}{8}

of what was left, so

\frac{8}{8}

-

\frac{6}{8}

=

\frac{2}{8}

of the remainder stays. The

\frac{2}{8}

applies to the

\frac{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

\frac{2}{8}

of

\frac{8}{15}

. Split the

\frac{8}{15}

into 8 equal parts of

\frac{1}{15}

each; 2 of them are left, which is

\frac{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

\frac{1}{15}

, and keeping 2 of them is

\frac{2}{15}

.

Answer:

\frac{2}{15}

4 · Reviewdoes it hold up?

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

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

Standardsmin grade 3

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

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

Variant 11 hard answer: $\frac{3}{16}$

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 $\frac{5}{16}$ of her money on snacks, then spends $\frac{8}{11}$ 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 $\frac{5}{16}$ of the whole on snacks
Then she spends $\frac{8}{11}$ 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 $\frac{8}{11}$ 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 $\frac{5}{16}$ ), then the fraction of that remainder still left after spending $\frac{8}{11}$ 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 =

\frac{16}{16}

. Spending

\frac{5}{16}

leaves

\frac{16}{16}

-

\frac{5}{16}

=

\frac{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

\frac{8}{11}

of what was left, so

\frac{11}{11}

-

\frac{8}{11}

=

\frac{3}{11}

of the remainder stays. The

\frac{3}{11}

applies to the

\frac{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

\frac{3}{11}

of

\frac{11}{16}

. Split the

\frac{11}{16}

into 11 equal parts of

\frac{1}{16}

each; 3 of them are left, which is

\frac{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

\frac{1}{16}

, and keeping 3 of them is

\frac{3}{16}

.

Answer:

\frac{3}{16}

4 · Reviewdoes it hold up?

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

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

Standardsmin grade 3

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

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

Variant 12 hard answer: $\frac{7}{20}$

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 $\frac{8}{20}$ of her money on snacks, then spends $\frac{5}{12}$ 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 $\frac{8}{20}$ of the whole on snacks
Then she spends $\frac{5}{12}$ 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 $\frac{5}{12}$ 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 $\frac{8}{20}$ ), then the fraction of that remainder still left after spending $\frac{5}{12}$ 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 =

\frac{20}{20}

. Spending

\frac{8}{20}

leaves

\frac{20}{20}

-

\frac{8}{20}

=

\frac{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

\frac{5}{12}

of what was left, so

\frac{12}{12}

-

\frac{5}{12}

=

\frac{7}{12}

of the remainder stays. The

\frac{7}{12}

applies to the

\frac{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

\frac{7}{12}

of

\frac{12}{20}

. Split the

\frac{12}{20}

into 12 equal parts of

\frac{1}{20}

each; 7 of them are left, which is

\frac{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

\frac{1}{20}

, and keeping 7 of them is

\frac{7}{20}

.

Answer:

\frac{7}{20}

4 · Reviewdoes it hold up?

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

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

Standardsmin grade 3

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

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

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

Generated variants — 12

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