Practice · Recover the whole from a fractional part · Sensim Math

Variant 1 easy answer: 8

Find $\frac{1}{5}$ of the number $\star$ that satisfies the following.

$\frac{3}{10} \text{ of } \star \text{ is } 12.$

Show solution

1 · Understandwhat's really being asked

A mystery number $\star$ has the property that $\frac{3}{10}$ of it equals 12. I first need to find $\star$ , then report $\frac{1}{5}$ of $\star$ .

Givens

$\frac{3}{10}$ of $\star$ is 12.
$\star$ is the whole quantity (the 'whole') being split into 10ths.

Unknowns

The value of $\frac{1}{5}$ of $\star$ .

Constraints

$\frac{3}{10}$ of $\star$ means $\star$ split into 10 equal parts, with 3 of those parts totaling 12.
$\star$ must come out a whole number so $\frac{1}{5}$ of it makes sense.

2 · Planchoose the strategy

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

I work backwards from the part to the whole: if 3 10ths equal 12, then one 10th equals $\frac{12}{3}$ , and the whole is 10 of those. Then taking $\frac{1}{5}$ of the recovered whole is a second small subproblem.

3 · Execute3 carry out the plan

1Find one 10th of $\star$

#11 Work Backwards 3.OA.A.4

3 equal 10ths make 12, so one 10th is 12 divided by 3.

12 \div 3 = 4

If 3 equal parts total 12, each part is found by sharing 12 into 3 equal pieces.

2Build the whole $\star$

#11 Work Backwards 3.NF.A.1

\star

is made of 10 such 10ths, so multiply one 10th (4) by 10.

4 \times 10 = 40

The whole is just all of its equal parts put back together.

3Take $\frac{1}{5}$ of $\star$

#7 Identify Subproblems 3.NF.A.1

Now find

\frac{1}{5}

of 40 by dividing 40 into 5 equal parts.

40 \div 5 = 8

A fraction

\frac{1}{5}

of a number is that number shared into 5 equal parts, taking 1.

Answer: 8

4 · Reviewdoes it hold up?

Check: $\frac{3}{10}$ of 40 is 40 / 10 x 3 = 4 x 3 = 12, which matches the given. And $\frac{1}{5}$ of 40 is 8. Both come out as clean whole numbers, so 40 is the right whole and 8 is the answer.

Another way: Convert to algebra (tool 13): ( $\frac{3}{10}$ ) x $\star$ = 12 gives $\star$ = 12 x 10 / 3 = 40, then $\frac{1}{5}$ of it = 8 - same result, but the work-backwards unit-fraction route avoids algebra.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding one 10th from 3 10ths = 12 (an unknown-part division).
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Rebuilding the whole from 10ths and taking $\frac{1}{5}$ of it.

💡Takeaway. This only needs Grade 3 fraction sense: find one equal part, build the whole, then split again!

Variant 2 easy answer: 6

Find $\frac{1}{5}$ of the number $\star$ that satisfies the following.

$\frac{4}{15} \text{ of } \star \text{ is } 8.$

Show solution

1 · Understandwhat's really being asked

A mystery number $\star$ has the property that $\frac{4}{15}$ of it equals 8. I first need to find $\star$ , then report $\frac{1}{5}$ of $\star$ .

Givens

$\frac{4}{15}$ of $\star$ is 8.
$\star$ is the whole quantity (the 'whole') being split into 15ths.

Unknowns

The value of $\frac{1}{5}$ of $\star$ .

Constraints

$\frac{4}{15}$ of $\star$ means $\star$ split into 15 equal parts, with 4 of those parts totaling 8.
$\star$ must come out a whole number so $\frac{1}{5}$ of it makes sense.

2 · Planchoose the strategy

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

I work backwards from the part to the whole: if 4 15ths equal 8, then one 15th equals $\frac{8}{4}$ , and the whole is 15 of those. Then taking $\frac{1}{5}$ of the recovered whole is a second small subproblem.

3 · Execute3 carry out the plan

1Find one 15th of $\star$

#11 Work Backwards 3.OA.A.4

4 equal 15ths make 8, so one 15th is 8 divided by 4.

8 \div 4 = 2

If 4 equal parts total 8, each part is found by sharing 8 into 4 equal pieces.

2Build the whole $\star$

#11 Work Backwards 3.NF.A.1

\star

is made of 15 such 15ths, so multiply one 15th (2) by 15.

2 \times 15 = 30

The whole is just all of its equal parts put back together.

3Take $\frac{1}{5}$ of $\star$

#7 Identify Subproblems 3.NF.A.1

Now find

\frac{1}{5}

of 30 by dividing 30 into 5 equal parts.

30 \div 5 = 6

A fraction

\frac{1}{5}

of a number is that number shared into 5 equal parts, taking 1.

Answer: 6

4 · Reviewdoes it hold up?

Check: $\frac{4}{15}$ of 30 is $\frac{30}{15}$ x 4 = 2 x 4 = 8, which matches the given. And $\frac{1}{5}$ of 30 is 6. Both come out as clean whole numbers, so 30 is the right whole and 6 is the answer.

Another way: Convert to algebra (tool 13): ( $\frac{4}{15}$ ) x $\star$ = 8 gives $\star$ = 8 x $\frac{15}{4}$ = 30, then $\frac{1}{5}$ of it = 6 - same result, but the work-backwards unit-fraction route avoids algebra.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding one 15th from 4 15ths = 8 (an unknown-part division).
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Rebuilding the whole from 15ths and taking $\frac{1}{5}$ of it.

💡Takeaway. This only needs Grade 3 fraction sense: find one equal part, build the whole, then split again!

Variant 3 easy answer: 4

Find $\frac{1}{6}$ of the number $\star$ that satisfies the following.

$\frac{5}{8} \text{ of } \star \text{ is } 15.$

Show solution

1 · Understandwhat's really being asked

A mystery number $\star$ has the property that $\frac{5}{8}$ of it equals 15. I first need to find $\star$ , then report $\frac{1}{6}$ of $\star$ .

Givens

$\frac{5}{8}$ of $\star$ is 15.
$\star$ is the whole quantity (the 'whole') being split into 8ths.

Unknowns

The value of $\frac{1}{6}$ of $\star$ .

Constraints

$\frac{5}{8}$ of $\star$ means $\star$ split into 8 equal parts, with 5 of those parts totaling 15.
$\star$ must come out a whole number so $\frac{1}{6}$ of it makes sense.

2 · Planchoose the strategy

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

I work backwards from the part to the whole: if 5 8ths equal 15, then one 8th equals $\frac{15}{5}$ , and the whole is 8 of those. Then taking $\frac{1}{6}$ of the recovered whole is a second small subproblem.

3 · Execute3 carry out the plan

1Find one 8th of $\star$

#11 Work Backwards 3.OA.A.4

5 equal 8ths make 15, so one 8th is 15 divided by 5.

15 \div 5 = 3

If 5 equal parts total 15, each part is found by sharing 15 into 5 equal pieces.

2Build the whole $\star$

#11 Work Backwards 3.NF.A.1

\star

is made of 8 such 8ths, so multiply one 8th (3) by 8.

3 \times 8 = 24

The whole is just all of its equal parts put back together.

3Take $\frac{1}{6}$ of $\star$

#7 Identify Subproblems 3.NF.A.1

Now find

\frac{1}{6}

of 24 by dividing 24 into 6 equal parts.

24 \div 6 = 4

A fraction

\frac{1}{6}

of a number is that number shared into 6 equal parts, taking 1.

Answer: 4

4 · Reviewdoes it hold up?

Check: $\frac{5}{8}$ of 24 is 24 / 8 x 5 = 3 x 5 = 15, which matches the given. And $\frac{1}{6}$ of 24 is 4. Both come out as clean whole numbers, so 24 is the right whole and 4 is the answer.

Another way: Convert to algebra (tool 13): ( $\frac{5}{8}$ ) x $\star$ = 15 gives $\star$ = 15 x 8 / 5 = 24, then $\frac{1}{6}$ of it = 4 - same result, but the work-backwards unit-fraction route avoids algebra.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding one 8th from 5 8ths = 15 (an unknown-part division).
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Rebuilding the whole from 8ths and taking $\frac{1}{6}$ of it.

💡Takeaway. This only needs Grade 3 fraction sense: find one equal part, build the whole, then split again!

Variant 4 easy answer: 9

Find $\frac{1}{5}$ of the number $\star$ that satisfies the following.

$\frac{4}{9} \text{ of } \star \text{ is } 20.$

Show solution

1 · Understandwhat's really being asked

A mystery number $\star$ has the property that $\frac{4}{9}$ of it equals 20. I first need to find $\star$ , then report $\frac{1}{5}$ of $\star$ .

Givens

$\frac{4}{9}$ of $\star$ is 20.
$\star$ is the whole quantity (the 'whole') being split into 9ths.

Unknowns

The value of $\frac{1}{5}$ of $\star$ .

Constraints

$\frac{4}{9}$ of $\star$ means $\star$ split into 9 equal parts, with 4 of those parts totaling 20.
$\star$ must come out a whole number so $\frac{1}{5}$ of it makes sense.

2 · Planchoose the strategy

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

I work backwards from the part to the whole: if 4 9ths equal 20, then one 9th equals $\frac{20}{4}$ , and the whole is 9 of those. Then taking $\frac{1}{5}$ of the recovered whole is a second small subproblem.

3 · Execute3 carry out the plan

1Find one 9th of $\star$

#11 Work Backwards 3.OA.A.4

4 equal 9ths make 20, so one 9th is 20 divided by 4.

20 \div 4 = 5

If 4 equal parts total 20, each part is found by sharing 20 into 4 equal pieces.

2Build the whole $\star$

#11 Work Backwards 3.NF.A.1

\star

is made of 9 such 9ths, so multiply one 9th (5) by 9.

5 \times 9 = 45

The whole is just all of its equal parts put back together.

3Take $\frac{1}{5}$ of $\star$

#7 Identify Subproblems 3.NF.A.1

Now find

\frac{1}{5}

of 45 by dividing 45 into 5 equal parts.

45 \div 5 = 9

A fraction

\frac{1}{5}

of a number is that number shared into 5 equal parts, taking 1.

Answer: 9

4 · Reviewdoes it hold up?

Check: $\frac{4}{9}$ of 45 is $\frac{45}{9}$ x 4 = 5 x 4 = 20, which matches the given. And $\frac{1}{5}$ of 45 is 9. Both come out as clean whole numbers, so 45 is the right whole and 9 is the answer.

Another way: Convert to algebra (tool 13): ( $\frac{4}{9}$ ) x $\star$ = 20 gives $\star$ = 20 x $\frac{9}{4}$ = 45, then $\frac{1}{5}$ of it = 9 - same result, but the work-backwards unit-fraction route avoids algebra.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding one 9th from 4 9ths = 20 (an unknown-part division).
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Rebuilding the whole from 9ths and taking $\frac{1}{5}$ of it.

💡Takeaway. This only needs Grade 3 fraction sense: find one equal part, build the whole, then split again!

Variant 5 medium answer: 4

Find $\frac{1}{9}$ of the number $\star$ that satisfies the following.

$\frac{7}{12} \text{ of } \star \text{ is } 21.$

Show solution

1 · Understandwhat's really being asked

A mystery number $\star$ has the property that $\frac{7}{12}$ of it equals 21. I first need to find $\star$ , then report $\frac{1}{9}$ of $\star$ .

Givens

$\frac{7}{12}$ of $\star$ is 21.
$\star$ is the whole quantity (the 'whole') being split into 12ths.

Unknowns

The value of $\frac{1}{9}$ of $\star$ .

Constraints

$\frac{7}{12}$ of $\star$ means $\star$ split into 12 equal parts, with 7 of those parts totaling 21.
$\star$ must come out a whole number so $\frac{1}{9}$ of it makes sense.

2 · Planchoose the strategy

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

I work backwards from the part to the whole: if 7 12ths equal 21, then one 12th equals $\frac{21}{7}$ , and the whole is 12 of those. Then taking $\frac{1}{9}$ of the recovered whole is a second small subproblem.

3 · Execute3 carry out the plan

1Find one 12th of $\star$

#11 Work Backwards 3.OA.A.4

7 equal 12ths make 21, so one 12th is 21 divided by 7.

21 \div 7 = 3

If 7 equal parts total 21, each part is found by sharing 21 into 7 equal pieces.

2Build the whole $\star$

#11 Work Backwards 3.NF.A.1

\star

is made of 12 such 12ths, so multiply one 12th (3) by 12.

3 \times 12 = 36

The whole is just all of its equal parts put back together.

3Take $\frac{1}{9}$ of $\star$

#7 Identify Subproblems 3.NF.A.1

Now find

\frac{1}{9}

of 36 by dividing 36 into 9 equal parts.

36 \div 9 = 4

A fraction

\frac{1}{9}

of a number is that number shared into 9 equal parts, taking 1.

Answer: 4

4 · Reviewdoes it hold up?

Check: $\frac{7}{12}$ of 36 is $\frac{36}{12}$ x 7 = 3 x 7 = 21, which matches the given. And $\frac{1}{9}$ of 36 is 4. Both come out as clean whole numbers, so 36 is the right whole and 4 is the answer.

Another way: Convert to algebra (tool 13): ( $\frac{7}{12}$ ) x $\star$ = 21 gives $\star$ = 21 x $\frac{12}{7}$ = 36, then $\frac{1}{9}$ of it = 4 - same result, but the work-backwards unit-fraction route avoids algebra.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding one 12th from 7 12ths = 21 (an unknown-part division).
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Rebuilding the whole from 12ths and taking $\frac{1}{9}$ of it.

💡Takeaway. This only needs Grade 3 fraction sense: find one equal part, build the whole, then split again!

Variant 6 medium answer: 10

Find $\frac{1}{8}$ of the number $\star$ that satisfies the following.

$\frac{3}{10} \text{ of } \star \text{ is } 24.$

Show solution

1 · Understandwhat's really being asked

A mystery number $\star$ has the property that $\frac{3}{10}$ of it equals 24. I first need to find $\star$ , then report $\frac{1}{8}$ of $\star$ .

Givens

$\frac{3}{10}$ of $\star$ is 24.
$\star$ is the whole quantity (the 'whole') being split into 10ths.

Unknowns

The value of $\frac{1}{8}$ of $\star$ .

Constraints

$\frac{3}{10}$ of $\star$ means $\star$ split into 10 equal parts, with 3 of those parts totaling 24.
$\star$ must come out a whole number so $\frac{1}{8}$ of it makes sense.

2 · Planchoose the strategy

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

I work backwards from the part to the whole: if 3 10ths equal 24, then one 10th equals $\frac{24}{3}$ , and the whole is 10 of those. Then taking $\frac{1}{8}$ of the recovered whole is a second small subproblem.

3 · Execute3 carry out the plan

1Find one 10th of $\star$

#11 Work Backwards 3.OA.A.4

3 equal 10ths make 24, so one 10th is 24 divided by 3.

24 \div 3 = 8

If 3 equal parts total 24, each part is found by sharing 24 into 3 equal pieces.

2Build the whole $\star$

#11 Work Backwards 3.NF.A.1

\star

is made of 10 such 10ths, so multiply one 10th (8) by 10.

8 \times 10 = 80

The whole is just all of its equal parts put back together.

3Take $\frac{1}{8}$ of $\star$

#7 Identify Subproblems 3.NF.A.1

Now find

\frac{1}{8}

of 80 by dividing 80 into 8 equal parts.

80 \div 8 = 10

A fraction

\frac{1}{8}

of a number is that number shared into 8 equal parts, taking 1.

Answer: 10

4 · Reviewdoes it hold up?

Check: $\frac{3}{10}$ of 80 is $\frac{80}{10}$ x 3 = 8 x 3 = 24, which matches the given. And $\frac{1}{8}$ of 80 is 10. Both come out as clean whole numbers, so 80 is the right whole and 10 is the answer.

Another way: Convert to algebra (tool 13): ( $\frac{3}{10}$ ) x $\star$ = 24 gives $\star$ = 24 x $\frac{10}{3}$ = 80, then $\frac{1}{8}$ of it = 10 - same result, but the work-backwards unit-fraction route avoids algebra.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding one 10th from 3 10ths = 24 (an unknown-part division).
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Rebuilding the whole from 10ths and taking $\frac{1}{8}$ of it.

💡Takeaway. This only needs Grade 3 fraction sense: find one equal part, build the whole, then split again!

Variant 7 medium answer: 15

Find $\frac{1}{4}$ of the number $\star$ that satisfies the following.

$\frac{5}{12} \text{ of } \star \text{ is } 25.$

Show solution

1 · Understandwhat's really being asked

A mystery number $\star$ has the property that $\frac{5}{12}$ of it equals 25. I first need to find $\star$ , then report $\frac{1}{4}$ of $\star$ .

Givens

$\frac{5}{12}$ of $\star$ is 25.
$\star$ is the whole quantity (the 'whole') being split into 12ths.

Unknowns

The value of $\frac{1}{4}$ of $\star$ .

Constraints

$\frac{5}{12}$ of $\star$ means $\star$ split into 12 equal parts, with 5 of those parts totaling 25.
$\star$ must come out a whole number so $\frac{1}{4}$ of it makes sense.

2 · Planchoose the strategy

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

I work backwards from the part to the whole: if 5 12ths equal 25, then one 12th equals $\frac{25}{5}$ , and the whole is 12 of those. Then taking $\frac{1}{4}$ of the recovered whole is a second small subproblem.

3 · Execute3 carry out the plan

1Find one 12th of $\star$

#11 Work Backwards 3.OA.A.4

5 equal 12ths make 25, so one 12th is 25 divided by 5.

25 \div 5 = 5

If 5 equal parts total 25, each part is found by sharing 25 into 5 equal pieces.

2Build the whole $\star$

#11 Work Backwards 3.NF.A.1

\star

is made of 12 such 12ths, so multiply one 12th (5) by 12.

5 \times 12 = 60

The whole is just all of its equal parts put back together.

3Take $\frac{1}{4}$ of $\star$

#7 Identify Subproblems 3.NF.A.1

Now find

\frac{1}{4}

of 60 by dividing 60 into 4 equal parts.

60 \div 4 = 15

A fraction

\frac{1}{4}

of a number is that number shared into 4 equal parts, taking 1.

Answer: 15

4 · Reviewdoes it hold up?

Check: $\frac{5}{12}$ of 60 is 60 / 12 x 5 = 5 x 5 = 25, which matches the given. And $\frac{1}{4}$ of 60 is 15. Both come out as clean whole numbers, so 60 is the right whole and 15 is the answer.

Another way: Convert to algebra (tool 13): ( $\frac{5}{12}$ ) x $\star$ = 25 gives $\star$ = 25 x 12 / 5 = 60, then $\frac{1}{4}$ of it = 15 - same result, but the work-backwards unit-fraction route avoids algebra.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding one 12th from 5 12ths = 25 (an unknown-part division).
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Rebuilding the whole from 12ths and taking $\frac{1}{4}$ of it.

💡Takeaway. This only needs Grade 3 fraction sense: find one equal part, build the whole, then split again!

Variant 8 hard answer: 9

Find $\frac{1}{6}$ of the number $\star$ that satisfies the following.

$\frac{5}{9} \text{ of } \star \text{ is } 30.$

Show solution

1 · Understandwhat's really being asked

A mystery number $\star$ has the property that $\frac{5}{9}$ of it equals 30. I first need to find $\star$ , then report $\frac{1}{6}$ of $\star$ .

Givens

$\frac{5}{9}$ of $\star$ is 30.
$\star$ is the whole quantity (the 'whole') being split into 9ths.

Unknowns

The value of $\frac{1}{6}$ of $\star$ .

Constraints

$\frac{5}{9}$ of $\star$ means $\star$ split into 9 equal parts, with 5 of those parts totaling 30.
$\star$ must come out a whole number so $\frac{1}{6}$ of it makes sense.

2 · Planchoose the strategy

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

I work backwards from the part to the whole: if 5 9ths equal 30, then one 9th equals $\frac{30}{5}$ , and the whole is 9 of those. Then taking $\frac{1}{6}$ of the recovered whole is a second small subproblem.

3 · Execute3 carry out the plan

1Find one 9th of $\star$

#11 Work Backwards 3.OA.A.4

5 equal 9ths make 30, so one 9th is 30 divided by 5.

30 \div 5 = 6

If 5 equal parts total 30, each part is found by sharing 30 into 5 equal pieces.

2Build the whole $\star$

#11 Work Backwards 3.NF.A.1

\star

is made of 9 such 9ths, so multiply one 9th (6) by 9.

6 \times 9 = 54

The whole is just all of its equal parts put back together.

3Take $\frac{1}{6}$ of $\star$

#7 Identify Subproblems 3.NF.A.1

Now find

\frac{1}{6}

of 54 by dividing 54 into 6 equal parts.

54 \div 6 = 9

A fraction

\frac{1}{6}

of a number is that number shared into 6 equal parts, taking 1.

Answer: 9

4 · Reviewdoes it hold up?

Check: $\frac{5}{9}$ of 54 is $\frac{54}{9}$ x 5 = 6 x 5 = 30, which matches the given. And $\frac{1}{6}$ of 54 is 9. Both come out as clean whole numbers, so 54 is the right whole and 9 is the answer.

Another way: Convert to algebra (tool 13): ( $\frac{5}{9}$ ) x $\star$ = 30 gives $\star$ = 30 x $\frac{9}{5}$ = 54, then $\frac{1}{6}$ of it = 9 - same result, but the work-backwards unit-fraction route avoids algebra.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding one 9th from 5 9ths = 30 (an unknown-part division).
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Rebuilding the whole from 9ths and taking $\frac{1}{6}$ of it.

💡Takeaway. This only needs Grade 3 fraction sense: find one equal part, build the whole, then split again!

Variant 9 hard answer: 9

Find $\frac{1}{4}$ of the number $\star$ that satisfies the following.

$\frac{5}{6} \text{ of } \star \text{ is } 30.$

Show solution

1 · Understandwhat's really being asked

A mystery number $\star$ has the property that $\frac{5}{6}$ of it equals 30. I first need to find $\star$ , then report $\frac{1}{4}$ of $\star$ .

Givens

$\frac{5}{6}$ of $\star$ is 30.
$\star$ is the whole quantity (the 'whole') being split into 6ths.

Unknowns

The value of $\frac{1}{4}$ of $\star$ .

Constraints

$\frac{5}{6}$ of $\star$ means $\star$ split into 6 equal parts, with 5 of those parts totaling 30.
$\star$ must come out a whole number so $\frac{1}{4}$ of it makes sense.

2 · Planchoose the strategy

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

I work backwards from the part to the whole: if 5 6ths equal 30, then one 6th equals $\frac{30}{5}$ , and the whole is 6 of those. Then taking $\frac{1}{4}$ of the recovered whole is a second small subproblem.

3 · Execute3 carry out the plan

1Find one 6th of $\star$

#11 Work Backwards 3.OA.A.4

5 equal 6ths make 30, so one 6th is 30 divided by 5.

30 \div 5 = 6

If 5 equal parts total 30, each part is found by sharing 30 into 5 equal pieces.

2Build the whole $\star$

#11 Work Backwards 3.NF.A.1

\star

is made of 6 such 6ths, so multiply one 6th (6) by 6.

6 \times 6 = 36

The whole is just all of its equal parts put back together.

3Take $\frac{1}{4}$ of $\star$

#7 Identify Subproblems 3.NF.A.1

Now find

\frac{1}{4}

of 36 by dividing 36 into 4 equal parts.

36 \div 4 = 9

A fraction

\frac{1}{4}

of a number is that number shared into 4 equal parts, taking 1.

Answer: 9

4 · Reviewdoes it hold up?

Check: $\frac{5}{6}$ of 36 is 36 / 6 x 5 = 6 x 5 = 30, which matches the given. And $\frac{1}{4}$ of 36 is 9. Both come out as clean whole numbers, so 36 is the right whole and 9 is the answer.

Another way: Convert to algebra (tool 13): ( $\frac{5}{6}$ ) x $\star$ = 30 gives $\star$ = 30 x 6 / 5 = 36, then $\frac{1}{4}$ of it = 9 - same result, but the work-backwards unit-fraction route avoids algebra.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding one 6th from 5 6ths = 30 (an unknown-part division).
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Rebuilding the whole from 6ths and taking $\frac{1}{4}$ of it.

💡Takeaway. This only needs Grade 3 fraction sense: find one equal part, build the whole, then split again!

Variant 10 medium answer: 8

Find $\frac{1}{6}$ of the number $\star$ that satisfies the following.

$\frac{5}{8} \text{ of } \star \text{ is } 30.$

Show solution

1 · Understandwhat's really being asked

A mystery number $\star$ has the property that $\frac{5}{8}$ of it equals 30. I first need to find $\star$ , then report $\frac{1}{6}$ of $\star$ .

Givens

$\frac{5}{8}$ of $\star$ is 30.
$\star$ is the whole quantity (the 'whole') being split into 8ths.

Unknowns

The value of $\frac{1}{6}$ of $\star$ .

Constraints

$\frac{5}{8}$ of $\star$ means $\star$ split into 8 equal parts, with 5 of those parts totaling 30.
$\star$ must come out a whole number so $\frac{1}{6}$ of it makes sense.

2 · Planchoose the strategy

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

I work backwards from the part to the whole: if 5 8ths equal 30, then one 8th equals $\frac{30}{5}$ , and the whole is 8 of those. Then taking $\frac{1}{6}$ of the recovered whole is a second small subproblem.

3 · Execute3 carry out the plan

1Find one 8th of $\star$

#11 Work Backwards 3.OA.A.4

5 equal 8ths make 30, so one 8th is 30 divided by 5.

30 \div 5 = 6

If 5 equal parts total 30, each part is found by sharing 30 into 5 equal pieces.

2Build the whole $\star$

#11 Work Backwards 3.NF.A.1

\star

is made of 8 such 8ths, so multiply one 8th (6) by 8.

6 \times 8 = 48

The whole is just all of its equal parts put back together.

3Take $\frac{1}{6}$ of $\star$

#7 Identify Subproblems 3.NF.A.1

Now find

\frac{1}{6}

of 48 by dividing 48 into 6 equal parts.

48 \div 6 = 8

A fraction

\frac{1}{6}

of a number is that number shared into 6 equal parts, taking 1.

Answer: 8

4 · Reviewdoes it hold up?

Check: $\frac{5}{8}$ of 48 is 48 / 8 x 5 = 6 x 5 = 30, which matches the given. And $\frac{1}{6}$ of 48 is 8. Both come out as clean whole numbers, so 48 is the right whole and 8 is the answer.

Another way: Convert to algebra (tool 13): ( $\frac{5}{8}$ ) x $\star$ = 30 gives $\star$ = 30 x 8 / 5 = 48, then $\frac{1}{6}$ of it = 8 - same result, but the work-backwards unit-fraction route avoids algebra.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding one 8th from 5 8ths = 30 (an unknown-part division).
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Rebuilding the whole from 8ths and taking $\frac{1}{6}$ of it.

💡Takeaway. This only needs Grade 3 fraction sense: find one equal part, build the whole, then split again!

Variant 11 hard answer: 10

Find $\frac{1}{5}$ of the number $\star$ that satisfies the following.

$\frac{7}{10} \text{ of } \star \text{ is } 35.$

Show solution

1 · Understandwhat's really being asked

A mystery number $\star$ has the property that $\frac{7}{10}$ of it equals 35. I first need to find $\star$ , then report $\frac{1}{5}$ of $\star$ .

Givens

$\frac{7}{10}$ of $\star$ is 35.
$\star$ is the whole quantity (the 'whole') being split into 10ths.

Unknowns

The value of $\frac{1}{5}$ of $\star$ .

Constraints

$\frac{7}{10}$ of $\star$ means $\star$ split into 10 equal parts, with 7 of those parts totaling 35.
$\star$ must come out a whole number so $\frac{1}{5}$ of it makes sense.

2 · Planchoose the strategy

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

I work backwards from the part to the whole: if 7 10ths equal 35, then one 10th equals $\frac{35}{7}$ , and the whole is 10 of those. Then taking $\frac{1}{5}$ of the recovered whole is a second small subproblem.

3 · Execute3 carry out the plan

1Find one 10th of $\star$

#11 Work Backwards 3.OA.A.4

7 equal 10ths make 35, so one 10th is 35 divided by 7.

35 \div 7 = 5

If 7 equal parts total 35, each part is found by sharing 35 into 7 equal pieces.

2Build the whole $\star$

#11 Work Backwards 3.NF.A.1

\star

is made of 10 such 10ths, so multiply one 10th (5) by 10.

5 \times 10 = 50

The whole is just all of its equal parts put back together.

3Take $\frac{1}{5}$ of $\star$

#7 Identify Subproblems 3.NF.A.1

Now find

\frac{1}{5}

of 50 by dividing 50 into 5 equal parts.

50 \div 5 = 10

A fraction

\frac{1}{5}

of a number is that number shared into 5 equal parts, taking 1.

Answer: 10

4 · Reviewdoes it hold up?

Check: $\frac{7}{10}$ of 50 is $\frac{50}{10}$ x 7 = 5 x 7 = 35, which matches the given. And $\frac{1}{5}$ of 50 is 10. Both come out as clean whole numbers, so 50 is the right whole and 10 is the answer.

Another way: Convert to algebra (tool 13): ( $\frac{7}{10}$ ) x $\star$ = 35 gives $\star$ = 35 x $\frac{10}{7}$ = 50, then $\frac{1}{5}$ of it = 10 - same result, but the work-backwards unit-fraction route avoids algebra.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding one 10th from 7 10ths = 35 (an unknown-part division).
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Rebuilding the whole from 10ths and taking $\frac{1}{5}$ of it.

💡Takeaway. This only needs Grade 3 fraction sense: find one equal part, build the whole, then split again!

Variant 12 hard answer: 9

Find $\frac{1}{8}$ of the number $\star$ that satisfies the following.

$\frac{5}{9} \text{ of } \star \text{ is } 40.$

Show solution

1 · Understandwhat's really being asked

A mystery number $\star$ has the property that $\frac{5}{9}$ of it equals 40. I first need to find $\star$ , then report $\frac{1}{8}$ of $\star$ .

Givens

$\frac{5}{9}$ of $\star$ is 40.
$\star$ is the whole quantity (the 'whole') being split into 9ths.

Unknowns

The value of $\frac{1}{8}$ of $\star$ .

Constraints

$\frac{5}{9}$ of $\star$ means $\star$ split into 9 equal parts, with 5 of those parts totaling 40.
$\star$ must come out a whole number so $\frac{1}{8}$ of it makes sense.

2 · Planchoose the strategy

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

I work backwards from the part to the whole: if 5 9ths equal 40, then one 9th equals $\frac{40}{5}$ , and the whole is 9 of those. Then taking $\frac{1}{8}$ of the recovered whole is a second small subproblem.

3 · Execute3 carry out the plan

1Find one 9th of $\star$

#11 Work Backwards 3.OA.A.4

5 equal 9ths make 40, so one 9th is 40 divided by 5.

40 \div 5 = 8

If 5 equal parts total 40, each part is found by sharing 40 into 5 equal pieces.

2Build the whole $\star$

#11 Work Backwards 3.NF.A.1

\star

is made of 9 such 9ths, so multiply one 9th (8) by 9.

8 \times 9 = 72

The whole is just all of its equal parts put back together.

3Take $\frac{1}{8}$ of $\star$

#7 Identify Subproblems 3.NF.A.1

Now find

\frac{1}{8}

of 72 by dividing 72 into 8 equal parts.

72 \div 8 = 9

A fraction

\frac{1}{8}

of a number is that number shared into 8 equal parts, taking 1.

Answer: 9

4 · Reviewdoes it hold up?

Check: $\frac{5}{9}$ of 72 is $\frac{72}{9}$ x 5 = 8 x 5 = 40, which matches the given. And $\frac{1}{8}$ of 72 is 9. Both come out as clean whole numbers, so 72 is the right whole and 9 is the answer.

Another way: Convert to algebra (tool 13): ( $\frac{5}{9}$ ) x $\star$ = 40 gives $\star$ = 40 x $\frac{9}{5}$ = 72, then $\frac{1}{8}$ of it = 9 - same result, but the work-backwards unit-fraction route avoids algebra.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding one 9th from 5 9ths = 40 (an unknown-part division).
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Rebuilding the whole from 9ths and taking $\frac{1}{8}$ of it.

💡Takeaway. This only needs Grade 3 fraction sense: find one equal part, build the whole, then split again!

Recover the whole from a fractional part · 12 practice problems

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