Practice · Recover each correct part from a wrong decimal · Sensim Math

Variant 1 easy

Mia and Liam were each asked to write the same fraction, which is in lowest terms, as a decimal. Mia misread the denominator and got $0.58\overline{3}$ . Liam misread the numerator and got $0.36\overline{1}$ . Write the original fraction as a repeating decimal.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A fraction in lowest terms was turned into a decimal twice, each time with one part misread. Mia kept the right numerator and got $0.58\overline{3}$ ; Liam kept the right denominator and got $0.36\overline{1}$ . Rebuild the true fraction and write it as a repeating decimal.

Givens

Mia (wrong denominator, right numerator) got $0.58\overline{3}$ .
Liam (wrong numerator, right denominator) got $0.36\overline{1}$ .
The original fraction is in lowest terms.

Unknowns

The original fraction, written as a repeating decimal.

Constraints

Read each wrong decimal back as a fraction in lowest terms; the correctly read part shows up there.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #15 Organize Information in More Ways#7 Identify Subproblems

Each wrong answer still carries one correct piece. Work backwards from each decimal to its fraction in lowest terms, take the numerator from Mia's result and the denominator from Liam's, then convert the rebuilt fraction to a decimal.

3 · Execute3 carry out the plan

1Undo Mia's decimal

#11 Work Backwards 8.NS.A.1

In

0.58\overline{3}

only the

3

repeats. Subtract the non-repeating part

58

from

583

and divide by

900

(one

9

for the repeating digit, two

0

s for the two digits that do not repeat).

0.58\overline{3} = \frac{583 - 58}{900} = \frac{525}{900} = \frac{7}{12}

Mia read the numerator correctly, so the numerator is

7

.

2Undo Liam's decimal

#11 Work Backwards 8.NS.A.1

In

0.36\overline{1}

only the

1

repeats, so use

900

as the denominator.

0.36\overline{1} = \frac{361 - 36}{900} = \frac{325}{900} = \frac{13}{36}

This reduces to

\frac{13}{36}

; Liam read the denominator correctly, so it is

36

.

3Rebuild the fraction and convert

#7 Identify Subproblems 7.NS.A.2

Put the correct numerator over the correct denominator.

\frac{7}{36}

is in lowest terms, and dividing gives

0.19444\ldots

.

\frac{7}{36} = 0.19444\ldots = 0.19\overline{4}

Only the

4

repeats, starting at the third decimal place.

Answer:

0.19\overline{4}

4 · Reviewdoes it hold up?

Check by converting back: $0.19\overline{4} = \frac{7}{36}$ . Its numerator matches Mia's $\frac{7}{12}$ and its denominator matches Liam's $\frac{13}{36}$ , as the story requires.

Another way: Use the $x$ method for Mia's decimal: if $x = 0.58\overline{3}$ , then $1000x - 100x = 583.\overline{3} - 58.\overline{3} = 525$ , so $x = \frac{525}{900} = \frac{7}{12}$ , the same numerator.

Standardsmin grade 8

8.NS.A.1 Know that numbers that are not rational are called irrational numbers — Converting the repeating decimals $0.58\overline{3}$ and $0.36\overline{1}$ back into fractions.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Converting the rebuilt fraction $\frac{7}{36}$ into its repeating decimal by long division.

💡Takeaway. A wrong answer still hides a right part -- work backwards to find it.

Variant 2 easy

Mia and Liam were each asked to write the same fraction, which is in lowest terms, as a decimal. Mia misread the denominator and got $0.8\overline{3}$ . Liam misread the numerator and got $0.6\overline{1}$ . Write the original fraction as a repeating decimal.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A fraction in lowest terms was turned into a decimal twice, each time with one part misread. Mia kept the right numerator and got $0.8\overline{3}$ ; Liam kept the right denominator and got $0.6\overline{1}$ . Rebuild the true fraction and write it as a repeating decimal.

Givens

Mia (wrong denominator, right numerator) got $0.8\overline{3}$ .
Liam (wrong numerator, right denominator) got $0.6\overline{1}$ .
The original fraction is in lowest terms.

Unknowns

The original fraction, written as a repeating decimal.

Constraints

Read each wrong decimal back as a fraction in lowest terms; the correctly read part shows up there.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #15 Organize Information in More Ways#7 Identify Subproblems

Each wrong answer still carries one correct piece. Work backwards from each decimal to its fraction in lowest terms, take the numerator from Mia's result and the denominator from Liam's, then convert the rebuilt fraction to a decimal.

3 · Execute3 carry out the plan

1Undo Mia's decimal

#11 Work Backwards 8.NS.A.1

In

0.8\overline{3}

only the

3

repeats. Subtract the non-repeating part

8

from

83

and divide by

90

(one

9

for the repeating digit, one

0

for the digit that does not repeat).

0.8\overline{3} = \frac{83 - 8}{90} = \frac{75}{90} = \frac{5}{6}

Mia read the numerator correctly, so the numerator is

5

.

2Undo Liam's decimal

#11 Work Backwards 8.NS.A.1

In

0.6\overline{1}

only the

1

repeats, so use

90

as the denominator.

0.6\overline{1} = \frac{61 - 6}{90} = \frac{55}{90} = \frac{11}{18}

This reduces to

\frac{11}{18}

; Liam read the denominator correctly, so it is

18

.

3Rebuild the fraction and convert

#7 Identify Subproblems 7.NS.A.2

Put the correct numerator over the correct denominator.

\frac{5}{18}

is in lowest terms, and dividing gives

0.2777\ldots

.

\frac{5}{18} = 0.2777\ldots = 0.2\overline{7}

Only the

7

repeats, starting at the second decimal place.

Answer:

0.2\overline{7}

4 · Reviewdoes it hold up?

Check by converting back: $0.2\overline{7} = \frac{5}{18}$ . Its numerator matches Mia's $\frac{5}{6}$ and its denominator matches Liam's $\frac{11}{18}$ , as the story requires.

Another way: Use the $x$ method for Mia's decimal: if $x = 0.8\overline{3}$ , then $100x - 10x = 83.\overline{3} - 8.\overline{3} = 75$ , so $x = \frac{75}{90} = \frac{5}{6}$ , the same numerator.

Standardsmin grade 8

8.NS.A.1 Know that numbers that are not rational are called irrational numbers — Converting the repeating decimals $0.8\overline{3}$ and $0.6\overline{1}$ back into fractions.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Converting the rebuilt fraction $\frac{5}{18}$ into its repeating decimal by long division.

💡Takeaway. A wrong answer still hides a right part -- work backwards to find it.

Variant 3 easy

Mia and Liam were each asked to write the same fraction, which is in lowest terms, as a decimal. Mia misread the denominator and got $0.58\overline{3}$ . Liam misread the numerator and got $0.8\overline{1}$ . Write the original fraction as a repeating decimal.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A fraction in lowest terms was turned into a decimal twice, each time with one part misread. Mia kept the right numerator and got $0.58\overline{3}$ ; Liam kept the right denominator and got $0.8\overline{1}$ . Rebuild the true fraction and write it as a repeating decimal.

Givens

Mia (wrong denominator, right numerator) got $0.58\overline{3}$ .
Liam (wrong numerator, right denominator) got $0.8\overline{1}$ .
The original fraction is in lowest terms.

Unknowns

The original fraction, written as a repeating decimal.

Constraints

Read each wrong decimal back as a fraction in lowest terms; the correctly read part shows up there.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #15 Organize Information in More Ways#7 Identify Subproblems

Each wrong answer still carries one correct piece. Work backwards from each decimal to its fraction in lowest terms, take the numerator from Mia's result and the denominator from Liam's, then convert the rebuilt fraction to a decimal.

3 · Execute3 carry out the plan

1Undo Mia's decimal

#11 Work Backwards 8.NS.A.1

In

0.58\overline{3}

only the

3

repeats. Subtract the non-repeating part

58

from

583

and divide by

900

(one

9

for the repeating digit, two

0

s for the two digits that do not repeat).

0.58\overline{3} = \frac{583 - 58}{900} = \frac{525}{900} = \frac{7}{12}

Mia read the numerator correctly, so the numerator is

7

.

2Undo Liam's decimal

#11 Work Backwards 8.NS.A.1

In

0.8\overline{1}

only the

1

repeats, so use

90

as the denominator.

0.8\overline{1} = \frac{81 - 8}{90} = \frac{73}{90}

\frac{73}{90}

is already in lowest terms; Liam read the denominator correctly, so it is

90

.

3Rebuild the fraction and convert

#7 Identify Subproblems 7.NS.A.2

Put the correct numerator over the correct denominator.

\frac{7}{90}

is in lowest terms, and dividing gives

0.0777\ldots

.

\frac{7}{90} = 0.0777\ldots = 0.0\overline{7}

Only the

7

repeats, starting at the second decimal place.

Answer:

0.0\overline{7}

4 · Reviewdoes it hold up?

Check by converting back: $0.0\overline{7} = \frac{7}{90}$ . Its numerator matches Mia's $\frac{7}{12}$ and its denominator matches Liam's $\frac{73}{90}$ , as the story requires.

Another way: Use the $x$ method for Mia's decimal: if $x = 0.58\overline{3}$ , then $1000x - 100x = 583.\overline{3} - 58.\overline{3} = 525$ , so $x = \frac{525}{900} = \frac{7}{12}$ , the same numerator.

Standardsmin grade 8

8.NS.A.1 Know that numbers that are not rational are called irrational numbers — Converting the repeating decimals $0.58\overline{3}$ and $0.8\overline{1}$ back into fractions.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Converting the rebuilt fraction $\frac{7}{90}$ into its repeating decimal by long division.

💡Takeaway. A wrong answer still hides a right part -- work backwards to find it.

Variant 4 medium

Mia and Liam were each asked to write the same fraction, which is in lowest terms, as a decimal. Mia misread the denominator and got $0.8\overline{6}$ . Liam misread the numerator and got $0.18\overline{3}$ . Write the original fraction as a repeating decimal.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A fraction in lowest terms was turned into a decimal twice, each time with one part misread. Mia kept the right numerator and got $0.8\overline{6}$ ; Liam kept the right denominator and got $0.18\overline{3}$ . Rebuild the true fraction and write it as a repeating decimal.

Givens

Mia (wrong denominator, right numerator) got $0.8\overline{6}$ .
Liam (wrong numerator, right denominator) got $0.18\overline{3}$ .
The original fraction is in lowest terms.

Unknowns

The original fraction, written as a repeating decimal.

Constraints

Read each wrong decimal back as a fraction in lowest terms; the correctly read part shows up there.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #15 Organize Information in More Ways#7 Identify Subproblems

Each wrong answer still carries one correct piece. Work backwards from each decimal to its fraction in lowest terms, take the numerator from Mia's result and the denominator from Liam's, then convert the rebuilt fraction to a decimal.

3 · Execute3 carry out the plan

1Undo Mia's decimal

#11 Work Backwards 8.NS.A.1

In

0.8\overline{6}

only the

6

repeats. Subtract the non-repeating part

8

from

86

and divide by

90

(one

9

for the repeating digit, one

0

for the digit that does not repeat).

0.8\overline{6} = \frac{86 - 8}{90} = \frac{78}{90} = \frac{13}{15}

Mia read the numerator correctly, so the numerator is

13

.

2Undo Liam's decimal

#11 Work Backwards 8.NS.A.1

In

0.18\overline{3}

only the

3

repeats, so use

900

as the denominator.

0.18\overline{3} = \frac{183 - 18}{900} = \frac{165}{900} = \frac{11}{60}

This reduces to

\frac{11}{60}

; Liam read the denominator correctly, so it is

60

.

3Rebuild the fraction and convert

#7 Identify Subproblems 7.NS.A.2

Put the correct numerator over the correct denominator.

\frac{13}{60}

is in lowest terms, and dividing gives

0.21666\ldots

.

\frac{13}{60} = 0.21666\ldots = 0.21\overline{6}

Only the

6

repeats, starting at the third decimal place.

Answer:

0.21\overline{6}

4 · Reviewdoes it hold up?

Check by converting back: $0.21\overline{6} = \frac{13}{60}$ . Its numerator matches Mia's $\frac{13}{15}$ and its denominator matches Liam's $\frac{11}{60}$ , as the story requires.

Another way: Use the $x$ method for Mia's decimal: if $x = 0.8\overline{6}$ , then $100x - 10x = 86.\overline{6} - 8.\overline{6} = 78$ , so $x = \frac{78}{90} = \frac{13}{15}$ , the same numerator.

Standardsmin grade 8

8.NS.A.1 Know that numbers that are not rational are called irrational numbers — Converting the repeating decimals $0.8\overline{6}$ and $0.18\overline{3}$ back into fractions.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Converting the rebuilt fraction $\frac{13}{60}$ into its repeating decimal by long division.

💡Takeaway. A wrong answer still hides a right part -- work backwards to find it.

Variant 5 easy

Mia and Liam were each asked to write the same fraction, which is in lowest terms, as a decimal. Mia misread the denominator and got $0.9\overline{6}$ . Liam misread the numerator and got $0.8\overline{2}$ . Write the original fraction as a repeating decimal.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A fraction in lowest terms was turned into a decimal twice, each time with one part misread. Mia kept the right numerator and got $0.9\overline{6}$ ; Liam kept the right denominator and got $0.8\overline{2}$ . Rebuild the true fraction and write it as a repeating decimal.

Givens

Mia (wrong denominator, right numerator) got $0.9\overline{6}$ .
Liam (wrong numerator, right denominator) got $0.8\overline{2}$ .
The original fraction is in lowest terms.

Unknowns

The original fraction, written as a repeating decimal.

Constraints

Read each wrong decimal back as a fraction in lowest terms; the correctly read part shows up there.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #15 Organize Information in More Ways#7 Identify Subproblems

Each wrong answer still carries one correct piece. Work backwards from each decimal to its fraction in lowest terms, take the numerator from Mia's result and the denominator from Liam's, then convert the rebuilt fraction to a decimal.

3 · Execute3 carry out the plan

1Undo Mia's decimal

#11 Work Backwards 8.NS.A.1

In

0.9\overline{6}

only the

6

repeats. Subtract the non-repeating part

9

from

96

and divide by

90

(one

9

for the repeating digit, one

0

for the digit that does not repeat).

0.9\overline{6} = \frac{96 - 9}{90} = \frac{87}{90} = \frac{29}{30}

Mia read the numerator correctly, so the numerator is

29

.

2Undo Liam's decimal

#11 Work Backwards 8.NS.A.1

In

0.8\overline{2}

only the

2

repeats, so use

90

as the denominator.

0.8\overline{2} = \frac{82 - 8}{90} = \frac{74}{90} = \frac{37}{45}

This reduces to

\frac{37}{45}

; Liam read the denominator correctly, so it is

45

.

3Rebuild the fraction and convert

#7 Identify Subproblems 7.NS.A.2

Put the correct numerator over the correct denominator.

\frac{29}{45}

is in lowest terms, and dividing gives

0.6444\ldots

.

\frac{29}{45} = 0.6444\ldots = 0.6\overline{4}

Only the

4

repeats, starting at the second decimal place.

Answer:

0.6\overline{4}

4 · Reviewdoes it hold up?

Check by converting back: $0.6\overline{4} = \frac{29}{45}$ . Its numerator matches Mia's $\frac{29}{30}$ and its denominator matches Liam's $\frac{37}{45}$ , as the story requires.

Another way: Use the $x$ method for Mia's decimal: if $x = 0.9\overline{6}$ , then $100x - 10x = 96.\overline{6} - 9.\overline{6} = 87$ , so $x = \frac{87}{90} = \frac{29}{30}$ , the same numerator.

Standardsmin grade 8

8.NS.A.1 Know that numbers that are not rational are called irrational numbers — Converting the repeating decimals $0.9\overline{6}$ and $0.8\overline{2}$ back into fractions.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Converting the rebuilt fraction $\frac{29}{45}$ into its repeating decimal by long division.

💡Takeaway. A wrong answer still hides a right part -- work backwards to find it.

Variant 6 medium

Mia and Liam were each asked to write the same fraction, which is in lowest terms, as a decimal. Mia misread the denominator and got $0.5\overline{6}$ . Liam misread the numerator and got $0.57\overline{3}$ . Write the original fraction as a repeating decimal.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A fraction in lowest terms was turned into a decimal twice, each time with one part misread. Mia kept the right numerator and got $0.5\overline{6}$ ; Liam kept the right denominator and got $0.57\overline{3}$ . Rebuild the true fraction and write it as a repeating decimal.

Givens

Mia (wrong denominator, right numerator) got $0.5\overline{6}$ .
Liam (wrong numerator, right denominator) got $0.57\overline{3}$ .
The original fraction is in lowest terms.

Unknowns

The original fraction, written as a repeating decimal.

Constraints

Read each wrong decimal back as a fraction in lowest terms; the correctly read part shows up there.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #15 Organize Information in More Ways#7 Identify Subproblems

Each wrong answer still carries one correct piece. Work backwards from each decimal to its fraction in lowest terms, take the numerator from Mia's result and the denominator from Liam's, then convert the rebuilt fraction to a decimal.

3 · Execute3 carry out the plan

1Undo Mia's decimal

#11 Work Backwards 8.NS.A.1

In

0.5\overline{6}

only the

6

repeats. Subtract the non-repeating part

5

from

56

and divide by

90

(one

9

for the repeating digit, one

0

for the digit that does not repeat).

0.5\overline{6} = \frac{56 - 5}{90} = \frac{51}{90} = \frac{17}{30}

Mia read the numerator correctly, so the numerator is

17

.

2Undo Liam's decimal

#11 Work Backwards 8.NS.A.1

In

0.57\overline{3}

only the

3

repeats, so use

900

as the denominator.

0.57\overline{3} = \frac{573 - 57}{900} = \frac{516}{900} = \frac{43}{75}

This reduces to

\frac{43}{75}

; Liam read the denominator correctly, so it is

75

.

3Rebuild the fraction and convert

#7 Identify Subproblems 7.NS.A.2

Put the correct numerator over the correct denominator.

\frac{17}{75}

is in lowest terms, and dividing gives

0.22666\ldots

.

\frac{17}{75} = 0.22666\ldots = 0.22\overline{6}

Only the

6

repeats, starting at the third decimal place.

Answer:

0.22\overline{6}

4 · Reviewdoes it hold up?

Check by converting back: $0.22\overline{6} = \frac{17}{75}$ . Its numerator matches Mia's $\frac{17}{30}$ and its denominator matches Liam's $\frac{43}{75}$ , as the story requires.

Another way: Use the $x$ method for Mia's decimal: if $x = 0.5\overline{6}$ , then $100x - 10x = 56.\overline{6} - 5.\overline{6} = 51$ , so $x = \frac{51}{90} = \frac{17}{30}$ , the same numerator.

Standardsmin grade 8

8.NS.A.1 Know that numbers that are not rational are called irrational numbers — Converting the repeating decimals $0.5\overline{6}$ and $0.57\overline{3}$ back into fractions.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Converting the rebuilt fraction $\frac{17}{75}$ into its repeating decimal by long division.

💡Takeaway. A wrong answer still hides a right part -- work backwards to find it.

Variant 7 medium

Mia and Liam were each asked to write the same fraction, which is in lowest terms, as a decimal. Mia misread the denominator and got $0.4\overline{3}$ . Liam misread the numerator and got $0.2\overline{6}$ . Write the original fraction as a repeating decimal.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A fraction in lowest terms was turned into a decimal twice, each time with one part misread. Mia kept the right numerator and got $0.4\overline{3}$ ; Liam kept the right denominator and got $0.2\overline{6}$ . Rebuild the true fraction and write it as a repeating decimal.

Givens

Mia (wrong denominator, right numerator) got $0.4\overline{3}$ .
Liam (wrong numerator, right denominator) got $0.2\overline{6}$ .
The original fraction is in lowest terms.

Unknowns

The original fraction, written as a repeating decimal.

Constraints

Read each wrong decimal back as a fraction in lowest terms; the correctly read part shows up there.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #15 Organize Information in More Ways#7 Identify Subproblems

Each wrong answer still carries one correct piece. Work backwards from each decimal to its fraction in lowest terms, take the numerator from Mia's result and the denominator from Liam's, then convert the rebuilt fraction to a decimal.

3 · Execute3 carry out the plan

1Undo Mia's decimal

#11 Work Backwards 8.NS.A.1

In

0.4\overline{3}

only the

3

repeats. Subtract the non-repeating part

4

from

43

and divide by

90

(one

9

for the repeating digit, one

0

for the digit that does not repeat).

0.4\overline{3} = \frac{43 - 4}{90} = \frac{39}{90} = \frac{13}{30}

Mia read the numerator correctly, so the numerator is

13

.

2Undo Liam's decimal

#11 Work Backwards 8.NS.A.1

In

0.2\overline{6}

only the

6

repeats, so use

90

as the denominator.

0.2\overline{6} = \frac{26 - 2}{90} = \frac{24}{90} = \frac{4}{15}

This reduces to

\frac{4}{15}

; Liam read the denominator correctly, so it is

15

.

3Rebuild the fraction and convert

#7 Identify Subproblems 7.NS.A.2

Put the correct numerator over the correct denominator.

\frac{13}{15}

is in lowest terms, and dividing gives

0.8666\ldots

.

\frac{13}{15} = 0.8666\ldots = 0.8\overline{6}

Only the

6

repeats, starting at the second decimal place.

Answer:

0.8\overline{6}

4 · Reviewdoes it hold up?

Check by converting back: $0.8\overline{6} = \frac{13}{15}$ . Its numerator matches Mia's $\frac{13}{30}$ and its denominator matches Liam's $\frac{4}{15}$ , as the story requires.

Another way: Use the $x$ method for Mia's decimal: if $x = 0.4\overline{3}$ , then $100x - 10x = 43.\overline{3} - 4.\overline{3} = 39$ , so $x = \frac{39}{90} = \frac{13}{30}$ , the same numerator.

Standardsmin grade 8

8.NS.A.1 Know that numbers that are not rational are called irrational numbers — Converting the repeating decimals $0.4\overline{3}$ and $0.2\overline{6}$ back into fractions.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Converting the rebuilt fraction $\frac{13}{15}$ into its repeating decimal by long division.

💡Takeaway. A wrong answer still hides a right part -- work backwards to find it.

Variant 8 medium

Mia and Liam were each asked to write the same fraction, which is in lowest terms, as a decimal. Mia misread the denominator and got $0.63\overline{8}$ . Liam misread the numerator and got $0.3\overline{5}$ . Write the original fraction as a repeating decimal.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A fraction in lowest terms was turned into a decimal twice, each time with one part misread. Mia kept the right numerator and got $0.63\overline{8}$ ; Liam kept the right denominator and got $0.3\overline{5}$ . Rebuild the true fraction and write it as a repeating decimal.

Givens

Mia (wrong denominator, right numerator) got $0.63\overline{8}$ .
Liam (wrong numerator, right denominator) got $0.3\overline{5}$ .
The original fraction is in lowest terms.

Unknowns

The original fraction, written as a repeating decimal.

Constraints

Read each wrong decimal back as a fraction in lowest terms; the correctly read part shows up there.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #15 Organize Information in More Ways#7 Identify Subproblems

Each wrong answer still carries one correct piece. Work backwards from each decimal to its fraction in lowest terms, take the numerator from Mia's result and the denominator from Liam's, then convert the rebuilt fraction to a decimal.

3 · Execute3 carry out the plan

1Undo Mia's decimal

#11 Work Backwards 8.NS.A.1

In

0.63\overline{8}

only the

8

repeats. Subtract the non-repeating part

63

from

638

and divide by

900

(one

9

for the repeating digit, two

0

s for the two digits that do not repeat).

0.63\overline{8} = \frac{638 - 63}{900} = \frac{575}{900} = \frac{23}{36}

Mia read the numerator correctly, so the numerator is

23

.

2Undo Liam's decimal

#11 Work Backwards 8.NS.A.1

In

0.3\overline{5}

only the

5

repeats, so use

90

as the denominator.

0.3\overline{5} = \frac{35 - 3}{90} = \frac{32}{90} = \frac{16}{45}

This reduces to

\frac{16}{45}

; Liam read the denominator correctly, so it is

45

.

3Rebuild the fraction and convert

#7 Identify Subproblems 7.NS.A.2

Put the correct numerator over the correct denominator.

\frac{23}{45}

is in lowest terms, and dividing gives

0.5111\ldots

.

\frac{23}{45} = 0.5111\ldots = 0.5\overline{1}

Only the

1

repeats, starting at the second decimal place.

Answer:

0.5\overline{1}

4 · Reviewdoes it hold up?

Check by converting back: $0.5\overline{1} = \frac{23}{45}$ . Its numerator matches Mia's $\frac{23}{36}$ and its denominator matches Liam's $\frac{16}{45}$ , as the story requires.

Another way: Use the $x$ method for Mia's decimal: if $x = 0.63\overline{8}$ , then $1000x - 100x = 638.\overline{8} - 63.\overline{8} = 575$ , so $x = \frac{575}{900} = \frac{23}{36}$ , the same numerator.

Standardsmin grade 8

8.NS.A.1 Know that numbers that are not rational are called irrational numbers — Converting the repeating decimals $0.63\overline{8}$ and $0.3\overline{5}$ back into fractions.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Converting the rebuilt fraction $\frac{23}{45}$ into its repeating decimal by long division.

💡Takeaway. A wrong answer still hides a right part -- work backwards to find it.

Variant 9 hard

Mia and Liam were each asked to write the same fraction, which is in lowest terms, as a decimal. Mia misread the denominator and got $0.4\overline{6}$ . Liam misread the numerator and got $0.4\overline{09}$ . Write the original fraction as a repeating decimal.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A fraction in lowest terms was turned into a decimal twice, each time with one part misread. Mia kept the right numerator and got $0.4\overline{6}$ ; Liam kept the right denominator and got $0.4\overline{09}$ . Rebuild the true fraction and write it as a repeating decimal.

Givens

Mia (wrong denominator, right numerator) got $0.4\overline{6}$ .
Liam (wrong numerator, right denominator) got $0.4\overline{09}$ .
The original fraction is in lowest terms.

Unknowns

The original fraction, written as a repeating decimal.

Constraints

Read each wrong decimal back as a fraction in lowest terms; the correctly read part shows up there.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #15 Organize Information in More Ways#7 Identify Subproblems

Each wrong answer still carries one correct piece. Work backwards from each decimal to its fraction in lowest terms, take the numerator from Mia's result and the denominator from Liam's, then convert the rebuilt fraction to a decimal.

3 · Execute3 carry out the plan

1Undo Mia's decimal

#11 Work Backwards 8.NS.A.1

In

0.4\overline{6}

only the

6

repeats. Subtract the non-repeating part

4

from

46

and divide by

90

(one

9

for the repeating digit, one

0

for the digit that does not repeat).

0.4\overline{6} = \frac{46 - 4}{90} = \frac{42}{90} = \frac{7}{15}

Mia read the numerator correctly, so the numerator is

7

.

2Undo Liam's decimal

#11 Work Backwards 8.NS.A.1

In

0.4\overline{09}

the block

09

repeats, so use

990

as the denominator.

0.4\overline{09} = \frac{409 - 4}{990} = \frac{405}{990} = \frac{9}{22}

This reduces to

\frac{9}{22}

; Liam read the denominator correctly, so it is

22

.

3Rebuild the fraction and convert

#7 Identify Subproblems 7.NS.A.2

Put the correct numerator over the correct denominator.

\frac{7}{22}

is in lowest terms, and dividing gives

0.31818\ldots

.

\frac{7}{22} = 0.31818\ldots = 0.3\overline{18}

The block

18

repeats, starting at the second decimal place.

Answer:

0.3\overline{18}

4 · Reviewdoes it hold up?

Check by converting back: $0.3\overline{18} = \frac{7}{22}$ . Its numerator matches Mia's $\frac{7}{15}$ and its denominator matches Liam's $\frac{9}{22}$ , as the story requires.

Another way: Use the $x$ method for Mia's decimal: if $x = 0.4\overline{6}$ , then $100x - 10x = 46.\overline{6} - 4.\overline{6} = 42$ , so $x = \frac{42}{90} = \frac{7}{15}$ , the same numerator.

Standardsmin grade 8

8.NS.A.1 Know that numbers that are not rational are called irrational numbers — Converting the repeating decimals $0.4\overline{6}$ and $0.4\overline{09}$ back into fractions.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Converting the rebuilt fraction $\frac{7}{22}$ into its repeating decimal by long division.

💡Takeaway. A wrong answer still hides a right part -- work backwards to find it.

Variant 10 hard

Mia and Liam were each asked to write the same fraction, which is in lowest terms, as a decimal. Mia misread the denominator and got $0.2\overline{6}$ . Liam misread the numerator and got $0.6\overline{72}$ . Write the original fraction as a repeating decimal.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A fraction in lowest terms was turned into a decimal twice, each time with one part misread. Mia kept the right numerator and got $0.2\overline{6}$ ; Liam kept the right denominator and got $0.6\overline{72}$ . Rebuild the true fraction and write it as a repeating decimal.

Givens

Mia (wrong denominator, right numerator) got $0.2\overline{6}$ .
Liam (wrong numerator, right denominator) got $0.6\overline{72}$ .
The original fraction is in lowest terms.

Unknowns

The original fraction, written as a repeating decimal.

Constraints

Read each wrong decimal back as a fraction in lowest terms; the correctly read part shows up there.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #15 Organize Information in More Ways#7 Identify Subproblems

Each wrong answer still carries one correct piece. Work backwards from each decimal to its fraction in lowest terms, take the numerator from Mia's result and the denominator from Liam's, then convert the rebuilt fraction to a decimal.

3 · Execute3 carry out the plan

1Undo Mia's decimal

#11 Work Backwards 8.NS.A.1

In

0.2\overline{6}

only the

6

repeats. Subtract the non-repeating part

2

from

26

and divide by

90

(one

9

for the repeating digit, one

0

for the digit that does not repeat).

0.2\overline{6} = \frac{26 - 2}{90} = \frac{24}{90} = \frac{4}{15}

Mia read the numerator correctly, so the numerator is

4

.

2Undo Liam's decimal

#11 Work Backwards 8.NS.A.1

In

0.6\overline{72}

the block

72

repeats, so use

990

as the denominator.

0.6\overline{72} = \frac{672 - 6}{990} = \frac{666}{990} = \frac{37}{55}

This reduces to

\frac{37}{55}

; Liam read the denominator correctly, so it is

55

.

3Rebuild the fraction and convert

#7 Identify Subproblems 7.NS.A.2

Put the correct numerator over the correct denominator.

\frac{4}{55}

is in lowest terms, and dividing gives

0.07272\ldots

.

\frac{4}{55} = 0.07272\ldots = 0.0\overline{72}

The block

72

repeats, starting at the second decimal place.

Answer:

0.0\overline{72}

4 · Reviewdoes it hold up?

Check by converting back: $0.0\overline{72} = \frac{4}{55}$ . Its numerator matches Mia's $\frac{4}{15}$ and its denominator matches Liam's $\frac{37}{55}$ , as the story requires.

Another way: Use the $x$ method for Mia's decimal: if $x = 0.2\overline{6}$ , then $100x - 10x = 26.\overline{6} - 2.\overline{6} = 24$ , so $x = \frac{24}{90} = \frac{4}{15}$ , the same numerator.

Standardsmin grade 8

8.NS.A.1 Know that numbers that are not rational are called irrational numbers — Converting the repeating decimals $0.2\overline{6}$ and $0.6\overline{72}$ back into fractions.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Converting the rebuilt fraction $\frac{4}{55}$ into its repeating decimal by long division.

💡Takeaway. A wrong answer still hides a right part -- work backwards to find it.

Variant 11 hard

Mia and Liam were each asked to write the same fraction, which is in lowest terms, as a decimal. Mia misread the denominator and got $0.6\overline{81}$ . Liam misread the numerator and got $0.61\overline{36}$ . Write the original fraction as a repeating decimal.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A fraction in lowest terms was turned into a decimal twice, each time with one part misread. Mia kept the right numerator and got $0.6\overline{81}$ ; Liam kept the right denominator and got $0.61\overline{36}$ . Rebuild the true fraction and write it as a repeating decimal.

Givens

Mia (wrong denominator, right numerator) got $0.6\overline{81}$ .
Liam (wrong numerator, right denominator) got $0.61\overline{36}$ .
The original fraction is in lowest terms.

Unknowns

The original fraction, written as a repeating decimal.

Constraints

Read each wrong decimal back as a fraction in lowest terms; the correctly read part shows up there.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #15 Organize Information in More Ways#7 Identify Subproblems

Each wrong answer still carries one correct piece. Work backwards from each decimal to its fraction in lowest terms, take the numerator from Mia's result and the denominator from Liam's, then convert the rebuilt fraction to a decimal.

3 · Execute3 carry out the plan

1Undo Mia's decimal

#11 Work Backwards 8.NS.A.1

In

0.6\overline{81}

the block

81

repeats. Subtract the non-repeating part

6

from

681

and divide by

990

(two

9

s for the two repeating digits, one

0

for the digit that does not repeat).

0.6\overline{81} = \frac{681 - 6}{990} = \frac{675}{990} = \frac{15}{22}

Mia read the numerator correctly, so the numerator is

15

.

2Undo Liam's decimal

#11 Work Backwards 8.NS.A.1

In

0.61\overline{36}

the block

36

repeats, so use

9900

as the denominator.

0.61\overline{36} = \frac{6136 - 61}{9900} = \frac{6075}{9900} = \frac{27}{44}

This reduces to

\frac{27}{44}

; Liam read the denominator correctly, so it is

44

.

3Rebuild the fraction and convert

#7 Identify Subproblems 7.NS.A.2

Put the correct numerator over the correct denominator.

\frac{15}{44}

is in lowest terms, and dividing gives

0.340909\ldots

.

\frac{15}{44} = 0.340909\ldots = 0.34\overline{09}

The block

09

repeats, starting at the third decimal place.

Answer:

0.34\overline{09}

4 · Reviewdoes it hold up?

Check by converting back: $0.34\overline{09} = \frac{15}{44}$ . Its numerator matches Mia's $\frac{15}{22}$ and its denominator matches Liam's $\frac{27}{44}$ , as the story requires.

Another way: Use the $x$ method for Mia's decimal: if $x = 0.6\overline{81}$ , then $1000x - 10x = 681.\overline{81} - 6.\overline{81} = 675$ , so $x = \frac{675}{990} = \frac{15}{22}$ , the same numerator.

Standardsmin grade 8

8.NS.A.1 Know that numbers that are not rational are called irrational numbers — Converting the repeating decimals $0.6\overline{81}$ and $0.61\overline{36}$ back into fractions.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Converting the rebuilt fraction $\frac{15}{44}$ into its repeating decimal by long division.

💡Takeaway. A wrong answer still hides a right part -- work backwards to find it.

Variant 12 hard

Mia and Liam were each asked to write the same fraction, which is in lowest terms, as a decimal. Mia misread the denominator and got $0.5\overline{90}$ . Liam misread the numerator and got $0.6\overline{5}$ . Write the original fraction as a repeating decimal.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A fraction in lowest terms was turned into a decimal twice, each time with one part misread. Mia kept the right numerator and got $0.5\overline{90}$ ; Liam kept the right denominator and got $0.6\overline{5}$ . Rebuild the true fraction and write it as a repeating decimal.

Givens

Mia (wrong denominator, right numerator) got $0.5\overline{90}$ .
Liam (wrong numerator, right denominator) got $0.6\overline{5}$ .
The original fraction is in lowest terms.

Unknowns

The original fraction, written as a repeating decimal.

Constraints

Read each wrong decimal back as a fraction in lowest terms; the correctly read part shows up there.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #15 Organize Information in More Ways#7 Identify Subproblems

Each wrong answer still carries one correct piece. Work backwards from each decimal to its fraction in lowest terms, take the numerator from Mia's result and the denominator from Liam's, then convert the rebuilt fraction to a decimal.

3 · Execute3 carry out the plan

1Undo Mia's decimal

#11 Work Backwards 8.NS.A.1

In

0.5\overline{90}

the block

90

repeats. Subtract the non-repeating part

5

from

590

and divide by

990

(two

9

s for the two repeating digits, one

0

for the digit that does not repeat).

0.5\overline{90} = \frac{590 - 5}{990} = \frac{585}{990} = \frac{13}{22}

Mia read the numerator correctly, so the numerator is

13

.

2Undo Liam's decimal

#11 Work Backwards 8.NS.A.1

In

0.6\overline{5}

only the

5

repeats, so use

90

as the denominator.

0.6\overline{5} = \frac{65 - 6}{90} = \frac{59}{90}

\frac{59}{90}

is already in lowest terms; Liam read the denominator correctly, so it is

90

.

3Rebuild the fraction and convert

#7 Identify Subproblems 7.NS.A.2

Put the correct numerator over the correct denominator.

\frac{13}{90}

is in lowest terms, and dividing gives

0.1444\ldots

.

\frac{13}{90} = 0.1444\ldots = 0.1\overline{4}

Only the

4

repeats, starting at the second decimal place.

Answer:

0.1\overline{4}

4 · Reviewdoes it hold up?

Check by converting back: $0.1\overline{4} = \frac{13}{90}$ . Its numerator matches Mia's $\frac{13}{22}$ and its denominator matches Liam's $\frac{59}{90}$ , as the story requires.

Another way: Use the $x$ method for Mia's decimal: if $x = 0.5\overline{90}$ , then $1000x - 10x = 590.\overline{90} - 5.\overline{90} = 585$ , so $x = \frac{585}{990} = \frac{13}{22}$ , the same numerator.

Standardsmin grade 8

8.NS.A.1 Know that numbers that are not rational are called irrational numbers — Converting the repeating decimals $0.5\overline{90}$ and $0.6\overline{5}$ back into fractions.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Converting the rebuilt fraction $\frac{13}{90}$ into its repeating decimal by long division.

💡Takeaway. A wrong answer still hides a right part -- work backwards to find it.

Recover each correct part from a wrong decimal · 12 practice problems

Generated variants — 12

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Where next?