Practice · Peel off whole parts and flip the remainder · Sensim Math

Variant 1 easy

Suppose

$\frac{41}{18}=a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d+\cfrac{1}{2}}}}$

where $a$ , $b$ , $c$ , $d$ are positive whole numbers. Find the value of $a+b+c-d$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Write $\frac{41}{18}$ as a stack of fractions: a whole number $a$ plus $1$ over ( $b$ plus $1$ over ( $c$ plus $1$ over ( $d + \frac{1}{2}$ ))). Find the whole numbers and then $a+b+c-d$ .

Givens

$\frac{41}{18}=a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d+\cfrac{1}{2}}}}$
$a$ , $b$ , $c$ , $d$ are positive whole numbers.

Unknowns

$a+b+c-d$

Constraints

Each part after a whole number is $1$ over something bigger than $1$ , so it is between $0$ and $1$ . That makes each whole number the whole-number part at its level.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #5 Look for a Pattern

Break the problem into levels. At each level: take the whole-number part, keep the leftover proper fraction, and flip it (a proper fraction $\frac{p}{q}$ equals $1 \div \frac{q}{p}$ ). Repeat the same move until the last level shows $d + \frac{1}{2}$ .

3 · Execute5 carry out the plan

1Level 1: find $a$

#7 Identify Subproblems 6.NS.A.1

Divide

41

by

18

:

18 \times 2 = 36

with remainder

5

.

\frac{41}{18} = 2 + \frac{5}{18}

The leftover

\frac{5}{18}

is less than

1

, so

a = 2

.

2Level 2: flip and find $b$

#5 Look for a Pattern 7.NS.A.2

A fraction equals

1

divided by its reciprocal.

\frac{5}{18} = \frac{1}{\frac{18}{5}},\quad \frac{18}{5} = 3 + \frac{3}{5}

So

b = 3

and the leftover is

\frac{3}{5}

.

3Level 3: flip again and find $c$

#5 Look for a Pattern 7.NS.A.2

Repeat the same move with

\frac{3}{5}

.

\frac{3}{5} = \frac{1}{\frac{5}{3}},\quad \frac{5}{3} = 1 + \frac{2}{3}

So

c = 1

and the leftover is

\frac{2}{3}

.

4Level 4: find $d$

#5 Look for a Pattern 7.NS.A.2

Flip

\frac{2}{3}

once more.

\frac{2}{3} = \frac{1}{\frac{3}{2}},\quad \frac{3}{2} = 1 + \frac{1}{2}

This matches

d + \frac{1}{2}

, so

d = 1

.

5Combine

#7 Identify Subproblems 6.NS.A.1

Substitute

a = 2

,

b = 3

,

c = 1

,

d = 1

.

2 + 3 + 1 - 1 = 5

a+b+c-d = 5

.

Answer: 5

4 · Reviewdoes it hold up?

Rebuild from the bottom: $1 + \frac{1}{2} = \frac{3}{2}$ ; $1 + \frac{2}{3} = \frac{5}{3}$ ; $3 + \frac{3}{5} = \frac{18}{5}$ ; $2 + \frac{5}{18} = \frac{41}{18}$ . It returns the original fraction.

Another way: Run the division algorithm on the numerator and denominator: $41 = 2 \times 18 + 5$ , $18 = 3 \times 5 + 3$ , $5 = 1 \times 3 + 2$ , $3 = 1 \times 2 + 1$ . The quotients $2, 3, 1, 1$ are $a, b, c, d$ .

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing to split an improper fraction into a whole number and a proper fraction.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Rewriting a fraction as $1$ divided by its reciprocal.

💡Takeaway. Take the whole part, flip what is left, repeat: that is all a stacked fraction asks.

Variant 2 easy

Suppose

$\frac{52}{11}=a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d+\cfrac{1}{2}}}}$

where $a$ , $b$ , $c$ , $d$ are positive whole numbers. Find the value of $a+b+c-d$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Write $\frac{52}{11}$ as a stack of fractions: a whole number $a$ plus $1$ over ( $b$ plus $1$ over ( $c$ plus $1$ over ( $d + \frac{1}{2}$ ))). Find the whole numbers and then $a+b+c-d$ .

Givens

$\frac{52}{11}=a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d+\cfrac{1}{2}}}}$
$a$ , $b$ , $c$ , $d$ are positive whole numbers.

Unknowns

$a+b+c-d$

Constraints

Each part after a whole number is $1$ over something bigger than $1$ , so it is between $0$ and $1$ . That makes each whole number the whole-number part at its level.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #5 Look for a Pattern

Break the problem into levels. At each level: take the whole-number part, keep the leftover proper fraction, and flip it (a proper fraction $\frac{p}{q}$ equals $1 \div \frac{q}{p}$ ). Repeat the same move until the last level shows $d + \frac{1}{2}$ .

3 · Execute5 carry out the plan

1Level 1: find $a$

#7 Identify Subproblems 6.NS.A.1

Divide

52

by

11

:

11 \times 4 = 44

with remainder

8

.

\frac{52}{11} = 4 + \frac{8}{11}

The leftover

\frac{8}{11}

is less than

1

, so

a = 4

.

2Level 2: flip and find $b$

#5 Look for a Pattern 7.NS.A.2

A fraction equals

1

divided by its reciprocal.

\frac{8}{11} = \frac{1}{\frac{11}{8}},\quad \frac{11}{8} = 1 + \frac{3}{8}

So

b = 1

and the leftover is

\frac{3}{8}

.

3Level 3: flip again and find $c$

#5 Look for a Pattern 7.NS.A.2

Repeat the same move with

\frac{3}{8}

.

\frac{3}{8} = \frac{1}{\frac{8}{3}},\quad \frac{8}{3} = 2 + \frac{2}{3}

So

c = 2

and the leftover is

\frac{2}{3}

.

4Level 4: find $d$

#5 Look for a Pattern 7.NS.A.2

Flip

\frac{2}{3}

once more.

\frac{2}{3} = \frac{1}{\frac{3}{2}},\quad \frac{3}{2} = 1 + \frac{1}{2}

This matches

d + \frac{1}{2}

, so

d = 1

.

5Combine

#7 Identify Subproblems 6.NS.A.1

Substitute

a = 4

,

b = 1

,

c = 2

,

d = 1

.

4 + 1 + 2 - 1 = 6

a+b+c-d = 6

.

Answer: 6

4 · Reviewdoes it hold up?

Rebuild from the bottom: $1 + \frac{1}{2} = \frac{3}{2}$ ; $2 + \frac{2}{3} = \frac{8}{3}$ ; $1 + \frac{3}{8} = \frac{11}{8}$ ; $4 + \frac{8}{11} = \frac{52}{11}$ . It returns the original fraction.

Another way: Run the division algorithm on the numerator and denominator: $52 = 4 \times 11 + 8$ , $11 = 1 \times 8 + 3$ , $8 = 2 \times 3 + 2$ , $3 = 1 \times 2 + 1$ . The quotients $4, 1, 2, 1$ are $a, b, c, d$ .

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing to split an improper fraction into a whole number and a proper fraction.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Rewriting a fraction as $1$ divided by its reciprocal.

💡Takeaway. Take the whole part, flip what is left, repeat: that is all a stacked fraction asks.

Variant 3 easy

Suppose

$\frac{53}{14}=a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d+\cfrac{1}{2}}}}$

where $a$ , $b$ , $c$ , $d$ are positive whole numbers. Find the value of $a+b+c-d$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Write $\frac{53}{14}$ as a stack of fractions: a whole number $a$ plus $1$ over ( $b$ plus $1$ over ( $c$ plus $1$ over ( $d + \frac{1}{2}$ ))). Find the whole numbers and then $a+b+c-d$ .

Givens

$\frac{53}{14}=a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d+\cfrac{1}{2}}}}$
$a$ , $b$ , $c$ , $d$ are positive whole numbers.

Unknowns

$a+b+c-d$

Constraints

Each part after a whole number is $1$ over something bigger than $1$ , so it is between $0$ and $1$ . That makes each whole number the whole-number part at its level.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #5 Look for a Pattern

Break the problem into levels. At each level: take the whole-number part, keep the leftover proper fraction, and flip it (a proper fraction $\frac{p}{q}$ equals $1 \div \frac{q}{p}$ ). Repeat the same move until the last level shows $d + \frac{1}{2}$ .

3 · Execute5 carry out the plan

1Level 1: find $a$

#7 Identify Subproblems 6.NS.A.1

Divide

53

by

14

:

14 \times 3 = 42

with remainder

11

.

\frac{53}{14} = 3 + \frac{11}{14}

The leftover

\frac{11}{14}

is less than

1

, so

a = 3

.

2Level 2: flip and find $b$

#5 Look for a Pattern 7.NS.A.2

A fraction equals

1

divided by its reciprocal.

\frac{11}{14} = \frac{1}{\frac{14}{11}},\quad \frac{14}{11} = 1 + \frac{3}{11}

So

b = 1

and the leftover is

\frac{3}{11}

.

3Level 3: flip again and find $c$

#5 Look for a Pattern 7.NS.A.2

Repeat the same move with

\frac{3}{11}

.

\frac{3}{11} = \frac{1}{\frac{11}{3}},\quad \frac{11}{3} = 3 + \frac{2}{3}

So

c = 3

and the leftover is

\frac{2}{3}

.

4Level 4: find $d$

#5 Look for a Pattern 7.NS.A.2

Flip

\frac{2}{3}

once more.

\frac{2}{3} = \frac{1}{\frac{3}{2}},\quad \frac{3}{2} = 1 + \frac{1}{2}

This matches

d + \frac{1}{2}

, so

d = 1

.

5Combine

#7 Identify Subproblems 6.NS.A.1

Substitute

a = 3

,

b = 1

,

c = 3

,

d = 1

.

3 + 1 + 3 - 1 = 6

a+b+c-d = 6

.

Answer: 6

4 · Reviewdoes it hold up?

Rebuild from the bottom: $1 + \frac{1}{2} = \frac{3}{2}$ ; $3 + \frac{2}{3} = \frac{11}{3}$ ; $1 + \frac{3}{11} = \frac{14}{11}$ ; $3 + \frac{11}{14} = \frac{53}{14}$ . It returns the original fraction.

Another way: Run the division algorithm on the numerator and denominator: $53 = 3 \times 14 + 11$ , $14 = 1 \times 11 + 3$ , $11 = 3 \times 3 + 2$ , $3 = 1 \times 2 + 1$ . The quotients $3, 1, 3, 1$ are $a, b, c, d$ .

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing to split an improper fraction into a whole number and a proper fraction.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Rewriting a fraction as $1$ divided by its reciprocal.

💡Takeaway. Take the whole part, flip what is left, repeat: that is all a stacked fraction asks.

Variant 4 easy

Suppose

$\frac{65}{19}=a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d+\cfrac{1}{2}}}}$

where $a$ , $b$ , $c$ , $d$ are positive whole numbers. Find the value of $a+b+c-d$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Write $\frac{65}{19}$ as a stack of fractions: a whole number $a$ plus $1$ over ( $b$ plus $1$ over ( $c$ plus $1$ over ( $d + \frac{1}{2}$ ))). Find the whole numbers and then $a+b+c-d$ .

Givens

$\frac{65}{19}=a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d+\cfrac{1}{2}}}}$
$a$ , $b$ , $c$ , $d$ are positive whole numbers.

Unknowns

$a+b+c-d$

Constraints

Each part after a whole number is $1$ over something bigger than $1$ , so it is between $0$ and $1$ . That makes each whole number the whole-number part at its level.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #5 Look for a Pattern

Break the problem into levels. At each level: take the whole-number part, keep the leftover proper fraction, and flip it (a proper fraction $\frac{p}{q}$ equals $1 \div \frac{q}{p}$ ). Repeat the same move until the last level shows $d + \frac{1}{2}$ .

3 · Execute5 carry out the plan

1Level 1: find $a$

#7 Identify Subproblems 6.NS.A.1

Divide

65

by

19

:

19 \times 3 = 57

with remainder

8

.

\frac{65}{19} = 3 + \frac{8}{19}

The leftover

\frac{8}{19}

is less than

1

, so

a = 3

.

2Level 2: flip and find $b$

#5 Look for a Pattern 7.NS.A.2

A fraction equals

1

divided by its reciprocal.

\frac{8}{19} = \frac{1}{\frac{19}{8}},\quad \frac{19}{8} = 2 + \frac{3}{8}

So

b = 2

and the leftover is

\frac{3}{8}

.

3Level 3: flip again and find $c$

#5 Look for a Pattern 7.NS.A.2

Repeat the same move with

\frac{3}{8}

.

\frac{3}{8} = \frac{1}{\frac{8}{3}},\quad \frac{8}{3} = 2 + \frac{2}{3}

So

c = 2

and the leftover is

\frac{2}{3}

.

4Level 4: find $d$

#5 Look for a Pattern 7.NS.A.2

Flip

\frac{2}{3}

once more.

\frac{2}{3} = \frac{1}{\frac{3}{2}},\quad \frac{3}{2} = 1 + \frac{1}{2}

This matches

d + \frac{1}{2}

, so

d = 1

.

5Combine

#7 Identify Subproblems 6.NS.A.1

Substitute

a = 3

,

b = 2

,

c = 2

,

d = 1

.

3 + 2 + 2 - 1 = 6

a+b+c-d = 6

.

Answer: 6

4 · Reviewdoes it hold up?

Rebuild from the bottom: $1 + \frac{1}{2} = \frac{3}{2}$ ; $2 + \frac{2}{3} = \frac{8}{3}$ ; $2 + \frac{3}{8} = \frac{19}{8}$ ; $3 + \frac{8}{19} = \frac{65}{19}$ . It returns the original fraction.

Another way: Run the division algorithm on the numerator and denominator: $65 = 3 \times 19 + 8$ , $19 = 2 \times 8 + 3$ , $8 = 2 \times 3 + 2$ , $3 = 1 \times 2 + 1$ . The quotients $3, 2, 2, 1$ are $a, b, c, d$ .

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing to split an improper fraction into a whole number and a proper fraction.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Rewriting a fraction as $1$ divided by its reciprocal.

💡Takeaway. Take the whole part, flip what is left, repeat: that is all a stacked fraction asks.

Variant 5 medium

Suppose

$\frac{67}{12}=a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d+\cfrac{1}{2}}}}$

where $a$ , $b$ , $c$ , $d$ are positive whole numbers. Find the value of $a+b+c-d$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Write $\frac{67}{12}$ as a stack of fractions: a whole number $a$ plus $1$ over ( $b$ plus $1$ over ( $c$ plus $1$ over ( $d + \frac{1}{2}$ ))). Find the whole numbers and then $a+b+c-d$ .

Givens

$\frac{67}{12}=a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d+\cfrac{1}{2}}}}$
$a$ , $b$ , $c$ , $d$ are positive whole numbers.

Unknowns

$a+b+c-d$

Constraints

Each part after a whole number is $1$ over something bigger than $1$ , so it is between $0$ and $1$ . That makes each whole number the whole-number part at its level.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #5 Look for a Pattern

Break the problem into levels. At each level: take the whole-number part, keep the leftover proper fraction, and flip it (a proper fraction $\frac{p}{q}$ equals $1 \div \frac{q}{p}$ ). Repeat the same move until the last level shows $d + \frac{1}{2}$ .

3 · Execute5 carry out the plan

1Level 1: find $a$

#7 Identify Subproblems 6.NS.A.1

Divide

67

by

12

:

12 \times 5 = 60

with remainder

7

.

\frac{67}{12} = 5 + \frac{7}{12}

The leftover

\frac{7}{12}

is less than

1

, so

a = 5

.

2Level 2: flip and find $b$

#5 Look for a Pattern 7.NS.A.2

A fraction equals

1

divided by its reciprocal.

\frac{7}{12} = \frac{1}{\frac{12}{7}},\quad \frac{12}{7} = 1 + \frac{5}{7}

So

b = 1

and the leftover is

\frac{5}{7}

.

3Level 3: flip again and find $c$

#5 Look for a Pattern 7.NS.A.2

Repeat the same move with

\frac{5}{7}

.

\frac{5}{7} = \frac{1}{\frac{7}{5}},\quad \frac{7}{5} = 1 + \frac{2}{5}

So

c = 1

and the leftover is

\frac{2}{5}

.

4Level 4: find $d$

#5 Look for a Pattern 7.NS.A.2

Flip

\frac{2}{5}

once more.

\frac{2}{5} = \frac{1}{\frac{5}{2}},\quad \frac{5}{2} = 2 + \frac{1}{2}

This matches

d + \frac{1}{2}

, so

d = 2

.

5Combine

#7 Identify Subproblems 6.NS.A.1

Substitute

a = 5

,

b = 1

,

c = 1

,

d = 2

.

5 + 1 + 1 - 2 = 5

a+b+c-d = 5

.

Answer: 5

4 · Reviewdoes it hold up?

Rebuild from the bottom: $2 + \frac{1}{2} = \frac{5}{2}$ ; $1 + \frac{2}{5} = \frac{7}{5}$ ; $1 + \frac{5}{7} = \frac{12}{7}$ ; $5 + \frac{7}{12} = \frac{67}{12}$ . It returns the original fraction.

Another way: Run the division algorithm on the numerator and denominator: $67 = 5 \times 12 + 7$ , $12 = 1 \times 7 + 5$ , $7 = 1 \times 5 + 2$ , $5 = 2 \times 2 + 1$ . The quotients $5, 1, 1, 2$ are $a, b, c, d$ .

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing to split an improper fraction into a whole number and a proper fraction.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Rewriting a fraction as $1$ divided by its reciprocal.

💡Takeaway. Take the whole part, flip what is left, repeat: that is all a stacked fraction asks.

Variant 6 medium

Suppose

$\frac{69}{8}=a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d+\cfrac{1}{2}}}}$

where $a$ , $b$ , $c$ , $d$ are positive whole numbers. Find the value of $a+b+c-d$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Write $\frac{69}{8}$ as a stack of fractions: a whole number $a$ plus $1$ over ( $b$ plus $1$ over ( $c$ plus $1$ over ( $d + \frac{1}{2}$ ))). Find the whole numbers and then $a+b+c-d$ .

Givens

$\frac{69}{8}=a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d+\cfrac{1}{2}}}}$
$a$ , $b$ , $c$ , $d$ are positive whole numbers.

Unknowns

$a+b+c-d$

Constraints

Each part after a whole number is $1$ over something bigger than $1$ , so it is between $0$ and $1$ . That makes each whole number the whole-number part at its level.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #5 Look for a Pattern

Break the problem into levels. At each level: take the whole-number part, keep the leftover proper fraction, and flip it (a proper fraction $\frac{p}{q}$ equals $1 \div \frac{q}{p}$ ). Repeat the same move until the last level shows $d + \frac{1}{2}$ .

3 · Execute5 carry out the plan

1Level 1: find $a$

#7 Identify Subproblems 6.NS.A.1

Divide

69

by

8

:

8 \times 8 = 64

with remainder

5

.

\frac{69}{8} = 8 + \frac{5}{8}

The leftover

\frac{5}{8}

is less than

1

, so

a = 8

.

2Level 2: flip and find $b$

#5 Look for a Pattern 7.NS.A.2

A fraction equals

1

divided by its reciprocal.

\frac{5}{8} = \frac{1}{\frac{8}{5}},\quad \frac{8}{5} = 1 + \frac{3}{5}

So

b = 1

and the leftover is

\frac{3}{5}

.

3Level 3: flip again and find $c$

#5 Look for a Pattern 7.NS.A.2

Repeat the same move with

\frac{3}{5}

.

\frac{3}{5} = \frac{1}{\frac{5}{3}},\quad \frac{5}{3} = 1 + \frac{2}{3}

So

c = 1

and the leftover is

\frac{2}{3}

.

4Level 4: find $d$

#5 Look for a Pattern 7.NS.A.2

Flip

\frac{2}{3}

once more.

\frac{2}{3} = \frac{1}{\frac{3}{2}},\quad \frac{3}{2} = 1 + \frac{1}{2}

This matches

d + \frac{1}{2}

, so

d = 1

.

5Combine

#7 Identify Subproblems 6.NS.A.1

Substitute

a = 8

,

b = 1

,

c = 1

,

d = 1

.

8 + 1 + 1 - 1 = 9

a+b+c-d = 9

.

Answer: 9

4 · Reviewdoes it hold up?

Rebuild from the bottom: $1 + \frac{1}{2} = \frac{3}{2}$ ; $1 + \frac{2}{3} = \frac{5}{3}$ ; $1 + \frac{3}{5} = \frac{8}{5}$ ; $8 + \frac{5}{8} = \frac{69}{8}$ . It returns the original fraction.

Another way: Run the division algorithm on the numerator and denominator: $69 = 8 \times 8 + 5$ , $8 = 1 \times 5 + 3$ , $5 = 1 \times 3 + 2$ , $3 = 1 \times 2 + 1$ . The quotients $8, 1, 1, 1$ are $a, b, c, d$ .

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing to split an improper fraction into a whole number and a proper fraction.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Rewriting a fraction as $1$ divided by its reciprocal.

💡Takeaway. Take the whole part, flip what is left, repeat: that is all a stacked fraction asks.

Variant 7 medium

Suppose

$\frac{83}{13}=a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d+\cfrac{1}{2}}}}$

where $a$ , $b$ , $c$ , $d$ are positive whole numbers. Find the value of $a+b+c-d$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Write $\frac{83}{13}$ as a stack of fractions: a whole number $a$ plus $1$ over ( $b$ plus $1$ over ( $c$ plus $1$ over ( $d + \frac{1}{2}$ ))). Find the whole numbers and then $a+b+c-d$ .

Givens

$\frac{83}{13}=a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d+\cfrac{1}{2}}}}$
$a$ , $b$ , $c$ , $d$ are positive whole numbers.

Unknowns

$a+b+c-d$

Constraints

Each part after a whole number is $1$ over something bigger than $1$ , so it is between $0$ and $1$ . That makes each whole number the whole-number part at its level.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #5 Look for a Pattern

Break the problem into levels. At each level: take the whole-number part, keep the leftover proper fraction, and flip it (a proper fraction $\frac{p}{q}$ equals $1 \div \frac{q}{p}$ ). Repeat the same move until the last level shows $d + \frac{1}{2}$ .

3 · Execute5 carry out the plan

1Level 1: find $a$

#7 Identify Subproblems 6.NS.A.1

Divide

83

by

13

:

13 \times 6 = 78

with remainder

5

.

\frac{83}{13} = 6 + \frac{5}{13}

The leftover

\frac{5}{13}

is less than

1

, so

a = 6

.

2Level 2: flip and find $b$

#5 Look for a Pattern 7.NS.A.2

A fraction equals

1

divided by its reciprocal.

\frac{5}{13} = \frac{1}{\frac{13}{5}},\quad \frac{13}{5} = 2 + \frac{3}{5}

So

b = 2

and the leftover is

\frac{3}{5}

.

3Level 3: flip again and find $c$

#5 Look for a Pattern 7.NS.A.2

Repeat the same move with

\frac{3}{5}

.

\frac{3}{5} = \frac{1}{\frac{5}{3}},\quad \frac{5}{3} = 1 + \frac{2}{3}

So

c = 1

and the leftover is

\frac{2}{3}

.

4Level 4: find $d$

#5 Look for a Pattern 7.NS.A.2

Flip

\frac{2}{3}

once more.

\frac{2}{3} = \frac{1}{\frac{3}{2}},\quad \frac{3}{2} = 1 + \frac{1}{2}

This matches

d + \frac{1}{2}

, so

d = 1

.

5Combine

#7 Identify Subproblems 6.NS.A.1

Substitute

a = 6

,

b = 2

,

c = 1

,

d = 1

.

6 + 2 + 1 - 1 = 8

a+b+c-d = 8

.

Answer: 8

4 · Reviewdoes it hold up?

Rebuild from the bottom: $1 + \frac{1}{2} = \frac{3}{2}$ ; $1 + \frac{2}{3} = \frac{5}{3}$ ; $2 + \frac{3}{5} = \frac{13}{5}$ ; $6 + \frac{5}{13} = \frac{83}{13}$ . It returns the original fraction.

Another way: Run the division algorithm on the numerator and denominator: $83 = 6 \times 13 + 5$ , $13 = 2 \times 5 + 3$ , $5 = 1 \times 3 + 2$ , $3 = 1 \times 2 + 1$ . The quotients $6, 2, 1, 1$ are $a, b, c, d$ .

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing to split an improper fraction into a whole number and a proper fraction.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Rewriting a fraction as $1$ divided by its reciprocal.

💡Takeaway. Take the whole part, flip what is left, repeat: that is all a stacked fraction asks.

Variant 8 medium

Suppose

$\frac{95}{18}=a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d+\cfrac{1}{2}}}}$

where $a$ , $b$ , $c$ , $d$ are positive whole numbers. Find the value of $a+b+c-d$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Write $\frac{95}{18}$ as a stack of fractions: a whole number $a$ plus $1$ over ( $b$ plus $1$ over ( $c$ plus $1$ over ( $d + \frac{1}{2}$ ))). Find the whole numbers and then $a+b+c-d$ .

Givens

$\frac{95}{18}=a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d+\cfrac{1}{2}}}}$
$a$ , $b$ , $c$ , $d$ are positive whole numbers.

Unknowns

$a+b+c-d$

Constraints

Each part after a whole number is $1$ over something bigger than $1$ , so it is between $0$ and $1$ . That makes each whole number the whole-number part at its level.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #5 Look for a Pattern

Break the problem into levels. At each level: take the whole-number part, keep the leftover proper fraction, and flip it (a proper fraction $\frac{p}{q}$ equals $1 \div \frac{q}{p}$ ). Repeat the same move until the last level shows $d + \frac{1}{2}$ .

3 · Execute5 carry out the plan

1Level 1: find $a$

#7 Identify Subproblems 6.NS.A.1

Divide

95

by

18

:

18 \times 5 = 90

with remainder

5

.

\frac{95}{18} = 5 + \frac{5}{18}

The leftover

\frac{5}{18}

is less than

1

, so

a = 5

.

2Level 2: flip and find $b$

#5 Look for a Pattern 7.NS.A.2

A fraction equals

1

divided by its reciprocal.

\frac{5}{18} = \frac{1}{\frac{18}{5}},\quad \frac{18}{5} = 3 + \frac{3}{5}

So

b = 3

and the leftover is

\frac{3}{5}

.

3Level 3: flip again and find $c$

#5 Look for a Pattern 7.NS.A.2

Repeat the same move with

\frac{3}{5}

.

\frac{3}{5} = \frac{1}{\frac{5}{3}},\quad \frac{5}{3} = 1 + \frac{2}{3}

So

c = 1

and the leftover is

\frac{2}{3}

.

4Level 4: find $d$

#5 Look for a Pattern 7.NS.A.2

Flip

\frac{2}{3}

once more.

\frac{2}{3} = \frac{1}{\frac{3}{2}},\quad \frac{3}{2} = 1 + \frac{1}{2}

This matches

d + \frac{1}{2}

, so

d = 1

.

5Combine

#7 Identify Subproblems 6.NS.A.1

Substitute

a = 5

,

b = 3

,

c = 1

,

d = 1

.

5 + 3 + 1 - 1 = 8

a+b+c-d = 8

.

Answer: 8

4 · Reviewdoes it hold up?

Rebuild from the bottom: $1 + \frac{1}{2} = \frac{3}{2}$ ; $1 + \frac{2}{3} = \frac{5}{3}$ ; $3 + \frac{3}{5} = \frac{18}{5}$ ; $5 + \frac{5}{18} = \frac{95}{18}$ . It returns the original fraction.

Another way: Run the division algorithm on the numerator and denominator: $95 = 5 \times 18 + 5$ , $18 = 3 \times 5 + 3$ , $5 = 1 \times 3 + 2$ , $3 = 1 \times 2 + 1$ . The quotients $5, 3, 1, 1$ are $a, b, c, d$ .

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing to split an improper fraction into a whole number and a proper fraction.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Rewriting a fraction as $1$ divided by its reciprocal.

💡Takeaway. Take the whole part, flip what is left, repeat: that is all a stacked fraction asks.

Variant 9 hard

Suppose

$\frac{105}{38}=a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d+\cfrac{1}{2}}}}$

where $a$ , $b$ , $c$ , $d$ are positive whole numbers. Find the value of $a+b+c-d$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Write $\frac{105}{38}$ as a stack of fractions: a whole number $a$ plus $1$ over ( $b$ plus $1$ over ( $c$ plus $1$ over ( $d + \frac{1}{2}$ ))). Find the whole numbers and then $a+b+c-d$ .

Givens

$\frac{105}{38}=a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d+\cfrac{1}{2}}}}$
$a$ , $b$ , $c$ , $d$ are positive whole numbers.

Unknowns

$a+b+c-d$

Constraints

Each part after a whole number is $1$ over something bigger than $1$ , so it is between $0$ and $1$ . That makes each whole number the whole-number part at its level.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #5 Look for a Pattern

Break the problem into levels. At each level: take the whole-number part, keep the leftover proper fraction, and flip it (a proper fraction $\frac{p}{q}$ equals $1 \div \frac{q}{p}$ ). Repeat the same move until the last level shows $d + \frac{1}{2}$ .

3 · Execute5 carry out the plan

1Level 1: find $a$

#7 Identify Subproblems 6.NS.A.1

Divide

105

by

38

:

38 \times 2 = 76

with remainder

29

.

\frac{105}{38} = 2 + \frac{29}{38}

The leftover

\frac{29}{38}

is less than

1

, so

a = 2

.

2Level 2: flip and find $b$

#5 Look for a Pattern 7.NS.A.2

A fraction equals

1

divided by its reciprocal.

\frac{29}{38} = \frac{1}{\frac{38}{29}},\quad \frac{38}{29} = 1 + \frac{9}{29}

So

b = 1

and the leftover is

\frac{9}{29}

.

3Level 3: flip again and find $c$

#5 Look for a Pattern 7.NS.A.2

Repeat the same move with

\frac{9}{29}

.

\frac{9}{29} = \frac{1}{\frac{29}{9}},\quad \frac{29}{9} = 3 + \frac{2}{9}

So

c = 3

and the leftover is

\frac{2}{9}

.

4Level 4: find $d$

#5 Look for a Pattern 7.NS.A.2

Flip

\frac{2}{9}

once more.

\frac{2}{9} = \frac{1}{\frac{9}{2}},\quad \frac{9}{2} = 4 + \frac{1}{2}

This matches

d + \frac{1}{2}

, so

d = 4

.

5Combine

#7 Identify Subproblems 6.NS.A.1

Substitute

a = 2

,

b = 1

,

c = 3

,

d = 4

.

2 + 1 + 3 - 4 = 2

a+b+c-d = 2

.

Answer: 2

4 · Reviewdoes it hold up?

Rebuild from the bottom: $4 + \frac{1}{2} = \frac{9}{2}$ ; $3 + \frac{2}{9} = \frac{29}{9}$ ; $1 + \frac{9}{29} = \frac{38}{29}$ ; $2 + \frac{29}{38} = \frac{105}{38}$ . It returns the original fraction.

Another way: Run the division algorithm on the numerator and denominator: $105 = 2 \times 38 + 29$ , $38 = 1 \times 29 + 9$ , $29 = 3 \times 9 + 2$ , $9 = 4 \times 2 + 1$ . The quotients $2, 1, 3, 4$ are $a, b, c, d$ .

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing to split an improper fraction into a whole number and a proper fraction.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Rewriting a fraction as $1$ divided by its reciprocal.

💡Takeaway. Take the whole part, flip what is left, repeat: that is all a stacked fraction asks.

Variant 10 hard

Suppose

$\frac{106}{33}=a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d+\cfrac{1}{2}}}}$

where $a$ , $b$ , $c$ , $d$ are positive whole numbers. Find the value of $a+b+c-d$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Write $\frac{106}{33}$ as a stack of fractions: a whole number $a$ plus $1$ over ( $b$ plus $1$ over ( $c$ plus $1$ over ( $d + \frac{1}{2}$ ))). Find the whole numbers and then $a+b+c-d$ .

Givens

$\frac{106}{33}=a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d+\cfrac{1}{2}}}}$
$a$ , $b$ , $c$ , $d$ are positive whole numbers.

Unknowns

$a+b+c-d$

Constraints

Each part after a whole number is $1$ over something bigger than $1$ , so it is between $0$ and $1$ . That makes each whole number the whole-number part at its level.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #5 Look for a Pattern

Break the problem into levels. At each level: take the whole-number part, keep the leftover proper fraction, and flip it (a proper fraction $\frac{p}{q}$ equals $1 \div \frac{q}{p}$ ). Repeat the same move until the last level shows $d + \frac{1}{2}$ .

3 · Execute5 carry out the plan

1Level 1: find $a$

#7 Identify Subproblems 6.NS.A.1

Divide

106

by

33

:

33 \times 3 = 99

with remainder

7

.

\frac{106}{33} = 3 + \frac{7}{33}

The leftover

\frac{7}{33}

is less than

1

, so

a = 3

.

2Level 2: flip and find $b$

#5 Look for a Pattern 7.NS.A.2

A fraction equals

1

divided by its reciprocal.

\frac{7}{33} = \frac{1}{\frac{33}{7}},\quad \frac{33}{7} = 4 + \frac{5}{7}

So

b = 4

and the leftover is

\frac{5}{7}

.

3Level 3: flip again and find $c$

#5 Look for a Pattern 7.NS.A.2

Repeat the same move with

\frac{5}{7}

.

\frac{5}{7} = \frac{1}{\frac{7}{5}},\quad \frac{7}{5} = 1 + \frac{2}{5}

So

c = 1

and the leftover is

\frac{2}{5}

.

4Level 4: find $d$

#5 Look for a Pattern 7.NS.A.2

Flip

\frac{2}{5}

once more.

\frac{2}{5} = \frac{1}{\frac{5}{2}},\quad \frac{5}{2} = 2 + \frac{1}{2}

This matches

d + \frac{1}{2}

, so

d = 2

.

5Combine

#7 Identify Subproblems 6.NS.A.1

Substitute

a = 3

,

b = 4

,

c = 1

,

d = 2

.

3 + 4 + 1 - 2 = 6

a+b+c-d = 6

.

Answer: 6

4 · Reviewdoes it hold up?

Rebuild from the bottom: $2 + \frac{1}{2} = \frac{5}{2}$ ; $1 + \frac{2}{5} = \frac{7}{5}$ ; $4 + \frac{5}{7} = \frac{33}{7}$ ; $3 + \frac{7}{33} = \frac{106}{33}$ . It returns the original fraction.

Another way: Run the division algorithm on the numerator and denominator: $106 = 3 \times 33 + 7$ , $33 = 4 \times 7 + 5$ , $7 = 1 \times 5 + 2$ , $5 = 2 \times 2 + 1$ . The quotients $3, 4, 1, 2$ are $a, b, c, d$ .

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing to split an improper fraction into a whole number and a proper fraction.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Rewriting a fraction as $1$ divided by its reciprocal.

💡Takeaway. Take the whole part, flip what is left, repeat: that is all a stacked fraction asks.

Variant 11 hard

Suppose

$\frac{109}{25}=a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d+\cfrac{1}{2}}}}$

where $a$ , $b$ , $c$ , $d$ are positive whole numbers. Find the value of $a+b+c-d$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Write $\frac{109}{25}$ as a stack of fractions: a whole number $a$ plus $1$ over ( $b$ plus $1$ over ( $c$ plus $1$ over ( $d + \frac{1}{2}$ ))). Find the whole numbers and then $a+b+c-d$ .

Givens

$\frac{109}{25}=a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d+\cfrac{1}{2}}}}$
$a$ , $b$ , $c$ , $d$ are positive whole numbers.

Unknowns

$a+b+c-d$

Constraints

Each part after a whole number is $1$ over something bigger than $1$ , so it is between $0$ and $1$ . That makes each whole number the whole-number part at its level.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #5 Look for a Pattern

Break the problem into levels. At each level: take the whole-number part, keep the leftover proper fraction, and flip it (a proper fraction $\frac{p}{q}$ equals $1 \div \frac{q}{p}$ ). Repeat the same move until the last level shows $d + \frac{1}{2}$ .

3 · Execute5 carry out the plan

1Level 1: find $a$

#7 Identify Subproblems 6.NS.A.1

Divide

109

by

25

:

25 \times 4 = 100

with remainder

9

.

\frac{109}{25} = 4 + \frac{9}{25}

The leftover

\frac{9}{25}

is less than

1

, so

a = 4

.

2Level 2: flip and find $b$

#5 Look for a Pattern 7.NS.A.2

A fraction equals

1

divided by its reciprocal.

\frac{9}{25} = \frac{1}{\frac{25}{9}},\quad \frac{25}{9} = 2 + \frac{7}{9}

So

b = 2

and the leftover is

\frac{7}{9}

.

3Level 3: flip again and find $c$

#5 Look for a Pattern 7.NS.A.2

Repeat the same move with

\frac{7}{9}

.

\frac{7}{9} = \frac{1}{\frac{9}{7}},\quad \frac{9}{7} = 1 + \frac{2}{7}

So

c = 1

and the leftover is

\frac{2}{7}

.

4Level 4: find $d$

#5 Look for a Pattern 7.NS.A.2

Flip

\frac{2}{7}

once more.

\frac{2}{7} = \frac{1}{\frac{7}{2}},\quad \frac{7}{2} = 3 + \frac{1}{2}

This matches

d + \frac{1}{2}

, so

d = 3

.

5Combine

#7 Identify Subproblems 6.NS.A.1

Substitute

a = 4

,

b = 2

,

c = 1

,

d = 3

.

4 + 2 + 1 - 3 = 4

a+b+c-d = 4

.

Answer: 4

4 · Reviewdoes it hold up?

Rebuild from the bottom: $3 + \frac{1}{2} = \frac{7}{2}$ ; $1 + \frac{2}{7} = \frac{9}{7}$ ; $2 + \frac{7}{9} = \frac{25}{9}$ ; $4 + \frac{9}{25} = \frac{109}{25}$ . It returns the original fraction.

Another way: Run the division algorithm on the numerator and denominator: $109 = 4 \times 25 + 9$ , $25 = 2 \times 9 + 7$ , $9 = 1 \times 7 + 2$ , $7 = 3 \times 2 + 1$ . The quotients $4, 2, 1, 3$ are $a, b, c, d$ .

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing to split an improper fraction into a whole number and a proper fraction.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Rewriting a fraction as $1$ divided by its reciprocal.

💡Takeaway. Take the whole part, flip what is left, repeat: that is all a stacked fraction asks.

Variant 12 hard

Suppose

$\frac{131}{17}=a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d+\cfrac{1}{2}}}}$

where $a$ , $b$ , $c$ , $d$ are positive whole numbers. Find the value of $a+b+c-d$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Write $\frac{131}{17}$ as a stack of fractions: a whole number $a$ plus $1$ over ( $b$ plus $1$ over ( $c$ plus $1$ over ( $d + \frac{1}{2}$ ))). Find the whole numbers and then $a+b+c-d$ .

Givens

$\frac{131}{17}=a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d+\cfrac{1}{2}}}}$
$a$ , $b$ , $c$ , $d$ are positive whole numbers.

Unknowns

$a+b+c-d$

Constraints

Each part after a whole number is $1$ over something bigger than $1$ , so it is between $0$ and $1$ . That makes each whole number the whole-number part at its level.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #5 Look for a Pattern

Break the problem into levels. At each level: take the whole-number part, keep the leftover proper fraction, and flip it (a proper fraction $\frac{p}{q}$ equals $1 \div \frac{q}{p}$ ). Repeat the same move until the last level shows $d + \frac{1}{2}$ .

3 · Execute5 carry out the plan

1Level 1: find $a$

#7 Identify Subproblems 6.NS.A.1

Divide

131

by

17

:

17 \times 7 = 119

with remainder

12

.

\frac{131}{17} = 7 + \frac{12}{17}

The leftover

\frac{12}{17}

is less than

1

, so

a = 7

.

2Level 2: flip and find $b$

#5 Look for a Pattern 7.NS.A.2

A fraction equals

1

divided by its reciprocal.

\frac{12}{17} = \frac{1}{\frac{17}{12}},\quad \frac{17}{12} = 1 + \frac{5}{12}

So

b = 1

and the leftover is

\frac{5}{12}

.

3Level 3: flip again and find $c$

#5 Look for a Pattern 7.NS.A.2

Repeat the same move with

\frac{5}{12}

.

\frac{5}{12} = \frac{1}{\frac{12}{5}},\quad \frac{12}{5} = 2 + \frac{2}{5}

So

c = 2

and the leftover is

\frac{2}{5}

.

4Level 4: find $d$

#5 Look for a Pattern 7.NS.A.2

Flip

\frac{2}{5}

once more.

\frac{2}{5} = \frac{1}{\frac{5}{2}},\quad \frac{5}{2} = 2 + \frac{1}{2}

This matches

d + \frac{1}{2}

, so

d = 2

.

5Combine

#7 Identify Subproblems 6.NS.A.1

Substitute

a = 7

,

b = 1

,

c = 2

,

d = 2

.

7 + 1 + 2 - 2 = 8

a+b+c-d = 8

.

Answer: 8

4 · Reviewdoes it hold up?

Rebuild from the bottom: $2 + \frac{1}{2} = \frac{5}{2}$ ; $2 + \frac{2}{5} = \frac{12}{5}$ ; $1 + \frac{5}{12} = \frac{17}{12}$ ; $7 + \frac{12}{17} = \frac{131}{17}$ . It returns the original fraction.

Another way: Run the division algorithm on the numerator and denominator: $131 = 7 \times 17 + 12$ , $17 = 1 \times 12 + 5$ , $12 = 2 \times 5 + 2$ , $5 = 2 \times 2 + 1$ . The quotients $7, 1, 2, 2$ are $a, b, c, d$ .

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing to split an improper fraction into a whole number and a proper fraction.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Rewriting a fraction as $1$ divided by its reciprocal.

💡Takeaway. Take the whole part, flip what is left, repeat: that is all a stacked fraction asks.

Peel off whole parts and flip the remainder · 12 practice problems

Generated variants — 12

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