Practice · Skip the lead-in, then add whole blocks · Sensim Math

Variant 1 easy

As a decimal, $\frac{1}{22} = 0.0\overline{45}$ . Let $A_x$ be the digit in the $x$ th place after the decimal point of $\frac{1}{22}$ . Find the value of $A_1 + A_2 + A_3 + A_4 + \cdots + A_{21}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The decimal for $\frac{1}{22}$ is $0.0\overline{45}$ . Add its first $21$ digits after the decimal point.

Givens

$\frac{1}{22} = 0.0\overline{45}$ .
$A_x$ is the digit in the $x$ th decimal place.

Unknowns

$A_1 + A_2 + \cdots + A_{21}$ .

Constraints

The first decimal place does not repeat; the block $45$ starts at the second place.

2 · Planchoose the strategy

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

Split the $21$ places into the non-repeating start and the repeating part. The repeating part is a pattern of whole $2$ -digit blocks, so add one block and multiply.

3 · Execute3 carry out the plan

1Separate the lead-in

#7 Identify Subproblems 8.NS.A.1

Place

1

holds

0

. That leaves places

2

through

21

.

21 - 1 = 20

The zero adds nothing, and

20

digits remain.

2Count whole blocks

#5 Look for a Pattern 4.OA.C.5

The repeating block

45

has

2

digits, and

20

splits evenly into blocks of

2

.

20 \div 2 = 10

Places

2

to

21

are exactly

10

copies of the block.

3Add one block, then multiply

#5 Look for a Pattern 4.OA.C.5

One block's digits add to

9

, and there are

10

blocks.

4 + 5 = 9, \quad 9 \times 10 = 90

The total digit sum is

90

.

Answer: 90

4 · Reviewdoes it hold up?

The average digit in a block is $\frac{9}{2} = 4.5$ , and $20$ digits at that average give $20 \times 4.5 = 90$ , matching the block count.

Another way: Use the remainder view: $21 = 2 \times 10 + 1$ , so the first $21$ places are the $1$ lead-in zero plus $10$ full blocks, again $0 + 10 \times 9 = 90$ .

Standardsmin grade 8

8.NS.A.1 Know that numbers that are not rational are called irrational numbers — Reading the decimal expansion of the rational number $\frac{1}{22}$ and its repeating block.
4.OA.C.5 Generate a number or shape pattern following a given rule — Using the repeating pattern of digits to add by whole blocks.

💡Takeaway. Peel off the digits that don't repeat, then count in full blocks.

Variant 2 easy

As a decimal, $\frac{1}{74} = 0.0\overline{135}$ . Let $A_x$ be the digit in the $x$ th place after the decimal point of $\frac{1}{74}$ . Find the value of $A_1 + A_2 + A_3 + A_4 + \cdots + A_{31}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The decimal for $\frac{1}{74}$ is $0.0\overline{135}$ . Add its first $31$ digits after the decimal point.

Givens

$\frac{1}{74} = 0.0\overline{135}$ .
$A_x$ is the digit in the $x$ th decimal place.

Unknowns

$A_1 + A_2 + \cdots + A_{31}$ .

Constraints

The first decimal place does not repeat; the block $135$ starts at the second place.

2 · Planchoose the strategy

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

Split the $31$ places into the non-repeating start and the repeating part. The repeating part is a pattern of whole $3$ -digit blocks, so add one block and multiply.

3 · Execute3 carry out the plan

1Separate the lead-in

#7 Identify Subproblems 8.NS.A.1

Place

1

holds

0

. That leaves places

2

through

31

.

31 - 1 = 30

The zero adds nothing, and

30

digits remain.

2Count whole blocks

#5 Look for a Pattern 4.OA.C.5

The repeating block

135

has

3

digits, and

30

splits evenly into blocks of

3

.

30 \div 3 = 10

Places

2

to

31

are exactly

10

copies of the block.

3Add one block, then multiply

#5 Look for a Pattern 4.OA.C.5

One block's digits add to

9

, and there are

10

blocks.

1 + 3 + 5 = 9, \quad 9 \times 10 = 90

The total digit sum is

90

.

Answer: 90

4 · Reviewdoes it hold up?

The average digit in a block is $\frac{9}{3} = 3$ , and $30$ digits at that average give $30 \times 3 = 90$ , matching the block count.

Another way: Use the remainder view: $31 = 3 \times 10 + 1$ , so the first $31$ places are the $1$ lead-in zero plus $10$ full blocks, again $0 + 10 \times 9 = 90$ .

Standardsmin grade 8

8.NS.A.1 Know that numbers that are not rational are called irrational numbers — Reading the decimal expansion of the rational number $\frac{1}{74}$ and its repeating block.
4.OA.C.5 Generate a number or shape pattern following a given rule — Using the repeating pattern of digits to add by whole blocks.

💡Takeaway. Peel off the digits that don't repeat, then count in full blocks.

Variant 3 easy

As a decimal, $\frac{1}{740} = 0.00\overline{135}$ . Let $A_x$ be the digit in the $x$ th place after the decimal point of $\frac{1}{740}$ . Find the value of $A_1 + A_2 + A_3 + A_4 + \cdots + A_{26}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The decimal for $\frac{1}{740}$ is $0.00\overline{135}$ . Add its first $26$ digits after the decimal point.

Givens

$\frac{1}{740} = 0.00\overline{135}$ .
$A_x$ is the digit in the $x$ th decimal place.

Unknowns

$A_1 + A_2 + \cdots + A_{26}$ .

Constraints

The first two decimal places do not repeat; the block $135$ starts at the third place.

2 · Planchoose the strategy

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

Split the $26$ places into the non-repeating start and the repeating part. The repeating part is a pattern of whole $3$ -digit blocks, so add one block and multiply.

3 · Execute3 carry out the plan

1Separate the lead-in

#7 Identify Subproblems 8.NS.A.1

Places

1

,

2

hold

0

,

0

. That leaves places

3

through

26

.

26 - 2 = 24

The zeros add nothing, and

24

digits remain.

2Count whole blocks

#5 Look for a Pattern 4.OA.C.5

The repeating block

135

has

3

digits, and

24

splits evenly into blocks of

3

.

24 \div 3 = 8

Places

3

to

26

are exactly

8

copies of the block.

3Add one block, then multiply

#5 Look for a Pattern 4.OA.C.5

One block's digits add to

9

, and there are

8

blocks.

1 + 3 + 5 = 9, \quad 9 \times 8 = 72

The total digit sum is

72

.

Answer: 72

4 · Reviewdoes it hold up?

The average digit in a block is $\frac{9}{3} = 3$ , and $24$ digits at that average give $24 \times 3 = 72$ , matching the block count.

Another way: Use the remainder view: $26 = 3 \times 8 + 2$ , so the first $26$ places are the $2$ lead-in zeros plus $8$ full blocks, again $0 + 8 \times 9 = 72$ .

Standardsmin grade 8

8.NS.A.1 Know that numbers that are not rational are called irrational numbers — Reading the decimal expansion of the rational number $\frac{1}{740}$ and its repeating block.
4.OA.C.5 Generate a number or shape pattern following a given rule — Using the repeating pattern of digits to add by whole blocks.

💡Takeaway. Peel off the digits that don't repeat, then count in full blocks.

Variant 4 medium

As a decimal, $\frac{1}{7000} = 0.000\overline{142857}$ . Let $A_x$ be the digit in the $x$ th place after the decimal point of $\frac{1}{7000}$ . Find the value of $A_1 + A_2 + A_3 + A_4 + \cdots + A_{39}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The decimal for $\frac{1}{7000}$ is $0.000\overline{142857}$ . Add its first $39$ digits after the decimal point.

Givens

$\frac{1}{7000} = 0.000\overline{142857}$ .
$A_x$ is the digit in the $x$ th decimal place.

Unknowns

$A_1 + A_2 + \cdots + A_{39}$ .

Constraints

The first three decimal places do not repeat; the block $142857$ starts at the fourth place.

2 · Planchoose the strategy

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

Split the $39$ places into the non-repeating start and the repeating part. The repeating part is a pattern of whole $6$ -digit blocks, so add one block and multiply.

3 · Execute3 carry out the plan

1Separate the lead-in

#7 Identify Subproblems 8.NS.A.1

Places

1

,

2

,

3

hold

0

,

0

,

0

. That leaves places

4

through

39

.

39 - 3 = 36

The zeros add nothing, and

36

digits remain.

2Count whole blocks

#5 Look for a Pattern 4.OA.C.5

The repeating block

142857

has

6

digits, and

36

splits evenly into blocks of

6

.

36 \div 6 = 6

Places

4

to

39

are exactly

6

copies of the block.

3Add one block, then multiply

#5 Look for a Pattern 4.OA.C.5

One block's digits add to

27

, and there are

6

blocks.

1 + 4 + 2 + 8 + 5 + 7 = 27, \quad 27 \times 6 = 162

The total digit sum is

162

.

Answer: 162

4 · Reviewdoes it hold up?

The average digit in a block is $\frac{27}{6} = 4.5$ , and $36$ digits at that average give $36 \times 4.5 = 162$ , matching the block count.

Another way: Use the remainder view: $39 = 6 \times 6 + 3$ , so the first $39$ places are the $3$ lead-in zeros plus $6$ full blocks, again $0 + 6 \times 27 = 162$ .

Standardsmin grade 8

8.NS.A.1 Know that numbers that are not rational are called irrational numbers — Reading the decimal expansion of the rational number $\frac{1}{7000}$ and its repeating block.
4.OA.C.5 Generate a number or shape pattern following a given rule — Using the repeating pattern of digits to add by whole blocks.

💡Takeaway. Peel off the digits that don't repeat, then count in full blocks.

Variant 5 easy

As a decimal, $\frac{1}{70} = 0.0\overline{142857}$ . Let $A_x$ be the digit in the $x$ th place after the decimal point of $\frac{1}{70}$ . Find the value of $A_1 + A_2 + A_3 + A_4 + \cdots + A_{37}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The decimal for $\frac{1}{70}$ is $0.0\overline{142857}$ . Add its first $37$ digits after the decimal point.

Givens

$\frac{1}{70} = 0.0\overline{142857}$ .
$A_x$ is the digit in the $x$ th decimal place.

Unknowns

$A_1 + A_2 + \cdots + A_{37}$ .

Constraints

The first decimal place does not repeat; the block $142857$ starts at the second place.

2 · Planchoose the strategy

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

Split the $37$ places into the non-repeating start and the repeating part. The repeating part is a pattern of whole $6$ -digit blocks, so add one block and multiply.

3 · Execute3 carry out the plan

1Separate the lead-in

#7 Identify Subproblems 8.NS.A.1

Place

1

holds

0

. That leaves places

2

through

37

.

37 - 1 = 36

The zero adds nothing, and

36

digits remain.

2Count whole blocks

#5 Look for a Pattern 4.OA.C.5

The repeating block

142857

has

6

digits, and

36

splits evenly into blocks of

6

.

36 \div 6 = 6

Places

2

to

37

are exactly

6

copies of the block.

3Add one block, then multiply

#5 Look for a Pattern 4.OA.C.5

One block's digits add to

27

, and there are

6

blocks.

1 + 4 + 2 + 8 + 5 + 7 = 27, \quad 27 \times 6 = 162

The total digit sum is

162

.

Answer: 162

4 · Reviewdoes it hold up?

The average digit in a block is $\frac{27}{6} = 4.5$ , and $36$ digits at that average give $36 \times 4.5 = 162$ , matching the block count.

Another way: Use the remainder view: $37 = 6 \times 6 + 1$ , so the first $37$ places are the $1$ lead-in zero plus $6$ full blocks, again $0 + 6 \times 27 = 162$ .

Standardsmin grade 8

8.NS.A.1 Know that numbers that are not rational are called irrational numbers — Reading the decimal expansion of the rational number $\frac{1}{70}$ and its repeating block.
4.OA.C.5 Generate a number or shape pattern following a given rule — Using the repeating pattern of digits to add by whole blocks.

💡Takeaway. Peel off the digits that don't repeat, then count in full blocks.

Variant 6 medium

As a decimal, $\frac{1}{65} = 0.0\overline{153846}$ . Let $A_x$ be the digit in the $x$ th place after the decimal point of $\frac{1}{65}$ . Find the value of $A_1 + A_2 + A_3 + A_4 + \cdots + A_{37}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The decimal for $\frac{1}{65}$ is $0.0\overline{153846}$ . Add its first $37$ digits after the decimal point.

Givens

$\frac{1}{65} = 0.0\overline{153846}$ .
$A_x$ is the digit in the $x$ th decimal place.

Unknowns

$A_1 + A_2 + \cdots + A_{37}$ .

Constraints

The first decimal place does not repeat; the block $153846$ starts at the second place.

2 · Planchoose the strategy

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

Split the $37$ places into the non-repeating start and the repeating part. The repeating part is a pattern of whole $6$ -digit blocks, so add one block and multiply.

3 · Execute3 carry out the plan

1Separate the lead-in

#7 Identify Subproblems 8.NS.A.1

Place

1

holds

0

. That leaves places

2

through

37

.

37 - 1 = 36

The zero adds nothing, and

36

digits remain.

2Count whole blocks

#5 Look for a Pattern 4.OA.C.5

The repeating block

153846

has

6

digits, and

36

splits evenly into blocks of

6

.

36 \div 6 = 6

Places

2

to

37

are exactly

6

copies of the block.

3Add one block, then multiply

#5 Look for a Pattern 4.OA.C.5

One block's digits add to

27

, and there are

6

blocks.

1 + 5 + 3 + 8 + 4 + 6 = 27, \quad 27 \times 6 = 162

The total digit sum is

162

.

Answer: 162

4 · Reviewdoes it hold up?

The average digit in a block is $\frac{27}{6} = 4.5$ , and $36$ digits at that average give $36 \times 4.5 = 162$ , matching the block count.

Another way: Use the remainder view: $37 = 6 \times 6 + 1$ , so the first $37$ places are the $1$ lead-in zero plus $6$ full blocks, again $0 + 6 \times 27 = 162$ .

Standardsmin grade 8

8.NS.A.1 Know that numbers that are not rational are called irrational numbers — Reading the decimal expansion of the rational number $\frac{1}{65}$ and its repeating block.
4.OA.C.5 Generate a number or shape pattern following a given rule — Using the repeating pattern of digits to add by whole blocks.

💡Takeaway. Peel off the digits that don't repeat, then count in full blocks.

Variant 7 medium

As a decimal, $\frac{1}{3500} = 0.000\overline{285714}$ . Let $A_x$ be the digit in the $x$ th place after the decimal point of $\frac{1}{3500}$ . Find the value of $A_1 + A_2 + A_3 + A_4 + \cdots + A_{45}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The decimal for $\frac{1}{3500}$ is $0.000\overline{285714}$ . Add its first $45$ digits after the decimal point.

Givens

$\frac{1}{3500} = 0.000\overline{285714}$ .
$A_x$ is the digit in the $x$ th decimal place.

Unknowns

$A_1 + A_2 + \cdots + A_{45}$ .

Constraints

The first three decimal places do not repeat; the block $285714$ starts at the fourth place.

2 · Planchoose the strategy

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

Split the $45$ places into the non-repeating start and the repeating part. The repeating part is a pattern of whole $6$ -digit blocks, so add one block and multiply.

3 · Execute3 carry out the plan

1Separate the lead-in

#7 Identify Subproblems 8.NS.A.1

Places

1

,

2

,

3

hold

0

,

0

,

0

. That leaves places

4

through

45

.

45 - 3 = 42

The zeros add nothing, and

42

digits remain.

2Count whole blocks

#5 Look for a Pattern 4.OA.C.5

The repeating block

285714

has

6

digits, and

42

splits evenly into blocks of

6

.

42 \div 6 = 7

Places

4

to

45

are exactly

7

copies of the block.

3Add one block, then multiply

#5 Look for a Pattern 4.OA.C.5

One block's digits add to

27

, and there are

7

blocks.

2 + 8 + 5 + 7 + 1 + 4 = 27, \quad 27 \times 7 = 189

The total digit sum is

189

.

Answer: 189

4 · Reviewdoes it hold up?

The average digit in a block is $\frac{27}{6} = 4.5$ , and $42$ digits at that average give $42 \times 4.5 = 189$ , matching the block count.

Another way: Use the remainder view: $45 = 6 \times 7 + 3$ , so the first $45$ places are the $3$ lead-in zeros plus $7$ full blocks, again $0 + 7 \times 27 = 189$ .

Standardsmin grade 8

8.NS.A.1 Know that numbers that are not rational are called irrational numbers — Reading the decimal expansion of the rational number $\frac{1}{3500}$ and its repeating block.
4.OA.C.5 Generate a number or shape pattern following a given rule — Using the repeating pattern of digits to add by whole blocks.

💡Takeaway. Peel off the digits that don't repeat, then count in full blocks.

Variant 8 medium

As a decimal, $\frac{1}{35} = 0.0\overline{285714}$ . Let $A_x$ be the digit in the $x$ th place after the decimal point of $\frac{1}{35}$ . Find the value of $A_1 + A_2 + A_3 + A_4 + \cdots + A_{31}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The decimal for $\frac{1}{35}$ is $0.0\overline{285714}$ . Add its first $31$ digits after the decimal point.

Givens

$\frac{1}{35} = 0.0\overline{285714}$ .
$A_x$ is the digit in the $x$ th decimal place.

Unknowns

$A_1 + A_2 + \cdots + A_{31}$ .

Constraints

The first decimal place does not repeat; the block $285714$ starts at the second place.

2 · Planchoose the strategy

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

Split the $31$ places into the non-repeating start and the repeating part. The repeating part is a pattern of whole $6$ -digit blocks, so add one block and multiply.

3 · Execute3 carry out the plan

1Separate the lead-in

#7 Identify Subproblems 8.NS.A.1

Place

1

holds

0

. That leaves places

2

through

31

.

31 - 1 = 30

The zero adds nothing, and

30

digits remain.

2Count whole blocks

#5 Look for a Pattern 4.OA.C.5

The repeating block

285714

has

6

digits, and

30

splits evenly into blocks of

6

.

30 \div 6 = 5

Places

2

to

31

are exactly

5

copies of the block.

3Add one block, then multiply

#5 Look for a Pattern 4.OA.C.5

One block's digits add to

27

, and there are

5

blocks.

2 + 8 + 5 + 7 + 1 + 4 = 27, \quad 27 \times 5 = 135

The total digit sum is

135

.

Answer: 135

4 · Reviewdoes it hold up?

The average digit in a block is $\frac{27}{6} = 4.5$ , and $30$ digits at that average give $30 \times 4.5 = 135$ , matching the block count.

Another way: Use the remainder view: $31 = 6 \times 5 + 1$ , so the first $31$ places are the $1$ lead-in zero plus $5$ full blocks, again $0 + 5 \times 27 = 135$ .

Standardsmin grade 8

8.NS.A.1 Know that numbers that are not rational are called irrational numbers — Reading the decimal expansion of the rational number $\frac{1}{35}$ and its repeating block.
4.OA.C.5 Generate a number or shape pattern following a given rule — Using the repeating pattern of digits to add by whole blocks.

💡Takeaway. Peel off the digits that don't repeat, then count in full blocks.

Variant 9 hard

As a decimal, $\frac{1}{350} = 0.00\overline{285714}$ . Let $A_x$ be the digit in the $x$ th place after the decimal point of $\frac{1}{350}$ . Find the value of $A_1 + A_2 + A_3 + A_4 + \cdots + A_{50}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The decimal for $\frac{1}{350}$ is $0.00\overline{285714}$ . Add its first $50$ digits after the decimal point.

Givens

$\frac{1}{350} = 0.00\overline{285714}$ .
$A_x$ is the digit in the $x$ th decimal place.

Unknowns

$A_1 + A_2 + \cdots + A_{50}$ .

Constraints

The first two decimal places do not repeat; the block $285714$ starts at the third place.

2 · Planchoose the strategy

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

Split the $50$ places into the non-repeating start and the repeating part. The repeating part is a pattern of whole $6$ -digit blocks, so add one block and multiply.

3 · Execute3 carry out the plan

1Separate the lead-in

#7 Identify Subproblems 8.NS.A.1

Places

1

,

2

hold

0

,

0

. That leaves places

3

through

50

.

50 - 2 = 48

The zeros add nothing, and

48

digits remain.

2Count whole blocks

#5 Look for a Pattern 4.OA.C.5

The repeating block

285714

has

6

digits, and

48

splits evenly into blocks of

6

.

48 \div 6 = 8

Places

3

to

50

are exactly

8

copies of the block.

3Add one block, then multiply

#5 Look for a Pattern 4.OA.C.5

One block's digits add to

27

, and there are

8

blocks.

2 + 8 + 5 + 7 + 1 + 4 = 27, \quad 27 \times 8 = 216

The total digit sum is

216

.

Answer: 216

4 · Reviewdoes it hold up?

The average digit in a block is $\frac{27}{6} = 4.5$ , and $48$ digits at that average give $48 \times 4.5 = 216$ , matching the block count.

Another way: Use the remainder view: $50 = 6 \times 8 + 2$ , so the first $50$ places are the $2$ lead-in zeros plus $8$ full blocks, again $0 + 8 \times 27 = 216$ .

Standardsmin grade 8

8.NS.A.1 Know that numbers that are not rational are called irrational numbers — Reading the decimal expansion of the rational number $\frac{1}{350}$ and its repeating block.
4.OA.C.5 Generate a number or shape pattern following a given rule — Using the repeating pattern of digits to add by whole blocks.

💡Takeaway. Peel off the digits that don't repeat, then count in full blocks.

Variant 10 hard

As a decimal, $\frac{1}{260} = 0.00\overline{384615}$ . Let $A_x$ be the digit in the $x$ th place after the decimal point of $\frac{1}{260}$ . Find the value of $A_1 + A_2 + A_3 + A_4 + \cdots + A_{32}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The decimal for $\frac{1}{260}$ is $0.00\overline{384615}$ . Add its first $32$ digits after the decimal point.

Givens

$\frac{1}{260} = 0.00\overline{384615}$ .
$A_x$ is the digit in the $x$ th decimal place.

Unknowns

$A_1 + A_2 + \cdots + A_{32}$ .

Constraints

The first two decimal places do not repeat; the block $384615$ starts at the third place.

2 · Planchoose the strategy

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

Split the $32$ places into the non-repeating start and the repeating part. The repeating part is a pattern of whole $6$ -digit blocks, so add one block and multiply.

3 · Execute3 carry out the plan

1Separate the lead-in

#7 Identify Subproblems 8.NS.A.1

Places

1

,

2

hold

0

,

0

. That leaves places

3

through

32

.

32 - 2 = 30

The zeros add nothing, and

30

digits remain.

2Count whole blocks

#5 Look for a Pattern 4.OA.C.5

The repeating block

384615

has

6

digits, and

30

splits evenly into blocks of

6

.

30 \div 6 = 5

Places

3

to

32

are exactly

5

copies of the block.

3Add one block, then multiply

#5 Look for a Pattern 4.OA.C.5

One block's digits add to

27

, and there are

5

blocks.

3 + 8 + 4 + 6 + 1 + 5 = 27, \quad 27 \times 5 = 135

The total digit sum is

135

.

Answer: 135

4 · Reviewdoes it hold up?

The average digit in a block is $\frac{27}{6} = 4.5$ , and $30$ digits at that average give $30 \times 4.5 = 135$ , matching the block count.

Another way: Use the remainder view: $32 = 6 \times 5 + 2$ , so the first $32$ places are the $2$ lead-in zeros plus $5$ full blocks, again $0 + 5 \times 27 = 135$ .

Standardsmin grade 8

8.NS.A.1 Know that numbers that are not rational are called irrational numbers — Reading the decimal expansion of the rational number $\frac{1}{260}$ and its repeating block.
4.OA.C.5 Generate a number or shape pattern following a given rule — Using the repeating pattern of digits to add by whole blocks.

💡Takeaway. Peel off the digits that don't repeat, then count in full blocks.

Variant 11 hard

As a decimal, $\frac{1}{26} = 0.0\overline{384615}$ . Let $A_x$ be the digit in the $x$ th place after the decimal point of $\frac{1}{26}$ . Find the value of $A_1 + A_2 + A_3 + A_4 + \cdots + A_{25}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The decimal for $\frac{1}{26}$ is $0.0\overline{384615}$ . Add its first $25$ digits after the decimal point.

Givens

$\frac{1}{26} = 0.0\overline{384615}$ .
$A_x$ is the digit in the $x$ th decimal place.

Unknowns

$A_1 + A_2 + \cdots + A_{25}$ .

Constraints

The first decimal place does not repeat; the block $384615$ starts at the second place.

2 · Planchoose the strategy

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

Split the $25$ places into the non-repeating start and the repeating part. The repeating part is a pattern of whole $6$ -digit blocks, so add one block and multiply.

3 · Execute3 carry out the plan

1Separate the lead-in

#7 Identify Subproblems 8.NS.A.1

Place

1

holds

0

. That leaves places

2

through

25

.

25 - 1 = 24

The zero adds nothing, and

24

digits remain.

2Count whole blocks

#5 Look for a Pattern 4.OA.C.5

The repeating block

384615

has

6

digits, and

24

splits evenly into blocks of

6

.

24 \div 6 = 4

Places

2

to

25

are exactly

4

copies of the block.

3Add one block, then multiply

#5 Look for a Pattern 4.OA.C.5

One block's digits add to

27

, and there are

4

blocks.

3 + 8 + 4 + 6 + 1 + 5 = 27, \quad 27 \times 4 = 108

The total digit sum is

108

.

Answer: 108

4 · Reviewdoes it hold up?

The average digit in a block is $\frac{27}{6} = 4.5$ , and $24$ digits at that average give $24 \times 4.5 = 108$ , matching the block count.

Another way: Use the remainder view: $25 = 6 \times 4 + 1$ , so the first $25$ places are the $1$ lead-in zero plus $4$ full blocks, again $0 + 4 \times 27 = 108$ .

Standardsmin grade 8

8.NS.A.1 Know that numbers that are not rational are called irrational numbers — Reading the decimal expansion of the rational number $\frac{1}{26}$ and its repeating block.
4.OA.C.5 Generate a number or shape pattern following a given rule — Using the repeating pattern of digits to add by whole blocks.

💡Takeaway. Peel off the digits that don't repeat, then count in full blocks.

Variant 12 hard

As a decimal, $\frac{1}{14} = 0.0\overline{714285}$ . Let $A_x$ be the digit in the $x$ th place after the decimal point of $\frac{1}{14}$ . Find the value of $A_1 + A_2 + A_3 + A_4 + \cdots + A_{43}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The decimal for $\frac{1}{14}$ is $0.0\overline{714285}$ . Add its first $43$ digits after the decimal point.

Givens

$\frac{1}{14} = 0.0\overline{714285}$ .
$A_x$ is the digit in the $x$ th decimal place.

Unknowns

$A_1 + A_2 + \cdots + A_{43}$ .

Constraints

The first decimal place does not repeat; the block $714285$ starts at the second place.

2 · Planchoose the strategy

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

Split the $43$ places into the non-repeating start and the repeating part. The repeating part is a pattern of whole $6$ -digit blocks, so add one block and multiply.

3 · Execute3 carry out the plan

1Separate the lead-in

#7 Identify Subproblems 8.NS.A.1

Place

1

holds

0

. That leaves places

2

through

43

.

43 - 1 = 42

The zero adds nothing, and

42

digits remain.

2Count whole blocks

#5 Look for a Pattern 4.OA.C.5

The repeating block

714285

has

6

digits, and

42

splits evenly into blocks of

6

.

42 \div 6 = 7

Places

2

to

43

are exactly

7

copies of the block.

3Add one block, then multiply

#5 Look for a Pattern 4.OA.C.5

One block's digits add to

27

, and there are

7

blocks.

7 + 1 + 4 + 2 + 8 + 5 = 27, \quad 27 \times 7 = 189

The total digit sum is

189

.

Answer: 189

4 · Reviewdoes it hold up?

The average digit in a block is $\frac{27}{6} = 4.5$ , and $42$ digits at that average give $42 \times 4.5 = 189$ , matching the block count.

Another way: Use the remainder view: $43 = 6 \times 7 + 1$ , so the first $43$ places are the $1$ lead-in zero plus $7$ full blocks, again $0 + 7 \times 27 = 189$ .

Standardsmin grade 8

8.NS.A.1 Know that numbers that are not rational are called irrational numbers — Reading the decimal expansion of the rational number $\frac{1}{14}$ and its repeating block.
4.OA.C.5 Generate a number or shape pattern following a given rule — Using the repeating pattern of digits to add by whole blocks.

💡Takeaway. Peel off the digits that don't repeat, then count in full blocks.

Skip the lead-in, then add whole blocks · 12 practice problems

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