Practice · Turn decimal-digit clues into inequalities · Sensim Math

Variant 1 easy

A natural number $n$ is less than $45$ . When $\frac{n}{45}$ is written as a decimal, its tenths digit is $2$ and its thousandths digit is $2$ . When $\frac{n+1}{45}$ is written as a decimal, its hundredths digit is $4$ . Find $n$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Each digit clue says the fraction lies in a certain range. Write those ranges as inequalities in $n$ , combine them to pin down $n$ , and confirm with the remaining digit clue.

Givens

$n$ is a natural number less than $45$ .
$\frac{n}{45}$ has tenths digit $2$ and thousandths digit $2$ .
$\frac{n+1}{45}$ has hundredths digit $4$ .

Unknowns

The value of $n$ .

Constraints

A tenths digit of $2$ means the number is at least $0.2$ and less than $0.3$ (it is less than $1$ here).

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#6 Guess and Check

A digit is a range in disguise: "tenths digit $2$ " means $0.2\le\frac{n}{45}<0.3$ . Turn the two strongest clues into inequalities, multiply by $45$ , and the integer $n$ is forced; then check the last clue.

3 · Execute4 carry out the plan

1Use the tenths digit of $\frac{n}{45}$

#13 Convert to Algebra 7.EE.B.4

Since

n<45

,

\frac{n}{45}<1

, so a tenths digit of

2

means

0.2\le\frac{n}{45}<0.3

. Multiply by

45

.

0.2\times 45\le n<0.3\times 45 \;\Rightarrow\; 9\le n<13.5

So

n

is one of

9, 10, \dots, 13

.

2Find the tenths digit of $\frac{n+1}{45}$

#3 Eliminate Possibilities 7.NS.A.2

From

9\le n\le 13

we get

10\le n+1\le 14

, so

\frac{n+1}{45}

is at least

\frac{10}{45}

and at most

\frac{14}{45}

, which is less than

0.34

. A tenths digit of

3

with hundredths digit

4

would need at least

0.34

, so its tenths digit is

2

.

\frac{10}{45}\le\frac{n+1}{45}\le\frac{14}{45}

A tenths digit of

3

with hundredths

4

would need

0.34

, too big.

3Use the hundredths digit of $\frac{n+1}{45}$

#13 Convert to Algebra 7.EE.B.4

Tenths digit

2

and hundredths digit

4

mean

0.24\le\frac{n+1}{45}<0.25

. Multiply by

45

.

0.24\times 45\le n+1<0.25\times 45 \;\Rightarrow\; 10.8\le n+1<11.25 \;\Rightarrow\; n+1=11

The only whole number in that short range is

11

, so

n=10

.

4Check the thousandths digit

#6 Guess and Check 7.NS.A.2

Divide to confirm the clue we have not used yet.

\frac{10}{45}=0.22222\ldots

The tenths digit is

2

and the thousandths digit is

2

, as required.

Answer:

10

4 · Reviewdoes it hold up?

$\frac{10}{45}=0.2222\ldots$ (tenths $2$ , thousandths $2$ ) and $\frac{11}{45}=0.2444\ldots$ (hundredths $4$ ). All three clues hold.

Another way: List the candidates $n=9, \dots, 13$ from the tenths clue and compute: $\frac{n}{45}$ has thousandths digit $2$ only for $n=10$ ( $0.2222\ldots$ ); then $\frac{11}{45}=0.2444\ldots$ works, so $n=10$ .

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing each digit clue as an inequality and solving for $n$.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Converting the fractions $\frac{n}{45}$ to decimals to read and check their digits.

💡Takeaway. A single digit of a decimal is really a range of values -- and ranges are what inequalities are for.

Variant 2 easy

A natural number $n$ is less than $48$ . When $\frac{n}{48}$ is written as a decimal, its tenths digit is $6$ and its thousandths digit is $5$ . When $\frac{n+1}{48}$ is written as a decimal, its hundredths digit is $4$ . Find $n$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Each digit clue says the fraction lies in a certain range. Write those ranges as inequalities in $n$ , combine them to pin down $n$ , and confirm with the remaining digit clue.

Givens

$n$ is a natural number less than $48$ .
$\frac{n}{48}$ has tenths digit $6$ and thousandths digit $5$ .
$\frac{n+1}{48}$ has hundredths digit $4$ .

Unknowns

The value of $n$ .

Constraints

A tenths digit of $6$ means the number is at least $0.6$ and less than $0.7$ (it is less than $1$ here).

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#6 Guess and Check

A digit is a range in disguise: "tenths digit $6$ " means $0.6\le\frac{n}{48}<0.7$ . Turn the two strongest clues into inequalities, multiply by $48$ , and the integer $n$ is forced; then check the last clue.

3 · Execute4 carry out the plan

1Use the tenths digit of $\frac{n}{48}$

#13 Convert to Algebra 7.EE.B.4

Since

n<48

,

\frac{n}{48}<1

, so a tenths digit of

6

means

0.6\le\frac{n}{48}<0.7

. Multiply by

48

.

0.6\times 48\le n<0.7\times 48 \;\Rightarrow\; 28.8\le n<33.6

So

n

is one of

29, 30, \dots, 33

.

2Find the tenths digit of $\frac{n+1}{48}$

#3 Eliminate Possibilities 7.NS.A.2

From

29\le n\le 33

we get

30\le n+1\le 34

, so

\frac{n+1}{48}

is at least

\frac{30}{48}

and at most

\frac{34}{48}

, which is less than

0.74

. A tenths digit of

7

with hundredths digit

4

would need at least

0.74

, so its tenths digit is

6

.

\frac{30}{48}=0.625\le\frac{n+1}{48}\le\frac{34}{48}

A tenths digit of

7

with hundredths

4

would need

0.74

, too big.

3Use the hundredths digit of $\frac{n+1}{48}$

#13 Convert to Algebra 7.EE.B.4

Tenths digit

6

and hundredths digit

4

mean

0.64\le\frac{n+1}{48}<0.65

. Multiply by

48

.

0.64\times 48\le n+1<0.65\times 48 \;\Rightarrow\; 30.72\le n+1<31.2 \;\Rightarrow\; n+1=31

The only whole number in that short range is

31

, so

n=30

.

4Check the thousandths digit

#6 Guess and Check 7.NS.A.2

Divide to confirm the clue we have not used yet.

\frac{30}{48}=0.625

The tenths digit is

6

and the thousandths digit is

5

, as required.

Answer:

30

4 · Reviewdoes it hold up?

$\frac{30}{48}=0.625$ (tenths $6$ , thousandths $5$ ) and $\frac{31}{48}=0.6458\ldots$ (hundredths $4$ ). All three clues hold.

Another way: List the candidates $n=29, \dots, 33$ from the tenths clue and compute: $\frac{n}{48}$ has thousandths digit $5$ only for $n=30$ ( $0.625$ ) and $n=31$ ( $0.6458\ldots$ ); then $\frac{31}{48}=0.6458\ldots$ works; $\frac{32}{48}=0.6666\ldots$ fails, so $n=30$ .

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing each digit clue as an inequality and solving for $n$.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Converting the fractions $\frac{n}{48}$ to decimals to read and check their digits.

💡Takeaway. A single digit of a decimal is really a range of values -- and ranges are what inequalities are for.

Variant 3 easy

A natural number $n$ is less than $55$ . When $\frac{n}{55}$ is written as a decimal, its tenths digit is $6$ and its thousandths digit is $8$ . When $\frac{n+1}{55}$ is written as a decimal, its hundredths digit is $3$ . Find $n$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Each digit clue says the fraction lies in a certain range. Write those ranges as inequalities in $n$ , combine them to pin down $n$ , and confirm with the remaining digit clue.

Givens

$n$ is a natural number less than $55$ .
$\frac{n}{55}$ has tenths digit $6$ and thousandths digit $8$ .
$\frac{n+1}{55}$ has hundredths digit $3$ .

Unknowns

The value of $n$ .

Constraints

A tenths digit of $6$ means the number is at least $0.6$ and less than $0.7$ (it is less than $1$ here).

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#6 Guess and Check

A digit is a range in disguise: "tenths digit $6$ " means $0.6\le\frac{n}{55}<0.7$ . Turn the two strongest clues into inequalities, multiply by $55$ , and the integer $n$ is forced; then check the last clue.

3 · Execute4 carry out the plan

1Use the tenths digit of $\frac{n}{55}$

#13 Convert to Algebra 7.EE.B.4

Since

n<55

,

\frac{n}{55}<1

, so a tenths digit of

6

means

0.6\le\frac{n}{55}<0.7

. Multiply by

55

.

0.6\times 55\le n<0.7\times 55 \;\Rightarrow\; 33\le n<38.5

So

n

is one of

33, 34, \dots, 38

.

2Find the tenths digit of $\frac{n+1}{55}$

#3 Eliminate Possibilities 7.NS.A.2

From

33\le n\le 38

we get

34\le n+1\le 39

, so

\frac{n+1}{55}

is at least

\frac{34}{55}

and at most

\frac{39}{55}

, which is less than

0.73

. A tenths digit of

7

with hundredths digit

3

would need at least

0.73

, so its tenths digit is

6

.

\frac{34}{55}\le\frac{n+1}{55}\le\frac{39}{55}

A tenths digit of

7

with hundredths

3

would need

0.73

, too big.

3Use the hundredths digit of $\frac{n+1}{55}$

#13 Convert to Algebra 7.EE.B.4

Tenths digit

6

and hundredths digit

3

mean

0.63\le\frac{n+1}{55}<0.64

. Multiply by

55

.

0.63\times 55\le n+1<0.64\times 55 \;\Rightarrow\; 34.65\le n+1<35.2 \;\Rightarrow\; n+1=35

The only whole number in that short range is

35

, so

n=34

.

4Check the thousandths digit

#6 Guess and Check 7.NS.A.2

Divide to confirm the clue we have not used yet.

\frac{34}{55}=0.61818\ldots

The tenths digit is

6

and the thousandths digit is

8

, as required.

Answer:

34

4 · Reviewdoes it hold up?

$\frac{34}{55}=0.6181\ldots$ (tenths $6$ , thousandths $8$ ) and $\frac{35}{55}=0.6363\ldots$ (hundredths $3$ ). All three clues hold.

Another way: List the candidates $n=33, \dots, 38$ from the tenths clue and compute: $\frac{n}{55}$ has thousandths digit $8$ only for $n=34$ ( $0.6181\ldots$ ); then $\frac{35}{55}=0.6363\ldots$ works, so $n=34$ .

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing each digit clue as an inequality and solving for $n$.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Converting the fractions $\frac{n}{55}$ to decimals to read and check their digits.

💡Takeaway. A single digit of a decimal is really a range of values -- and ranges are what inequalities are for.

Variant 4 easy

A natural number $n$ is less than $60$ . When $\frac{n}{60}$ is written as a decimal, its tenths digit is $3$ and its thousandths digit is $6$ . When $\frac{n+1}{60}$ is written as a decimal, its hundredths digit is $8$ . Find $n$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Each digit clue says the fraction lies in a certain range. Write those ranges as inequalities in $n$ , combine them to pin down $n$ , and confirm with the remaining digit clue.

Givens

$n$ is a natural number less than $60$ .
$\frac{n}{60}$ has tenths digit $3$ and thousandths digit $6$ .
$\frac{n+1}{60}$ has hundredths digit $8$ .

Unknowns

The value of $n$ .

Constraints

A tenths digit of $3$ means the number is at least $0.3$ and less than $0.4$ (it is less than $1$ here).

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#6 Guess and Check

A digit is a range in disguise: "tenths digit $3$ " means $0.3\le\frac{n}{60}<0.4$ . Turn the two strongest clues into inequalities, multiply by $60$ , and the integer $n$ is forced; then check the last clue.

3 · Execute4 carry out the plan

1Use the tenths digit of $\frac{n}{60}$

#13 Convert to Algebra 7.EE.B.4

Since

n<60

,

\frac{n}{60}<1

, so a tenths digit of

3

means

0.3\le\frac{n}{60}<0.4

. Multiply by

60

.

0.3\times 60\le n<0.4\times 60 \;\Rightarrow\; 18\le n<24

So

n

is one of

18, 19, \dots, 23

.

2Find the tenths digit of $\frac{n+1}{60}$

#3 Eliminate Possibilities 7.NS.A.2

From

18\le n\le 23

we get

19\le n+1\le 24

, so

\frac{n+1}{60}

is at least

\frac{19}{60}

and at most

\frac{24}{60}

, which is less than

0.48

. A tenths digit of

4

with hundredths digit

8

would need at least

0.48

, so its tenths digit is

3

.

\frac{19}{60}\le\frac{n+1}{60}\le\frac{24}{60}=0.4

A tenths digit of

4

with hundredths

8

would need

0.48

, too big.

3Use the hundredths digit of $\frac{n+1}{60}$

#13 Convert to Algebra 7.EE.B.4

Tenths digit

3

and hundredths digit

8

mean

0.38\le\frac{n+1}{60}<0.39

. Multiply by

60

.

0.38\times 60\le n+1<0.39\times 60 \;\Rightarrow\; 22.8\le n+1<23.4 \;\Rightarrow\; n+1=23

The only whole number in that short range is

23

, so

n=22

.

4Check the thousandths digit

#6 Guess and Check 7.NS.A.2

Divide to confirm the clue we have not used yet.

\frac{22}{60}=0.36666\ldots

The tenths digit is

3

and the thousandths digit is

6

, as required.

Answer:

22

4 · Reviewdoes it hold up?

$\frac{22}{60}=0.3666\ldots$ (tenths $3$ , thousandths $6$ ) and $\frac{23}{60}=0.3833\ldots$ (hundredths $8$ ). All three clues hold.

Another way: List the candidates $n=18, \dots, 23$ from the tenths clue and compute: $\frac{n}{60}$ has thousandths digit $6$ only for $n=19$ ( $0.3166\ldots$ ) and $n=22$ ( $0.3666\ldots$ ); then $\frac{20}{60}=0.3333\ldots$ fails; $\frac{23}{60}=0.3833\ldots$ works, so $n=22$ .

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing each digit clue as an inequality and solving for $n$.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Converting the fractions $\frac{n}{60}$ to decimals to read and check their digits.

💡Takeaway. A single digit of a decimal is really a range of values -- and ranges are what inequalities are for.

Variant 5 medium

A natural number $n$ is less than $64$ . When $\frac{n}{64}$ is written as a decimal, its tenths digit is $3$ and its thousandths digit is $9$ . When $\frac{n+1}{64}$ is written as a decimal, its hundredths digit is $7$ . Find $n$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Each digit clue says the fraction lies in a certain range. Write those ranges as inequalities in $n$ , combine them to pin down $n$ , and confirm with the remaining digit clue.

Givens

$n$ is a natural number less than $64$ .
$\frac{n}{64}$ has tenths digit $3$ and thousandths digit $9$ .
$\frac{n+1}{64}$ has hundredths digit $7$ .

Unknowns

The value of $n$ .

Constraints

A tenths digit of $3$ means the number is at least $0.3$ and less than $0.4$ (it is less than $1$ here).

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#6 Guess and Check

A digit is a range in disguise: "tenths digit $3$ " means $0.3\le\frac{n}{64}<0.4$ . Turn the two strongest clues into inequalities, multiply by $64$ , and the integer $n$ is forced; then check the last clue.

3 · Execute4 carry out the plan

1Use the tenths digit of $\frac{n}{64}$

#13 Convert to Algebra 7.EE.B.4

Since

n<64

,

\frac{n}{64}<1

, so a tenths digit of

3

means

0.3\le\frac{n}{64}<0.4

. Multiply by

64

.

0.3\times 64\le n<0.4\times 64 \;\Rightarrow\; 19.2\le n<25.6

So

n

is one of

20, 21, \dots, 25

.

2Find the tenths digit of $\frac{n+1}{64}$

#3 Eliminate Possibilities 7.NS.A.2

From

20\le n\le 25

we get

21\le n+1\le 26

, so

\frac{n+1}{64}

is at least

\frac{21}{64}

and at most

\frac{26}{64}

, which is less than

0.47

. A tenths digit of

4

with hundredths digit

7

would need at least

0.47

, so its tenths digit is

3

.

\frac{21}{64}=0.328125\le\frac{n+1}{64}\le\frac{26}{64}=0.40625

A tenths digit of

4

with hundredths

7

would need

0.47

, too big.

3Use the hundredths digit of $\frac{n+1}{64}$

#13 Convert to Algebra 7.EE.B.4

Tenths digit

3

and hundredths digit

7

mean

0.37\le\frac{n+1}{64}<0.38

. Multiply by

64

.

0.37\times 64\le n+1<0.38\times 64 \;\Rightarrow\; 23.68\le n+1<24.32 \;\Rightarrow\; n+1=24

The only whole number in that short range is

24

, so

n=23

.

4Check the thousandths digit

#6 Guess and Check 7.NS.A.2

Divide to confirm the clue we have not used yet.

\frac{23}{64}=0.35937\ldots

The tenths digit is

3

and the thousandths digit is

9

, as required.

Answer:

23

4 · Reviewdoes it hold up?

$\frac{23}{64}=0.3593\ldots$ (tenths $3$ , thousandths $9$ ) and $\frac{24}{64}=0.375$ (hundredths $7$ ). All three clues hold.

Another way: List the candidates $n=20, \dots, 25$ from the tenths clue and compute: $\frac{n}{64}$ has thousandths digit $9$ only for $n=23$ ( $0.3593\ldots$ ); then $\frac{24}{64}=0.375$ works, so $n=23$ .

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing each digit clue as an inequality and solving for $n$.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Converting the fractions $\frac{n}{64}$ to decimals to read and check their digits.

💡Takeaway. A single digit of a decimal is really a range of values -- and ranges are what inequalities are for.

Variant 6 medium

A natural number $n$ is less than $65$ . When $\frac{n}{65}$ is written as a decimal, its tenths digit is $4$ and its thousandths digit is $6$ . When $\frac{n+1}{65}$ is written as a decimal, its hundredths digit is $9$ . Find $n$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Each digit clue says the fraction lies in a certain range. Write those ranges as inequalities in $n$ , combine them to pin down $n$ , and confirm with the remaining digit clue.

Givens

$n$ is a natural number less than $65$ .
$\frac{n}{65}$ has tenths digit $4$ and thousandths digit $6$ .
$\frac{n+1}{65}$ has hundredths digit $9$ .

Unknowns

The value of $n$ .

Constraints

A tenths digit of $4$ means the number is at least $0.4$ and less than $0.5$ (it is less than $1$ here).

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#6 Guess and Check

A digit is a range in disguise: "tenths digit $4$ " means $0.4\le\frac{n}{65}<0.5$ . Turn the two strongest clues into inequalities, multiply by $65$ , and the integer $n$ is forced; then check the last clue.

3 · Execute4 carry out the plan

1Use the tenths digit of $\frac{n}{65}$

#13 Convert to Algebra 7.EE.B.4

Since

n<65

,

\frac{n}{65}<1

, so a tenths digit of

4

means

0.4\le\frac{n}{65}<0.5

. Multiply by

65

.

0.4\times 65\le n<0.5\times 65 \;\Rightarrow\; 26\le n<32.5

So

n

is one of

26, 27, \dots, 32

.

2Find the tenths digit of $\frac{n+1}{65}$

#3 Eliminate Possibilities 7.NS.A.2

From

26\le n\le 32

we get

27\le n+1\le 33

, so

\frac{n+1}{65}

is at least

\frac{27}{65}

and at most

\frac{33}{65}

, which is less than

0.59

. A tenths digit of

5

with hundredths digit

9

would need at least

0.59

, so its tenths digit is

4

.

\frac{27}{65}\le\frac{n+1}{65}\le\frac{33}{65}

A tenths digit of

5

with hundredths

9

would need

0.59

, too big.

3Use the hundredths digit of $\frac{n+1}{65}$

#13 Convert to Algebra 7.EE.B.4

Tenths digit

4

and hundredths digit

9

mean

0.49\le\frac{n+1}{65}<0.5

. Multiply by

65

.

0.49\times 65\le n+1<0.5\times 65 \;\Rightarrow\; 31.85\le n+1<32.5 \;\Rightarrow\; n+1=32

The only whole number in that short range is

32

, so

n=31

.

4Check the thousandths digit

#6 Guess and Check 7.NS.A.2

Divide to confirm the clue we have not used yet.

\frac{31}{65}=0.47692\ldots

The tenths digit is

4

and the thousandths digit is

6

, as required.

Answer:

31

4 · Reviewdoes it hold up?

$\frac{31}{65}=0.4769\ldots$ (tenths $4$ , thousandths $6$ ) and $\frac{32}{65}=0.4923\ldots$ (hundredths $9$ ). All three clues hold.

Another way: List the candidates $n=26, \dots, 32$ from the tenths clue and compute: $\frac{n}{65}$ has thousandths digit $6$ only for $n=29$ ( $0.4461\ldots$ ) and $n=31$ ( $0.4769\ldots$ ); then $\frac{30}{65}=0.4615\ldots$ fails; $\frac{32}{65}=0.4923\ldots$ works, so $n=31$ .

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing each digit clue as an inequality and solving for $n$.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Converting the fractions $\frac{n}{65}$ to decimals to read and check their digits.

💡Takeaway. A single digit of a decimal is really a range of values -- and ranges are what inequalities are for.

Variant 7 medium

A natural number $n$ is less than $70$ . When $\frac{n}{70}$ is written as a decimal, its tenths digit is $6$ and its thousandths digit is $1$ . When $\frac{n+1}{70}$ is written as a decimal, its hundredths digit is $8$ . Find $n$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Each digit clue says the fraction lies in a certain range. Write those ranges as inequalities in $n$ , combine them to pin down $n$ , and confirm with the remaining digit clue.

Givens

$n$ is a natural number less than $70$ .
$\frac{n}{70}$ has tenths digit $6$ and thousandths digit $1$ .
$\frac{n+1}{70}$ has hundredths digit $8$ .

Unknowns

The value of $n$ .

Constraints

A tenths digit of $6$ means the number is at least $0.6$ and less than $0.7$ (it is less than $1$ here).

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#6 Guess and Check

A digit is a range in disguise: "tenths digit $6$ " means $0.6\le\frac{n}{70}<0.7$ . Turn the two strongest clues into inequalities, multiply by $70$ , and the integer $n$ is forced; then check the last clue.

3 · Execute4 carry out the plan

1Use the tenths digit of $\frac{n}{70}$

#13 Convert to Algebra 7.EE.B.4

Since

n<70

,

\frac{n}{70}<1

, so a tenths digit of

6

means

0.6\le\frac{n}{70}<0.7

. Multiply by

70

.

0.6\times 70\le n<0.7\times 70 \;\Rightarrow\; 42\le n<49

So

n

is one of

42, 43, \dots, 48

.

2Find the tenths digit of $\frac{n+1}{70}$

#3 Eliminate Possibilities 7.NS.A.2

From

42\le n\le 48

we get

43\le n+1\le 49

, so

\frac{n+1}{70}

is at least

\frac{43}{70}

and at most

\frac{49}{70}

, which is less than

0.78

. A tenths digit of

7

with hundredths digit

8

would need at least

0.78

, so its tenths digit is

6

.

\frac{43}{70}\le\frac{n+1}{70}\le\frac{49}{70}=0.7

A tenths digit of

7

with hundredths

8

would need

0.78

, too big.

3Use the hundredths digit of $\frac{n+1}{70}$

#13 Convert to Algebra 7.EE.B.4

Tenths digit

6

and hundredths digit

8

mean

0.68\le\frac{n+1}{70}<0.69

. Multiply by

70

.

0.68\times 70\le n+1<0.69\times 70 \;\Rightarrow\; 47.6\le n+1<48.3 \;\Rightarrow\; n+1=48

The only whole number in that short range is

48

, so

n=47

.

4Check the thousandths digit

#6 Guess and Check 7.NS.A.2

Divide to confirm the clue we have not used yet.

\frac{47}{70}=0.67142\ldots

The tenths digit is

6

and the thousandths digit is

1

, as required.

Answer:

47

4 · Reviewdoes it hold up?

$\frac{47}{70}=0.6714\ldots$ (tenths $6$ , thousandths $1$ ) and $\frac{48}{70}=0.6857\ldots$ (hundredths $8$ ). All three clues hold.

Another way: List the candidates $n=42, \dots, 48$ from the tenths clue and compute: $\frac{n}{70}$ has thousandths digit $1$ only for $n=47$ ( $0.6714\ldots$ ); then $\frac{48}{70}=0.6857\ldots$ works, so $n=47$ .

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing each digit clue as an inequality and solving for $n$.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Converting the fractions $\frac{n}{70}$ to decimals to read and check their digits.

💡Takeaway. A single digit of a decimal is really a range of values -- and ranges are what inequalities are for.

Variant 8 medium

A natural number $n$ is less than $75$ . When $\frac{n}{75}$ is written as a decimal, its tenths digit is $1$ and its thousandths digit is $3$ . When $\frac{n+1}{75}$ is written as a decimal, its hundredths digit is $8$ . Find $n$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Each digit clue says the fraction lies in a certain range. Write those ranges as inequalities in $n$ , combine them to pin down $n$ , and confirm with the remaining digit clue.

Givens

$n$ is a natural number less than $75$ .
$\frac{n}{75}$ has tenths digit $1$ and thousandths digit $3$ .
$\frac{n+1}{75}$ has hundredths digit $8$ .

Unknowns

The value of $n$ .

Constraints

A tenths digit of $1$ means the number is at least $0.1$ and less than $0.2$ (it is less than $1$ here).

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#6 Guess and Check

A digit is a range in disguise: "tenths digit $1$ " means $0.1\le\frac{n}{75}<0.2$ . Turn the two strongest clues into inequalities, multiply by $75$ , and the integer $n$ is forced; then check the last clue.

3 · Execute4 carry out the plan

1Use the tenths digit of $\frac{n}{75}$

#13 Convert to Algebra 7.EE.B.4

Since

n<75

,

\frac{n}{75}<1

, so a tenths digit of

1

means

0.1\le\frac{n}{75}<0.2

. Multiply by

75

.

0.1\times 75\le n<0.2\times 75 \;\Rightarrow\; 7.5\le n<15

So

n

is one of

8, 9, \dots, 14

.

2Find the tenths digit of $\frac{n+1}{75}$

#3 Eliminate Possibilities 7.NS.A.2

From

8\le n\le 14

we get

9\le n+1\le 15

, so

\frac{n+1}{75}

is at least

\frac{9}{75}

and at most

\frac{15}{75}

, which is less than

0.28

. A tenths digit of

2

with hundredths digit

8

would need at least

0.28

, so its tenths digit is

1

.

\frac{9}{75}=0.12\le\frac{n+1}{75}\le\frac{15}{75}=0.2

A tenths digit of

2

with hundredths

8

would need

0.28

, too big.

3Use the hundredths digit of $\frac{n+1}{75}$

#13 Convert to Algebra 7.EE.B.4

Tenths digit

1

and hundredths digit

8

mean

0.18\le\frac{n+1}{75}<0.19

. Multiply by

75

.

0.18\times 75\le n+1<0.19\times 75 \;\Rightarrow\; 13.5\le n+1<14.25 \;\Rightarrow\; n+1=14

The only whole number in that short range is

14

, so

n=13

.

4Check the thousandths digit

#6 Guess and Check 7.NS.A.2

Divide to confirm the clue we have not used yet.

\frac{13}{75}=0.17333\ldots

The tenths digit is

1

and the thousandths digit is

3

, as required.

Answer:

13

4 · Reviewdoes it hold up?

$\frac{13}{75}=0.1733\ldots$ (tenths $1$ , thousandths $3$ ) and $\frac{14}{75}=0.1866\ldots$ (hundredths $8$ ). All three clues hold.

Another way: List the candidates $n=8, \dots, 14$ from the tenths clue and compute: $\frac{n}{75}$ has thousandths digit $3$ only for $n=10$ ( $0.1333\ldots$ ) and $n=13$ ( $0.1733\ldots$ ); then $\frac{11}{75}=0.1466\ldots$ fails; $\frac{14}{75}=0.1866\ldots$ works, so $n=13$ .

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing each digit clue as an inequality and solving for $n$.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Converting the fractions $\frac{n}{75}$ to decimals to read and check their digits.

💡Takeaway. A single digit of a decimal is really a range of values -- and ranges are what inequalities are for.

Variant 9 hard

A natural number $n$ is less than $80$ . When $\frac{n}{80}$ is written as a decimal, its tenths digit is $5$ and its thousandths digit is $7$ . When $\frac{n+1}{80}$ is written as a decimal, its hundredths digit is $5$ . Find $n$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Each digit clue says the fraction lies in a certain range. Write those ranges as inequalities in $n$ , combine them to pin down $n$ , and confirm with the remaining digit clue.

Givens

$n$ is a natural number less than $80$ .
$\frac{n}{80}$ has tenths digit $5$ and thousandths digit $7$ .
$\frac{n+1}{80}$ has hundredths digit $5$ .

Unknowns

The value of $n$ .

Constraints

A tenths digit of $5$ means the number is at least $0.5$ and less than $0.6$ (it is less than $1$ here).

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#6 Guess and Check

A digit is a range in disguise: "tenths digit $5$ " means $0.5\le\frac{n}{80}<0.6$ . Turn the two strongest clues into inequalities, multiply by $80$ , and the integer $n$ is forced; then check the last clue.

3 · Execute4 carry out the plan

1Use the tenths digit of $\frac{n}{80}$

#13 Convert to Algebra 7.EE.B.4

Since

n<80

,

\frac{n}{80}<1

, so a tenths digit of

5

means

0.5\le\frac{n}{80}<0.6

. Multiply by

80

.

0.5\times 80\le n<0.6\times 80 \;\Rightarrow\; 40\le n<48

So

n

is one of

40, 41, \dots, 47

.

2Find the tenths digit of $\frac{n+1}{80}$

#3 Eliminate Possibilities 7.NS.A.2

From

40\le n\le 47

we get

41\le n+1\le 48

, so

\frac{n+1}{80}

is at least

\frac{41}{80}

and at most

\frac{48}{80}

, which is less than

0.65

. A tenths digit of

6

with hundredths digit

5

would need at least

0.65

, so its tenths digit is

5

.

\frac{41}{80}=0.5125\le\frac{n+1}{80}\le\frac{48}{80}=0.6

A tenths digit of

6

with hundredths

5

would need

0.65

, too big.

3Use the hundredths digit of $\frac{n+1}{80}$

#13 Convert to Algebra 7.EE.B.4

Tenths digit

5

and hundredths digit

5

mean

0.55\le\frac{n+1}{80}<0.56

. Multiply by

80

.

0.55\times 80\le n+1<0.56\times 80 \;\Rightarrow\; 44\le n+1<44.8 \;\Rightarrow\; n+1=44

The only whole number in that short range is

44

, so

n=43

.

4Check the thousandths digit

#6 Guess and Check 7.NS.A.2

Divide to confirm the clue we have not used yet.

\frac{43}{80}=0.5375

The tenths digit is

5

and the thousandths digit is

7

, as required.

Answer:

43

4 · Reviewdoes it hold up?

$\frac{43}{80}=0.5375$ (tenths $5$ , thousandths $7$ ) and $\frac{44}{80}=0.55$ (hundredths $5$ ). All three clues hold.

Another way: List the candidates $n=40, \dots, 47$ from the tenths clue and compute: $\frac{n}{80}$ has thousandths digit $7$ only for $n=43$ ( $0.5375$ ) and $n=47$ ( $0.5875$ ); then $\frac{44}{80}=0.55$ works; $\frac{48}{80}=0.6$ fails, so $n=43$ .

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing each digit clue as an inequality and solving for $n$.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Converting the fractions $\frac{n}{80}$ to decimals to read and check their digits.

💡Takeaway. A single digit of a decimal is really a range of values -- and ranges are what inequalities are for.

Variant 10 hard

A natural number $n$ is less than $85$ . When $\frac{n}{85}$ is written as a decimal, its tenths digit is $3$ and its thousandths digit is $5$ . When $\frac{n+1}{85}$ is written as a decimal, its hundredths digit is $1$ . Find $n$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Each digit clue says the fraction lies in a certain range. Write those ranges as inequalities in $n$ , combine them to pin down $n$ , and confirm with the remaining digit clue.

Givens

$n$ is a natural number less than $85$ .
$\frac{n}{85}$ has tenths digit $3$ and thousandths digit $5$ .
$\frac{n+1}{85}$ has hundredths digit $1$ .

Unknowns

The value of $n$ .

Constraints

A tenths digit of $3$ means the number is at least $0.3$ and less than $0.4$ (it is less than $1$ here).

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#6 Guess and Check

A digit is a range in disguise: "tenths digit $3$ " means $0.3\le\frac{n}{85}<0.4$ . Turn the two strongest clues into inequalities, multiply by $85$ , and the integer $n$ is forced; then check the last clue.

3 · Execute4 carry out the plan

1Use the tenths digit of $\frac{n}{85}$

#13 Convert to Algebra 7.EE.B.4

Since

n<85

,

\frac{n}{85}<1

, so a tenths digit of

3

means

0.3\le\frac{n}{85}<0.4

. Multiply by

85

.

0.3\times 85\le n<0.4\times 85 \;\Rightarrow\; 25.5\le n<34

So

n

is one of

26, 27, \dots, 33

.

2Find the tenths digit of $\frac{n+1}{85}$

#3 Eliminate Possibilities 7.NS.A.2

From

26\le n\le 33

we get

27\le n+1\le 34

, so

\frac{n+1}{85}

is at least

\frac{27}{85}

and at most

\frac{34}{85}

, which is less than

0.41

. A tenths digit of

4

with hundredths digit

1

would need at least

0.41

, so its tenths digit is

3

.

\frac{27}{85}\le\frac{n+1}{85}\le\frac{34}{85}=0.4

A tenths digit of

4

with hundredths

1

would need

0.41

, too big.

3Use the hundredths digit of $\frac{n+1}{85}$

#13 Convert to Algebra 7.EE.B.4

Tenths digit

3

and hundredths digit

1

mean

0.31\le\frac{n+1}{85}<0.32

. Multiply by

85

.

0.31\times 85\le n+1<0.32\times 85 \;\Rightarrow\; 26.35\le n+1<27.2 \;\Rightarrow\; n+1=27

The only whole number in that short range is

27

, so

n=26

.

4Check the thousandths digit

#6 Guess and Check 7.NS.A.2

Divide to confirm the clue we have not used yet.

\frac{26}{85}=0.30588\ldots

The tenths digit is

3

and the thousandths digit is

5

, as required.

Answer:

26

4 · Reviewdoes it hold up?

$\frac{26}{85}=0.3058\ldots$ (tenths $3$ , thousandths $5$ ) and $\frac{27}{85}=0.3176\ldots$ (hundredths $1$ ). All three clues hold.

Another way: List the candidates $n=26, \dots, 33$ from the tenths clue and compute: $\frac{n}{85}$ has thousandths digit $5$ only for $n=26$ ( $0.3058\ldots$ ); then $\frac{27}{85}=0.3176\ldots$ works, so $n=26$ .

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing each digit clue as an inequality and solving for $n$.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Converting the fractions $\frac{n}{85}$ to decimals to read and check their digits.

💡Takeaway. A single digit of a decimal is really a range of values -- and ranges are what inequalities are for.

Variant 11 hard

A natural number $n$ is less than $90$ . When $\frac{n}{90}$ is written as a decimal, its tenths digit is $4$ and its thousandths digit is $1$ . When $\frac{n+1}{90}$ is written as a decimal, its hundredths digit is $2$ . Find $n$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Each digit clue says the fraction lies in a certain range. Write those ranges as inequalities in $n$ , combine them to pin down $n$ , and confirm with the remaining digit clue.

Givens

$n$ is a natural number less than $90$ .
$\frac{n}{90}$ has tenths digit $4$ and thousandths digit $1$ .
$\frac{n+1}{90}$ has hundredths digit $2$ .

Unknowns

The value of $n$ .

Constraints

A tenths digit of $4$ means the number is at least $0.4$ and less than $0.5$ (it is less than $1$ here).

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#6 Guess and Check

A digit is a range in disguise: "tenths digit $4$ " means $0.4\le\frac{n}{90}<0.5$ . Turn the two strongest clues into inequalities, multiply by $90$ , and the integer $n$ is forced; then check the last clue.

3 · Execute4 carry out the plan

1Use the tenths digit of $\frac{n}{90}$

#13 Convert to Algebra 7.EE.B.4

Since

n<90

,

\frac{n}{90}<1

, so a tenths digit of

4

means

0.4\le\frac{n}{90}<0.5

. Multiply by

90

.

0.4\times 90\le n<0.5\times 90 \;\Rightarrow\; 36\le n<45

So

n

is one of

36, 37, \dots, 44

.

2Find the tenths digit of $\frac{n+1}{90}$

#3 Eliminate Possibilities 7.NS.A.2

From

36\le n\le 44

we get

37\le n+1\le 45

, so

\frac{n+1}{90}

is at least

\frac{37}{90}

and at most

\frac{45}{90}

, which is less than

0.52

. A tenths digit of

5

with hundredths digit

2

would need at least

0.52

, so its tenths digit is

4

.

\frac{37}{90}\le\frac{n+1}{90}\le\frac{45}{90}=0.5

A tenths digit of

5

with hundredths

2

would need

0.52

, too big.

3Use the hundredths digit of $\frac{n+1}{90}$

#13 Convert to Algebra 7.EE.B.4

Tenths digit

4

and hundredths digit

2

mean

0.42\le\frac{n+1}{90}<0.43

. Multiply by

90

.

0.42\times 90\le n+1<0.43\times 90 \;\Rightarrow\; 37.8\le n+1<38.7 \;\Rightarrow\; n+1=38

The only whole number in that short range is

38

, so

n=37

.

4Check the thousandths digit

#6 Guess and Check 7.NS.A.2

Divide to confirm the clue we have not used yet.

\frac{37}{90}=0.41111\ldots

The tenths digit is

4

and the thousandths digit is

1

, as required.

Answer:

37

4 · Reviewdoes it hold up?

$\frac{37}{90}=0.4111\ldots$ (tenths $4$ , thousandths $1$ ) and $\frac{38}{90}=0.4222\ldots$ (hundredths $2$ ). All three clues hold.

Another way: List the candidates $n=36, \dots, 44$ from the tenths clue and compute: $\frac{n}{90}$ has thousandths digit $1$ only for $n=37$ ( $0.4111\ldots$ ); then $\frac{38}{90}=0.4222\ldots$ works, so $n=37$ .

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing each digit clue as an inequality and solving for $n$.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Converting the fractions $\frac{n}{90}$ to decimals to read and check their digits.

💡Takeaway. A single digit of a decimal is really a range of values -- and ranges are what inequalities are for.

Variant 12 hard

A natural number $n$ is less than $95$ . When $\frac{n}{95}$ is written as a decimal, its tenths digit is $6$ and its thousandths digit is $2$ . When $\frac{n+1}{95}$ is written as a decimal, its hundredths digit is $5$ . Find $n$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Each digit clue says the fraction lies in a certain range. Write those ranges as inequalities in $n$ , combine them to pin down $n$ , and confirm with the remaining digit clue.

Givens

$n$ is a natural number less than $95$ .
$\frac{n}{95}$ has tenths digit $6$ and thousandths digit $2$ .
$\frac{n+1}{95}$ has hundredths digit $5$ .

Unknowns

The value of $n$ .

Constraints

A tenths digit of $6$ means the number is at least $0.6$ and less than $0.7$ (it is less than $1$ here).

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#6 Guess and Check

A digit is a range in disguise: "tenths digit $6$ " means $0.6\le\frac{n}{95}<0.7$ . Turn the two strongest clues into inequalities, multiply by $95$ , and the integer $n$ is forced; then check the last clue.

3 · Execute4 carry out the plan

1Use the tenths digit of $\frac{n}{95}$

#13 Convert to Algebra 7.EE.B.4

Since

n<95

,

\frac{n}{95}<1

, so a tenths digit of

6

means

0.6\le\frac{n}{95}<0.7

. Multiply by

95

.

0.6\times 95\le n<0.7\times 95 \;\Rightarrow\; 57\le n<66.5

So

n

is one of

57, 58, \dots, 66

.

2Find the tenths digit of $\frac{n+1}{95}$

#3 Eliminate Possibilities 7.NS.A.2

From

57\le n\le 66

we get

58\le n+1\le 67

, so

\frac{n+1}{95}

is at least

\frac{58}{95}

and at most

\frac{67}{95}

, which is less than

0.75

. A tenths digit of

7

with hundredths digit

5

would need at least

0.75

, so its tenths digit is

6

.

\frac{58}{95}\le\frac{n+1}{95}\le\frac{67}{95}

A tenths digit of

7

with hundredths

5

would need

0.75

, too big.

3Use the hundredths digit of $\frac{n+1}{95}$

#13 Convert to Algebra 7.EE.B.4

Tenths digit

6

and hundredths digit

5

mean

0.65\le\frac{n+1}{95}<0.66

. Multiply by

95

.

0.65\times 95\le n+1<0.66\times 95 \;\Rightarrow\; 61.75\le n+1<62.7 \;\Rightarrow\; n+1=62

The only whole number in that short range is

62

, so

n=61

.

4Check the thousandths digit

#6 Guess and Check 7.NS.A.2

Divide to confirm the clue we have not used yet.

\frac{61}{95}=0.64210\ldots

The tenths digit is

6

and the thousandths digit is

2

, as required.

Answer:

61

4 · Reviewdoes it hold up?

$\frac{61}{95}=0.6421\ldots$ (tenths $6$ , thousandths $2$ ) and $\frac{62}{95}=0.6526\ldots$ (hundredths $5$ ). All three clues hold.

Another way: List the candidates $n=57, \dots, 66$ from the tenths clue and compute: $\frac{n}{95}$ has thousandths digit $2$ only for $n=61$ ( $0.6421\ldots$ ) and $n=62$ ( $0.6526\ldots$ ); then $\frac{62}{95}=0.6526\ldots$ works; $\frac{63}{95}=0.6631\ldots$ fails, so $n=61$ .

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing each digit clue as an inequality and solving for $n$.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Converting the fractions $\frac{n}{95}$ to decimals to read and check their digits.

💡Takeaway. A single digit of a decimal is really a range of values -- and ranges are what inequalities are for.

Turn decimal-digit clues into inequalities · 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?