← Rounding cannot be undone, but it can be bounded · Inequality Range Membership

Rounding cannot be undone, but it can be bounded · 12 practice problems

6.EE.B.86.NS.B.35.NBT.A.4

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Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy

A number was divided by $9$ , and the quotient rounded to the nearest whole number came to $7$ . How many natural numbers could that number be?

Your answer

Show solution

1 · Understandwhat's really being asked

Some natural number divided by 9 rounds to 7. We want how many different numbers it could have been.

Givens

The number was divided by 9.
The quotient rounded to the nearest whole number is 7.

Unknowns

How many natural numbers fit that description.

Constraints

Rounding is to the nearest whole number, so a quotient of exactly 6.5 rounds up to 7.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #3 Eliminate Possibilities#8 Analyze the Units

Rounding loses information, so the original quotient cannot be recovered -- but it can be trapped between two bounds. Find those, multiply them back up, and count the whole numbers that survive.

3 · Execute4 carry out the plan

1Bound the quotient

#11 Work Backwards 5.NBT.A.4

Anything from 6.5 up to but not including 7.5 rounds to 7.

6.5 \le \text{quotient} < 7.5

Half a unit either side of the rounded value.

2Multiply both bounds by 9

#8 Analyze the Units 6.EE.B.8

The quotient came from dividing by 9, so multiplying undoes it -- and the bounds move with it.

58.5 \le \text{number} < 67.5

The number lives between 58.5 and 67.5.

3Find the first and last whole number inside

#3 Eliminate Possibilities 6.EE.B.8

The smallest natural number at least 58.5 is 59, and the largest below 67.5 is 67.

59 \le \text{number} \le 67

From 59 to 67.

4Count them

#3 Eliminate Possibilities 6.NS.B.3

Both ends are included, so subtract and add one.

67 - 59 + 1 = 9

9 numbers work.

Answer: 9

4 · Reviewdoes it hold up?

Checking the ends without dividing: 59 is at least 58.5 and below 67.5, while 58 falls short of 58.5 and 68 reaches 67.5 -- so the run really does stop where it stops.

Another way: The window is 9 wide, so it holds 9 or so whole numbers -- 9 here. That is a quick check on the count without redoing the work.

Standardsmin grade 6

6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Writing the possible values as an inequality and scaling it.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Counting the whole numbers the range contains.
5.NBT.A.4 Round decimals to any place — Reading what 'rounds to the nearest whole number' allows.

💡Takeaway. A rounded number does not tell you what it was. It tells you the smallest and largest it could have been.

Variant 2 easy

A number was divided by $4$ , and the quotient rounded to the nearest whole number came to $9$ . How many natural numbers could that number be?

Your answer

Show solution

1 · Understandwhat's really being asked

Some natural number divided by 4 rounds to 9. We want how many different numbers it could have been.

Givens

The number was divided by 4.
The quotient rounded to the nearest whole number is 9.

Unknowns

How many natural numbers fit that description.

Constraints

Rounding is to the nearest whole number, so a quotient of exactly 8.5 rounds up to 9.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #3 Eliminate Possibilities#8 Analyze the Units

Rounding loses information, so the original quotient cannot be recovered -- but it can be trapped between two bounds. Find those, multiply them back up, and count the whole numbers that survive.

3 · Execute4 carry out the plan

1Bound the quotient

#11 Work Backwards 5.NBT.A.4

Anything from 8.5 up to but not including 9.5 rounds to 9.

8.5 \le \text{quotient} < 9.5

Half a unit either side of the rounded value.

2Multiply both bounds by 4

#8 Analyze the Units 6.EE.B.8

The quotient came from dividing by 4, so multiplying undoes it -- and the bounds move with it.

34 \le \text{number} < 38

The number lives between 34 and 38.

3Find the first and last whole number inside

#3 Eliminate Possibilities 6.EE.B.8

The smallest natural number at least 34 is 34, and the largest below 38 is 37.

34 \le \text{number} \le 37

From 34 to 37.

4Count them

#3 Eliminate Possibilities 6.NS.B.3

Both ends are included, so subtract and add one.

37 - 34 + 1 = 4

4 numbers work.

Answer: 4

4 · Reviewdoes it hold up?

Checking the ends without dividing: 34 is at least 34 and below 38, while 33 falls short of 34 and 38 reaches 38 -- so the run really does stop where it stops.

Another way: The window is 4 wide, so it holds 4 or so whole numbers -- 4 here. That is a quick check on the count without redoing the work.

Standardsmin grade 6

6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Writing the possible values as an inequality and scaling it.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Counting the whole numbers the range contains.
5.NBT.A.4 Round decimals to any place — Reading what 'rounds to the nearest whole number' allows.

💡Takeaway. A rounded number does not tell you what it was. It tells you the smallest and largest it could have been.

Variant 3 easy

A number was divided by $6$ , and the quotient rounded to the nearest whole number came to $11$ . How many natural numbers could that number be?

Your answer

Show solution

1 · Understandwhat's really being asked

Some natural number divided by 6 rounds to 11. We want how many different numbers it could have been.

Givens

The number was divided by 6.
The quotient rounded to the nearest whole number is 11.

Unknowns

How many natural numbers fit that description.

Constraints

Rounding is to the nearest whole number, so a quotient of exactly 10.5 rounds up to 11.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #3 Eliminate Possibilities#8 Analyze the Units

Rounding loses information, so the original quotient cannot be recovered -- but it can be trapped between two bounds. Find those, multiply them back up, and count the whole numbers that survive.

3 · Execute4 carry out the plan

1Bound the quotient

#11 Work Backwards 5.NBT.A.4

Anything from 10.5 up to but not including 11.5 rounds to 11.

10.5 \le \text{quotient} < 11.5

Half a unit either side of the rounded value.

2Multiply both bounds by 6

#8 Analyze the Units 6.EE.B.8

The quotient came from dividing by 6, so multiplying undoes it -- and the bounds move with it.

63 \le \text{number} < 69

The number lives between 63 and 69.

3Find the first and last whole number inside

#3 Eliminate Possibilities 6.EE.B.8

The smallest natural number at least 63 is 63, and the largest below 69 is 68.

63 \le \text{number} \le 68

From 63 to 68.

4Count them

#3 Eliminate Possibilities 6.NS.B.3

Both ends are included, so subtract and add one.

68 - 63 + 1 = 6

6 numbers work.

Answer: 6

4 · Reviewdoes it hold up?

Checking the ends without dividing: 63 is at least 63 and below 69, while 62 falls short of 63 and 69 reaches 69 -- so the run really does stop where it stops.

Another way: The window is 6 wide, so it holds 6 or so whole numbers -- 6 here. That is a quick check on the count without redoing the work.

Standardsmin grade 6

6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Writing the possible values as an inequality and scaling it.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Counting the whole numbers the range contains.
5.NBT.A.4 Round decimals to any place — Reading what 'rounds to the nearest whole number' allows.

💡Takeaway. A rounded number does not tell you what it was. It tells you the smallest and largest it could have been.

Variant 4 easy

A number was divided by $11$ , and the quotient rounded to the nearest whole number came to $8$ . How many natural numbers could that number be?

Your answer

Show solution

1 · Understandwhat's really being asked

Some natural number divided by 11 rounds to 8. We want how many different numbers it could have been.

Givens

The number was divided by 11.
The quotient rounded to the nearest whole number is 8.

Unknowns

How many natural numbers fit that description.

Constraints

Rounding is to the nearest whole number, so a quotient of exactly 7.5 rounds up to 8.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #3 Eliminate Possibilities#8 Analyze the Units

Rounding loses information, so the original quotient cannot be recovered -- but it can be trapped between two bounds. Find those, multiply them back up, and count the whole numbers that survive.

3 · Execute4 carry out the plan

1Bound the quotient

#11 Work Backwards 5.NBT.A.4

Anything from 7.5 up to but not including 8.5 rounds to 8.

7.5 \le \text{quotient} < 8.5

Half a unit either side of the rounded value.

2Multiply both bounds by 11

#8 Analyze the Units 6.EE.B.8

The quotient came from dividing by 11, so multiplying undoes it -- and the bounds move with it.

82.5 \le \text{number} < 93.5

The number lives between 82.5 and 93.5.

3Find the first and last whole number inside

#3 Eliminate Possibilities 6.EE.B.8

The smallest natural number at least 82.5 is 83, and the largest below 93.5 is 93.

83 \le \text{number} \le 93

From 83 to 93.

4Count them

#3 Eliminate Possibilities 6.NS.B.3

Both ends are included, so subtract and add one.

93 - 83 + 1 = 11

11 numbers work.

Answer: 11

4 · Reviewdoes it hold up?

Checking the ends without dividing: 83 is at least 82.5 and below 93.5, while 82 falls short of 82.5 and 94 reaches 93.5 -- so the run really does stop where it stops.

Another way: The window is 11 wide, so it holds 11 or so whole numbers -- 11 here. That is a quick check on the count without redoing the work.

Standardsmin grade 6

6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Writing the possible values as an inequality and scaling it.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Counting the whole numbers the range contains.
5.NBT.A.4 Round decimals to any place — Reading what 'rounds to the nearest whole number' allows.

💡Takeaway. A rounded number does not tell you what it was. It tells you the smallest and largest it could have been.

Variant 5 medium

A number was divided by $12$ , and the quotient rounded to the nearest whole number came to $6$ . How many natural numbers could that number be?

Your answer

Show solution

1 · Understandwhat's really being asked

Some natural number divided by 12 rounds to 6. We want how many different numbers it could have been.

Givens

The number was divided by 12.
The quotient rounded to the nearest whole number is 6.

Unknowns

How many natural numbers fit that description.

Constraints

Rounding is to the nearest whole number, so a quotient of exactly 5.5 rounds up to 6.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #3 Eliminate Possibilities#8 Analyze the Units

Rounding loses information, so the original quotient cannot be recovered -- but it can be trapped between two bounds. Find those, multiply them back up, and count the whole numbers that survive.

3 · Execute4 carry out the plan

1Bound the quotient

#11 Work Backwards 5.NBT.A.4

Anything from 5.5 up to but not including 6.5 rounds to 6.

5.5 \le \text{quotient} < 6.5

Half a unit either side of the rounded value.

2Multiply both bounds by 12

#8 Analyze the Units 6.EE.B.8

The quotient came from dividing by 12, so multiplying undoes it -- and the bounds move with it.

66 \le \text{number} < 78

The number lives between 66 and 78.

3Find the first and last whole number inside

#3 Eliminate Possibilities 6.EE.B.8

The smallest natural number at least 66 is 66, and the largest below 78 is 77.

66 \le \text{number} \le 77

From 66 to 77.

4Count them

#3 Eliminate Possibilities 6.NS.B.3

Both ends are included, so subtract and add one.

77 - 66 + 1 = 12

12 numbers work.

Answer: 12

4 · Reviewdoes it hold up?

Checking the ends without dividing: 66 is at least 66 and below 78, while 65 falls short of 66 and 78 reaches 78 -- so the run really does stop where it stops.

Another way: The window is 12 wide, so it holds 12 or so whole numbers -- 12 here. That is a quick check on the count without redoing the work.

Standardsmin grade 6

6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Writing the possible values as an inequality and scaling it.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Counting the whole numbers the range contains.
5.NBT.A.4 Round decimals to any place — Reading what 'rounds to the nearest whole number' allows.

💡Takeaway. A rounded number does not tell you what it was. It tells you the smallest and largest it could have been.

Variant 6 medium

A number was divided by $5$ , and the quotient rounded to the nearest whole number came to $12$ . How many natural numbers could that number be?

Your answer

Show solution

1 · Understandwhat's really being asked

Some natural number divided by 5 rounds to 12. We want how many different numbers it could have been.

Givens

The number was divided by 5.
The quotient rounded to the nearest whole number is 12.

Unknowns

How many natural numbers fit that description.

Constraints

Rounding is to the nearest whole number, so a quotient of exactly 11.5 rounds up to 12.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #3 Eliminate Possibilities#8 Analyze the Units

Rounding loses information, so the original quotient cannot be recovered -- but it can be trapped between two bounds. Find those, multiply them back up, and count the whole numbers that survive.

3 · Execute4 carry out the plan

1Bound the quotient

#11 Work Backwards 5.NBT.A.4

Anything from 11.5 up to but not including 12.5 rounds to 12.

11.5 \le \text{quotient} < 12.5

Half a unit either side of the rounded value.

2Multiply both bounds by 5

#8 Analyze the Units 6.EE.B.8

The quotient came from dividing by 5, so multiplying undoes it -- and the bounds move with it.

57.5 \le \text{number} < 62.5

The number lives between 57.5 and 62.5.

3Find the first and last whole number inside

#3 Eliminate Possibilities 6.EE.B.8

The smallest natural number at least 57.5 is 58, and the largest below 62.5 is 62.

58 \le \text{number} \le 62

From 58 to 62.

4Count them

#3 Eliminate Possibilities 6.NS.B.3

Both ends are included, so subtract and add one.

62 - 58 + 1 = 5

5 numbers work.

Answer: 5

4 · Reviewdoes it hold up?

Checking the ends without dividing: 58 is at least 57.5 and below 62.5, while 57 falls short of 57.5 and 63 reaches 62.5 -- so the run really does stop where it stops.

Another way: The window is 5 wide, so it holds 5 or so whole numbers -- 5 here. That is a quick check on the count without redoing the work.

Standardsmin grade 6

6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Writing the possible values as an inequality and scaling it.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Counting the whole numbers the range contains.
5.NBT.A.4 Round decimals to any place — Reading what 'rounds to the nearest whole number' allows.

💡Takeaway. A rounded number does not tell you what it was. It tells you the smallest and largest it could have been.

Variant 7 medium

A number was divided by $7$ , and the quotient rounded to the nearest whole number came to $13$ . How many natural numbers could that number be?

Your answer

Show solution

1 · Understandwhat's really being asked

Some natural number divided by 7 rounds to 13. We want how many different numbers it could have been.

Givens

The number was divided by 7.
The quotient rounded to the nearest whole number is 13.

Unknowns

How many natural numbers fit that description.

Constraints

Rounding is to the nearest whole number, so a quotient of exactly 12.5 rounds up to 13.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #3 Eliminate Possibilities#8 Analyze the Units

Rounding loses information, so the original quotient cannot be recovered -- but it can be trapped between two bounds. Find those, multiply them back up, and count the whole numbers that survive.

3 · Execute4 carry out the plan

1Bound the quotient

#11 Work Backwards 5.NBT.A.4

Anything from 12.5 up to but not including 13.5 rounds to 13.

12.5 \le \text{quotient} < 13.5

Half a unit either side of the rounded value.

2Multiply both bounds by 7

#8 Analyze the Units 6.EE.B.8

The quotient came from dividing by 7, so multiplying undoes it -- and the bounds move with it.

87.5 \le \text{number} < 94.5

The number lives between 87.5 and 94.5.

3Find the first and last whole number inside

#3 Eliminate Possibilities 6.EE.B.8

The smallest natural number at least 87.5 is 88, and the largest below 94.5 is 94.

88 \le \text{number} \le 94

From 88 to 94.

4Count them

#3 Eliminate Possibilities 6.NS.B.3

Both ends are included, so subtract and add one.

94 - 88 + 1 = 7

7 numbers work.

Answer: 7

4 · Reviewdoes it hold up?

Checking the ends without dividing: 88 is at least 87.5 and below 94.5, while 87 falls short of 87.5 and 95 reaches 94.5 -- so the run really does stop where it stops.

Another way: The window is 7 wide, so it holds 7 or so whole numbers -- 7 here. That is a quick check on the count without redoing the work.

Standardsmin grade 6

6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Writing the possible values as an inequality and scaling it.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Counting the whole numbers the range contains.
5.NBT.A.4 Round decimals to any place — Reading what 'rounds to the nearest whole number' allows.

💡Takeaway. A rounded number does not tell you what it was. It tells you the smallest and largest it could have been.

Variant 8 medium

A number was divided by $10$ , and the quotient rounded to the nearest whole number came to $14$ . How many natural numbers could that number be?

Your answer

Show solution

1 · Understandwhat's really being asked

Some natural number divided by 10 rounds to 14. We want how many different numbers it could have been.

Givens

The number was divided by 10.
The quotient rounded to the nearest whole number is 14.

Unknowns

How many natural numbers fit that description.

Constraints

Rounding is to the nearest whole number, so a quotient of exactly 13.5 rounds up to 14.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #3 Eliminate Possibilities#8 Analyze the Units

Rounding loses information, so the original quotient cannot be recovered -- but it can be trapped between two bounds. Find those, multiply them back up, and count the whole numbers that survive.

3 · Execute4 carry out the plan

1Bound the quotient

#11 Work Backwards 5.NBT.A.4

Anything from 13.5 up to but not including 14.5 rounds to 14.

13.5 \le \text{quotient} < 14.5

Half a unit either side of the rounded value.

2Multiply both bounds by 10

#8 Analyze the Units 6.EE.B.8

The quotient came from dividing by 10, so multiplying undoes it -- and the bounds move with it.

135 \le \text{number} < 145

The number lives between 135 and 145.

3Find the first and last whole number inside

#3 Eliminate Possibilities 6.EE.B.8

The smallest natural number at least 135 is 135, and the largest below 145 is 144.

135 \le \text{number} \le 144

From 135 to 144.

4Count them

#3 Eliminate Possibilities 6.NS.B.3

Both ends are included, so subtract and add one.

144 - 135 + 1 = 10

10 numbers work.

Answer: 10

4 · Reviewdoes it hold up?

Checking the ends without dividing: 135 is at least 135 and below 145, while 134 falls short of 135 and 145 reaches 145 -- so the run really does stop where it stops.

Another way: The window is 10 wide, so it holds 10 or so whole numbers -- 10 here. That is a quick check on the count without redoing the work.

Standardsmin grade 6

6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Writing the possible values as an inequality and scaling it.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Counting the whole numbers the range contains.
5.NBT.A.4 Round decimals to any place — Reading what 'rounds to the nearest whole number' allows.

💡Takeaway. A rounded number does not tell you what it was. It tells you the smallest and largest it could have been.

Variant 9 hard

A number was divided by $14$ , and the quotient rounded to the nearest whole number came to $4$ . How many natural numbers could that number be?

Your answer

Show solution

1 · Understandwhat's really being asked

Some natural number divided by 14 rounds to 4. We want how many different numbers it could have been.

Givens

The number was divided by 14.
The quotient rounded to the nearest whole number is 4.

Unknowns

How many natural numbers fit that description.

Constraints

Rounding is to the nearest whole number, so a quotient of exactly 3.5 rounds up to 4.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #3 Eliminate Possibilities#8 Analyze the Units

Rounding loses information, so the original quotient cannot be recovered -- but it can be trapped between two bounds. Find those, multiply them back up, and count the whole numbers that survive.

3 · Execute4 carry out the plan

1Bound the quotient

#11 Work Backwards 5.NBT.A.4

Anything from 3.5 up to but not including 4.5 rounds to 4.

3.5 \le \text{quotient} < 4.5

Half a unit either side of the rounded value.

2Multiply both bounds by 14

#8 Analyze the Units 6.EE.B.8

The quotient came from dividing by 14, so multiplying undoes it -- and the bounds move with it.

49 \le \text{number} < 63

The number lives between 49 and 63.

3Find the first and last whole number inside

#3 Eliminate Possibilities 6.EE.B.8

The smallest natural number at least 49 is 49, and the largest below 63 is 62.

49 \le \text{number} \le 62

From 49 to 62.

4Count them

#3 Eliminate Possibilities 6.NS.B.3

Both ends are included, so subtract and add one.

62 - 49 + 1 = 14

14 numbers work.

Answer: 14

4 · Reviewdoes it hold up?

Checking the ends without dividing: 49 is at least 49 and below 63, while 48 falls short of 49 and 63 reaches 63 -- so the run really does stop where it stops.

Another way: The window is 14 wide, so it holds 14 or so whole numbers -- 14 here. That is a quick check on the count without redoing the work.

Standardsmin grade 6

6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Writing the possible values as an inequality and scaling it.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Counting the whole numbers the range contains.
5.NBT.A.4 Round decimals to any place — Reading what 'rounds to the nearest whole number' allows.

💡Takeaway. A rounded number does not tell you what it was. It tells you the smallest and largest it could have been.

Variant 10 hard

A number was divided by $8$ , and the quotient rounded to the nearest whole number came to $15$ . How many natural numbers could that number be?

Your answer

Show solution

1 · Understandwhat's really being asked

Some natural number divided by 8 rounds to 15. We want how many different numbers it could have been.

Givens

The number was divided by 8.
The quotient rounded to the nearest whole number is 15.

Unknowns

How many natural numbers fit that description.

Constraints

Rounding is to the nearest whole number, so a quotient of exactly 14.5 rounds up to 15.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #3 Eliminate Possibilities#8 Analyze the Units

Rounding loses information, so the original quotient cannot be recovered -- but it can be trapped between two bounds. Find those, multiply them back up, and count the whole numbers that survive.

3 · Execute4 carry out the plan

1Bound the quotient

#11 Work Backwards 5.NBT.A.4

Anything from 14.5 up to but not including 15.5 rounds to 15.

14.5 \le \text{quotient} < 15.5

Half a unit either side of the rounded value.

2Multiply both bounds by 8

#8 Analyze the Units 6.EE.B.8

The quotient came from dividing by 8, so multiplying undoes it -- and the bounds move with it.

116 \le \text{number} < 124

The number lives between 116 and 124.

3Find the first and last whole number inside

#3 Eliminate Possibilities 6.EE.B.8

The smallest natural number at least 116 is 116, and the largest below 124 is 123.

116 \le \text{number} \le 123

From 116 to 123.

4Count them

#3 Eliminate Possibilities 6.NS.B.3

Both ends are included, so subtract and add one.

123 - 116 + 1 = 8

8 numbers work.

Answer: 8

4 · Reviewdoes it hold up?

Checking the ends without dividing: 116 is at least 116 and below 124, while 115 falls short of 116 and 124 reaches 124 -- so the run really does stop where it stops.

Another way: The window is 8 wide, so it holds 8 or so whole numbers -- 8 here. That is a quick check on the count without redoing the work.

Standardsmin grade 6

6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Writing the possible values as an inequality and scaling it.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Counting the whole numbers the range contains.
5.NBT.A.4 Round decimals to any place — Reading what 'rounds to the nearest whole number' allows.

💡Takeaway. A rounded number does not tell you what it was. It tells you the smallest and largest it could have been.

Variant 11 hard

A number was divided by $16$ , and the quotient rounded to the nearest whole number came to $5$ . How many natural numbers could that number be?

Your answer

Show solution

1 · Understandwhat's really being asked

Some natural number divided by 16 rounds to 5. We want how many different numbers it could have been.

Givens

The number was divided by 16.
The quotient rounded to the nearest whole number is 5.

Unknowns

How many natural numbers fit that description.

Constraints

Rounding is to the nearest whole number, so a quotient of exactly 4.5 rounds up to 5.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #3 Eliminate Possibilities#8 Analyze the Units

Rounding loses information, so the original quotient cannot be recovered -- but it can be trapped between two bounds. Find those, multiply them back up, and count the whole numbers that survive.

3 · Execute4 carry out the plan

1Bound the quotient

#11 Work Backwards 5.NBT.A.4

Anything from 4.5 up to but not including 5.5 rounds to 5.

4.5 \le \text{quotient} < 5.5

Half a unit either side of the rounded value.

2Multiply both bounds by 16

#8 Analyze the Units 6.EE.B.8

The quotient came from dividing by 16, so multiplying undoes it -- and the bounds move with it.

72 \le \text{number} < 88

The number lives between 72 and 88.

3Find the first and last whole number inside

#3 Eliminate Possibilities 6.EE.B.8

The smallest natural number at least 72 is 72, and the largest below 88 is 87.

72 \le \text{number} \le 87

From 72 to 87.

4Count them

#3 Eliminate Possibilities 6.NS.B.3

Both ends are included, so subtract and add one.

87 - 72 + 1 = 16

16 numbers work.

Answer: 16

4 · Reviewdoes it hold up?

Checking the ends without dividing: 72 is at least 72 and below 88, while 71 falls short of 72 and 88 reaches 88 -- so the run really does stop where it stops.

Another way: The window is 16 wide, so it holds 16 or so whole numbers -- 16 here. That is a quick check on the count without redoing the work.

Standardsmin grade 6

6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Writing the possible values as an inequality and scaling it.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Counting the whole numbers the range contains.
5.NBT.A.4 Round decimals to any place — Reading what 'rounds to the nearest whole number' allows.

💡Takeaway. A rounded number does not tell you what it was. It tells you the smallest and largest it could have been.

Variant 12 hard

A number was divided by $3$ , and the quotient rounded to the nearest whole number came to $20$ . How many natural numbers could that number be?

Your answer

Show solution

1 · Understandwhat's really being asked

Some natural number divided by 3 rounds to 20. We want how many different numbers it could have been.

Givens

The number was divided by 3.
The quotient rounded to the nearest whole number is 20.

Unknowns

How many natural numbers fit that description.

Constraints

Rounding is to the nearest whole number, so a quotient of exactly 19.5 rounds up to 20.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #3 Eliminate Possibilities#8 Analyze the Units

Rounding loses information, so the original quotient cannot be recovered -- but it can be trapped between two bounds. Find those, multiply them back up, and count the whole numbers that survive.

3 · Execute4 carry out the plan

1Bound the quotient

#11 Work Backwards 5.NBT.A.4

Anything from 19.5 up to but not including 20.5 rounds to 20.

19.5 \le \text{quotient} < 20.5

Half a unit either side of the rounded value.

2Multiply both bounds by 3

#8 Analyze the Units 6.EE.B.8

The quotient came from dividing by 3, so multiplying undoes it -- and the bounds move with it.

58.5 \le \text{number} < 61.5

The number lives between 58.5 and 61.5.

3Find the first and last whole number inside

#3 Eliminate Possibilities 6.EE.B.8

The smallest natural number at least 58.5 is 59, and the largest below 61.5 is 61.

59 \le \text{number} \le 61

From 59 to 61.

4Count them

#3 Eliminate Possibilities 6.NS.B.3

Both ends are included, so subtract and add one.

61 - 59 + 1 = 3

3 numbers work.

Answer: 3

4 · Reviewdoes it hold up?

Checking the ends without dividing: 59 is at least 58.5 and below 61.5, while 58 falls short of 58.5 and 62 reaches 61.5 -- so the run really does stop where it stops.

Another way: The window is 3 wide, so it holds 3 or so whole numbers -- 3 here. That is a quick check on the count without redoing the work.

Standardsmin grade 6

6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Writing the possible values as an inequality and scaling it.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Counting the whole numbers the range contains.
5.NBT.A.4 Round decimals to any place — Reading what 'rounds to the nearest whole number' allows.

💡Takeaway. A rounded number does not tell you what it was. It tells you the smallest and largest it could have been.

Where next?

← Back to the problem: Rounding cannot be undone, but it can be bounded