← Overlap of two ranges on a number line · Inequality Range Membership

Overlap of two ranges on a number line · 12 practice problems

6.NS.C.66.NS.C.76.EE.B.8

Generated variants — 12

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

Variant 1 easy answer: 16

Two ranges of numbers are given.

  • numbers that are at least 1212 and at most 3939
  • numbers that are greater than 55 and less than 2828

How many whole numbers belong to both ranges?

Show solution
1 · Understandwhat's really being asked

One range runs from 12 up to 39, including both ends. The other runs between 5 and 28, including neither. We must count the whole numbers that are in both at once.

Givens
  • Range one: at least 12, at most 39.
  • Range two: greater than 5, less than 28.
  • Only whole numbers are counted.
Unknowns
  • How many whole numbers lie in both ranges.
Constraints
  • 'At least' and 'at most' include their endpoints; 'greater than' and 'less than' do not.
2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Draw both ranges on one number line, using a filled dot where the endpoint counts and a hollow one where it does not. The overlap is then visible, and its two edges can be settled one at a time.

3 · Execute4 carry out the plan

1Picture both ranges on a number line

#1 Draw a Diagram 6.NS.C.6
The first range is a solid bar from 12 to 39 with both ends filled. The second is a bar from 5 to 28 with both ends hollow, so 5 and 28 themselves are outside it.
12≤x≤39,5<x<2812 \le x \le 39,\quad 5 < x < 28
The overlap is where the two bars sit on top of each other.

2Find the left edge of the overlap

#7 Identify Subproblems 6.NS.C.7
To be in both, a number must clear the later of the two starts. The second range excludes 5 itself, so its first whole number is 6; the first range starts at 12. The first range decides.
max⁡(12, 5+1)=12\max(12,\ 5 + 1) = 12
The overlap begins at 12.

3Find the right edge of the overlap

#7 Identify Subproblems 6.NS.C.7
Likewise it must stop at the earlier of the two ends. The second range excludes 28, so its last whole number is 27; the first ends at 39. The second range decides.
min⁡(39, 28−1)=27\min(39,\ 28 - 1) = 27
The overlap ends at 27.

4Count the whole numbers from 12 to 27

#7 Identify Subproblems 6.EE.B.8
Counting inclusive means subtracting and adding one -- 12 itself has to be counted.
27−12+1=1627 - 12 + 1 = 16
There are 16 whole numbers in both ranges.
Answer: 16
4 · Reviewdoes it hold up?

Check the two edges belong: 12 and 27 both satisfy 12 <= x <= 39 and 5 < x < 28, while 11 and 28 each fail one of them.

Another way: Listing 12, 13, ..., 27 and counting gives the same 16; the subtraction is the same count done at once, and the '+1' is what keeps the first number from being dropped.

Standardsmin grade 6
  • 6.NS.C.6 Understand a rational number as a point on the number line — Placing both ranges on one number line to see the overlap.
  • 6.NS.C.7 Understand ordering and absolute value of rational numbers — Deciding which range's endpoint governs each edge of the overlap.
  • 6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Reading each range as an inequality and counting 12 through 27.
💡Takeaway. 'At least' keeps its endpoint and 'greater than' does not -- so read the words before you trust the numbers.
Variant 2 easy answer: 22

Two ranges of numbers are given.

  • numbers that are at least 2525 and at most 5858
  • numbers that are greater than 1919 and less than 4747

How many whole numbers belong to both ranges?

Show solution
1 · Understandwhat's really being asked

One range runs from 25 up to 58, including both ends. The other runs between 19 and 47, including neither. We must count the whole numbers that are in both at once.

Givens
  • Range one: at least 25, at most 58.
  • Range two: greater than 19, less than 47.
  • Only whole numbers are counted.
Unknowns
  • How many whole numbers lie in both ranges.
Constraints
  • 'At least' and 'at most' include their endpoints; 'greater than' and 'less than' do not.
2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Draw both ranges on one number line, using a filled dot where the endpoint counts and a hollow one where it does not. The overlap is then visible, and its two edges can be settled one at a time.

3 · Execute4 carry out the plan

1Picture both ranges on a number line

#1 Draw a Diagram 6.NS.C.6
The first range is a solid bar from 25 to 58 with both ends filled. The second is a bar from 19 to 47 with both ends hollow, so 19 and 47 themselves are outside it.
25≤x≤58,19<x<4725 \le x \le 58,\quad 19 < x < 47
The overlap is where the two bars sit on top of each other.

2Find the left edge of the overlap

#7 Identify Subproblems 6.NS.C.7
To be in both, a number must clear the later of the two starts. The second range excludes 19 itself, so its first whole number is 20; the first range starts at 25. The first range decides.
max⁡(25, 19+1)=25\max(25,\ 19 + 1) = 25
The overlap begins at 25.

3Find the right edge of the overlap

#7 Identify Subproblems 6.NS.C.7
Likewise it must stop at the earlier of the two ends. The second range excludes 47, so its last whole number is 46; the first ends at 58. The second range decides.
min⁡(58, 47−1)=46\min(58,\ 47 - 1) = 46
The overlap ends at 46.

4Count the whole numbers from 25 to 46

#7 Identify Subproblems 6.EE.B.8
Counting inclusive means subtracting and adding one -- 25 itself has to be counted.
46−25+1=2246 - 25 + 1 = 22
There are 22 whole numbers in both ranges.
Answer: 22
4 · Reviewdoes it hold up?

Check the two edges belong: 25 and 46 both satisfy 25 <= x <= 58 and 19 < x < 47, while 24 and 47 each fail one of them.

Another way: Listing 25, 26, ..., 46 and counting gives the same 22; the subtraction is the same count done at once, and the '+1' is what keeps the first number from being dropped.

Standardsmin grade 6
  • 6.NS.C.6 Understand a rational number as a point on the number line — Placing both ranges on one number line to see the overlap.
  • 6.NS.C.7 Understand ordering and absolute value of rational numbers — Deciding which range's endpoint governs each edge of the overlap.
  • 6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Reading each range as an inequality and counting 25 through 46.
💡Takeaway. 'At least' keeps its endpoint and 'greater than' does not -- so read the words before you trust the numbers.
Variant 3 easy answer: 22

Two ranges of numbers are given.

  • numbers that are at least 1515 and at most 4444
  • numbers that are greater than 2222 and less than 6060

How many whole numbers belong to both ranges?

Show solution
1 · Understandwhat's really being asked

One range runs from 15 up to 44, including both ends. The other runs between 22 and 60, including neither. We must count the whole numbers that are in both at once.

Givens
  • Range one: at least 15, at most 44.
  • Range two: greater than 22, less than 60.
  • Only whole numbers are counted.
Unknowns
  • How many whole numbers lie in both ranges.
Constraints
  • 'At least' and 'at most' include their endpoints; 'greater than' and 'less than' do not.
2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Draw both ranges on one number line, using a filled dot where the endpoint counts and a hollow one where it does not. The overlap is then visible, and its two edges can be settled one at a time.

3 · Execute4 carry out the plan

1Picture both ranges on a number line

#1 Draw a Diagram 6.NS.C.6
The first range is a solid bar from 15 to 44 with both ends filled. The second is a bar from 22 to 60 with both ends hollow, so 22 and 60 themselves are outside it.
15≤x≤44,22<x<6015 \le x \le 44,\quad 22 < x < 60
The overlap is where the two bars sit on top of each other.

2Find the left edge of the overlap

#7 Identify Subproblems 6.NS.C.7
To be in both, a number must clear the later of the two starts. The second range excludes 22 itself, so its first whole number is 23; the first range starts at 15. The second range decides.
max⁡(15, 22+1)=23\max(15,\ 22 + 1) = 23
The overlap begins at 23.

3Find the right edge of the overlap

#7 Identify Subproblems 6.NS.C.7
Likewise it must stop at the earlier of the two ends. The second range excludes 60, so its last whole number is 59; the first ends at 44. The first range decides.
min⁡(44, 60−1)=44\min(44,\ 60 - 1) = 44
The overlap ends at 44.

4Count the whole numbers from 23 to 44

#7 Identify Subproblems 6.EE.B.8
Counting inclusive means subtracting and adding one -- 23 itself has to be counted.
44−23+1=2244 - 23 + 1 = 22
There are 22 whole numbers in both ranges.
Answer: 22
4 · Reviewdoes it hold up?

Check the two edges belong: 23 and 44 both satisfy 15 <= x <= 44 and 22 < x < 60, while 22 and 45 each fail one of them.

Another way: Listing 23, 24, ..., 44 and counting gives the same 22; the subtraction is the same count done at once, and the '+1' is what keeps the first number from being dropped.

Standardsmin grade 6
  • 6.NS.C.6 Understand a rational number as a point on the number line — Placing both ranges on one number line to see the overlap.
  • 6.NS.C.7 Understand ordering and absolute value of rational numbers — Deciding which range's endpoint governs each edge of the overlap.
  • 6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Reading each range as an inequality and counting 23 through 44.
💡Takeaway. 'At least' keeps its endpoint and 'greater than' does not -- so read the words before you trust the numbers.
Variant 4 easy answer: 24

Two ranges of numbers are given.

  • numbers that are at least 3030 and at most 6565
  • numbers that are greater than 4141 and less than 8080

How many whole numbers belong to both ranges?

Show solution
1 · Understandwhat's really being asked

One range runs from 30 up to 65, including both ends. The other runs between 41 and 80, including neither. We must count the whole numbers that are in both at once.

Givens
  • Range one: at least 30, at most 65.
  • Range two: greater than 41, less than 80.
  • Only whole numbers are counted.
Unknowns
  • How many whole numbers lie in both ranges.
Constraints
  • 'At least' and 'at most' include their endpoints; 'greater than' and 'less than' do not.
2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Draw both ranges on one number line, using a filled dot where the endpoint counts and a hollow one where it does not. The overlap is then visible, and its two edges can be settled one at a time.

3 · Execute4 carry out the plan

1Picture both ranges on a number line

#1 Draw a Diagram 6.NS.C.6
The first range is a solid bar from 30 to 65 with both ends filled. The second is a bar from 41 to 80 with both ends hollow, so 41 and 80 themselves are outside it.
30≤x≤65,41<x<8030 \le x \le 65,\quad 41 < x < 80
The overlap is where the two bars sit on top of each other.

2Find the left edge of the overlap

#7 Identify Subproblems 6.NS.C.7
To be in both, a number must clear the later of the two starts. The second range excludes 41 itself, so its first whole number is 42; the first range starts at 30. The second range decides.
max⁡(30, 41+1)=42\max(30,\ 41 + 1) = 42
The overlap begins at 42.

3Find the right edge of the overlap

#7 Identify Subproblems 6.NS.C.7
Likewise it must stop at the earlier of the two ends. The second range excludes 80, so its last whole number is 79; the first ends at 65. The first range decides.
min⁡(65, 80−1)=65\min(65,\ 80 - 1) = 65
The overlap ends at 65.

4Count the whole numbers from 42 to 65

#7 Identify Subproblems 6.EE.B.8
Counting inclusive means subtracting and adding one -- 42 itself has to be counted.
65−42+1=2465 - 42 + 1 = 24
There are 24 whole numbers in both ranges.
Answer: 24
4 · Reviewdoes it hold up?

Check the two edges belong: 42 and 65 both satisfy 30 <= x <= 65 and 41 < x < 80, while 41 and 66 each fail one of them.

Another way: Listing 42, 43, ..., 65 and counting gives the same 24; the subtraction is the same count done at once, and the '+1' is what keeps the first number from being dropped.

Standardsmin grade 6
  • 6.NS.C.6 Understand a rational number as a point on the number line — Placing both ranges on one number line to see the overlap.
  • 6.NS.C.7 Understand ordering and absolute value of rational numbers — Deciding which range's endpoint governs each edge of the overlap.
  • 6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Reading each range as an inequality and counting 42 through 65.
💡Takeaway. 'At least' keeps its endpoint and 'greater than' does not -- so read the words before you trust the numbers.
Variant 5 medium answer: 16

Two ranges of numbers are given.

  • numbers that are at least 4848 and at most 7272
  • numbers that are greater than 5656 and less than 9393

How many whole numbers belong to both ranges?

Show solution
1 · Understandwhat's really being asked

One range runs from 48 up to 72, including both ends. The other runs between 56 and 93, including neither. We must count the whole numbers that are in both at once.

Givens
  • Range one: at least 48, at most 72.
  • Range two: greater than 56, less than 93.
  • Only whole numbers are counted.
Unknowns
  • How many whole numbers lie in both ranges.
Constraints
  • 'At least' and 'at most' include their endpoints; 'greater than' and 'less than' do not.
2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Draw both ranges on one number line, using a filled dot where the endpoint counts and a hollow one where it does not. The overlap is then visible, and its two edges can be settled one at a time.

3 · Execute4 carry out the plan

1Picture both ranges on a number line

#1 Draw a Diagram 6.NS.C.6
The first range is a solid bar from 48 to 72 with both ends filled. The second is a bar from 56 to 93 with both ends hollow, so 56 and 93 themselves are outside it.
48≤x≤72,56<x<9348 \le x \le 72,\quad 56 < x < 93
The overlap is where the two bars sit on top of each other.

2Find the left edge of the overlap

#7 Identify Subproblems 6.NS.C.7
To be in both, a number must clear the later of the two starts. The second range excludes 56 itself, so its first whole number is 57; the first range starts at 48. The second range decides.
max⁡(48, 56+1)=57\max(48,\ 56 + 1) = 57
The overlap begins at 57.

3Find the right edge of the overlap

#7 Identify Subproblems 6.NS.C.7
Likewise it must stop at the earlier of the two ends. The second range excludes 93, so its last whole number is 92; the first ends at 72. The first range decides.
min⁡(72, 93−1)=72\min(72,\ 93 - 1) = 72
The overlap ends at 72.

4Count the whole numbers from 57 to 72

#7 Identify Subproblems 6.EE.B.8
Counting inclusive means subtracting and adding one -- 57 itself has to be counted.
72−57+1=1672 - 57 + 1 = 16
There are 16 whole numbers in both ranges.
Answer: 16
4 · Reviewdoes it hold up?

Check the two edges belong: 57 and 72 both satisfy 48 <= x <= 72 and 56 < x < 93, while 56 and 73 each fail one of them.

Another way: Listing 57, 58, ..., 72 and counting gives the same 16; the subtraction is the same count done at once, and the '+1' is what keeps the first number from being dropped.

Standardsmin grade 6
  • 6.NS.C.6 Understand a rational number as a point on the number line — Placing both ranges on one number line to see the overlap.
  • 6.NS.C.7 Understand ordering and absolute value of rational numbers — Deciding which range's endpoint governs each edge of the overlap.
  • 6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Reading each range as an inequality and counting 57 through 72.
💡Takeaway. 'At least' keeps its endpoint and 'greater than' does not -- so read the words before you trust the numbers.
Variant 6 medium answer: 27

Two ranges of numbers are given.

  • numbers that are at least 6060 and at most 9595
  • numbers that are greater than 6060 and less than 8888

How many whole numbers belong to both ranges?

Show solution
1 · Understandwhat's really being asked

One range runs from 60 up to 95, including both ends. The other runs between 60 and 88, including neither. We must count the whole numbers that are in both at once.

Givens
  • Range one: at least 60, at most 95.
  • Range two: greater than 60, less than 88.
  • Only whole numbers are counted.
Unknowns
  • How many whole numbers lie in both ranges.
Constraints
  • 'At least' and 'at most' include their endpoints; 'greater than' and 'less than' do not.
2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Draw both ranges on one number line, using a filled dot where the endpoint counts and a hollow one where it does not. The overlap is then visible, and its two edges can be settled one at a time.

3 · Execute4 carry out the plan

1Picture both ranges on a number line

#1 Draw a Diagram 6.NS.C.6
The first range is a solid bar from 60 to 95 with both ends filled. The second is a bar from 60 to 88 with both ends hollow, so 60 and 88 themselves are outside it.
60≤x≤95,60<x<8860 \le x \le 95,\quad 60 < x < 88
The overlap is where the two bars sit on top of each other.

2Find the left edge of the overlap

#7 Identify Subproblems 6.NS.C.7
To be in both, a number must clear the later of the two starts. The second range excludes 60 itself, so its first whole number is 61; the first range starts at 60. The second range decides.
max⁡(60, 60+1)=61\max(60,\ 60 + 1) = 61
The overlap begins at 61.

3Find the right edge of the overlap

#7 Identify Subproblems 6.NS.C.7
Likewise it must stop at the earlier of the two ends. The second range excludes 88, so its last whole number is 87; the first ends at 95. The second range decides.
min⁡(95, 88−1)=87\min(95,\ 88 - 1) = 87
The overlap ends at 87.

4Count the whole numbers from 61 to 87

#7 Identify Subproblems 6.EE.B.8
Counting inclusive means subtracting and adding one -- 61 itself has to be counted.
87−61+1=2787 - 61 + 1 = 27
There are 27 whole numbers in both ranges.
Answer: 27
4 · Reviewdoes it hold up?

Check the two edges belong: 61 and 87 both satisfy 60 <= x <= 95 and 60 < x < 88, while 60 and 88 each fail one of them.

Another way: Listing 61, 62, ..., 87 and counting gives the same 27; the subtraction is the same count done at once, and the '+1' is what keeps the first number from being dropped.

Standardsmin grade 6
  • 6.NS.C.6 Understand a rational number as a point on the number line — Placing both ranges on one number line to see the overlap.
  • 6.NS.C.7 Understand ordering and absolute value of rational numbers — Deciding which range's endpoint governs each edge of the overlap.
  • 6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Reading each range as an inequality and counting 61 through 87.
💡Takeaway. 'At least' keeps its endpoint and 'greater than' does not -- so read the words before you trust the numbers.
Variant 7 medium answer: 11

Two ranges of numbers are given.

  • numbers that are at least 7070 and at most 110110
  • numbers that are greater than 8484 and less than 9696

How many whole numbers belong to both ranges?

Show solution
1 · Understandwhat's really being asked

One range runs from 70 up to 110, including both ends. The other runs between 84 and 96, including neither. We must count the whole numbers that are in both at once.

Givens
  • Range one: at least 70, at most 110.
  • Range two: greater than 84, less than 96.
  • Only whole numbers are counted.
Unknowns
  • How many whole numbers lie in both ranges.
Constraints
  • 'At least' and 'at most' include their endpoints; 'greater than' and 'less than' do not.
2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Draw both ranges on one number line, using a filled dot where the endpoint counts and a hollow one where it does not. The overlap is then visible, and its two edges can be settled one at a time.

3 · Execute4 carry out the plan

1Picture both ranges on a number line

#1 Draw a Diagram 6.NS.C.6
The first range is a solid bar from 70 to 110 with both ends filled. The second is a bar from 84 to 96 with both ends hollow, so 84 and 96 themselves are outside it.
70≤x≤110,84<x<9670 \le x \le 110,\quad 84 < x < 96
The overlap is where the two bars sit on top of each other.

2Find the left edge of the overlap

#7 Identify Subproblems 6.NS.C.7
To be in both, a number must clear the later of the two starts. The second range excludes 84 itself, so its first whole number is 85; the first range starts at 70. The second range decides.
max⁡(70, 84+1)=85\max(70,\ 84 + 1) = 85
The overlap begins at 85.

3Find the right edge of the overlap

#7 Identify Subproblems 6.NS.C.7
Likewise it must stop at the earlier of the two ends. The second range excludes 96, so its last whole number is 95; the first ends at 110. The second range decides.
min⁡(110, 96−1)=95\min(110,\ 96 - 1) = 95
The overlap ends at 95.

4Count the whole numbers from 85 to 95

#7 Identify Subproblems 6.EE.B.8
Counting inclusive means subtracting and adding one -- 85 itself has to be counted.
95−85+1=1195 - 85 + 1 = 11
There are 11 whole numbers in both ranges.
Answer: 11
4 · Reviewdoes it hold up?

Check the two edges belong: 85 and 95 both satisfy 70 <= x <= 110 and 84 < x < 96, while 84 and 96 each fail one of them.

Another way: Listing 85, 86, ..., 95 and counting gives the same 11; the subtraction is the same count done at once, and the '+1' is what keeps the first number from being dropped.

Standardsmin grade 6
  • 6.NS.C.6 Understand a rational number as a point on the number line — Placing both ranges on one number line to see the overlap.
  • 6.NS.C.7 Understand ordering and absolute value of rational numbers — Deciding which range's endpoint governs each edge of the overlap.
  • 6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Reading each range as an inequality and counting 85 through 95.
💡Takeaway. 'At least' keeps its endpoint and 'greater than' does not -- so read the words before you trust the numbers.
Variant 8 medium answer: 27

Two ranges of numbers are given.

  • numbers that are at least 3333 and at most 7777
  • numbers that are greater than 5050 and less than 120120

How many whole numbers belong to both ranges?

Show solution
1 · Understandwhat's really being asked

One range runs from 33 up to 77, including both ends. The other runs between 50 and 120, including neither. We must count the whole numbers that are in both at once.

Givens
  • Range one: at least 33, at most 77.
  • Range two: greater than 50, less than 120.
  • Only whole numbers are counted.
Unknowns
  • How many whole numbers lie in both ranges.
Constraints
  • 'At least' and 'at most' include their endpoints; 'greater than' and 'less than' do not.
2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Draw both ranges on one number line, using a filled dot where the endpoint counts and a hollow one where it does not. The overlap is then visible, and its two edges can be settled one at a time.

3 · Execute4 carry out the plan

1Picture both ranges on a number line

#1 Draw a Diagram 6.NS.C.6
The first range is a solid bar from 33 to 77 with both ends filled. The second is a bar from 50 to 120 with both ends hollow, so 50 and 120 themselves are outside it.
33≤x≤77,50<x<12033 \le x \le 77,\quad 50 < x < 120
The overlap is where the two bars sit on top of each other.

2Find the left edge of the overlap

#7 Identify Subproblems 6.NS.C.7
To be in both, a number must clear the later of the two starts. The second range excludes 50 itself, so its first whole number is 51; the first range starts at 33. The second range decides.
max⁡(33, 50+1)=51\max(33,\ 50 + 1) = 51
The overlap begins at 51.

3Find the right edge of the overlap

#7 Identify Subproblems 6.NS.C.7
Likewise it must stop at the earlier of the two ends. The second range excludes 120, so its last whole number is 119; the first ends at 77. The first range decides.
min⁡(77, 120−1)=77\min(77,\ 120 - 1) = 77
The overlap ends at 77.

4Count the whole numbers from 51 to 77

#7 Identify Subproblems 6.EE.B.8
Counting inclusive means subtracting and adding one -- 51 itself has to be counted.
77−51+1=2777 - 51 + 1 = 27
There are 27 whole numbers in both ranges.
Answer: 27
4 · Reviewdoes it hold up?

Check the two edges belong: 51 and 77 both satisfy 33 <= x <= 77 and 50 < x < 120, while 50 and 78 each fail one of them.

Another way: Listing 51, 52, ..., 77 and counting gives the same 27; the subtraction is the same count done at once, and the '+1' is what keeps the first number from being dropped.

Standardsmin grade 6
  • 6.NS.C.6 Understand a rational number as a point on the number line — Placing both ranges on one number line to see the overlap.
  • 6.NS.C.7 Understand ordering and absolute value of rational numbers — Deciding which range's endpoint governs each edge of the overlap.
  • 6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Reading each range as an inequality and counting 51 through 77.
💡Takeaway. 'At least' keeps its endpoint and 'greater than' does not -- so read the words before you trust the numbers.
Variant 9 hard answer: 27

Two ranges of numbers are given.

  • numbers that are at least 8181 and at most 129129
  • numbers that are greater than 9090 and less than 118118

How many whole numbers belong to both ranges?

Show solution
1 · Understandwhat's really being asked

One range runs from 81 up to 129, including both ends. The other runs between 90 and 118, including neither. We must count the whole numbers that are in both at once.

Givens
  • Range one: at least 81, at most 129.
  • Range two: greater than 90, less than 118.
  • Only whole numbers are counted.
Unknowns
  • How many whole numbers lie in both ranges.
Constraints
  • 'At least' and 'at most' include their endpoints; 'greater than' and 'less than' do not.
2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Draw both ranges on one number line, using a filled dot where the endpoint counts and a hollow one where it does not. The overlap is then visible, and its two edges can be settled one at a time.

3 · Execute4 carry out the plan

1Picture both ranges on a number line

#1 Draw a Diagram 6.NS.C.6
The first range is a solid bar from 81 to 129 with both ends filled. The second is a bar from 90 to 118 with both ends hollow, so 90 and 118 themselves are outside it.
81≤x≤129,90<x<11881 \le x \le 129,\quad 90 < x < 118
The overlap is where the two bars sit on top of each other.

2Find the left edge of the overlap

#7 Identify Subproblems 6.NS.C.7
To be in both, a number must clear the later of the two starts. The second range excludes 90 itself, so its first whole number is 91; the first range starts at 81. The second range decides.
max⁡(81, 90+1)=91\max(81,\ 90 + 1) = 91
The overlap begins at 91.

3Find the right edge of the overlap

#7 Identify Subproblems 6.NS.C.7
Likewise it must stop at the earlier of the two ends. The second range excludes 118, so its last whole number is 117; the first ends at 129. The second range decides.
min⁡(129, 118−1)=117\min(129,\ 118 - 1) = 117
The overlap ends at 117.

4Count the whole numbers from 91 to 117

#7 Identify Subproblems 6.EE.B.8
Counting inclusive means subtracting and adding one -- 91 itself has to be counted.
117−91+1=27117 - 91 + 1 = 27
There are 27 whole numbers in both ranges.
Answer: 27
4 · Reviewdoes it hold up?

Check the two edges belong: 91 and 117 both satisfy 81 <= x <= 129 and 90 < x < 118, while 90 and 118 each fail one of them.

Another way: Listing 91, 92, ..., 117 and counting gives the same 27; the subtraction is the same count done at once, and the '+1' is what keeps the first number from being dropped.

Standardsmin grade 6
  • 6.NS.C.6 Understand a rational number as a point on the number line — Placing both ranges on one number line to see the overlap.
  • 6.NS.C.7 Understand ordering and absolute value of rational numbers — Deciding which range's endpoint governs each edge of the overlap.
  • 6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Reading each range as an inequality and counting 91 through 117.
💡Takeaway. 'At least' keeps its endpoint and 'greater than' does not -- so read the words before you trust the numbers.
Variant 10 hard answer: 28

Two ranges of numbers are given.

  • numbers that are at least 100100 and at most 140140
  • numbers that are greater than 112112 and less than 175175

How many whole numbers belong to both ranges?

Show solution
1 · Understandwhat's really being asked

One range runs from 100 up to 140, including both ends. The other runs between 112 and 175, including neither. We must count the whole numbers that are in both at once.

Givens
  • Range one: at least 100, at most 140.
  • Range two: greater than 112, less than 175.
  • Only whole numbers are counted.
Unknowns
  • How many whole numbers lie in both ranges.
Constraints
  • 'At least' and 'at most' include their endpoints; 'greater than' and 'less than' do not.
2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Draw both ranges on one number line, using a filled dot where the endpoint counts and a hollow one where it does not. The overlap is then visible, and its two edges can be settled one at a time.

3 · Execute4 carry out the plan

1Picture both ranges on a number line

#1 Draw a Diagram 6.NS.C.6
The first range is a solid bar from 100 to 140 with both ends filled. The second is a bar from 112 to 175 with both ends hollow, so 112 and 175 themselves are outside it.
100≤x≤140,112<x<175100 \le x \le 140,\quad 112 < x < 175
The overlap is where the two bars sit on top of each other.

2Find the left edge of the overlap

#7 Identify Subproblems 6.NS.C.7
To be in both, a number must clear the later of the two starts. The second range excludes 112 itself, so its first whole number is 113; the first range starts at 100. The second range decides.
max⁡(100, 112+1)=113\max(100,\ 112 + 1) = 113
The overlap begins at 113.

3Find the right edge of the overlap

#7 Identify Subproblems 6.NS.C.7
Likewise it must stop at the earlier of the two ends. The second range excludes 175, so its last whole number is 174; the first ends at 140. The first range decides.
min⁡(140, 175−1)=140\min(140,\ 175 - 1) = 140
The overlap ends at 140.

4Count the whole numbers from 113 to 140

#7 Identify Subproblems 6.EE.B.8
Counting inclusive means subtracting and adding one -- 113 itself has to be counted.
140−113+1=28140 - 113 + 1 = 28
There are 28 whole numbers in both ranges.
Answer: 28
4 · Reviewdoes it hold up?

Check the two edges belong: 113 and 140 both satisfy 100 <= x <= 140 and 112 < x < 175, while 112 and 141 each fail one of them.

Another way: Listing 113, 114, ..., 140 and counting gives the same 28; the subtraction is the same count done at once, and the '+1' is what keeps the first number from being dropped.

Standardsmin grade 6
  • 6.NS.C.6 Understand a rational number as a point on the number line — Placing both ranges on one number line to see the overlap.
  • 6.NS.C.7 Understand ordering and absolute value of rational numbers — Deciding which range's endpoint governs each edge of the overlap.
  • 6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Reading each range as an inequality and counting 113 through 140.
💡Takeaway. 'At least' keeps its endpoint and 'greater than' does not -- so read the words before you trust the numbers.
Variant 11 hard answer: 27

Two ranges of numbers are given.

  • numbers that are at least 150150 and at most 190190
  • numbers that are greater than 163163 and less than 205205

How many whole numbers belong to both ranges?

Show solution
1 · Understandwhat's really being asked

One range runs from 150 up to 190, including both ends. The other runs between 163 and 205, including neither. We must count the whole numbers that are in both at once.

Givens
  • Range one: at least 150, at most 190.
  • Range two: greater than 163, less than 205.
  • Only whole numbers are counted.
Unknowns
  • How many whole numbers lie in both ranges.
Constraints
  • 'At least' and 'at most' include their endpoints; 'greater than' and 'less than' do not.
2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Draw both ranges on one number line, using a filled dot where the endpoint counts and a hollow one where it does not. The overlap is then visible, and its two edges can be settled one at a time.

3 · Execute4 carry out the plan

1Picture both ranges on a number line

#1 Draw a Diagram 6.NS.C.6
The first range is a solid bar from 150 to 190 with both ends filled. The second is a bar from 163 to 205 with both ends hollow, so 163 and 205 themselves are outside it.
150≤x≤190,163<x<205150 \le x \le 190,\quad 163 < x < 205
The overlap is where the two bars sit on top of each other.

2Find the left edge of the overlap

#7 Identify Subproblems 6.NS.C.7
To be in both, a number must clear the later of the two starts. The second range excludes 163 itself, so its first whole number is 164; the first range starts at 150. The second range decides.
max⁡(150, 163+1)=164\max(150,\ 163 + 1) = 164
The overlap begins at 164.

3Find the right edge of the overlap

#7 Identify Subproblems 6.NS.C.7
Likewise it must stop at the earlier of the two ends. The second range excludes 205, so its last whole number is 204; the first ends at 190. The first range decides.
min⁡(190, 205−1)=190\min(190,\ 205 - 1) = 190
The overlap ends at 190.

4Count the whole numbers from 164 to 190

#7 Identify Subproblems 6.EE.B.8
Counting inclusive means subtracting and adding one -- 164 itself has to be counted.
190−164+1=27190 - 164 + 1 = 27
There are 27 whole numbers in both ranges.
Answer: 27
4 · Reviewdoes it hold up?

Check the two edges belong: 164 and 190 both satisfy 150 <= x <= 190 and 163 < x < 205, while 163 and 191 each fail one of them.

Another way: Listing 164, 165, ..., 190 and counting gives the same 27; the subtraction is the same count done at once, and the '+1' is what keeps the first number from being dropped.

Standardsmin grade 6
  • 6.NS.C.6 Understand a rational number as a point on the number line — Placing both ranges on one number line to see the overlap.
  • 6.NS.C.7 Understand ordering and absolute value of rational numbers — Deciding which range's endpoint governs each edge of the overlap.
  • 6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Reading each range as an inequality and counting 164 through 190.
💡Takeaway. 'At least' keeps its endpoint and 'greater than' does not -- so read the words before you trust the numbers.
Variant 12 hard answer: 30

Two ranges of numbers are given.

  • numbers that are at least 200200 and at most 245245
  • numbers that are greater than 188188 and less than 230230

How many whole numbers belong to both ranges?

Show solution
1 · Understandwhat's really being asked

One range runs from 200 up to 245, including both ends. The other runs between 188 and 230, including neither. We must count the whole numbers that are in both at once.

Givens
  • Range one: at least 200, at most 245.
  • Range two: greater than 188, less than 230.
  • Only whole numbers are counted.
Unknowns
  • How many whole numbers lie in both ranges.
Constraints
  • 'At least' and 'at most' include their endpoints; 'greater than' and 'less than' do not.
2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Draw both ranges on one number line, using a filled dot where the endpoint counts and a hollow one where it does not. The overlap is then visible, and its two edges can be settled one at a time.

3 · Execute4 carry out the plan

1Picture both ranges on a number line

#1 Draw a Diagram 6.NS.C.6
The first range is a solid bar from 200 to 245 with both ends filled. The second is a bar from 188 to 230 with both ends hollow, so 188 and 230 themselves are outside it.
200≤x≤245,188<x<230200 \le x \le 245,\quad 188 < x < 230
The overlap is where the two bars sit on top of each other.

2Find the left edge of the overlap

#7 Identify Subproblems 6.NS.C.7
To be in both, a number must clear the later of the two starts. The second range excludes 188 itself, so its first whole number is 189; the first range starts at 200. The first range decides.
max⁡(200, 188+1)=200\max(200,\ 188 + 1) = 200
The overlap begins at 200.

3Find the right edge of the overlap

#7 Identify Subproblems 6.NS.C.7
Likewise it must stop at the earlier of the two ends. The second range excludes 230, so its last whole number is 229; the first ends at 245. The second range decides.
min⁡(245, 230−1)=229\min(245,\ 230 - 1) = 229
The overlap ends at 229.

4Count the whole numbers from 200 to 229

#7 Identify Subproblems 6.EE.B.8
Counting inclusive means subtracting and adding one -- 200 itself has to be counted.
229−200+1=30229 - 200 + 1 = 30
There are 30 whole numbers in both ranges.
Answer: 30
4 · Reviewdoes it hold up?

Check the two edges belong: 200 and 229 both satisfy 200 <= x <= 245 and 188 < x < 230, while 199 and 230 each fail one of them.

Another way: Listing 200, 201, ..., 229 and counting gives the same 30; the subtraction is the same count done at once, and the '+1' is what keeps the first number from being dropped.

Standardsmin grade 6
  • 6.NS.C.6 Understand a rational number as a point on the number line — Placing both ranges on one number line to see the overlap.
  • 6.NS.C.7 Understand ordering and absolute value of rational numbers — Deciding which range's endpoint governs each edge of the overlap.
  • 6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Reading each range as an inequality and counting 200 through 229.
💡Takeaway. 'At least' keeps its endpoint and 'greater than' does not -- so read the words before you trust the numbers.