← Compare the next digit to fill a blank in decimals · Compare Fractions and Decimals by Structure

Compare the next digit to fill a blank in decimals · 9 practice problems

4.NF.C.7

Generated variants — 9

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

Variant 1 medium answer: C, B, A

Each □\square can be any digit from 0 to 9. List the labels in order from least to greatest.

(A) 89.□98(B) 8□.096(C) 80.0□2\text{(A)}\ 89.\square 98 \qquad \text{(B)}\ 8\square .096 \qquad \text{(C)}\ 80.0\square 2

Show solution
1 · Understandwhat's really being asked

Each □\square can hold any digit 0-9. The labelled numbers are (A) = 89.□\square 98 (the □\square is a tenths digit); (B) = 8□\square .096 (the □\square is a ones digit); (C) = 80.0□\square 2 (the □\square is a hundredths digit). No matter what digits fill the boxes, order the labels from least to greatest.

Givens
  • (A) = 89.□\square 98 (the □\square is a tenths digit)
  • (B) = 8□\square .096 (the □\square is a ones digit)
  • (C) = 80.0□\square 2 (the □\square is a hundredths digit)
  • Each □\square can be any digit from 0 to 9.
Unknowns
  • The order of labels A, B, C from least to greatest.
Constraints
  • The ordering must hold for every allowed choice of the □\square digits.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Each number can only range over a small interval as its □\square runs 0-9. If those intervals do not overlap, the order is fixed for all choices. So we find the smallest-and-largest possible value of each label and compare the ranges.

3 · Execute3 carry out the plan

1Find each label's possible range

#2 Make a Systematic List 5.NBT.A.3
Let each □\square run from 0 to 9. (C) 80.0#2 ranges from 80.002 to 80.092. (B) 8#.096 ranges from 80.096 to 89.096. (A) 89.#98 ranges from 89.098 to 89.998.
C∈[80.002,80.092], B∈[80.096,89.096], A∈[89.098,89.998]C\in[80.002,80.092],\ B\in[80.096,89.096],\ A\in[89.098,89.998]
Plugging in the smallest and largest digit shows the whole span each number can cover.

2Compare C with B

#6 Guess and Check 4.NF.C.7
C's largest value is 80.092, and B's smallest value is 80.096. Since 80.092 < 80.096, every C is less than every B - the ranges do not overlap.
80.092<80.096⇒C<B80.092 < 80.096 \Rightarrow C < B
If C can never reach as high as B's lowest, C is always smaller.

3Compare B with A

#6 Guess and Check 4.NF.C.7
B's largest value is 89.096, and A's smallest value is 89.098. Since 89.096 < 89.098, every B is less than every A - the ranges do not overlap.
89.096<89.098⇒B<A89.096 < 89.098 \Rightarrow B < A
If B can never reach as high as A's lowest, B is always smaller.
Answer: C, B, A
4 · Reviewdoes it hold up?

The ranges [80.002, 80.092], [80.096, 89.096], [89.098, 89.998] are disjoint and line up in the order C, then B, then A, so the ordering holds for every digit choice.

Another way: Guess and Check the extreme cases: even when each □\square takes the digit that pushes it hardest toward its neighbour, the ranges still keep the order C, B, A.

Standardsmin grade 5
  • 4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Comparing the labels using place value.
  • 5.NBT.A.3 Read, write, and compare decimals to thousandths — Reading the thousandths-place numbers and finding each label's range.
💡Takeaway. Find the lowest and highest each number can be - if the ranges don't overlap, the order is locked in!
Variant 2 hard answer: C, B, A

Each □\square can be any digit from 0 to 9. List the labels in order from least to greatest.

(A) 19.□98(B) 1□.096(C) 10.0□2\text{(A)}\ 19.\square 98 \qquad \text{(B)}\ 1\square .096 \qquad \text{(C)}\ 10.0\square 2

Show solution
1 · Understandwhat's really being asked

Each □\square can hold any digit 0-9. The labelled numbers are (A) = 19.□\square 98 (the □\square is a tenths digit); (B) = 1□\square .096 (the □\square is a ones digit); (C) = 10.0□\square 2 (the □\square is a hundredths digit). No matter what digits fill the boxes, order the labels from least to greatest.

Givens
  • (A) = 19.□\square 98 (the □\square is a tenths digit)
  • (B) = 1□\square .096 (the □\square is a ones digit)
  • (C) = 10.0□\square 2 (the □\square is a hundredths digit)
  • Each □\square can be any digit from 0 to 9.
Unknowns
  • The order of labels A, B, C from least to greatest.
Constraints
  • The ordering must hold for every allowed choice of the □\square digits.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Each number can only range over a small interval as its □\square runs 0-9. If those intervals do not overlap, the order is fixed for all choices. So we find the smallest-and-largest possible value of each label and compare the ranges.

3 · Execute3 carry out the plan

1Find each label's possible range

#2 Make a Systematic List 5.NBT.A.3
Let each □\square run from 0 to 9. (C) 10.0#2 ranges from 10.002 to 10.092. (B) 1#.096 ranges from 10.096 to 19.096. (A) 19.#98 ranges from 19.098 to 19.998.
C∈[10.002,10.092], B∈[10.096,19.096], A∈[19.098,19.998]C\in[10.002,10.092],\ B\in[10.096,19.096],\ A\in[19.098,19.998]
Plugging in the smallest and largest digit shows the whole span each number can cover.

2Compare C with B

#6 Guess and Check 4.NF.C.7
C's largest value is 10.092, and B's smallest value is 10.096. Since 10.092 < 10.096, every C is less than every B - the ranges do not overlap.
10.092<10.096⇒C<B10.092 < 10.096 \Rightarrow C < B
If C can never reach as high as B's lowest, C is always smaller.

3Compare B with A

#6 Guess and Check 4.NF.C.7
B's largest value is 19.096, and A's smallest value is 19.098. Since 19.096 < 19.098, every B is less than every A - the ranges do not overlap.
19.096<19.098⇒B<A19.096 < 19.098 \Rightarrow B < A
If B can never reach as high as A's lowest, B is always smaller.
Answer: C, B, A
4 · Reviewdoes it hold up?

The ranges [10.002, 10.092], [10.096, 19.096], [19.098, 19.998] are disjoint and line up in the order C, then B, then A, so the ordering holds for every digit choice.

Another way: Guess and Check the extreme cases: even when each □\square takes the digit that pushes it hardest toward its neighbour, the ranges still keep the order C, B, A.

Standardsmin grade 5
  • 4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Comparing the labels using place value.
  • 5.NBT.A.3 Read, write, and compare decimals to thousandths — Reading the thousandths-place numbers and finding each label's range.
💡Takeaway. Find the lowest and highest each number can be - if the ranges don't overlap, the order is locked in!
Variant 3 medium answer: C, B, A

Each □\square can be any digit from 0 to 9. List the labels in order from least to greatest.

(A) 59.□98(B) 5□.096(C) 50.0□2\text{(A)}\ 59.\square 98 \qquad \text{(B)}\ 5\square .096 \qquad \text{(C)}\ 50.0\square 2

Show solution
1 · Understandwhat's really being asked

Each □\square can hold any digit 0-9. The labelled numbers are (A) = 59.□\square 98 (the □\square is a tenths digit); (B) = 5□\square .096 (the □\square is a ones digit); (C) = 50.0□\square 2 (the □\square is a hundredths digit). No matter what digits fill the boxes, order the labels from least to greatest.

Givens
  • (A) = 59.□\square 98 (the □\square is a tenths digit)
  • (B) = 5□\square .096 (the □\square is a ones digit)
  • (C) = 50.0□\square 2 (the □\square is a hundredths digit)
  • Each □\square can be any digit from 0 to 9.
Unknowns
  • The order of labels A, B, C from least to greatest.
Constraints
  • The ordering must hold for every allowed choice of the □\square digits.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Each number can only range over a small interval as its □\square runs 0-9. If those intervals do not overlap, the order is fixed for all choices. So we find the smallest-and-largest possible value of each label and compare the ranges.

3 · Execute3 carry out the plan

1Find each label's possible range

#2 Make a Systematic List 5.NBT.A.3
Let each □\square run from 0 to 9. (C) 50.0#2 ranges from 50.002 to 50.092. (B) 5#.096 ranges from 50.096 to 59.096. (A) 59.#98 ranges from 59.098 to 59.998.
C∈[50.002,50.092], B∈[50.096,59.096], A∈[59.098,59.998]C\in[50.002,50.092],\ B\in[50.096,59.096],\ A\in[59.098,59.998]
Plugging in the smallest and largest digit shows the whole span each number can cover.

2Compare C with B

#6 Guess and Check 4.NF.C.7
C's largest value is 50.092, and B's smallest value is 50.096. Since 50.092 < 50.096, every C is less than every B - the ranges do not overlap.
50.092<50.096⇒C<B50.092 < 50.096 \Rightarrow C < B
If C can never reach as high as B's lowest, C is always smaller.

3Compare B with A

#6 Guess and Check 4.NF.C.7
B's largest value is 59.096, and A's smallest value is 59.098. Since 59.096 < 59.098, every B is less than every A - the ranges do not overlap.
59.096<59.098⇒B<A59.096 < 59.098 \Rightarrow B < A
If B can never reach as high as A's lowest, B is always smaller.
Answer: C, B, A
4 · Reviewdoes it hold up?

The ranges [50.002, 50.092], [50.096, 59.096], [59.098, 59.998] are disjoint and line up in the order C, then B, then A, so the ordering holds for every digit choice.

Another way: Guess and Check the extreme cases: even when each □\square takes the digit that pushes it hardest toward its neighbour, the ranges still keep the order C, B, A.

Standardsmin grade 5
  • 4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Comparing the labels using place value.
  • 5.NBT.A.3 Read, write, and compare decimals to thousandths — Reading the thousandths-place numbers and finding each label's range.
💡Takeaway. Find the lowest and highest each number can be - if the ranges don't overlap, the order is locked in!
Variant 4 easy answer: C, B, A

Each □\square can be any digit from 0 to 9. List the labels in order from least to greatest.

(A) 49.□98(B) 4□.096(C) 40.0□2\text{(A)}\ 49.\square 98 \qquad \text{(B)}\ 4\square .096 \qquad \text{(C)}\ 40.0\square 2

Show solution
1 · Understandwhat's really being asked

Each □\square can hold any digit 0-9. The labelled numbers are (A) = 49.□\square 98 (the □\square is a tenths digit); (B) = 4□\square .096 (the □\square is a ones digit); (C) = 40.0□\square 2 (the □\square is a hundredths digit). No matter what digits fill the boxes, order the labels from least to greatest.

Givens
  • (A) = 49.□\square 98 (the □\square is a tenths digit)
  • (B) = 4□\square .096 (the □\square is a ones digit)
  • (C) = 40.0□\square 2 (the □\square is a hundredths digit)
  • Each □\square can be any digit from 0 to 9.
Unknowns
  • The order of labels A, B, C from least to greatest.
Constraints
  • The ordering must hold for every allowed choice of the □\square digits.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Each number can only range over a small interval as its □\square runs 0-9. If those intervals do not overlap, the order is fixed for all choices. So we find the smallest-and-largest possible value of each label and compare the ranges.

3 · Execute3 carry out the plan

1Find each label's possible range

#2 Make a Systematic List 5.NBT.A.3
Let each □\square run from 0 to 9. (C) 40.0#2 ranges from 40.002 to 40.092. (B) 4#.096 ranges from 40.096 to 49.096. (A) 49.#98 ranges from 49.098 to 49.998.
C∈[40.002,40.092], B∈[40.096,49.096], A∈[49.098,49.998]C\in[40.002,40.092],\ B\in[40.096,49.096],\ A\in[49.098,49.998]
Plugging in the smallest and largest digit shows the whole span each number can cover.

2Compare C with B

#6 Guess and Check 4.NF.C.7
C's largest value is 40.092, and B's smallest value is 40.096. Since 40.092 < 40.096, every C is less than every B - the ranges do not overlap.
40.092<40.096⇒C<B40.092 < 40.096 \Rightarrow C < B
If C can never reach as high as B's lowest, C is always smaller.

3Compare B with A

#6 Guess and Check 4.NF.C.7
B's largest value is 49.096, and A's smallest value is 49.098. Since 49.096 < 49.098, every B is less than every A - the ranges do not overlap.
49.096<49.098⇒B<A49.096 < 49.098 \Rightarrow B < A
If B can never reach as high as A's lowest, B is always smaller.
Answer: C, B, A
4 · Reviewdoes it hold up?

The ranges [40.002, 40.092], [40.096, 49.096], [49.098, 49.998] are disjoint and line up in the order C, then B, then A, so the ordering holds for every digit choice.

Another way: Guess and Check the extreme cases: even when each □\square takes the digit that pushes it hardest toward its neighbour, the ranges still keep the order C, B, A.

Standardsmin grade 5
  • 4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Comparing the labels using place value.
  • 5.NBT.A.3 Read, write, and compare decimals to thousandths — Reading the thousandths-place numbers and finding each label's range.
💡Takeaway. Find the lowest and highest each number can be - if the ranges don't overlap, the order is locked in!
Variant 5 hard answer: C, B, A

Each □\square can be any digit from 0 to 9. List the labels in order from least to greatest.

(A) 39.□98(B) 3□.096(C) 30.0□2\text{(A)}\ 39.\square 98 \qquad \text{(B)}\ 3\square .096 \qquad \text{(C)}\ 30.0\square 2

Show solution
1 · Understandwhat's really being asked

Each □\square can hold any digit 0-9. The labelled numbers are (A) = 39.□\square 98 (the □\square is a tenths digit); (B) = 3□\square .096 (the □\square is a ones digit); (C) = 30.0□\square 2 (the □\square is a hundredths digit). No matter what digits fill the boxes, order the labels from least to greatest.

Givens
  • (A) = 39.□\square 98 (the □\square is a tenths digit)
  • (B) = 3□\square .096 (the □\square is a ones digit)
  • (C) = 30.0□\square 2 (the □\square is a hundredths digit)
  • Each □\square can be any digit from 0 to 9.
Unknowns
  • The order of labels A, B, C from least to greatest.
Constraints
  • The ordering must hold for every allowed choice of the □\square digits.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Each number can only range over a small interval as its □\square runs 0-9. If those intervals do not overlap, the order is fixed for all choices. So we find the smallest-and-largest possible value of each label and compare the ranges.

3 · Execute3 carry out the plan

1Find each label's possible range

#2 Make a Systematic List 5.NBT.A.3
Let each □\square run from 0 to 9. (C) 30.0#2 ranges from 30.002 to 30.092. (B) 3#.096 ranges from 30.096 to 39.096. (A) 39.#98 ranges from 39.098 to 39.998.
C∈[30.002,30.092], B∈[30.096,39.096], A∈[39.098,39.998]C\in[30.002,30.092],\ B\in[30.096,39.096],\ A\in[39.098,39.998]
Plugging in the smallest and largest digit shows the whole span each number can cover.

2Compare C with B

#6 Guess and Check 4.NF.C.7
C's largest value is 30.092, and B's smallest value is 30.096. Since 30.092 < 30.096, every C is less than every B - the ranges do not overlap.
30.092<30.096⇒C<B30.092 < 30.096 \Rightarrow C < B
If C can never reach as high as B's lowest, C is always smaller.

3Compare B with A

#6 Guess and Check 4.NF.C.7
B's largest value is 39.096, and A's smallest value is 39.098. Since 39.096 < 39.098, every B is less than every A - the ranges do not overlap.
39.096<39.098⇒B<A39.096 < 39.098 \Rightarrow B < A
If B can never reach as high as A's lowest, B is always smaller.
Answer: C, B, A
4 · Reviewdoes it hold up?

The ranges [30.002, 30.092], [30.096, 39.096], [39.098, 39.998] are disjoint and line up in the order C, then B, then A, so the ordering holds for every digit choice.

Another way: Guess and Check the extreme cases: even when each □\square takes the digit that pushes it hardest toward its neighbour, the ranges still keep the order C, B, A.

Standardsmin grade 5
  • 4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Comparing the labels using place value.
  • 5.NBT.A.3 Read, write, and compare decimals to thousandths — Reading the thousandths-place numbers and finding each label's range.
💡Takeaway. Find the lowest and highest each number can be - if the ranges don't overlap, the order is locked in!
Variant 6 easy answer: C, B, A

Each □\square can be any digit from 0 to 9. List the labels in order from least to greatest.

(A) 79.□98(B) 7□.096(C) 70.0□2\text{(A)}\ 79.\square 98 \qquad \text{(B)}\ 7\square .096 \qquad \text{(C)}\ 70.0\square 2

Show solution
1 · Understandwhat's really being asked

Each □\square can hold any digit 0-9. The labelled numbers are (A) = 79.□\square 98 (the □\square is a tenths digit); (B) = 7□\square .096 (the □\square is a ones digit); (C) = 70.0□\square 2 (the □\square is a hundredths digit). No matter what digits fill the boxes, order the labels from least to greatest.

Givens
  • (A) = 79.□\square 98 (the □\square is a tenths digit)
  • (B) = 7□\square .096 (the □\square is a ones digit)
  • (C) = 70.0□\square 2 (the □\square is a hundredths digit)
  • Each □\square can be any digit from 0 to 9.
Unknowns
  • The order of labels A, B, C from least to greatest.
Constraints
  • The ordering must hold for every allowed choice of the □\square digits.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Each number can only range over a small interval as its □\square runs 0-9. If those intervals do not overlap, the order is fixed for all choices. So we find the smallest-and-largest possible value of each label and compare the ranges.

3 · Execute3 carry out the plan

1Find each label's possible range

#2 Make a Systematic List 5.NBT.A.3
Let each □\square run from 0 to 9. (C) 70.0#2 ranges from 70.002 to 70.092. (B) 7#.096 ranges from 70.096 to 79.096. (A) 79.#98 ranges from 79.098 to 79.998.
C∈[70.002,70.092], B∈[70.096,79.096], A∈[79.098,79.998]C\in[70.002,70.092],\ B\in[70.096,79.096],\ A\in[79.098,79.998]
Plugging in the smallest and largest digit shows the whole span each number can cover.

2Compare C with B

#6 Guess and Check 4.NF.C.7
C's largest value is 70.092, and B's smallest value is 70.096. Since 70.092 < 70.096, every C is less than every B - the ranges do not overlap.
70.092<70.096⇒C<B70.092 < 70.096 \Rightarrow C < B
If C can never reach as high as B's lowest, C is always smaller.

3Compare B with A

#6 Guess and Check 4.NF.C.7
B's largest value is 79.096, and A's smallest value is 79.098. Since 79.096 < 79.098, every B is less than every A - the ranges do not overlap.
79.096<79.098⇒B<A79.096 < 79.098 \Rightarrow B < A
If B can never reach as high as A's lowest, B is always smaller.
Answer: C, B, A
4 · Reviewdoes it hold up?

The ranges [70.002, 70.092], [70.096, 79.096], [79.098, 79.998] are disjoint and line up in the order C, then B, then A, so the ordering holds for every digit choice.

Another way: Guess and Check the extreme cases: even when each □\square takes the digit that pushes it hardest toward its neighbour, the ranges still keep the order C, B, A.

Standardsmin grade 5
  • 4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Comparing the labels using place value.
  • 5.NBT.A.3 Read, write, and compare decimals to thousandths — Reading the thousandths-place numbers and finding each label's range.
💡Takeaway. Find the lowest and highest each number can be - if the ranges don't overlap, the order is locked in!
Variant 7 medium answer: C, B, A

Each □\square can be any digit from 0 to 9. List the labels in order from least to greatest.

(A) 29.□98(B) 2□.096(C) 20.0□2\text{(A)}\ 29.\square 98 \qquad \text{(B)}\ 2\square .096 \qquad \text{(C)}\ 20.0\square 2

Show solution
1 · Understandwhat's really being asked

Each □\square can hold any digit 0-9. The labelled numbers are (A) = 29.□\square 98 (the □\square is a tenths digit); (B) = 2□\square .096 (the □\square is a ones digit); (C) = 20.0□\square 2 (the □\square is a hundredths digit). No matter what digits fill the boxes, order the labels from least to greatest.

Givens
  • (A) = 29.□\square 98 (the □\square is a tenths digit)
  • (B) = 2□\square .096 (the □\square is a ones digit)
  • (C) = 20.0□\square 2 (the □\square is a hundredths digit)
  • Each □\square can be any digit from 0 to 9.
Unknowns
  • The order of labels A, B, C from least to greatest.
Constraints
  • The ordering must hold for every allowed choice of the □\square digits.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Each number can only range over a small interval as its □\square runs 0-9. If those intervals do not overlap, the order is fixed for all choices. So we find the smallest-and-largest possible value of each label and compare the ranges.

3 · Execute3 carry out the plan

1Find each label's possible range

#2 Make a Systematic List 5.NBT.A.3
Let each □\square run from 0 to 9. (C) 20.0#2 ranges from 20.002 to 20.092. (B) 2#.096 ranges from 20.096 to 29.096. (A) 29.#98 ranges from 29.098 to 29.998.
C∈[20.002,20.092], B∈[20.096,29.096], A∈[29.098,29.998]C\in[20.002,20.092],\ B\in[20.096,29.096],\ A\in[29.098,29.998]
Plugging in the smallest and largest digit shows the whole span each number can cover.

2Compare C with B

#6 Guess and Check 4.NF.C.7
C's largest value is 20.092, and B's smallest value is 20.096. Since 20.092 < 20.096, every C is less than every B - the ranges do not overlap.
20.092<20.096⇒C<B20.092 < 20.096 \Rightarrow C < B
If C can never reach as high as B's lowest, C is always smaller.

3Compare B with A

#6 Guess and Check 4.NF.C.7
B's largest value is 29.096, and A's smallest value is 29.098. Since 29.096 < 29.098, every B is less than every A - the ranges do not overlap.
29.096<29.098⇒B<A29.096 < 29.098 \Rightarrow B < A
If B can never reach as high as A's lowest, B is always smaller.
Answer: C, B, A
4 · Reviewdoes it hold up?

The ranges [20.002, 20.092], [20.096, 29.096], [29.098, 29.998] are disjoint and line up in the order C, then B, then A, so the ordering holds for every digit choice.

Another way: Guess and Check the extreme cases: even when each □\square takes the digit that pushes it hardest toward its neighbour, the ranges still keep the order C, B, A.

Standardsmin grade 5
  • 4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Comparing the labels using place value.
  • 5.NBT.A.3 Read, write, and compare decimals to thousandths — Reading the thousandths-place numbers and finding each label's range.
💡Takeaway. Find the lowest and highest each number can be - if the ranges don't overlap, the order is locked in!
Variant 8 easy answer: C, B, A

Each □\square can be any digit from 0 to 9. List the labels in order from least to greatest.

(A) 69.□98(B) 6□.096(C) 60.0□2\text{(A)}\ 69.\square 98 \qquad \text{(B)}\ 6\square .096 \qquad \text{(C)}\ 60.0\square 2

Show solution
1 · Understandwhat's really being asked

Each □\square can hold any digit 0-9. The labelled numbers are (A) = 69.□\square 98 (the □\square is a tenths digit); (B) = 6□\square .096 (the □\square is a ones digit); (C) = 60.0□\square 2 (the □\square is a hundredths digit). No matter what digits fill the boxes, order the labels from least to greatest.

Givens
  • (A) = 69.□\square 98 (the □\square is a tenths digit)
  • (B) = 6□\square .096 (the □\square is a ones digit)
  • (C) = 60.0□\square 2 (the □\square is a hundredths digit)
  • Each □\square can be any digit from 0 to 9.
Unknowns
  • The order of labels A, B, C from least to greatest.
Constraints
  • The ordering must hold for every allowed choice of the □\square digits.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Each number can only range over a small interval as its □\square runs 0-9. If those intervals do not overlap, the order is fixed for all choices. So we find the smallest-and-largest possible value of each label and compare the ranges.

3 · Execute3 carry out the plan

1Find each label's possible range

#2 Make a Systematic List 5.NBT.A.3
Let each □\square run from 0 to 9. (C) 60.0#2 ranges from 60.002 to 60.092. (B) 6#.096 ranges from 60.096 to 69.096. (A) 69.#98 ranges from 69.098 to 69.998.
C∈[60.002,60.092], B∈[60.096,69.096], A∈[69.098,69.998]C\in[60.002,60.092],\ B\in[60.096,69.096],\ A\in[69.098,69.998]
Plugging in the smallest and largest digit shows the whole span each number can cover.

2Compare C with B

#6 Guess and Check 4.NF.C.7
C's largest value is 60.092, and B's smallest value is 60.096. Since 60.092 < 60.096, every C is less than every B - the ranges do not overlap.
60.092<60.096⇒C<B60.092 < 60.096 \Rightarrow C < B
If C can never reach as high as B's lowest, C is always smaller.

3Compare B with A

#6 Guess and Check 4.NF.C.7
B's largest value is 69.096, and A's smallest value is 69.098. Since 69.096 < 69.098, every B is less than every A - the ranges do not overlap.
69.096<69.098⇒B<A69.096 < 69.098 \Rightarrow B < A
If B can never reach as high as A's lowest, B is always smaller.
Answer: C, B, A
4 · Reviewdoes it hold up?

The ranges [60.002, 60.092], [60.096, 69.096], [69.098, 69.998] are disjoint and line up in the order C, then B, then A, so the ordering holds for every digit choice.

Another way: Guess and Check the extreme cases: even when each □\square takes the digit that pushes it hardest toward its neighbour, the ranges still keep the order C, B, A.

Standardsmin grade 5
  • 4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Comparing the labels using place value.
  • 5.NBT.A.3 Read, write, and compare decimals to thousandths — Reading the thousandths-place numbers and finding each label's range.
💡Takeaway. Find the lowest and highest each number can be - if the ranges don't overlap, the order is locked in!
Variant 9 hard answer: C, B, A

Each □\square can be any digit from 0 to 9. List the labels in order from least to greatest.

(A) 99.□98(B) 9□.096(C) 90.0□2\text{(A)}\ 99.\square 98 \qquad \text{(B)}\ 9\square .096 \qquad \text{(C)}\ 90.0\square 2

Show solution
1 · Understandwhat's really being asked

Each □\square can hold any digit 0-9. The labelled numbers are (A) = 99.□\square 98 (the □\square is a tenths digit); (B) = 9□\square .096 (the □\square is a ones digit); (C) = 90.0□\square 2 (the □\square is a hundredths digit). No matter what digits fill the boxes, order the labels from least to greatest.

Givens
  • (A) = 99.□\square 98 (the □\square is a tenths digit)
  • (B) = 9□\square .096 (the □\square is a ones digit)
  • (C) = 90.0□\square 2 (the □\square is a hundredths digit)
  • Each □\square can be any digit from 0 to 9.
Unknowns
  • The order of labels A, B, C from least to greatest.
Constraints
  • The ordering must hold for every allowed choice of the □\square digits.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Each number can only range over a small interval as its □\square runs 0-9. If those intervals do not overlap, the order is fixed for all choices. So we find the smallest-and-largest possible value of each label and compare the ranges.

3 · Execute3 carry out the plan

1Find each label's possible range

#2 Make a Systematic List 5.NBT.A.3
Let each □\square run from 0 to 9. (C) 90.0#2 ranges from 90.002 to 90.092. (B) 9#.096 ranges from 90.096 to 99.096. (A) 99.#98 ranges from 99.098 to 99.998.
C∈[90.002,90.092], B∈[90.096,99.096], A∈[99.098,99.998]C\in[90.002,90.092],\ B\in[90.096,99.096],\ A\in[99.098,99.998]
Plugging in the smallest and largest digit shows the whole span each number can cover.

2Compare C with B

#6 Guess and Check 4.NF.C.7
C's largest value is 90.092, and B's smallest value is 90.096. Since 90.092 < 90.096, every C is less than every B - the ranges do not overlap.
90.092<90.096⇒C<B90.092 < 90.096 \Rightarrow C < B
If C can never reach as high as B's lowest, C is always smaller.

3Compare B with A

#6 Guess and Check 4.NF.C.7
B's largest value is 99.096, and A's smallest value is 99.098. Since 99.096 < 99.098, every B is less than every A - the ranges do not overlap.
99.096<99.098⇒B<A99.096 < 99.098 \Rightarrow B < A
If B can never reach as high as A's lowest, B is always smaller.
Answer: C, B, A
4 · Reviewdoes it hold up?

The ranges [90.002, 90.092], [90.096, 99.096], [99.098, 99.998] are disjoint and line up in the order C, then B, then A, so the ordering holds for every digit choice.

Another way: Guess and Check the extreme cases: even when each □\square takes the digit that pushes it hardest toward its neighbour, the ranges still keep the order C, B, A.

Standardsmin grade 5
  • 4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Comparing the labels using place value.
  • 5.NBT.A.3 Read, write, and compare decimals to thousandths — Reading the thousandths-place numbers and finding each label's range.
💡Takeaway. Find the lowest and highest each number can be - if the ranges don't overlap, the order is locked in!