← Compare place by place to bound an unknown digit · Pin Down a Number from Digit and Range Conditions

Compare place by place to bound an unknown digit · 10 practice problems

4.NBT.A.2

Generated variants — 10

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

Variant 1 easy answer: 0, 1, 2, 3, 4, 5, 6

Find every digit 00 to 99 that can go in the □\square to make the statement true.

3,271,045>3,2□8,4003{,}271{,}045 > 3{,}2\square8{,}400

Show solution
1 · Understandwhat's really being asked

Find every single digit 0-9 that can replace the box so that 3,271,045 is greater than the boxed number.

Givens
  • The fixed number is 3,271,045 (7 digits).
  • The compared number has one unknown digit in the ten-thousands place.
  • We need 3,271,045 > the compared number.
Unknowns
  • All digits 0-9 that make the inequality true
Constraints
  • The box holds a single digit from 0 to 9.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

Both numbers have the same number of digits, so I line them up place by place from the left and compare; the first place where they differ decides which is larger, which bounds the box digit. I then list every digit that fits.

3 · Execute4 carry out the plan

1Line up the digits

#1 Draw a Diagram 4.NBT.A.2
Both numbers are 7 digits long. Compare from the highest place, moving right until a place differs.
3 2 7 1 0 4 5  vs  3 2 □ 8 4 0 03\,2\,7\,1\,0\,4\,5 \;\text{vs}\; 3\,2\,\square\,8\,4\,0\,0
When two numbers have the same length, you compare digit by digit from the left, exactly like reading them aloud.

2Reach the deciding place

#2 Make a Systematic List 4.NBT.A.2
The places to the left of the box match, so the ten-thousands place (the box) decides. The fixed number has 7 there. To keep 3,271,045 the bigger number, 7 must be at least as large as the box.
7  ?  □7 \;?\; \square
The leftmost place where two numbers differ is the one that decides which is larger.

3Test the tie case box = 7

#2 Make a Systematic List 4.NBT.A.2
If the box is 7, the compared number is 3,278,400. Comparing 3,271,045 with 3,278,400, the fixed number is < the compared one, so box = 7 fails.
3,271,045<3,278,4003{,}271{,}045 < 3{,}278{,}400
If the deciding digits tie, you peek at the next place to break the tie.

4List the working digits

#2 Make a Systematic List 4.NBT.A.2
Trying each digit 0-9 in the box, the inequality holds exactly for 0, 1, 2, 3, 4, 5, 6. The largest that works is 6, giving 3,268,400 < 3,271,045.
□∈{0,1,2,3,4,5,6}\square \in \{0,1,2,3,4,5,6\}
Any box digit small enough keeps the compared number below the fixed number.
Answer: 0, 1, 2, 3, 4, 5, 6
4 · Reviewdoes it hold up?

With box = 6, the compared number 3,268,400 < 3,271,045 (true). With box = 7 it is 3,278,400, which is < 3,271,045. The cutoff is correct, so 0, 1, 2, 3, 4, 5, 6 are exactly the digits that work.

Another way: Guess and check (tool 6): plug each digit 0-9 into the box and compare; the inequality holds only for the listed digits.

Standardsmin grade 4
  • 4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing two equal-length numbers place by place to bound the unknown digit.
💡Takeaway. This only needs Grade 4 comparing: line the numbers up and the first place they differ tells you the answer!
Variant 2 easy answer: 0, 1, 2, 3, 4, 5

Find every digit 00 to 99 that can go in the □\square to make the statement true.

4,360,092>4,3□8,5174{,}360{,}092 > 4{,}3\square8{,}517

Show solution
1 · Understandwhat's really being asked

Find every single digit 0-9 that can replace the box so that 4,360,092 is greater than the boxed number.

Givens
  • The fixed number is 4,360,092 (7 digits).
  • The compared number has one unknown digit in the ten-thousands place.
  • We need 4,360,092 > the compared number.
Unknowns
  • All digits 0-9 that make the inequality true
Constraints
  • The box holds a single digit from 0 to 9.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

Both numbers have the same number of digits, so I line them up place by place from the left and compare; the first place where they differ decides which is larger, which bounds the box digit. I then list every digit that fits.

3 · Execute4 carry out the plan

1Line up the digits

#1 Draw a Diagram 4.NBT.A.2
Both numbers are 7 digits long. Compare from the highest place, moving right until a place differs.
4 3 6 0 0 9 2  vs  4 3 □ 8 5 1 74\,3\,6\,0\,0\,9\,2 \;\text{vs}\; 4\,3\,\square\,8\,5\,1\,7
When two numbers have the same length, you compare digit by digit from the left, exactly like reading them aloud.

2Reach the deciding place

#2 Make a Systematic List 4.NBT.A.2
The places to the left of the box match, so the ten-thousands place (the box) decides. The fixed number has 6 there. To keep 4,360,092 the bigger number, 6 must be at least as large as the box.
6  ?  □6 \;?\; \square
The leftmost place where two numbers differ is the one that decides which is larger.

3Test the tie case box = 6

#2 Make a Systematic List 4.NBT.A.2
If the box is 6, the compared number is 4,368,517. Comparing 4,360,092 with 4,368,517, the fixed number is < the compared one, so box = 6 fails.
4,360,092<4,368,5174{,}360{,}092 < 4{,}368{,}517
If the deciding digits tie, you peek at the next place to break the tie.

4List the working digits

#2 Make a Systematic List 4.NBT.A.2
Trying each digit 0-9 in the box, the inequality holds exactly for 0, 1, 2, 3, 4, 5. The largest that works is 5, giving 4,358,517 < 4,360,092.
□∈{0,1,2,3,4,5}\square \in \{0,1,2,3,4,5\}
Any box digit small enough keeps the compared number below the fixed number.
Answer: 0, 1, 2, 3, 4, 5
4 · Reviewdoes it hold up?

With box = 5, the compared number 4,358,517 < 4,360,092 (true). With box = 6 it is 4,368,517, which is < 4,360,092. The cutoff is correct, so 0, 1, 2, 3, 4, 5 are exactly the digits that work.

Another way: Guess and check (tool 6): plug each digit 0-9 into the box and compare; the inequality holds only for the listed digits.

Standardsmin grade 4
  • 4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing two equal-length numbers place by place to bound the unknown digit.
💡Takeaway. This only needs Grade 4 comparing: line the numbers up and the first place they differ tells you the answer!
Variant 3 easy answer: 0, 1, 2

Find every digit 00 to 99 that can go in the □\square to make the statement true.

7,430,556>7,4□9,0017{,}430{,}556 > 7{,}4\square9{,}001

Show solution
1 · Understandwhat's really being asked

Find every single digit 0-9 that can replace the box so that 7,430,556 is greater than the boxed number.

Givens
  • The fixed number is 7,430,556 (7 digits).
  • The compared number has one unknown digit in the ten-thousands place.
  • We need 7,430,556 > the compared number.
Unknowns
  • All digits 0-9 that make the inequality true
Constraints
  • The box holds a single digit from 0 to 9.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

Both numbers have the same number of digits, so I line them up place by place from the left and compare; the first place where they differ decides which is larger, which bounds the box digit. I then list every digit that fits.

3 · Execute4 carry out the plan

1Line up the digits

#1 Draw a Diagram 4.NBT.A.2
Both numbers are 7 digits long. Compare from the highest place, moving right until a place differs.
7 4 3 0 5 5 6  vs  7 4 □ 9 0 0 17\,4\,3\,0\,5\,5\,6 \;\text{vs}\; 7\,4\,\square\,9\,0\,0\,1
When two numbers have the same length, you compare digit by digit from the left, exactly like reading them aloud.

2Reach the deciding place

#2 Make a Systematic List 4.NBT.A.2
The places to the left of the box match, so the ten-thousands place (the box) decides. The fixed number has 3 there. To keep 7,430,556 the bigger number, 3 must be at least as large as the box.
3  ?  □3 \;?\; \square
The leftmost place where two numbers differ is the one that decides which is larger.

3Test the tie case box = 3

#2 Make a Systematic List 4.NBT.A.2
If the box is 3, the compared number is 7,439,001. Comparing 7,430,556 with 7,439,001, the fixed number is < the compared one, so box = 3 fails.
7,430,556<7,439,0017{,}430{,}556 < 7{,}439{,}001
If the deciding digits tie, you peek at the next place to break the tie.

4List the working digits

#2 Make a Systematic List 4.NBT.A.2
Trying each digit 0-9 in the box, the inequality holds exactly for 0, 1, 2. The largest that works is 2, giving 7,429,001 < 7,430,556.
□∈{0,1,2}\square \in \{0,1,2\}
Any box digit small enough keeps the compared number below the fixed number.
Answer: 0, 1, 2
4 · Reviewdoes it hold up?

With box = 2, the compared number 7,429,001 < 7,430,556 (true). With box = 3 it is 7,439,001, which is < 7,430,556. The cutoff is correct, so 0, 1, 2 are exactly the digits that work.

Another way: Guess and check (tool 6): plug each digit 0-9 into the box and compare; the inequality holds only for the listed digits.

Standardsmin grade 4
  • 4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing two equal-length numbers place by place to bound the unknown digit.
💡Takeaway. This only needs Grade 4 comparing: line the numbers up and the first place they differ tells you the answer!
Variant 4 medium answer: 0, 1, 2, 3, 4, 5, 6, 7, 8

Find every digit 00 to 99 that can go in the □\square to make the statement true.

1,592,604>1,5□7,0881{,}592{,}604 > 1{,}5\square7{,}088

Show solution
1 · Understandwhat's really being asked

Find every single digit 0-9 that can replace the box so that 1,592,604 is greater than the boxed number.

Givens
  • The fixed number is 1,592,604 (7 digits).
  • The compared number has one unknown digit in the ten-thousands place.
  • We need 1,592,604 > the compared number.
Unknowns
  • All digits 0-9 that make the inequality true
Constraints
  • The box holds a single digit from 0 to 9.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

Both numbers have the same number of digits, so I line them up place by place from the left and compare; the first place where they differ decides which is larger, which bounds the box digit. I then list every digit that fits.

3 · Execute4 carry out the plan

1Line up the digits

#1 Draw a Diagram 4.NBT.A.2
Both numbers are 7 digits long. Compare from the highest place, moving right until a place differs.
1 5 9 2 6 0 4  vs  1 5 □ 7 0 8 81\,5\,9\,2\,6\,0\,4 \;\text{vs}\; 1\,5\,\square\,7\,0\,8\,8
When two numbers have the same length, you compare digit by digit from the left, exactly like reading them aloud.

2Reach the deciding place

#2 Make a Systematic List 4.NBT.A.2
The places to the left of the box match, so the ten-thousands place (the box) decides. The fixed number has 9 there. To keep 1,592,604 the bigger number, 9 must be at least as large as the box.
9  ?  □9 \;?\; \square
The leftmost place where two numbers differ is the one that decides which is larger.

3Test the tie case box = 9

#2 Make a Systematic List 4.NBT.A.2
If the box is 9, the compared number is 1,597,088. Comparing 1,592,604 with 1,597,088, the fixed number is < the compared one, so box = 9 fails.
1,592,604<1,597,0881{,}592{,}604 < 1{,}597{,}088
If the deciding digits tie, you peek at the next place to break the tie.

4List the working digits

#2 Make a Systematic List 4.NBT.A.2
Trying each digit 0-9 in the box, the inequality holds exactly for 0, 1, 2, 3, 4, 5, 6, 7, 8. The largest that works is 8, giving 1,587,088 < 1,592,604.
□∈{0,1,2,3,4,5,6,7,8}\square \in \{0,1,2,3,4,5,6,7,8\}
Any box digit small enough keeps the compared number below the fixed number.
Answer: 0, 1, 2, 3, 4, 5, 6, 7, 8
4 · Reviewdoes it hold up?

With box = 8, the compared number 1,587,088 < 1,592,604 (true). With box = 9 it is 1,597,088, which is < 1,592,604. The cutoff is correct, so 0, 1, 2, 3, 4, 5, 6, 7, 8 are exactly the digits that work.

Another way: Guess and check (tool 6): plug each digit 0-9 into the box and compare; the inequality holds only for the listed digits.

Standardsmin grade 4
  • 4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing two equal-length numbers place by place to bound the unknown digit.
💡Takeaway. This only needs Grade 4 comparing: line the numbers up and the first place they differ tells you the answer!
Variant 5 medium answer: 0, 1, 2, 3

Find every digit 00 to 99 that can go in the □\square to make the statement true.

8,147,329>8,1□9,6208{,}147{,}329 > 8{,}1\square9{,}620

Show solution
1 · Understandwhat's really being asked

Find every single digit 0-9 that can replace the box so that 8,147,329 is greater than the boxed number.

Givens
  • The fixed number is 8,147,329 (7 digits).
  • The compared number has one unknown digit in the ten-thousands place.
  • We need 8,147,329 > the compared number.
Unknowns
  • All digits 0-9 that make the inequality true
Constraints
  • The box holds a single digit from 0 to 9.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

Both numbers have the same number of digits, so I line them up place by place from the left and compare; the first place where they differ decides which is larger, which bounds the box digit. I then list every digit that fits.

3 · Execute4 carry out the plan

1Line up the digits

#1 Draw a Diagram 4.NBT.A.2
Both numbers are 7 digits long. Compare from the highest place, moving right until a place differs.
8 1 4 7 3 2 9  vs  8 1 □ 9 6 2 08\,1\,4\,7\,3\,2\,9 \;\text{vs}\; 8\,1\,\square\,9\,6\,2\,0
When two numbers have the same length, you compare digit by digit from the left, exactly like reading them aloud.

2Reach the deciding place

#2 Make a Systematic List 4.NBT.A.2
The places to the left of the box match, so the ten-thousands place (the box) decides. The fixed number has 4 there. To keep 8,147,329 the bigger number, 4 must be at least as large as the box.
4  ?  □4 \;?\; \square
The leftmost place where two numbers differ is the one that decides which is larger.

3Test the tie case box = 4

#2 Make a Systematic List 4.NBT.A.2
If the box is 4, the compared number is 8,149,620. Comparing 8,147,329 with 8,149,620, the fixed number is < the compared one, so box = 4 fails.
8,147,329<8,149,6208{,}147{,}329 < 8{,}149{,}620
If the deciding digits tie, you peek at the next place to break the tie.

4List the working digits

#2 Make a Systematic List 4.NBT.A.2
Trying each digit 0-9 in the box, the inequality holds exactly for 0, 1, 2, 3. The largest that works is 3, giving 8,139,620 < 8,147,329.
□∈{0,1,2,3}\square \in \{0,1,2,3\}
Any box digit small enough keeps the compared number below the fixed number.
Answer: 0, 1, 2, 3
4 · Reviewdoes it hold up?

With box = 3, the compared number 8,139,620 < 8,147,329 (true). With box = 4 it is 8,149,620, which is < 8,147,329. The cutoff is correct, so 0, 1, 2, 3 are exactly the digits that work.

Another way: Guess and check (tool 6): plug each digit 0-9 into the box and compare; the inequality holds only for the listed digits.

Standardsmin grade 4
  • 4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing two equal-length numbers place by place to bound the unknown digit.
💡Takeaway. This only needs Grade 4 comparing: line the numbers up and the first place they differ tells you the answer!
Variant 6 medium answer: 0, 1, 2, 3, 4

Find every digit 00 to 99 that can go in the □\square to make the statement true.

9,052,678>9,0□6,1559{,}052{,}678 > 9{,}0\square6{,}155

Show solution
1 · Understandwhat's really being asked

Find every single digit 0-9 that can replace the box so that 9,052,678 is greater than the boxed number.

Givens
  • The fixed number is 9,052,678 (7 digits).
  • The compared number has one unknown digit in the ten-thousands place.
  • We need 9,052,678 > the compared number.
Unknowns
  • All digits 0-9 that make the inequality true
Constraints
  • The box holds a single digit from 0 to 9.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

Both numbers have the same number of digits, so I line them up place by place from the left and compare; the first place where they differ decides which is larger, which bounds the box digit. I then list every digit that fits.

3 · Execute4 carry out the plan

1Line up the digits

#1 Draw a Diagram 4.NBT.A.2
Both numbers are 7 digits long. Compare from the highest place, moving right until a place differs.
9 0 5 2 6 7 8  vs  9 0 □ 6 1 5 59\,0\,5\,2\,6\,7\,8 \;\text{vs}\; 9\,0\,\square\,6\,1\,5\,5
When two numbers have the same length, you compare digit by digit from the left, exactly like reading them aloud.

2Reach the deciding place

#2 Make a Systematic List 4.NBT.A.2
The places to the left of the box match, so the ten-thousands place (the box) decides. The fixed number has 5 there. To keep 9,052,678 the bigger number, 5 must be at least as large as the box.
5  ?  □5 \;?\; \square
The leftmost place where two numbers differ is the one that decides which is larger.

3Test the tie case box = 5

#2 Make a Systematic List 4.NBT.A.2
If the box is 5, the compared number is 9,056,155. Comparing 9,052,678 with 9,056,155, the fixed number is < the compared one, so box = 5 fails.
9,052,678<9,056,1559{,}052{,}678 < 9{,}056{,}155
If the deciding digits tie, you peek at the next place to break the tie.

4List the working digits

#2 Make a Systematic List 4.NBT.A.2
Trying each digit 0-9 in the box, the inequality holds exactly for 0, 1, 2, 3, 4. The largest that works is 4, giving 9,046,155 < 9,052,678.
□∈{0,1,2,3,4}\square \in \{0,1,2,3,4\}
Any box digit small enough keeps the compared number below the fixed number.
Answer: 0, 1, 2, 3, 4
4 · Reviewdoes it hold up?

With box = 4, the compared number 9,046,155 < 9,052,678 (true). With box = 5 it is 9,056,155, which is < 9,052,678. The cutoff is correct, so 0, 1, 2, 3, 4 are exactly the digits that work.

Another way: Guess and check (tool 6): plug each digit 0-9 into the box and compare; the inequality holds only for the listed digits.

Standardsmin grade 4
  • 4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing two equal-length numbers place by place to bound the unknown digit.
💡Takeaway. This only needs Grade 4 comparing: line the numbers up and the first place they differ tells you the answer!
Variant 7 medium answer: 0, 1, 2, 3, 4, 5, 6, 7

Find every digit 00 to 99 that can go in the □\square to make the statement true.

2,685,310>2,6□7,4442{,}685{,}310 > 2{,}6\square7{,}444

Show solution
1 · Understandwhat's really being asked

Find every single digit 0-9 that can replace the box so that 2,685,310 is greater than the boxed number.

Givens
  • The fixed number is 2,685,310 (7 digits).
  • The compared number has one unknown digit in the ten-thousands place.
  • We need 2,685,310 > the compared number.
Unknowns
  • All digits 0-9 that make the inequality true
Constraints
  • The box holds a single digit from 0 to 9.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

Both numbers have the same number of digits, so I line them up place by place from the left and compare; the first place where they differ decides which is larger, which bounds the box digit. I then list every digit that fits.

3 · Execute4 carry out the plan

1Line up the digits

#1 Draw a Diagram 4.NBT.A.2
Both numbers are 7 digits long. Compare from the highest place, moving right until a place differs.
2 6 8 5 3 1 0  vs  2 6 □ 7 4 4 42\,6\,8\,5\,3\,1\,0 \;\text{vs}\; 2\,6\,\square\,7\,4\,4\,4
When two numbers have the same length, you compare digit by digit from the left, exactly like reading them aloud.

2Reach the deciding place

#2 Make a Systematic List 4.NBT.A.2
The places to the left of the box match, so the ten-thousands place (the box) decides. The fixed number has 8 there. To keep 2,685,310 the bigger number, 8 must be at least as large as the box.
8  ?  □8 \;?\; \square
The leftmost place where two numbers differ is the one that decides which is larger.

3Test the tie case box = 8

#2 Make a Systematic List 4.NBT.A.2
If the box is 8, the compared number is 2,687,444. Comparing 2,685,310 with 2,687,444, the fixed number is < the compared one, so box = 8 fails.
2,685,310<2,687,4442{,}685{,}310 < 2{,}687{,}444
If the deciding digits tie, you peek at the next place to break the tie.

4List the working digits

#2 Make a Systematic List 4.NBT.A.2
Trying each digit 0-9 in the box, the inequality holds exactly for 0, 1, 2, 3, 4, 5, 6, 7. The largest that works is 7, giving 2,677,444 < 2,685,310.
□∈{0,1,2,3,4,5,6,7}\square \in \{0,1,2,3,4,5,6,7\}
Any box digit small enough keeps the compared number below the fixed number.
Answer: 0, 1, 2, 3, 4, 5, 6, 7
4 · Reviewdoes it hold up?

With box = 7, the compared number 2,677,444 < 2,685,310 (true). With box = 8 it is 2,687,444, which is < 2,685,310. The cutoff is correct, so 0, 1, 2, 3, 4, 5, 6, 7 are exactly the digits that work.

Another way: Guess and check (tool 6): plug each digit 0-9 into the box and compare; the inequality holds only for the listed digits.

Standardsmin grade 4
  • 4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing two equal-length numbers place by place to bound the unknown digit.
💡Takeaway. This only needs Grade 4 comparing: line the numbers up and the first place they differ tells you the answer!
Variant 8 hard answer: 0, 1, 2, 3

Find every digit 00 to 99 that can go in the □\square to make the statement true.

4,541,592>4,5□6,7194{,}541{,}592 > 4{,}5\square6{,}719

Show solution
1 · Understandwhat's really being asked

Find every single digit 0-9 that can replace the box so that 4,541,592 is greater than the boxed number.

Givens
  • The fixed number is 4,541,592 (7 digits).
  • The compared number has one unknown digit in the ten-thousands place.
  • We need 4,541,592 > the compared number.
Unknowns
  • All digits 0-9 that make the inequality true
Constraints
  • The box holds a single digit from 0 to 9.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

Both numbers have the same number of digits, so I line them up place by place from the left and compare; the first place where they differ decides which is larger, which bounds the box digit. I then list every digit that fits.

3 · Execute4 carry out the plan

1Line up the digits

#1 Draw a Diagram 4.NBT.A.2
Both numbers are 7 digits long. Compare from the highest place, moving right until a place differs.
4 5 4 1 5 9 2  vs  4 5 □ 6 7 1 94\,5\,4\,1\,5\,9\,2 \;\text{vs}\; 4\,5\,\square\,6\,7\,1\,9
When two numbers have the same length, you compare digit by digit from the left, exactly like reading them aloud.

2Reach the deciding place

#2 Make a Systematic List 4.NBT.A.2
The places to the left of the box match, so the ten-thousands place (the box) decides. The fixed number has 4 there. To keep 4,541,592 the bigger number, 4 must be at least as large as the box.
4  ?  □4 \;?\; \square
The leftmost place where two numbers differ is the one that decides which is larger.

3Test the tie case box = 4

#2 Make a Systematic List 4.NBT.A.2
If the box is 4, the compared number is 4,546,719. Comparing 4,541,592 with 4,546,719, the fixed number is < the compared one, so box = 4 fails.
4,541,592<4,546,7194{,}541{,}592 < 4{,}546{,}719
If the deciding digits tie, you peek at the next place to break the tie.

4List the working digits

#2 Make a Systematic List 4.NBT.A.2
Trying each digit 0-9 in the box, the inequality holds exactly for 0, 1, 2, 3. The largest that works is 3, giving 4,536,719 < 4,541,592.
□∈{0,1,2,3}\square \in \{0,1,2,3\}
Any box digit small enough keeps the compared number below the fixed number.
Answer: 0, 1, 2, 3
4 · Reviewdoes it hold up?

With box = 3, the compared number 4,536,719 < 4,541,592 (true). With box = 4 it is 4,546,719, which is < 4,541,592. The cutoff is correct, so 0, 1, 2, 3 are exactly the digits that work.

Another way: Guess and check (tool 6): plug each digit 0-9 into the box and compare; the inequality holds only for the listed digits.

Standardsmin grade 4
  • 4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing two equal-length numbers place by place to bound the unknown digit.
💡Takeaway. This only needs Grade 4 comparing: line the numbers up and the first place they differ tells you the answer!
Variant 9 hard answer: 0, 1

Find every digit 00 to 99 that can go in the □\square to make the statement true.

5,119,860>5,1□7,2035{,}119{,}860 > 5{,}1\square7{,}203

Show solution
1 · Understandwhat's really being asked

Find every single digit 0-9 that can replace the box so that 5,119,860 is greater than the boxed number.

Givens
  • The fixed number is 5,119,860 (7 digits).
  • The compared number has one unknown digit in the ten-thousands place.
  • We need 5,119,860 > the compared number.
Unknowns
  • All digits 0-9 that make the inequality true
Constraints
  • The box holds a single digit from 0 to 9.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

Both numbers have the same number of digits, so I line them up place by place from the left and compare; the first place where they differ decides which is larger, which bounds the box digit. I then list every digit that fits.

3 · Execute4 carry out the plan

1Line up the digits

#1 Draw a Diagram 4.NBT.A.2
Both numbers are 7 digits long. Compare from the highest place, moving right until a place differs.
5 1 1 9 8 6 0  vs  5 1 □ 7 2 0 35\,1\,1\,9\,8\,6\,0 \;\text{vs}\; 5\,1\,\square\,7\,2\,0\,3
When two numbers have the same length, you compare digit by digit from the left, exactly like reading them aloud.

2Reach the deciding place

#2 Make a Systematic List 4.NBT.A.2
The places to the left of the box match, so the ten-thousands place (the box) decides. The fixed number has 1 there. To keep 5,119,860 the bigger number, 1 must be at least as large as the box.
1  ?  □1 \;?\; \square
The leftmost place where two numbers differ is the one that decides which is larger.

3Test the tie case box = 1

#2 Make a Systematic List 4.NBT.A.2
If the box is 1, the compared number is 5,117,203. Comparing 5,119,860 with 5,117,203, the fixed number is > the compared one, so box = 1 works.
5,119,860>5,117,2035{,}119{,}860 > 5{,}117{,}203
If the deciding digits tie, you peek at the next place to break the tie.

4List the working digits

#2 Make a Systematic List 4.NBT.A.2
Trying each digit 0-9 in the box, the inequality holds exactly for 0, 1. The largest that works is 1, giving 5,117,203 < 5,119,860.
□∈{0,1}\square \in \{0,1\}
Any box digit small enough keeps the compared number below the fixed number.
Answer: 0, 1
4 · Reviewdoes it hold up?

With box = 1, the compared number 5,117,203 < 5,119,860 (true). With box = 1 it is 5,117,203, which is > 5,119,860. The cutoff is correct, so 0, 1 are exactly the digits that work.

Another way: Guess and check (tool 6): plug each digit 0-9 into the box and compare; the inequality holds only for the listed digits.

Standardsmin grade 4
  • 4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing two equal-length numbers place by place to bound the unknown digit.
💡Takeaway. This only needs Grade 4 comparing: line the numbers up and the first place they differ tells you the answer!
Variant 10 hard answer: 0, 1, 2, 3, 4

Find every digit 00 to 99 that can go in the □\square to make the statement true.

6,845,999>6,8□4,1136{,}845{,}999 > 6{,}8\square4{,}113

Show solution
1 · Understandwhat's really being asked

Find every single digit 0-9 that can replace the box so that 6,845,999 is greater than the boxed number.

Givens
  • The fixed number is 6,845,999 (7 digits).
  • The compared number has one unknown digit in the ten-thousands place.
  • We need 6,845,999 > the compared number.
Unknowns
  • All digits 0-9 that make the inequality true
Constraints
  • The box holds a single digit from 0 to 9.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

Both numbers have the same number of digits, so I line them up place by place from the left and compare; the first place where they differ decides which is larger, which bounds the box digit. I then list every digit that fits.

3 · Execute4 carry out the plan

1Line up the digits

#1 Draw a Diagram 4.NBT.A.2
Both numbers are 7 digits long. Compare from the highest place, moving right until a place differs.
6 8 4 5 9 9 9  vs  6 8 □ 4 1 1 36\,8\,4\,5\,9\,9\,9 \;\text{vs}\; 6\,8\,\square\,4\,1\,1\,3
When two numbers have the same length, you compare digit by digit from the left, exactly like reading them aloud.

2Reach the deciding place

#2 Make a Systematic List 4.NBT.A.2
The places to the left of the box match, so the ten-thousands place (the box) decides. The fixed number has 4 there. To keep 6,845,999 the bigger number, 4 must be at least as large as the box.
4  ?  □4 \;?\; \square
The leftmost place where two numbers differ is the one that decides which is larger.

3Test the tie case box = 4

#2 Make a Systematic List 4.NBT.A.2
If the box is 4, the compared number is 6,844,113. Comparing 6,845,999 with 6,844,113, the fixed number is > the compared one, so box = 4 works.
6,845,999>6,844,1136{,}845{,}999 > 6{,}844{,}113
If the deciding digits tie, you peek at the next place to break the tie.

4List the working digits

#2 Make a Systematic List 4.NBT.A.2
Trying each digit 0-9 in the box, the inequality holds exactly for 0, 1, 2, 3, 4. The largest that works is 4, giving 6,844,113 < 6,845,999.
□∈{0,1,2,3,4}\square \in \{0,1,2,3,4\}
Any box digit small enough keeps the compared number below the fixed number.
Answer: 0, 1, 2, 3, 4
4 · Reviewdoes it hold up?

With box = 4, the compared number 6,844,113 < 6,845,999 (true). With box = 4 it is 6,844,113, which is > 6,845,999. The cutoff is correct, so 0, 1, 2, 3, 4 are exactly the digits that work.

Another way: Guess and check (tool 6): plug each digit 0-9 into the box and compare; the inequality holds only for the listed digits.

Standardsmin grade 4
  • 4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing two equal-length numbers place by place to bound the unknown digit.
💡Takeaway. This only needs Grade 4 comparing: line the numbers up and the first place they differ tells you the answer!