Practice · Compare place by place to bound an unknown digit · Sensim Math

Variant 1 easy

Find every digit $0$ to $9$ that can go in the $\square$ to make the statement true.

$3{,}271{,}045 > 3{,}2\square8{,}400$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Find every single digit 0-9 that can replace the $\square$ 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 $\square$ 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 $\square$ 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 \;\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

\square

match, so the ten-thousands place (the

\square

) decides. The fixed number has 7 there. To keep 3,271,045 the bigger number, 7 must be at least as large as the

\square

.

7 \;?\; \square

The leftmost place where two numbers differ is the one that decides which is larger.

3Test the tie case $\square$ = 7

#2 Make a Systematic List 4.NBT.A.2

If the

\square

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

\square

= 7 fails.

3{,}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

\square

, 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.

\square \in \{0,1,2,3,4,5,6\}

Any

\square

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 $\square$ = 6, the compared number 3,268,400 < 3,271,045 (true). With $\square$ = 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 $\square$ 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

Find every digit $0$ to $9$ that can go in the $\square$ to make the statement true.

$4{,}360{,}092 > 4{,}3\square8{,}517$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Find every single digit 0-9 that can replace the $\square$ 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 $\square$ 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 $\square$ 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 \;\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

\square

match, so the ten-thousands place (the

\square

) decides. The fixed number has 6 there. To keep 4,360,092 the bigger number, 6 must be at least as large as the

\square

.

6 \;?\; \square

The leftmost place where two numbers differ is the one that decides which is larger.

3Test the tie case $\square$ = 6

#2 Make a Systematic List 4.NBT.A.2

If the

\square

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

\square

= 6 fails.

4{,}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

\square

, the inequality holds exactly for 0, 1, 2, 3, 4, 5. The largest that works is 5, giving 4,358,517 < 4,360,092.

\square \in \{0,1,2,3,4,5\}

Any

\square

digit small enough keeps the compared number below the fixed number.

Answer: 0, 1, 2, 3, 4, 5

4 · Reviewdoes it hold up?

With $\square$ = 5, the compared number 4,358,517 < 4,360,092 (true). With $\square$ = 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 $\square$ 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

Find every digit $0$ to $9$ that can go in the $\square$ to make the statement true.

$7{,}430{,}556 > 7{,}4\square9{,}001$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Find every single digit 0-9 that can replace the $\square$ 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 $\square$ 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 $\square$ 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 \;\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

\square

match, so the ten-thousands place (the

\square

) decides. The fixed number has 3 there. To keep 7,430,556 the bigger number, 3 must be at least as large as the

\square

.

3 \;?\; \square

The leftmost place where two numbers differ is the one that decides which is larger.

3Test the tie case $\square$ = 3

#2 Make a Systematic List 4.NBT.A.2

If the

\square

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

\square

= 3 fails.

7{,}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

\square

, the inequality holds exactly for 0, 1, 2. The largest that works is 2, giving 7,429,001 < 7,430,556.

\square \in \{0,1,2\}

Any

\square

digit small enough keeps the compared number below the fixed number.

Answer: 0, 1, 2

4 · Reviewdoes it hold up?

With $\square$ = 2, the compared number 7,429,001 < 7,430,556 (true). With $\square$ = 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 $\square$ 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

Find every digit $0$ to $9$ that can go in the $\square$ to make the statement true.

$1{,}592{,}604 > 1{,}5\square7{,}088$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Find every single digit 0-9 that can replace the $\square$ 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 $\square$ 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 $\square$ 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 \;\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

\square

match, so the ten-thousands place (the

\square

) decides. The fixed number has 9 there. To keep 1,592,604 the bigger number, 9 must be at least as large as the

\square

.

9 \;?\; \square

The leftmost place where two numbers differ is the one that decides which is larger.

3Test the tie case $\square$ = 9

#2 Make a Systematic List 4.NBT.A.2

If the

\square

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

\square

= 9 fails.

1{,}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

\square

, 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.

\square \in \{0,1,2,3,4,5,6,7,8\}

Any

\square

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 $\square$ = 8, the compared number 1,587,088 < 1,592,604 (true). With $\square$ = 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 $\square$ 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

Find every digit $0$ to $9$ that can go in the $\square$ to make the statement true.

$8{,}147{,}329 > 8{,}1\square9{,}620$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Find every single digit 0-9 that can replace the $\square$ 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 $\square$ 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 $\square$ 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 \;\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

\square

match, so the ten-thousands place (the

\square

) decides. The fixed number has 4 there. To keep 8,147,329 the bigger number, 4 must be at least as large as the

\square

.

4 \;?\; \square

The leftmost place where two numbers differ is the one that decides which is larger.

3Test the tie case $\square$ = 4

#2 Make a Systematic List 4.NBT.A.2

If the

\square

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

\square

= 4 fails.

8{,}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

\square

, the inequality holds exactly for 0, 1, 2, 3. The largest that works is 3, giving 8,139,620 < 8,147,329.

\square \in \{0,1,2,3\}

Any

\square

digit small enough keeps the compared number below the fixed number.

Answer: 0, 1, 2, 3

4 · Reviewdoes it hold up?

With $\square$ = 3, the compared number 8,139,620 < 8,147,329 (true). With $\square$ = 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 $\square$ 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

Find every digit $0$ to $9$ that can go in the $\square$ to make the statement true.

$9{,}052{,}678 > 9{,}0\square6{,}155$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Find every single digit 0-9 that can replace the $\square$ 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 $\square$ 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 $\square$ 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 \;\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

\square

match, so the ten-thousands place (the

\square

) decides. The fixed number has 5 there. To keep 9,052,678 the bigger number, 5 must be at least as large as the

\square

.

5 \;?\; \square

The leftmost place where two numbers differ is the one that decides which is larger.

3Test the tie case $\square$ = 5

#2 Make a Systematic List 4.NBT.A.2

If the

\square

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

\square

= 5 fails.

9{,}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

\square

, the inequality holds exactly for 0, 1, 2, 3, 4. The largest that works is 4, giving 9,046,155 < 9,052,678.

\square \in \{0,1,2,3,4\}

Any

\square

digit small enough keeps the compared number below the fixed number.

Answer: 0, 1, 2, 3, 4

4 · Reviewdoes it hold up?

With $\square$ = 4, the compared number 9,046,155 < 9,052,678 (true). With $\square$ = 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 $\square$ 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

Find every digit $0$ to $9$ that can go in the $\square$ to make the statement true.

$2{,}685{,}310 > 2{,}6\square7{,}444$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Find every single digit 0-9 that can replace the $\square$ 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 $\square$ 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 $\square$ 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 \;\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

\square

match, so the ten-thousands place (the

\square

) decides. The fixed number has 8 there. To keep 2,685,310 the bigger number, 8 must be at least as large as the

\square

.

8 \;?\; \square

The leftmost place where two numbers differ is the one that decides which is larger.

3Test the tie case $\square$ = 8

#2 Make a Systematic List 4.NBT.A.2

If the

\square

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

\square

= 8 fails.

2{,}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

\square

, 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.

\square \in \{0,1,2,3,4,5,6,7\}

Any

\square

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 $\square$ = 7, the compared number 2,677,444 < 2,685,310 (true). With $\square$ = 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 $\square$ 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

Find every digit $0$ to $9$ that can go in the $\square$ to make the statement true.

$4{,}541{,}592 > 4{,}5\square6{,}719$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Find every single digit 0-9 that can replace the $\square$ 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 $\square$ 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 $\square$ 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 \;\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

\square

match, so the ten-thousands place (the

\square

) decides. The fixed number has 4 there. To keep 4,541,592 the bigger number, 4 must be at least as large as the

\square

.

4 \;?\; \square

The leftmost place where two numbers differ is the one that decides which is larger.

3Test the tie case $\square$ = 4

#2 Make a Systematic List 4.NBT.A.2

If the

\square

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

\square

= 4 fails.

4{,}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

\square

, the inequality holds exactly for 0, 1, 2, 3. The largest that works is 3, giving 4,536,719 < 4,541,592.

\square \in \{0,1,2,3\}

Any

\square

digit small enough keeps the compared number below the fixed number.

Answer: 0, 1, 2, 3

4 · Reviewdoes it hold up?

With $\square$ = 3, the compared number 4,536,719 < 4,541,592 (true). With $\square$ = 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 $\square$ 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

Find every digit $0$ to $9$ that can go in the $\square$ to make the statement true.

$5{,}119{,}860 > 5{,}1\square7{,}203$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Find every single digit 0-9 that can replace the $\square$ 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 $\square$ 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 $\square$ 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 \;\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

\square

match, so the ten-thousands place (the

\square

) decides. The fixed number has 1 there. To keep 5,119,860 the bigger number, 1 must be at least as large as the

\square

.

1 \;?\; \square

The leftmost place where two numbers differ is the one that decides which is larger.

3Test the tie case $\square$ = 1

#2 Make a Systematic List 4.NBT.A.2

If the

\square

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

\square

= 1 works.

5{,}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

\square

, the inequality holds exactly for 0, 1. The largest that works is 1, giving 5,117,203 < 5,119,860.

\square \in \{0,1\}

Any

\square

digit small enough keeps the compared number below the fixed number.

Answer: 0, 1

4 · Reviewdoes it hold up?

With $\square$ = 1, the compared number 5,117,203 < 5,119,860 (true). With $\square$ = 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 $\square$ 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

Find every digit $0$ to $9$ that can go in the $\square$ to make the statement true.

$6{,}845{,}999 > 6{,}8\square4{,}113$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Find every single digit 0-9 that can replace the $\square$ 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 $\square$ 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 $\square$ 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 \;\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

\square

match, so the ten-thousands place (the

\square

) decides. The fixed number has 4 there. To keep 6,845,999 the bigger number, 4 must be at least as large as the

\square

.

4 \;?\; \square

The leftmost place where two numbers differ is the one that decides which is larger.

3Test the tie case $\square$ = 4

#2 Make a Systematic List 4.NBT.A.2

If the

\square

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

\square

= 4 works.

6{,}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

\square

, the inequality holds exactly for 0, 1, 2, 3, 4. The largest that works is 4, giving 6,844,113 < 6,845,999.

\square \in \{0,1,2,3,4\}

Any

\square

digit small enough keeps the compared number below the fixed number.

Answer: 0, 1, 2, 3, 4

4 · Reviewdoes it hold up?

With $\square$ = 4, the compared number 6,844,113 < 6,845,999 (true). With $\square$ = 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 $\square$ 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!

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

Generated variants — 10

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