← Find a number from digit clues · Pin Down a Number from Digit and Range Conditions

Find a number from digit clues · 12 practice problems

4.NBT.A.2

Generated variants — 12

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

Variant 1 easy answer: 1911

Find the four-digit number that satisfies all of the conditions below.

It is greater than $1000$ and less than $2000$ .
The hundreds digit is $9$ .
The sum of all of its digits is $12$ .
The tens digit and the ones digit are equal.

Show solution

1 · Understandwhat's really being asked

Find the four-digit number that is between 1000 and 2000, has hundreds digit 9, has digits adding to 12, and has equal tens and ones digits.

Givens

The number is greater than 1000 and less than 2000.
The hundreds digit is 9.
The sum of all four digits is 12.
The tens digit equals the ones digit.

Unknowns

The four-digit number meeting all conditions.

Constraints

Each digit is 0 through 9.
All four conditions must hold together.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems

Each clue pins down one place value, so I work through the conditions in order to nail the thousands and hundreds digits, then solve a small equation for the matching tens and ones digits.

3 · Execute3 carry out the plan

1Fix the thousands and hundreds digits

#7 Identify Subproblems 4.NBT.A.2

Being between 1000 and 2000 forces the thousands digit to be 1. The next clue says the hundreds digit is 9. So far the number looks like 1 9 _ _.

1000 < n < 2000 \Rightarrow \text{thousands} = 1,\quad \text{hundreds} = 9

A number in the 1000s must start with 1, and a stated hundreds digit is just read straight off.

2Use the digit sum to find the equal digits

#6 Guess and Check 4.NBT.A.2

The four digits add to 12. We already have 1 + 9 = 10, so the tens and ones digits must add to 2. Since they are equal, each is 1.

1 + 9 + e + e = 12 \Rightarrow 2e = 2 \Rightarrow e = 1

Two equal digits that total 2 can only both be 1.

3Assemble the number

#7 Identify Subproblems 4.NBT.A.2

Thousands 1, hundreds 9, tens 1, ones 1 gives 1911.

1911

Placing each found digit in its place builds the answer.

Answer: 1911

4 · Reviewdoes it hold up?

Check 1911: it is between 1000 and 2000, the hundreds digit is 9, the digits 1+9+1+1 = 12, and the tens and ones digits are both 1. All four conditions hold.

Another way: List every number 19ee between 1900 and 1999 with equal last two digits and pick the one whose digits sum to 12; only 1911 works.

Standardsmin grade 4

4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Reading place values and assembling a four-digit number from digit conditions.

💡Takeaway. Take the clues one at a time: each one fixes a digit, and two equal digits summing to 2 must both be 1!

Variant 2 easy answer: 2822

Find the four-digit number that satisfies all of the conditions below.

It is greater than $2000$ and less than $3000$ .
The hundreds digit is $8$ .
The sum of all of its digits is $14$ .
The tens digit and the ones digit are equal.

Show solution

1 · Understandwhat's really being asked

Find the four-digit number that is between 2000 and 3000, has hundreds digit 8, has digits adding to 14, and has equal tens and ones digits.

Givens

The number is greater than 2000 and less than 3000.
The hundreds digit is 8.
The sum of all four digits is 14.
The tens digit equals the ones digit.

Unknowns

The four-digit number meeting all conditions.

Constraints

Each digit is 0 through 9.
All four conditions must hold together.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems

Each clue pins down one place value, so I work through the conditions in order to nail the thousands and hundreds digits, then solve a small equation for the matching tens and ones digits.

3 · Execute3 carry out the plan

1Fix the thousands and hundreds digits

#7 Identify Subproblems 4.NBT.A.2

Being between 2000 and 3000 forces the thousands digit to be 2. The next clue says the hundreds digit is 8. So far the number looks like 2 8 _ _.

2000 < n < 3000 \Rightarrow \text{thousands} = 2,\quad \text{hundreds} = 8

A number in the 2000s must start with 2, and a stated hundreds digit is just read straight off.

2Use the digit sum to find the equal digits

#6 Guess and Check 4.NBT.A.2

The four digits add to 14. We already have 2 + 8 = 10, so the tens and ones digits must add to 4. Since they are equal, each is 2.

2 + 8 + e + e = 14 \Rightarrow 2e = 4 \Rightarrow e = 2

Two equal digits that total 4 can only both be 2.

3Assemble the number

#7 Identify Subproblems 4.NBT.A.2

Thousands 2, hundreds 8, tens 2, ones 2 gives 2822.

2822

Placing each found digit in its place builds the answer.

Answer: 2822

4 · Reviewdoes it hold up?

Check 2822: it is between 2000 and 3000, the hundreds digit is 8, the digits 2+8+2+2 = 14, and the tens and ones digits are both 2. All four conditions hold.

Another way: List every number 28ee between 2800 and 2899 with equal last two digits and pick the one whose digits sum to 14; only 2822 works.

Standardsmin grade 4

4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Reading place values and assembling a four-digit number from digit conditions.

💡Takeaway. Take the clues one at a time: each one fixes a digit, and two equal digits summing to 4 must both be 2!

Variant 3 easy answer: 3544

Find the four-digit number that satisfies all of the conditions below.

It is greater than $3000$ and less than $4000$ .
The hundreds digit is $5$ .
The sum of all of its digits is $16$ .
The tens digit and the ones digit are equal.

Show solution

1 · Understandwhat's really being asked

Find the four-digit number that is between 3000 and 4000, has hundreds digit 5, has digits adding to 16, and has equal tens and ones digits.

Givens

The number is greater than 3000 and less than 4000.
The hundreds digit is 5.
The sum of all four digits is 16.
The tens digit equals the ones digit.

Unknowns

The four-digit number meeting all conditions.

Constraints

Each digit is 0 through 9.
All four conditions must hold together.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems

Each clue pins down one place value, so I work through the conditions in order to nail the thousands and hundreds digits, then solve a small equation for the matching tens and ones digits.

3 · Execute3 carry out the plan

1Fix the thousands and hundreds digits

#7 Identify Subproblems 4.NBT.A.2

Being between 3000 and 4000 forces the thousands digit to be 3. The next clue says the hundreds digit is 5. So far the number looks like 3 5 _ _.

3000 < n < 4000 \Rightarrow \text{thousands} = 3,\quad \text{hundreds} = 5

A number in the 3000s must start with 3, and a stated hundreds digit is just read straight off.

2Use the digit sum to find the equal digits

#6 Guess and Check 4.NBT.A.2

The four digits add to 16. We already have 3 + 5 = 8, so the tens and ones digits must add to 8. Since they are equal, each is 4.

3 + 5 + e + e = 16 \Rightarrow 2e = 8 \Rightarrow e = 4

Two equal digits that total 8 can only both be 4.

3Assemble the number

#7 Identify Subproblems 4.NBT.A.2

Thousands 3, hundreds 5, tens 4, ones 4 gives 3544.

3544

Placing each found digit in its place builds the answer.

Answer: 3544

4 · Reviewdoes it hold up?

Check 3544: it is between 3000 and 4000, the hundreds digit is 5, the digits 3+5+4+4 = 16, and the tens and ones digits are both 4. All four conditions hold.

Another way: List every number 35ee between 3500 and 3599 with equal last two digits and pick the one whose digits sum to 16; only 3544 works.

Standardsmin grade 4

4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Reading place values and assembling a four-digit number from digit conditions.

💡Takeaway. Take the clues one at a time: each one fixes a digit, and two equal digits summing to 8 must both be 4!

Variant 4 easy answer: 3744

Find the four-digit number that satisfies all of the conditions below.

It is greater than $3000$ and less than $4000$ .
The hundreds digit is $7$ .
The sum of all of its digits is $18$ .
The tens digit and the ones digit are equal.

Show solution

1 · Understandwhat's really being asked

Find the four-digit number that is between 3000 and 4000, has hundreds digit 7, has digits adding to 18, and has equal tens and ones digits.

Givens

The number is greater than 3000 and less than 4000.
The hundreds digit is 7.
The sum of all four digits is 18.
The tens digit equals the ones digit.

Unknowns

The four-digit number meeting all conditions.

Constraints

Each digit is 0 through 9.
All four conditions must hold together.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems

Each clue pins down one place value, so I work through the conditions in order to nail the thousands and hundreds digits, then solve a small equation for the matching tens and ones digits.

3 · Execute3 carry out the plan

1Fix the thousands and hundreds digits

#7 Identify Subproblems 4.NBT.A.2

Being between 3000 and 4000 forces the thousands digit to be 3. The next clue says the hundreds digit is 7. So far the number looks like 3 7 _ _.

3000 < n < 4000 \Rightarrow \text{thousands} = 3,\quad \text{hundreds} = 7

A number in the 3000s must start with 3, and a stated hundreds digit is just read straight off.

2Use the digit sum to find the equal digits

#6 Guess and Check 4.NBT.A.2

The four digits add to 18. We already have 3 + 7 = 10, so the tens and ones digits must add to 8. Since they are equal, each is 4.

3 + 7 + e + e = 18 \Rightarrow 2e = 8 \Rightarrow e = 4

Two equal digits that total 8 can only both be 4.

3Assemble the number

#7 Identify Subproblems 4.NBT.A.2

Thousands 3, hundreds 7, tens 4, ones 4 gives 3744.

3744

Placing each found digit in its place builds the answer.

Answer: 3744

4 · Reviewdoes it hold up?

Check 3744: it is between 3000 and 4000, the hundreds digit is 7, the digits 3+7+4+4 = 18, and the tens and ones digits are both 4. All four conditions hold.

Another way: List every number 37ee between 3700 and 3799 with equal last two digits and pick the one whose digits sum to 18; only 3744 works.

Standardsmin grade 4

4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Reading place values and assembling a four-digit number from digit conditions.

💡Takeaway. Take the clues one at a time: each one fixes a digit, and two equal digits summing to 8 must both be 4!

Variant 5 medium answer: 4655

Find the four-digit number that satisfies all of the conditions below.

It is greater than $4000$ and less than $5000$ .
The hundreds digit is $6$ .
The sum of all of its digits is $20$ .
The tens digit and the ones digit are equal.

Show solution

1 · Understandwhat's really being asked

Find the four-digit number that is between 4000 and 5000, has hundreds digit 6, has digits adding to 20, and has equal tens and ones digits.

Givens

The number is greater than 4000 and less than 5000.
The hundreds digit is 6.
The sum of all four digits is 20.
The tens digit equals the ones digit.

Unknowns

The four-digit number meeting all conditions.

Constraints

Each digit is 0 through 9.
All four conditions must hold together.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems

Each clue pins down one place value, so I work through the conditions in order to nail the thousands and hundreds digits, then solve a small equation for the matching tens and ones digits.

3 · Execute3 carry out the plan

1Fix the thousands and hundreds digits

#7 Identify Subproblems 4.NBT.A.2

Being between 4000 and 5000 forces the thousands digit to be 4. The next clue says the hundreds digit is 6. So far the number looks like 4 6 _ _.

4000 < n < 5000 \Rightarrow \text{thousands} = 4,\quad \text{hundreds} = 6

A number in the 4000s must start with 4, and a stated hundreds digit is just read straight off.

2Use the digit sum to find the equal digits

#6 Guess and Check 4.NBT.A.2

The four digits add to 20. We already have 4 + 6 = 10, so the tens and ones digits must add to 10. Since they are equal, each is 5.

4 + 6 + e + e = 20 \Rightarrow 2e = 10 \Rightarrow e = 5

Two equal digits that total 10 can only both be 5.

3Assemble the number

#7 Identify Subproblems 4.NBT.A.2

Thousands 4, hundreds 6, tens 5, ones 5 gives 4655.

4655

Placing each found digit in its place builds the answer.

Answer: 4655

4 · Reviewdoes it hold up?

Check 4655: it is between 4000 and 5000, the hundreds digit is 6, the digits 4+6+5+5 = 20, and the tens and ones digits are both 5. All four conditions hold.

Another way: List every number 46ee between 4600 and 4699 with equal last two digits and pick the one whose digits sum to 20; only 4655 works.

Standardsmin grade 4

4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Reading place values and assembling a four-digit number from digit conditions.

💡Takeaway. Take the clues one at a time: each one fixes a digit, and two equal digits summing to 10 must both be 5!

Variant 6 medium answer: 4444

Find the four-digit number that satisfies all of the conditions below.

It is greater than $4000$ and less than $5000$ .
The hundreds digit is $4$ .
The sum of all of its digits is $16$ .
The tens digit and the ones digit are equal.

Show solution

1 · Understandwhat's really being asked

Find the four-digit number that is between 4000 and 5000, has hundreds digit 4, has digits adding to 16, and has equal tens and ones digits.

Givens

The number is greater than 4000 and less than 5000.
The hundreds digit is 4.
The sum of all four digits is 16.
The tens digit equals the ones digit.

Unknowns

The four-digit number meeting all conditions.

Constraints

Each digit is 0 through 9.
All four conditions must hold together.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems

Each clue pins down one place value, so I work through the conditions in order to nail the thousands and hundreds digits, then solve a small equation for the matching tens and ones digits.

3 · Execute3 carry out the plan

1Fix the thousands and hundreds digits

#7 Identify Subproblems 4.NBT.A.2

Being between 4000 and 5000 forces the thousands digit to be 4. The next clue says the hundreds digit is 4. So far the number looks like 4 4 _ _.

4000 < n < 5000 \Rightarrow \text{thousands} = 4,\quad \text{hundreds} = 4

A number in the 4000s must start with 4, and a stated hundreds digit is just read straight off.

2Use the digit sum to find the equal digits

#6 Guess and Check 4.NBT.A.2

The four digits add to 16. We already have 4 + 4 = 8, so the tens and ones digits must add to 8. Since they are equal, each is 4.

4 + 4 + e + e = 16 \Rightarrow 2e = 8 \Rightarrow e = 4

Two equal digits that total 8 can only both be 4.

3Assemble the number

#7 Identify Subproblems 4.NBT.A.2

Thousands 4, hundreds 4, tens 4, ones 4 gives 4444.

4444

Placing each found digit in its place builds the answer.

Answer: 4444

4 · Reviewdoes it hold up?

Check 4444: it is between 4000 and 5000, the hundreds digit is 4, the digits 4+4+4+4 = 16, and the tens and ones digits are both 4. All four conditions hold.

Another way: List every number 44ee between 4400 and 4499 with equal last two digits and pick the one whose digits sum to 16; only 4444 works.

Standardsmin grade 4

4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Reading place values and assembling a four-digit number from digit conditions.

💡Takeaway. Take the clues one at a time: each one fixes a digit, and two equal digits summing to 8 must both be 4!

Variant 7 medium answer: 5233

Find the four-digit number that satisfies all of the conditions below.

It is greater than $5000$ and less than $6000$ .
The hundreds digit is $2$ .
The sum of all of its digits is $13$ .
The tens digit and the ones digit are equal.

Show solution

1 · Understandwhat's really being asked

Find the four-digit number that is between 5000 and 6000, has hundreds digit 2, has digits adding to 13, and has equal tens and ones digits.

Givens

The number is greater than 5000 and less than 6000.
The hundreds digit is 2.
The sum of all four digits is 13.
The tens digit equals the ones digit.

Unknowns

The four-digit number meeting all conditions.

Constraints

Each digit is 0 through 9.
All four conditions must hold together.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems

Each clue pins down one place value, so I work through the conditions in order to nail the thousands and hundreds digits, then solve a small equation for the matching tens and ones digits.

3 · Execute3 carry out the plan

1Fix the thousands and hundreds digits

#7 Identify Subproblems 4.NBT.A.2

Being between 5000 and 6000 forces the thousands digit to be 5. The next clue says the hundreds digit is 2. So far the number looks like 5 2 _ _.

5000 < n < 6000 \Rightarrow \text{thousands} = 5,\quad \text{hundreds} = 2

A number in the 5000s must start with 5, and a stated hundreds digit is just read straight off.

2Use the digit sum to find the equal digits

#6 Guess and Check 4.NBT.A.2

The four digits add to 13. We already have 5 + 2 = 7, so the tens and ones digits must add to 6. Since they are equal, each is 3.

5 + 2 + e + e = 13 \Rightarrow 2e = 6 \Rightarrow e = 3

Two equal digits that total 6 can only both be 3.

3Assemble the number

#7 Identify Subproblems 4.NBT.A.2

Thousands 5, hundreds 2, tens 3, ones 3 gives 5233.

5233

Placing each found digit in its place builds the answer.

Answer: 5233

4 · Reviewdoes it hold up?

Check 5233: it is between 5000 and 6000, the hundreds digit is 2, the digits 5+2+3+3 = 13, and the tens and ones digits are both 3. All four conditions hold.

Another way: List every number 52ee between 5200 and 5299 with equal last two digits and pick the one whose digits sum to 13; only 5233 works.

Standardsmin grade 4

4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Reading place values and assembling a four-digit number from digit conditions.

💡Takeaway. Take the clues one at a time: each one fixes a digit, and two equal digits summing to 6 must both be 3!

Variant 8 medium answer: 5577

Find the four-digit number that satisfies all of the conditions below.

It is greater than $5000$ and less than $6000$ .
The hundreds digit is $5$ .
The sum of all of its digits is $24$ .
The tens digit and the ones digit are equal.

Show solution

1 · Understandwhat's really being asked

Find the four-digit number that is between 5000 and 6000, has hundreds digit 5, has digits adding to 24, and has equal tens and ones digits.

Givens

The number is greater than 5000 and less than 6000.
The hundreds digit is 5.
The sum of all four digits is 24.
The tens digit equals the ones digit.

Unknowns

The four-digit number meeting all conditions.

Constraints

Each digit is 0 through 9.
All four conditions must hold together.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems

Each clue pins down one place value, so I work through the conditions in order to nail the thousands and hundreds digits, then solve a small equation for the matching tens and ones digits.

3 · Execute3 carry out the plan

1Fix the thousands and hundreds digits

#7 Identify Subproblems 4.NBT.A.2

Being between 5000 and 6000 forces the thousands digit to be 5. The next clue says the hundreds digit is 5. So far the number looks like 5 5 _ _.

5000 < n < 6000 \Rightarrow \text{thousands} = 5,\quad \text{hundreds} = 5

A number in the 5000s must start with 5, and a stated hundreds digit is just read straight off.

2Use the digit sum to find the equal digits

#6 Guess and Check 4.NBT.A.2

The four digits add to 24. We already have 5 + 5 = 10, so the tens and ones digits must add to 14. Since they are equal, each is 7.

5 + 5 + e + e = 24 \Rightarrow 2e = 14 \Rightarrow e = 7

Two equal digits that total 14 can only both be 7.

3Assemble the number

#7 Identify Subproblems 4.NBT.A.2

Thousands 5, hundreds 5, tens 7, ones 7 gives 5577.

5577

Placing each found digit in its place builds the answer.

Answer: 5577

4 · Reviewdoes it hold up?

Check 5577: it is between 5000 and 6000, the hundreds digit is 5, the digits 5+5+7+7 = 24, and the tens and ones digits are both 7. All four conditions hold.

Another way: List every number 55ee between 5500 and 5599 with equal last two digits and pick the one whose digits sum to 24; only 5577 works.

Standardsmin grade 4

4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Reading place values and assembling a four-digit number from digit conditions.

💡Takeaway. Take the clues one at a time: each one fixes a digit, and two equal digits summing to 14 must both be 7!

Variant 9 hard answer: 6066

Find the four-digit number that satisfies all of the conditions below.

It is greater than $6000$ and less than $7000$ .
The hundreds digit is $0$ .
The sum of all of its digits is $18$ .
The tens digit and the ones digit are equal.

Show solution

1 · Understandwhat's really being asked

Find the four-digit number that is between 6000 and 7000, has hundreds digit 0, has digits adding to 18, and has equal tens and ones digits.

Givens

The number is greater than 6000 and less than 7000.
The hundreds digit is 0.
The sum of all four digits is 18.
The tens digit equals the ones digit.

Unknowns

The four-digit number meeting all conditions.

Constraints

Each digit is 0 through 9.
All four conditions must hold together.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems

Each clue pins down one place value, so I work through the conditions in order to nail the thousands and hundreds digits, then solve a small equation for the matching tens and ones digits.

3 · Execute3 carry out the plan

1Fix the thousands and hundreds digits

#7 Identify Subproblems 4.NBT.A.2

Being between 6000 and 7000 forces the thousands digit to be 6. The next clue says the hundreds digit is 0. So far the number looks like 6 0 _ _.

6000 < n < 7000 \Rightarrow \text{thousands} = 6,\quad \text{hundreds} = 0

A number in the 6000s must start with 6, and a stated hundreds digit is just read straight off.

2Use the digit sum to find the equal digits

#6 Guess and Check 4.NBT.A.2

The four digits add to 18. We already have 6 + 0 = 6, so the tens and ones digits must add to 12. Since they are equal, each is 6.

6 + 0 + e + e = 18 \Rightarrow 2e = 12 \Rightarrow e = 6

Two equal digits that total 12 can only both be 6.

3Assemble the number

#7 Identify Subproblems 4.NBT.A.2

Thousands 6, hundreds 0, tens 6, ones 6 gives 6066.

6066

Placing each found digit in its place builds the answer.

Answer: 6066

4 · Reviewdoes it hold up?

Check 6066: it is between 6000 and 7000, the hundreds digit is 0, the digits 6+0+6+6 = 18, and the tens and ones digits are both 6. All four conditions hold.

Another way: List every number 60ee between 6000 and 6099 with equal last two digits and pick the one whose digits sum to 18; only 6066 works.

Standardsmin grade 4

4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Reading place values and assembling a four-digit number from digit conditions.

💡Takeaway. Take the clues one at a time: each one fixes a digit, and two equal digits summing to 12 must both be 6!

Variant 10 hard answer: 7322

Find the four-digit number that satisfies all of the conditions below.

It is greater than $7000$ and less than $8000$ .
The hundreds digit is $3$ .
The sum of all of its digits is $14$ .
The tens digit and the ones digit are equal.

Show solution

1 · Understandwhat's really being asked

Find the four-digit number that is between 7000 and 8000, has hundreds digit 3, has digits adding to 14, and has equal tens and ones digits.

Givens

The number is greater than 7000 and less than 8000.
The hundreds digit is 3.
The sum of all four digits is 14.
The tens digit equals the ones digit.

Unknowns

The four-digit number meeting all conditions.

Constraints

Each digit is 0 through 9.
All four conditions must hold together.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems

Each clue pins down one place value, so I work through the conditions in order to nail the thousands and hundreds digits, then solve a small equation for the matching tens and ones digits.

3 · Execute3 carry out the plan

1Fix the thousands and hundreds digits

#7 Identify Subproblems 4.NBT.A.2

Being between 7000 and 8000 forces the thousands digit to be 7. The next clue says the hundreds digit is 3. So far the number looks like 7 3 _ _.

7000 < n < 8000 \Rightarrow \text{thousands} = 7,\quad \text{hundreds} = 3

A number in the 7000s must start with 7, and a stated hundreds digit is just read straight off.

2Use the digit sum to find the equal digits

#6 Guess and Check 4.NBT.A.2

The four digits add to 14. We already have 7 + 3 = 10, so the tens and ones digits must add to 4. Since they are equal, each is 2.

7 + 3 + e + e = 14 \Rightarrow 2e = 4 \Rightarrow e = 2

Two equal digits that total 4 can only both be 2.

3Assemble the number

#7 Identify Subproblems 4.NBT.A.2

Thousands 7, hundreds 3, tens 2, ones 2 gives 7322.

7322

Placing each found digit in its place builds the answer.

Answer: 7322

4 · Reviewdoes it hold up?

Check 7322: it is between 7000 and 8000, the hundreds digit is 3, the digits 7+3+2+2 = 14, and the tens and ones digits are both 2. All four conditions hold.

Another way: List every number 73ee between 7300 and 7399 with equal last two digits and pick the one whose digits sum to 14; only 7322 works.

Standardsmin grade 4

4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Reading place values and assembling a four-digit number from digit conditions.

💡Takeaway. Take the clues one at a time: each one fixes a digit, and two equal digits summing to 4 must both be 2!

Variant 11 hard answer: 8155

Find the four-digit number that satisfies all of the conditions below.

It is greater than $8000$ and less than $9000$ .
The hundreds digit is $1$ .
The sum of all of its digits is $19$ .
The tens digit and the ones digit are equal.

Show solution

1 · Understandwhat's really being asked

Find the four-digit number that is between 8000 and 9000, has hundreds digit 1, has digits adding to 19, and has equal tens and ones digits.

Givens

The number is greater than 8000 and less than 9000.
The hundreds digit is 1.
The sum of all four digits is 19.
The tens digit equals the ones digit.

Unknowns

The four-digit number meeting all conditions.

Constraints

Each digit is 0 through 9.
All four conditions must hold together.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems

Each clue pins down one place value, so I work through the conditions in order to nail the thousands and hundreds digits, then solve a small equation for the matching tens and ones digits.

3 · Execute3 carry out the plan

1Fix the thousands and hundreds digits

#7 Identify Subproblems 4.NBT.A.2

Being between 8000 and 9000 forces the thousands digit to be 8. The next clue says the hundreds digit is 1. So far the number looks like 8 1 _ _.

8000 < n < 9000 \Rightarrow \text{thousands} = 8,\quad \text{hundreds} = 1

A number in the 8000s must start with 8, and a stated hundreds digit is just read straight off.

2Use the digit sum to find the equal digits

#6 Guess and Check 4.NBT.A.2

The four digits add to 19. We already have 8 + 1 = 9, so the tens and ones digits must add to 10. Since they are equal, each is 5.

8 + 1 + e + e = 19 \Rightarrow 2e = 10 \Rightarrow e = 5

Two equal digits that total 10 can only both be 5.

3Assemble the number

#7 Identify Subproblems 4.NBT.A.2

Thousands 8, hundreds 1, tens 5, ones 5 gives 8155.

8155

Placing each found digit in its place builds the answer.

Answer: 8155

4 · Reviewdoes it hold up?

Check 8155: it is between 8000 and 9000, the hundreds digit is 1, the digits 8+1+5+5 = 19, and the tens and ones digits are both 5. All four conditions hold.

Another way: List every number 81ee between 8100 and 8199 with equal last two digits and pick the one whose digits sum to 19; only 8155 works.

Standardsmin grade 4

4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Reading place values and assembling a four-digit number from digit conditions.

💡Takeaway. Take the clues one at a time: each one fixes a digit, and two equal digits summing to 10 must both be 5!

Variant 12 hard answer: 9022

Find the four-digit number that satisfies all of the conditions below.

It is greater than $9000$ and less than $10000$ .
The hundreds digit is $0$ .
The sum of all of its digits is $13$ .
The tens digit and the ones digit are equal.

Show solution

1 · Understandwhat's really being asked

Find the four-digit number that is between 9000 and 10000, has hundreds digit 0, has digits adding to 13, and has equal tens and ones digits.

Givens

The number is greater than 9000 and less than 10000.
The hundreds digit is 0.
The sum of all four digits is 13.
The tens digit equals the ones digit.

Unknowns

The four-digit number meeting all conditions.

Constraints

Each digit is 0 through 9.
All four conditions must hold together.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems

Each clue pins down one place value, so I work through the conditions in order to nail the thousands and hundreds digits, then solve a small equation for the matching tens and ones digits.

3 · Execute3 carry out the plan

1Fix the thousands and hundreds digits

#7 Identify Subproblems 4.NBT.A.2

Being between 9000 and 10000 forces the thousands digit to be 9. The next clue says the hundreds digit is 0. So far the number looks like 9 0 _ _.

9000 < n < 10000 \Rightarrow \text{thousands} = 9,\quad \text{hundreds} = 0

A number in the 9000s must start with 9, and a stated hundreds digit is just read straight off.

2Use the digit sum to find the equal digits

#6 Guess and Check 4.NBT.A.2

The four digits add to 13. We already have 9 + 0 = 9, so the tens and ones digits must add to 4. Since they are equal, each is 2.

9 + 0 + e + e = 13 \Rightarrow 2e = 4 \Rightarrow e = 2

Two equal digits that total 4 can only both be 2.

3Assemble the number

#7 Identify Subproblems 4.NBT.A.2

Thousands 9, hundreds 0, tens 2, ones 2 gives 9022.

9022

Placing each found digit in its place builds the answer.

Answer: 9022

4 · Reviewdoes it hold up?

Check 9022: it is between 9000 and 10000, the hundreds digit is 0, the digits 9+0+2+2 = 13, and the tens and ones digits are both 2. All four conditions hold.

Another way: List every number 90ee between 9000 and 9099 with equal last two digits and pick the one whose digits sum to 13; only 9022 works.

Standardsmin grade 4

4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Reading place values and assembling a four-digit number from digit conditions.

💡Takeaway. Take the clues one at a time: each one fixes a digit, and two equal digits summing to 4 must both be 2!