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Closest number may be above or below target · 12 practice problems

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Generated variants — 12

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

Variant 1 easy

Using each of the $5$ number cards exactly once, you form the five-digit number that is closest to $20000$ . Round that five-digit number to the nearest hundred.

Number cards: $5$ , $1$ , $3$ , $0$ , $8$

Your answer

Show solution

1 · Understandwhat's really being asked

The cards 5, 1, 3, 0, 8 make many five-digit numbers. We want the one nearest 20000, and then that number rounded to the nearest hundred.

Givens

The cards are 5, 1, 3, 0, 8, each used once.
The target is 20000.
The result is then rounded to the nearest hundred.

Unknowns

The closest number, rounded to the nearest hundred.

Constraints

A five-digit number cannot start with 0.

2 · Planchoose the strategy

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

Closest does not mean below. Build the largest number under 20000 and the smallest at or above it, then see which gap is smaller -- two candidates instead of every arrangement.

3 · Execute4 carry out the plan

1Build the closest number below 20000

#6 Guess and Check 4.NBT.A.1

Make the leading digits as large as they can be while staying under 20000, then keep going down the places.

18530

18530 is the biggest the cards make below 20000.

2Build the closest number at or above 20000

#6 Guess and Check 4.NBT.A.1

Now the other side: as small as possible while not falling under the target.

30158

30158 is the smallest at or above 20000.

3Compare the two gaps

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

Distance decides, not which side of the target it is on.

20000 - 18530 = 1470,\quad 30158 - 20000 = 10158

The below one wins, so 18530 is closest.

4Round 18530 to the nearest hundred

#6 Guess and Check 3.NBT.A.1

The tens digit decides: 3 means round down.

18530 \to 18500

The answer is 18500.

Answer: 18500

4 · Reviewdoes it hold up?

No other arrangement is nearer: 18530 and 30158 are the tightest on each side, at 1470 and 10158 away.

Another way: Listing all arrangements and sorting by distance to 20000 gives the same 18530, at many more numbers to write.

Standardsmin grade 4

4.NBT.A.1 Recognize that a digit represents ten times what it represents in place to its right — Placing cards by how much each position is worth.
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing the two candidates by their distance to the target.
3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Rounding 18530 to the nearest hundred.

💡Takeaway. Nearest can be on either side. Build the best candidate below and the best above, then let the distances decide.

Variant 2 easy

Using each of the $5$ number cards exactly once, you form the five-digit number that is closest to $30000$ . Round that five-digit number to the nearest hundred.

Number cards: $9$ , $0$ , $4$ , $1$ , $7$

Your answer

Show solution

1 · Understandwhat's really being asked

The cards 9, 0, 4, 1, 7 make many five-digit numbers. We want the one nearest 30000, and then that number rounded to the nearest hundred.

Givens

The cards are 9, 0, 4, 1, 7, each used once.
The target is 30000.
The result is then rounded to the nearest hundred.

Unknowns

The closest number, rounded to the nearest hundred.

Constraints

A five-digit number cannot start with 0.

2 · Planchoose the strategy

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

Closest does not mean below. Build the largest number under 30000 and the smallest at or above it, then see which gap is smaller -- two candidates instead of every arrangement.

3 · Execute4 carry out the plan

1Build the closest number below 30000

#6 Guess and Check 4.NBT.A.1

Make the leading digits as large as they can be while staying under 30000, then keep going down the places.

19740

19740 is the biggest the cards make below 30000.

2Build the closest number at or above 30000

#6 Guess and Check 4.NBT.A.1

Now the other side: as small as possible while not falling under the target.

40179

40179 is the smallest at or above 30000.

3Compare the two gaps

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

Distance decides, not which side of the target it is on.

30000 - 19740 = 10260,\quad 40179 - 30000 = 10179

The at or above one wins, so 40179 is closest.

4Round 40179 to the nearest hundred

#6 Guess and Check 3.NBT.A.1

The tens digit decides: 7 means round up.

40179 \to 40200

The answer is 40200.

Answer: 40200

4 · Reviewdoes it hold up?

No other arrangement is nearer: 19740 and 40179 are the tightest on each side, at 10260 and 10179 away.

Another way: Listing all arrangements and sorting by distance to 30000 gives the same 40179, at many more numbers to write.

Standardsmin grade 4

4.NBT.A.1 Recognize that a digit represents ten times what it represents in place to its right — Placing cards by how much each position is worth.
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing the two candidates by their distance to the target.
3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Rounding 40179 to the nearest hundred.

💡Takeaway. Nearest can be on either side. Build the best candidate below and the best above, then let the distances decide.

Variant 3 easy

Using each of the $5$ number cards exactly once, you form the five-digit number that is closest to $30000$ . Round that five-digit number to the nearest hundred.

Number cards: $0$ , $2$ , $7$ , $3$ , $9$

Your answer

Show solution

1 · Understandwhat's really being asked

The cards 0, 2, 7, 3, 9 make many five-digit numbers. We want the one nearest 30000, and then that number rounded to the nearest hundred.

Givens

The cards are 0, 2, 7, 3, 9, each used once.
The target is 30000.
The result is then rounded to the nearest hundred.

Unknowns

The closest number, rounded to the nearest hundred.

Constraints

A five-digit number cannot start with 0.

2 · Planchoose the strategy

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

Closest does not mean below. Build the largest number under 30000 and the smallest at or above it, then see which gap is smaller -- two candidates instead of every arrangement.

3 · Execute4 carry out the plan

1Build the closest number below 30000

#6 Guess and Check 4.NBT.A.1

Make the leading digits as large as they can be while staying under 30000, then keep going down the places.

29730

29730 is the biggest the cards make below 30000.

2Build the closest number at or above 30000

#6 Guess and Check 4.NBT.A.1

Now the other side: as small as possible while not falling under the target.

30279

30279 is the smallest at or above 30000.

3Compare the two gaps

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

Distance decides, not which side of the target it is on.

30000 - 29730 = 270,\quad 30279 - 30000 = 279

The below one wins, so 29730 is closest.

4Round 29730 to the nearest hundred

#6 Guess and Check 3.NBT.A.1

The tens digit decides: 3 means round down.

29730 \to 29700

The answer is 29700.

Answer: 29700

4 · Reviewdoes it hold up?

No other arrangement is nearer: 29730 and 30279 are the tightest on each side, at 270 and 279 away.

Another way: Listing all arrangements and sorting by distance to 30000 gives the same 29730, at many more numbers to write.

Standardsmin grade 4

4.NBT.A.1 Recognize that a digit represents ten times what it represents in place to its right — Placing cards by how much each position is worth.
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing the two candidates by their distance to the target.
3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Rounding 29730 to the nearest hundred.

💡Takeaway. Nearest can be on either side. Build the best candidate below and the best above, then let the distances decide.

Variant 4 medium

Using each of the $5$ number cards exactly once, you form the five-digit number that is closest to $40000$ . Round that five-digit number to the nearest hundred.

Number cards: $4$ , $7$ , $1$ , $2$ , $9$

Your answer

Show solution

1 · Understandwhat's really being asked

The cards 4, 7, 1, 2, 9 make many five-digit numbers. We want the one nearest 40000, and then that number rounded to the nearest hundred.

Givens

The cards are 4, 7, 1, 2, 9, each used once.
The target is 40000.
The result is then rounded to the nearest hundred.

Unknowns

The closest number, rounded to the nearest hundred.

Constraints

A five-digit number cannot start with 0.

2 · Planchoose the strategy

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

Closest does not mean below. Build the largest number under 40000 and the smallest at or above it, then see which gap is smaller -- two candidates instead of every arrangement.

3 · Execute4 carry out the plan

1Build the closest number below 40000

#6 Guess and Check 4.NBT.A.1

Make the leading digits as large as they can be while staying under 40000, then keep going down the places.

29741

29741 is the biggest the cards make below 40000.

2Build the closest number at or above 40000

#6 Guess and Check 4.NBT.A.1

Now the other side: as small as possible while not falling under the target.

41279

41279 is the smallest at or above 40000.

3Compare the two gaps

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

Distance decides, not which side of the target it is on.

40000 - 29741 = 10259,\quad 41279 - 40000 = 1279

The at or above one wins, so 41279 is closest.

4Round 41279 to the nearest hundred

#6 Guess and Check 3.NBT.A.1

The tens digit decides: 7 means round up.

41279 \to 41300

The answer is 41300.

Answer: 41300

4 · Reviewdoes it hold up?

No other arrangement is nearer: 29741 and 41279 are the tightest on each side, at 10259 and 1279 away.

Another way: Listing all arrangements and sorting by distance to 40000 gives the same 41279, at many more numbers to write.

Standardsmin grade 4

4.NBT.A.1 Recognize that a digit represents ten times what it represents in place to its right — Placing cards by how much each position is worth.
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing the two candidates by their distance to the target.
3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Rounding 41279 to the nearest hundred.

💡Takeaway. Nearest can be on either side. Build the best candidate below and the best above, then let the distances decide.

Variant 5 easy

Using each of the $5$ number cards exactly once, you form the five-digit number that is closest to $40000$ . Round that five-digit number to the nearest hundred.

Number cards: $3$ , $5$ , $0$ , $7$ , $1$

Your answer

Show solution

1 · Understandwhat's really being asked

The cards 3, 5, 0, 7, 1 make many five-digit numbers. We want the one nearest 40000, and then that number rounded to the nearest hundred.

Givens

The cards are 3, 5, 0, 7, 1, each used once.
The target is 40000.
The result is then rounded to the nearest hundred.

Unknowns

The closest number, rounded to the nearest hundred.

Constraints

A five-digit number cannot start with 0.

2 · Planchoose the strategy

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

Closest does not mean below. Build the largest number under 40000 and the smallest at or above it, then see which gap is smaller -- two candidates instead of every arrangement.

3 · Execute4 carry out the plan

1Build the closest number below 40000

#6 Guess and Check 4.NBT.A.1

Make the leading digits as large as they can be while staying under 40000, then keep going down the places.

37510

37510 is the biggest the cards make below 40000.

2Build the closest number at or above 40000

#6 Guess and Check 4.NBT.A.1

Now the other side: as small as possible while not falling under the target.

50137

50137 is the smallest at or above 40000.

3Compare the two gaps

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

Distance decides, not which side of the target it is on.

40000 - 37510 = 2490,\quad 50137 - 40000 = 10137

The below one wins, so 37510 is closest.

4Round 37510 to the nearest hundred

#6 Guess and Check 3.NBT.A.1

The tens digit decides: 1 means round down.

37510 \to 37500

The answer is 37500.

Answer: 37500

4 · Reviewdoes it hold up?

No other arrangement is nearer: 37510 and 50137 are the tightest on each side, at 2490 and 10137 away.

Another way: Listing all arrangements and sorting by distance to 40000 gives the same 37510, at many more numbers to write.

Standardsmin grade 4

4.NBT.A.1 Recognize that a digit represents ten times what it represents in place to its right — Placing cards by how much each position is worth.
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing the two candidates by their distance to the target.
3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Rounding 37510 to the nearest hundred.

💡Takeaway. Nearest can be on either side. Build the best candidate below and the best above, then let the distances decide.

Variant 6 medium

Using each of the $5$ number cards exactly once, you form the five-digit number that is closest to $50000$ . Round that five-digit number to the nearest hundred.

Number cards: $3$ , $6$ , $9$ , $2$ , $5$

Your answer

Show solution

1 · Understandwhat's really being asked

The cards 3, 6, 9, 2, 5 make many five-digit numbers. We want the one nearest 50000, and then that number rounded to the nearest hundred.

Givens

The cards are 3, 6, 9, 2, 5, each used once.
The target is 50000.
The result is then rounded to the nearest hundred.

Unknowns

The closest number, rounded to the nearest hundred.

Constraints

A five-digit number cannot start with 0.

2 · Planchoose the strategy

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

Closest does not mean below. Build the largest number under 50000 and the smallest at or above it, then see which gap is smaller -- two candidates instead of every arrangement.

3 · Execute4 carry out the plan

1Build the closest number below 50000

#6 Guess and Check 4.NBT.A.1

Make the leading digits as large as they can be while staying under 50000, then keep going down the places.

39652

39652 is the biggest the cards make below 50000.

2Build the closest number at or above 50000

#6 Guess and Check 4.NBT.A.1

Now the other side: as small as possible while not falling under the target.

52369

52369 is the smallest at or above 50000.

3Compare the two gaps

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

Distance decides, not which side of the target it is on.

50000 - 39652 = 10348,\quad 52369 - 50000 = 2369

The at or above one wins, so 52369 is closest.

4Round 52369 to the nearest hundred

#6 Guess and Check 3.NBT.A.1

The tens digit decides: 6 means round up.

52369 \to 52400

The answer is 52400.

Answer: 52400

4 · Reviewdoes it hold up?

No other arrangement is nearer: 39652 and 52369 are the tightest on each side, at 10348 and 2369 away.

Another way: Listing all arrangements and sorting by distance to 50000 gives the same 52369, at many more numbers to write.

Standardsmin grade 4

4.NBT.A.1 Recognize that a digit represents ten times what it represents in place to its right — Placing cards by how much each position is worth.
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing the two candidates by their distance to the target.
3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Rounding 52369 to the nearest hundred.

💡Takeaway. Nearest can be on either side. Build the best candidate below and the best above, then let the distances decide.

Variant 7 medium

Using each of the $5$ number cards exactly once, you form the five-digit number that is closest to $50000$ . Round that five-digit number to the nearest hundred.

Number cards: $1$ , $4$ , $6$ , $8$ , $2$

Your answer

Show solution

1 · Understandwhat's really being asked

The cards 1, 4, 6, 8, 2 make many five-digit numbers. We want the one nearest 50000, and then that number rounded to the nearest hundred.

Givens

The cards are 1, 4, 6, 8, 2, each used once.
The target is 50000.
The result is then rounded to the nearest hundred.

Unknowns

The closest number, rounded to the nearest hundred.

Constraints

A five-digit number cannot start with 0.

2 · Planchoose the strategy

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

Closest does not mean below. Build the largest number under 50000 and the smallest at or above it, then see which gap is smaller -- two candidates instead of every arrangement.

3 · Execute4 carry out the plan

1Build the closest number below 50000

#6 Guess and Check 4.NBT.A.1

Make the leading digits as large as they can be while staying under 50000, then keep going down the places.

48621

48621 is the biggest the cards make below 50000.

2Build the closest number at or above 50000

#6 Guess and Check 4.NBT.A.1

Now the other side: as small as possible while not falling under the target.

61248

61248 is the smallest at or above 50000.

3Compare the two gaps

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

Distance decides, not which side of the target it is on.

50000 - 48621 = 1379,\quad 61248 - 50000 = 11248

The below one wins, so 48621 is closest.

4Round 48621 to the nearest hundred

#6 Guess and Check 3.NBT.A.1

The tens digit decides: 2 means round down.

48621 \to 48600

The answer is 48600.

Answer: 48600

4 · Reviewdoes it hold up?

No other arrangement is nearer: 48621 and 61248 are the tightest on each side, at 1379 and 11248 away.

Another way: Listing all arrangements and sorting by distance to 50000 gives the same 48621, at many more numbers to write.

Standardsmin grade 4

4.NBT.A.1 Recognize that a digit represents ten times what it represents in place to its right — Placing cards by how much each position is worth.
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing the two candidates by their distance to the target.
3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Rounding 48621 to the nearest hundred.

💡Takeaway. Nearest can be on either side. Build the best candidate below and the best above, then let the distances decide.

Variant 8 medium

Using each of the $5$ number cards exactly once, you form the five-digit number that is closest to $50000$ . Round that five-digit number to the nearest hundred.

Number cards: $6$ , $3$ , $8$ , $0$ , $5$

Your answer

Show solution

1 · Understandwhat's really being asked

The cards 6, 3, 8, 0, 5 make many five-digit numbers. We want the one nearest 50000, and then that number rounded to the nearest hundred.

Givens

The cards are 6, 3, 8, 0, 5, each used once.
The target is 50000.
The result is then rounded to the nearest hundred.

Unknowns

The closest number, rounded to the nearest hundred.

Constraints

A five-digit number cannot start with 0.

2 · Planchoose the strategy

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

Closest does not mean below. Build the largest number under 50000 and the smallest at or above it, then see which gap is smaller -- two candidates instead of every arrangement.

3 · Execute4 carry out the plan

1Build the closest number below 50000

#6 Guess and Check 4.NBT.A.1

Make the leading digits as large as they can be while staying under 50000, then keep going down the places.

38650

38650 is the biggest the cards make below 50000.

2Build the closest number at or above 50000

#6 Guess and Check 4.NBT.A.1

Now the other side: as small as possible while not falling under the target.

50368

50368 is the smallest at or above 50000.

3Compare the two gaps

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

Distance decides, not which side of the target it is on.

50000 - 38650 = 11350,\quad 50368 - 50000 = 368

The at or above one wins, so 50368 is closest.

4Round 50368 to the nearest hundred

#6 Guess and Check 3.NBT.A.1

The tens digit decides: 6 means round up.

50368 \to 50400

The answer is 50400.

Answer: 50400

4 · Reviewdoes it hold up?

No other arrangement is nearer: 38650 and 50368 are the tightest on each side, at 11350 and 368 away.

Another way: Listing all arrangements and sorting by distance to 50000 gives the same 50368, at many more numbers to write.

Standardsmin grade 4

4.NBT.A.1 Recognize that a digit represents ten times what it represents in place to its right — Placing cards by how much each position is worth.
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing the two candidates by their distance to the target.
3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Rounding 50368 to the nearest hundred.

💡Takeaway. Nearest can be on either side. Build the best candidate below and the best above, then let the distances decide.

Variant 9 hard

Using each of the $5$ number cards exactly once, you form the five-digit number that is closest to $60000$ . Round that five-digit number to the nearest hundred.

Number cards: $8$ , $5$ , $2$ , $6$ , $3$

Your answer

Show solution

1 · Understandwhat's really being asked

The cards 8, 5, 2, 6, 3 make many five-digit numbers. We want the one nearest 60000, and then that number rounded to the nearest hundred.

Givens

The cards are 8, 5, 2, 6, 3, each used once.
The target is 60000.
The result is then rounded to the nearest hundred.

Unknowns

The closest number, rounded to the nearest hundred.

Constraints

A five-digit number cannot start with 0.

2 · Planchoose the strategy

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

Closest does not mean below. Build the largest number under 60000 and the smallest at or above it, then see which gap is smaller -- two candidates instead of every arrangement.

3 · Execute4 carry out the plan

1Build the closest number below 60000

#6 Guess and Check 4.NBT.A.1

Make the leading digits as large as they can be while staying under 60000, then keep going down the places.

58632

58632 is the biggest the cards make below 60000.

2Build the closest number at or above 60000

#6 Guess and Check 4.NBT.A.1

Now the other side: as small as possible while not falling under the target.

62358

62358 is the smallest at or above 60000.

3Compare the two gaps

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

Distance decides, not which side of the target it is on.

60000 - 58632 = 1368,\quad 62358 - 60000 = 2358

The below one wins, so 58632 is closest.

4Round 58632 to the nearest hundred

#6 Guess and Check 3.NBT.A.1

The tens digit decides: 3 means round down.

58632 \to 58600

The answer is 58600.

Answer: 58600

4 · Reviewdoes it hold up?

No other arrangement is nearer: 58632 and 62358 are the tightest on each side, at 1368 and 2358 away.

Another way: Listing all arrangements and sorting by distance to 60000 gives the same 58632, at many more numbers to write.

Standardsmin grade 4

4.NBT.A.1 Recognize that a digit represents ten times what it represents in place to its right — Placing cards by how much each position is worth.
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing the two candidates by their distance to the target.
3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Rounding 58632 to the nearest hundred.

💡Takeaway. Nearest can be on either side. Build the best candidate below and the best above, then let the distances decide.

Variant 10 hard

Using each of the $5$ number cards exactly once, you form the five-digit number that is closest to $60000$ . Round that five-digit number to the nearest hundred.

Number cards: $2$ , $9$ , $4$ , $6$ , $8$

Your answer

Show solution

1 · Understandwhat's really being asked

The cards 2, 9, 4, 6, 8 make many five-digit numbers. We want the one nearest 60000, and then that number rounded to the nearest hundred.

Givens

The cards are 2, 9, 4, 6, 8, each used once.
The target is 60000.
The result is then rounded to the nearest hundred.

Unknowns

The closest number, rounded to the nearest hundred.

Constraints

A five-digit number cannot start with 0.

2 · Planchoose the strategy

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

Closest does not mean below. Build the largest number under 60000 and the smallest at or above it, then see which gap is smaller -- two candidates instead of every arrangement.

3 · Execute4 carry out the plan

1Build the closest number below 60000

#6 Guess and Check 4.NBT.A.1

Make the leading digits as large as they can be while staying under 60000, then keep going down the places.

49862

49862 is the biggest the cards make below 60000.

2Build the closest number at or above 60000

#6 Guess and Check 4.NBT.A.1

Now the other side: as small as possible while not falling under the target.

62489

62489 is the smallest at or above 60000.

3Compare the two gaps

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

Distance decides, not which side of the target it is on.

60000 - 49862 = 10138,\quad 62489 - 60000 = 2489

The at or above one wins, so 62489 is closest.

4Round 62489 to the nearest hundred

#6 Guess and Check 3.NBT.A.1

The tens digit decides: 8 means round up.

62489 \to 62500

The answer is 62500.

Answer: 62500

4 · Reviewdoes it hold up?

No other arrangement is nearer: 49862 and 62489 are the tightest on each side, at 10138 and 2489 away.

Another way: Listing all arrangements and sorting by distance to 60000 gives the same 62489, at many more numbers to write.

Standardsmin grade 4

4.NBT.A.1 Recognize that a digit represents ten times what it represents in place to its right — Placing cards by how much each position is worth.
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing the two candidates by their distance to the target.
3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Rounding 62489 to the nearest hundred.

💡Takeaway. Nearest can be on either side. Build the best candidate below and the best above, then let the distances decide.

Variant 11 hard

Using each of the $5$ number cards exactly once, you form the five-digit number that is closest to $70000$ . Round that five-digit number to the nearest hundred.

Number cards: $7$ , $2$ , $4$ , $9$ , $1$

Your answer

Show solution

1 · Understandwhat's really being asked

The cards 7, 2, 4, 9, 1 make many five-digit numbers. We want the one nearest 70000, and then that number rounded to the nearest hundred.

Givens

The cards are 7, 2, 4, 9, 1, each used once.
The target is 70000.
The result is then rounded to the nearest hundred.

Unknowns

The closest number, rounded to the nearest hundred.

Constraints

A five-digit number cannot start with 0.

2 · Planchoose the strategy

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

Closest does not mean below. Build the largest number under 70000 and the smallest at or above it, then see which gap is smaller -- two candidates instead of every arrangement.

3 · Execute4 carry out the plan

1Build the closest number below 70000

#6 Guess and Check 4.NBT.A.1

Make the leading digits as large as they can be while staying under 70000, then keep going down the places.

49721

49721 is the biggest the cards make below 70000.

2Build the closest number at or above 70000

#6 Guess and Check 4.NBT.A.1

Now the other side: as small as possible while not falling under the target.

71249

71249 is the smallest at or above 70000.

3Compare the two gaps

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

Distance decides, not which side of the target it is on.

70000 - 49721 = 20279,\quad 71249 - 70000 = 1249

The at or above one wins, so 71249 is closest.

4Round 71249 to the nearest hundred

#6 Guess and Check 3.NBT.A.1

The tens digit decides: 4 means round down.

71249 \to 71200

The answer is 71200.

Answer: 71200

4 · Reviewdoes it hold up?

No other arrangement is nearer: 49721 and 71249 are the tightest on each side, at 20279 and 1249 away.

Another way: Listing all arrangements and sorting by distance to 70000 gives the same 71249, at many more numbers to write.

Standardsmin grade 4

4.NBT.A.1 Recognize that a digit represents ten times what it represents in place to its right — Placing cards by how much each position is worth.
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing the two candidates by their distance to the target.
3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Rounding 71249 to the nearest hundred.

💡Takeaway. Nearest can be on either side. Build the best candidate below and the best above, then let the distances decide.

Variant 12 hard

Using each of the $5$ number cards exactly once, you form the five-digit number that is closest to $70000$ . Round that five-digit number to the nearest hundred.

Number cards: $1$ , $8$ , $3$ , $7$ , $4$

Your answer

Show solution

1 · Understandwhat's really being asked

The cards 1, 8, 3, 7, 4 make many five-digit numbers. We want the one nearest 70000, and then that number rounded to the nearest hundred.

Givens

The cards are 1, 8, 3, 7, 4, each used once.
The target is 70000.
The result is then rounded to the nearest hundred.

Unknowns

The closest number, rounded to the nearest hundred.

Constraints

A five-digit number cannot start with 0.

2 · Planchoose the strategy

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

Closest does not mean below. Build the largest number under 70000 and the smallest at or above it, then see which gap is smaller -- two candidates instead of every arrangement.

3 · Execute4 carry out the plan

1Build the closest number below 70000

#6 Guess and Check 4.NBT.A.1

Make the leading digits as large as they can be while staying under 70000, then keep going down the places.

48731

48731 is the biggest the cards make below 70000.

2Build the closest number at or above 70000

#6 Guess and Check 4.NBT.A.1

Now the other side: as small as possible while not falling under the target.

71348

71348 is the smallest at or above 70000.

3Compare the two gaps

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

Distance decides, not which side of the target it is on.

70000 - 48731 = 21269,\quad 71348 - 70000 = 1348

The at or above one wins, so 71348 is closest.

4Round 71348 to the nearest hundred

#6 Guess and Check 3.NBT.A.1

The tens digit decides: 4 means round down.

71348 \to 71300

The answer is 71300.

Answer: 71300

4 · Reviewdoes it hold up?

No other arrangement is nearer: 48731 and 71348 are the tightest on each side, at 21269 and 1348 away.

Another way: Listing all arrangements and sorting by distance to 70000 gives the same 71348, at many more numbers to write.

Standardsmin grade 4

4.NBT.A.1 Recognize that a digit represents ten times what it represents in place to its right — Placing cards by how much each position is worth.
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing the two candidates by their distance to the target.
3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Rounding 71348 to the nearest hundred.

💡Takeaway. Nearest can be on either side. Build the best candidate below and the best above, then let the distances decide.

Where next?

← Back to the problem: Closest number may be above or below target