← Round by looking at lower-place digits · Pin Down a Number from Digit and Range Conditions

Round by looking at lower-place digits · 12 practice problems

3.NBT.A.15.NBT.A.4

Generated variants — 12

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

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

For the 4-digit number 18□118\square1, rounding it down to the nearest hundred gives the same result as rounding it to the nearest hundred (round half up).

Find every digit that can go in the □\square.

Show solution
1 · Understandwhat's really being asked

A number has one digit hidden, in the place that decides rounding to the nearest hundred. Rounding it down must land on the same answer as rounding it the normal half-up way. We need every digit for which that happens.

Givens
  • The number is 18, then the hidden digit, then 1.
  • Rounding down to the nearest hundred and rounding half up must agree.
Unknowns
  • Every digit the box can hold.
Constraints
  • The hidden digit is one of 0 through 9.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #5 Look for a Pattern

Work out what each rounding gives, digit by digit. Rounding down barely depends on the digit; rounding half up depends on it entirely, so comparing them shows exactly where the two agree.

3 · Execute3 carry out the plan

1Find what 'round down' gives

#2 Make a Systematic List 3.NBT.A.1
Rounding down always drops to the hundred below, no matter which digit is in the box.
1801→1800,1891→18001801 \to 1800,\quad 1891 \to 1800
That side of the comparison is fixed at 1800.

2See when 'round half up' also gives 1800

#2 Make a Systematic List 5.NBT.A.4
Half up looks at whether the part below the hundred reaches halfway, which is 50. The box digit is what decides that.
1801→1800,1831→1800,1861→1900,1891→19001801 \to 1800,\quad 1831 \to 1800,\quad 1861 \to 1900,\quad 1891 \to 1900
Half up gives 1800 depending on the digit.

3Match the two results

#5 Look for a Pattern 3.NBT.A.1
Keep the digits where both roundings land on the same hundred: 0, 1, 2, 3, 4. The others (5, 6, 7, 8, 9) disagree.
□∈{0,1,2,3,4}\square \in \{0, 1, 2, 3, 4\}
The answer is 0, 1, 2, 3, 4.
Answer: 0, 1, 2, 3, 4
4 · Reviewdoes it hold up?

Spot-check a boundary: with 0 in the box the number is 1801 and both roundings give 1800; with 5 they part company.

Another way: Mark the halfway point on a number line at 1850 and see which digits put the number on the far side; the same digits survive.

Standardsmin grade 5
  • 3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Rounding down to the nearest hundred for each digit.
  • 5.NBT.A.4 Round decimals to any place — Applying round-half-up and comparing the two results.
💡Takeaway. Rounding half up asks one question: has the part below reached halfway? Find the halfway mark and the digits sort themselves.
Variant 2 easy answer: 5, 6, 7, 8, 9

For the 4-digit number 35□835\square8, rounding it up to the nearest hundred gives the same result as rounding it to the nearest hundred (round half up).

Find every digit that can go in the □\square.

Show solution
1 · Understandwhat's really being asked

A number has one digit hidden, in the place that decides rounding to the nearest hundred. Rounding it up must land on the same answer as rounding it the normal half-up way. We need every digit for which that happens.

Givens
  • The number is 35, then the hidden digit, then 8.
  • Rounding up to the nearest hundred and rounding half up must agree.
Unknowns
  • Every digit the box can hold.
Constraints
  • The hidden digit is one of 0 through 9.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #5 Look for a Pattern

Work out what each rounding gives, digit by digit. Rounding up barely depends on the digit; rounding half up depends on it entirely, so comparing them shows exactly where the two agree.

3 · Execute3 carry out the plan

1Find what 'round up' gives

#2 Make a Systematic List 3.NBT.A.1
Rounding up always goes to the next hundred above, no matter which digit is in the box, because the lower digits are never all zero.
3508→3600,3598→36003508 \to 3600,\quad 3598 \to 3600
That side of the comparison is fixed at 3600.

2See when 'round half up' also gives 3600

#2 Make a Systematic List 5.NBT.A.4
Half up looks at whether the part below the hundred reaches halfway, which is 50. The box digit is what decides that.
3508→3500,3538→3500,3568→3600,3598→36003508 \to 3500,\quad 3538 \to 3500,\quad 3568 \to 3600,\quad 3598 \to 3600
Half up gives 3600 depending on the digit.

3Match the two results

#5 Look for a Pattern 3.NBT.A.1
Keep the digits where both roundings land on the same hundred: 5, 6, 7, 8, 9. The others (0, 1, 2, 3, 4) disagree.
□∈{5,6,7,8,9}\square \in \{5, 6, 7, 8, 9\}
The answer is 5, 6, 7, 8, 9.
Answer: 5, 6, 7, 8, 9
4 · Reviewdoes it hold up?

Spot-check a boundary: with 5 in the box the number is 3558 and both roundings give 3600; with 0 they part company.

Another way: Mark the halfway point on a number line at 3550 and see which digits put the number on the far side; the same digits survive.

Standardsmin grade 5
  • 3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Rounding up to the nearest hundred for each digit.
  • 5.NBT.A.4 Round decimals to any place — Applying round-half-up and comparing the two results.
💡Takeaway. Rounding half up asks one question: has the part below reached halfway? Find the halfway mark and the digits sort themselves.
Variant 3 easy answer: 5, 6, 7, 8, 9

For the 5-digit number 29□4729\square47, rounding it up to the nearest thousand gives the same result as rounding it to the nearest thousand (round half up).

Find every digit that can go in the □\square.

Show solution
1 · Understandwhat's really being asked

A number has one digit hidden, in the place that decides rounding to the nearest thousand. Rounding it up must land on the same answer as rounding it the normal half-up way. We need every digit for which that happens.

Givens
  • The number is 29, then the hidden digit, then 47.
  • Rounding up to the nearest thousand and rounding half up must agree.
Unknowns
  • Every digit the box can hold.
Constraints
  • The hidden digit is one of 0 through 9.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #5 Look for a Pattern

Work out what each rounding gives, digit by digit. Rounding up barely depends on the digit; rounding half up depends on it entirely, so comparing them shows exactly where the two agree.

3 · Execute3 carry out the plan

1Find what 'round up' gives

#2 Make a Systematic List 3.NBT.A.1
Rounding up always goes to the next thousand above, no matter which digit is in the box, because the lower digits are never all zero.
29047→30000,29947→3000029047 \to 30000,\quad 29947 \to 30000
That side of the comparison is fixed at 30000.

2See when 'round half up' also gives 30000

#2 Make a Systematic List 5.NBT.A.4
Half up looks at whether the part below the thousand reaches halfway, which is 500. The box digit is what decides that.
29047→29000,29347→29000,29647→30000,29947→3000029047 \to 29000,\quad 29347 \to 29000,\quad 29647 \to 30000,\quad 29947 \to 30000
Half up gives 30000 depending on the digit.

3Match the two results

#5 Look for a Pattern 3.NBT.A.1
Keep the digits where both roundings land on the same thousand: 5, 6, 7, 8, 9. The others (0, 1, 2, 3, 4) disagree.
□∈{5,6,7,8,9}\square \in \{5, 6, 7, 8, 9\}
The answer is 5, 6, 7, 8, 9.
Answer: 5, 6, 7, 8, 9
4 · Reviewdoes it hold up?

Spot-check a boundary: with 5 in the box the number is 29547 and both roundings give 30000; with 0 they part company.

Another way: Mark the halfway point on a number line at 29500 and see which digits put the number on the far side; the same digits survive.

Standardsmin grade 5
  • 3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Rounding up to the nearest thousand for each digit.
  • 5.NBT.A.4 Round decimals to any place — Applying round-half-up and comparing the two results.
💡Takeaway. Rounding half up asks one question: has the part below reached halfway? Find the halfway mark and the digits sort themselves.
Variant 4 easy answer: 0, 1, 2, 3, 4

For the 4-digit number 47□347\square3, rounding it down to the nearest hundred gives the same result as rounding it to the nearest hundred (round half up).

Find every digit that can go in the □\square.

Show solution
1 · Understandwhat's really being asked

A number has one digit hidden, in the place that decides rounding to the nearest hundred. Rounding it down must land on the same answer as rounding it the normal half-up way. We need every digit for which that happens.

Givens
  • The number is 47, then the hidden digit, then 3.
  • Rounding down to the nearest hundred and rounding half up must agree.
Unknowns
  • Every digit the box can hold.
Constraints
  • The hidden digit is one of 0 through 9.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #5 Look for a Pattern

Work out what each rounding gives, digit by digit. Rounding down barely depends on the digit; rounding half up depends on it entirely, so comparing them shows exactly where the two agree.

3 · Execute3 carry out the plan

1Find what 'round down' gives

#2 Make a Systematic List 3.NBT.A.1
Rounding down always drops to the hundred below, no matter which digit is in the box.
4703→4700,4793→47004703 \to 4700,\quad 4793 \to 4700
That side of the comparison is fixed at 4700.

2See when 'round half up' also gives 4700

#2 Make a Systematic List 5.NBT.A.4
Half up looks at whether the part below the hundred reaches halfway, which is 50. The box digit is what decides that.
4703→4700,4733→4700,4763→4800,4793→48004703 \to 4700,\quad 4733 \to 4700,\quad 4763 \to 4800,\quad 4793 \to 4800
Half up gives 4700 depending on the digit.

3Match the two results

#5 Look for a Pattern 3.NBT.A.1
Keep the digits where both roundings land on the same hundred: 0, 1, 2, 3, 4. The others (5, 6, 7, 8, 9) disagree.
□∈{0,1,2,3,4}\square \in \{0, 1, 2, 3, 4\}
The answer is 0, 1, 2, 3, 4.
Answer: 0, 1, 2, 3, 4
4 · Reviewdoes it hold up?

Spot-check a boundary: with 0 in the box the number is 4703 and both roundings give 4700; with 5 they part company.

Another way: Mark the halfway point on a number line at 4750 and see which digits put the number on the far side; the same digits survive.

Standardsmin grade 5
  • 3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Rounding down to the nearest hundred for each digit.
  • 5.NBT.A.4 Round decimals to any place — Applying round-half-up and comparing the two results.
💡Takeaway. Rounding half up asks one question: has the part below reached halfway? Find the halfway mark and the digits sort themselves.
Variant 5 medium answer: 5, 6, 7, 8, 9

For the 4-digit number 62□662\square6, rounding it up to the nearest hundred gives the same result as rounding it to the nearest hundred (round half up).

Find every digit that can go in the □\square.

Show solution
1 · Understandwhat's really being asked

A number has one digit hidden, in the place that decides rounding to the nearest hundred. Rounding it up must land on the same answer as rounding it the normal half-up way. We need every digit for which that happens.

Givens
  • The number is 62, then the hidden digit, then 6.
  • Rounding up to the nearest hundred and rounding half up must agree.
Unknowns
  • Every digit the box can hold.
Constraints
  • The hidden digit is one of 0 through 9.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #5 Look for a Pattern

Work out what each rounding gives, digit by digit. Rounding up barely depends on the digit; rounding half up depends on it entirely, so comparing them shows exactly where the two agree.

3 · Execute3 carry out the plan

1Find what 'round up' gives

#2 Make a Systematic List 3.NBT.A.1
Rounding up always goes to the next hundred above, no matter which digit is in the box, because the lower digits are never all zero.
6206→6300,6296→63006206 \to 6300,\quad 6296 \to 6300
That side of the comparison is fixed at 6300.

2See when 'round half up' also gives 6300

#2 Make a Systematic List 5.NBT.A.4
Half up looks at whether the part below the hundred reaches halfway, which is 50. The box digit is what decides that.
6206→6200,6236→6200,6266→6300,6296→63006206 \to 6200,\quad 6236 \to 6200,\quad 6266 \to 6300,\quad 6296 \to 6300
Half up gives 6300 depending on the digit.

3Match the two results

#5 Look for a Pattern 3.NBT.A.1
Keep the digits where both roundings land on the same hundred: 5, 6, 7, 8, 9. The others (0, 1, 2, 3, 4) disagree.
□∈{5,6,7,8,9}\square \in \{5, 6, 7, 8, 9\}
The answer is 5, 6, 7, 8, 9.
Answer: 5, 6, 7, 8, 9
4 · Reviewdoes it hold up?

Spot-check a boundary: with 5 in the box the number is 6256 and both roundings give 6300; with 0 they part company.

Another way: Mark the halfway point on a number line at 6250 and see which digits put the number on the far side; the same digits survive.

Standardsmin grade 5
  • 3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Rounding up to the nearest hundred for each digit.
  • 5.NBT.A.4 Round decimals to any place — Applying round-half-up and comparing the two results.
💡Takeaway. Rounding half up asks one question: has the part below reached halfway? Find the halfway mark and the digits sort themselves.
Variant 6 medium answer: 5, 6, 7, 8, 9

For the 5-digit number 68□2568\square25, rounding it up to the nearest thousand gives the same result as rounding it to the nearest thousand (round half up).

Find every digit that can go in the □\square.

Show solution
1 · Understandwhat's really being asked

A number has one digit hidden, in the place that decides rounding to the nearest thousand. Rounding it up must land on the same answer as rounding it the normal half-up way. We need every digit for which that happens.

Givens
  • The number is 68, then the hidden digit, then 25.
  • Rounding up to the nearest thousand and rounding half up must agree.
Unknowns
  • Every digit the box can hold.
Constraints
  • The hidden digit is one of 0 through 9.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #5 Look for a Pattern

Work out what each rounding gives, digit by digit. Rounding up barely depends on the digit; rounding half up depends on it entirely, so comparing them shows exactly where the two agree.

3 · Execute3 carry out the plan

1Find what 'round up' gives

#2 Make a Systematic List 3.NBT.A.1
Rounding up always goes to the next thousand above, no matter which digit is in the box, because the lower digits are never all zero.
68025→69000,68925→6900068025 \to 69000,\quad 68925 \to 69000
That side of the comparison is fixed at 69000.

2See when 'round half up' also gives 69000

#2 Make a Systematic List 5.NBT.A.4
Half up looks at whether the part below the thousand reaches halfway, which is 500. The box digit is what decides that.
68025→68000,68325→68000,68625→69000,68925→6900068025 \to 68000,\quad 68325 \to 68000,\quad 68625 \to 69000,\quad 68925 \to 69000
Half up gives 69000 depending on the digit.

3Match the two results

#5 Look for a Pattern 3.NBT.A.1
Keep the digits where both roundings land on the same thousand: 5, 6, 7, 8, 9. The others (0, 1, 2, 3, 4) disagree.
□∈{5,6,7,8,9}\square \in \{5, 6, 7, 8, 9\}
The answer is 5, 6, 7, 8, 9.
Answer: 5, 6, 7, 8, 9
4 · Reviewdoes it hold up?

Spot-check a boundary: with 5 in the box the number is 68525 and both roundings give 69000; with 0 they part company.

Another way: Mark the halfway point on a number line at 68500 and see which digits put the number on the far side; the same digits survive.

Standardsmin grade 5
  • 3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Rounding up to the nearest thousand for each digit.
  • 5.NBT.A.4 Round decimals to any place — Applying round-half-up and comparing the two results.
💡Takeaway. Rounding half up asks one question: has the part below reached halfway? Find the halfway mark and the digits sort themselves.
Variant 7 medium answer: 5, 6, 7, 8, 9

For the 4-digit number 86□986\square9, rounding it up to the nearest hundred gives the same result as rounding it to the nearest hundred (round half up).

Find every digit that can go in the □\square.

Show solution
1 · Understandwhat's really being asked

A number has one digit hidden, in the place that decides rounding to the nearest hundred. Rounding it up must land on the same answer as rounding it the normal half-up way. We need every digit for which that happens.

Givens
  • The number is 86, then the hidden digit, then 9.
  • Rounding up to the nearest hundred and rounding half up must agree.
Unknowns
  • Every digit the box can hold.
Constraints
  • The hidden digit is one of 0 through 9.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #5 Look for a Pattern

Work out what each rounding gives, digit by digit. Rounding up barely depends on the digit; rounding half up depends on it entirely, so comparing them shows exactly where the two agree.

3 · Execute3 carry out the plan

1Find what 'round up' gives

#2 Make a Systematic List 3.NBT.A.1
Rounding up always goes to the next hundred above, no matter which digit is in the box, because the lower digits are never all zero.
8609→8700,8699→87008609 \to 8700,\quad 8699 \to 8700
That side of the comparison is fixed at 8700.

2See when 'round half up' also gives 8700

#2 Make a Systematic List 5.NBT.A.4
Half up looks at whether the part below the hundred reaches halfway, which is 50. The box digit is what decides that.
8609→8600,8639→8600,8669→8700,8699→87008609 \to 8600,\quad 8639 \to 8600,\quad 8669 \to 8700,\quad 8699 \to 8700
Half up gives 8700 depending on the digit.

3Match the two results

#5 Look for a Pattern 3.NBT.A.1
Keep the digits where both roundings land on the same hundred: 5, 6, 7, 8, 9. The others (0, 1, 2, 3, 4) disagree.
□∈{5,6,7,8,9}\square \in \{5, 6, 7, 8, 9\}
The answer is 5, 6, 7, 8, 9.
Answer: 5, 6, 7, 8, 9
4 · Reviewdoes it hold up?

Spot-check a boundary: with 5 in the box the number is 8659 and both roundings give 8700; with 0 they part company.

Another way: Mark the halfway point on a number line at 8650 and see which digits put the number on the far side; the same digits survive.

Standardsmin grade 5
  • 3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Rounding up to the nearest hundred for each digit.
  • 5.NBT.A.4 Round decimals to any place — Applying round-half-up and comparing the two results.
💡Takeaway. Rounding half up asks one question: has the part below reached halfway? Find the halfway mark and the digits sort themselves.
Variant 8 medium answer: 0, 1, 2, 3, 4

For the 5-digit number 91□7291\square72, rounding it down to the nearest thousand gives the same result as rounding it to the nearest thousand (round half up).

Find every digit that can go in the □\square.

Show solution
1 · Understandwhat's really being asked

A number has one digit hidden, in the place that decides rounding to the nearest thousand. Rounding it down must land on the same answer as rounding it the normal half-up way. We need every digit for which that happens.

Givens
  • The number is 91, then the hidden digit, then 72.
  • Rounding down to the nearest thousand and rounding half up must agree.
Unknowns
  • Every digit the box can hold.
Constraints
  • The hidden digit is one of 0 through 9.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #5 Look for a Pattern

Work out what each rounding gives, digit by digit. Rounding down barely depends on the digit; rounding half up depends on it entirely, so comparing them shows exactly where the two agree.

3 · Execute3 carry out the plan

1Find what 'round down' gives

#2 Make a Systematic List 3.NBT.A.1
Rounding down always drops to the thousand below, no matter which digit is in the box.
91072→91000,91972→9100091072 \to 91000,\quad 91972 \to 91000
That side of the comparison is fixed at 91000.

2See when 'round half up' also gives 91000

#2 Make a Systematic List 5.NBT.A.4
Half up looks at whether the part below the thousand reaches halfway, which is 500. The box digit is what decides that.
91072→91000,91372→91000,91672→92000,91972→9200091072 \to 91000,\quad 91372 \to 91000,\quad 91672 \to 92000,\quad 91972 \to 92000
Half up gives 91000 depending on the digit.

3Match the two results

#5 Look for a Pattern 3.NBT.A.1
Keep the digits where both roundings land on the same thousand: 0, 1, 2, 3, 4. The others (5, 6, 7, 8, 9) disagree.
□∈{0,1,2,3,4}\square \in \{0, 1, 2, 3, 4\}
The answer is 0, 1, 2, 3, 4.
Answer: 0, 1, 2, 3, 4
4 · Reviewdoes it hold up?

Spot-check a boundary: with 0 in the box the number is 91072 and both roundings give 91000; with 5 they part company.

Another way: Mark the halfway point on a number line at 91500 and see which digits put the number on the far side; the same digits survive.

Standardsmin grade 5
  • 3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Rounding down to the nearest thousand for each digit.
  • 5.NBT.A.4 Round decimals to any place — Applying round-half-up and comparing the two results.
💡Takeaway. Rounding half up asks one question: has the part below reached halfway? Find the halfway mark and the digits sort themselves.
Variant 9 hard answer: 0, 1, 2, 3, 4

For the 5-digit number 240□4240\square4, rounding it down to the nearest hundred gives the same result as rounding it to the nearest hundred (round half up).

Find every digit that can go in the □\square.

Show solution
1 · Understandwhat's really being asked

A number has one digit hidden, in the place that decides rounding to the nearest hundred. Rounding it down must land on the same answer as rounding it the normal half-up way. We need every digit for which that happens.

Givens
  • The number is 240, then the hidden digit, then 4.
  • Rounding down to the nearest hundred and rounding half up must agree.
Unknowns
  • Every digit the box can hold.
Constraints
  • The hidden digit is one of 0 through 9.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #5 Look for a Pattern

Work out what each rounding gives, digit by digit. Rounding down barely depends on the digit; rounding half up depends on it entirely, so comparing them shows exactly where the two agree.

3 · Execute3 carry out the plan

1Find what 'round down' gives

#2 Make a Systematic List 3.NBT.A.1
Rounding down always drops to the hundred below, no matter which digit is in the box.
24004→24000,24094→2400024004 \to 24000,\quad 24094 \to 24000
That side of the comparison is fixed at 24000.

2See when 'round half up' also gives 24000

#2 Make a Systematic List 5.NBT.A.4
Half up looks at whether the part below the hundred reaches halfway, which is 50. The box digit is what decides that.
24004→24000,24034→24000,24064→24100,24094→2410024004 \to 24000,\quad 24034 \to 24000,\quad 24064 \to 24100,\quad 24094 \to 24100
Half up gives 24000 depending on the digit.

3Match the two results

#5 Look for a Pattern 3.NBT.A.1
Keep the digits where both roundings land on the same hundred: 0, 1, 2, 3, 4. The others (5, 6, 7, 8, 9) disagree.
□∈{0,1,2,3,4}\square \in \{0, 1, 2, 3, 4\}
The answer is 0, 1, 2, 3, 4.
Answer: 0, 1, 2, 3, 4
4 · Reviewdoes it hold up?

Spot-check a boundary: with 0 in the box the number is 24004 and both roundings give 24000; with 5 they part company.

Another way: Mark the halfway point on a number line at 24050 and see which digits put the number on the far side; the same digits survive.

Standardsmin grade 5
  • 3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Rounding down to the nearest hundred for each digit.
  • 5.NBT.A.4 Round decimals to any place — Applying round-half-up and comparing the two results.
💡Takeaway. Rounding half up asks one question: has the part below reached halfway? Find the halfway mark and the digits sort themselves.
Variant 10 hard answer: 5, 6, 7, 8, 9

For the 5-digit number 357□7357\square7, rounding it up to the nearest hundred gives the same result as rounding it to the nearest hundred (round half up).

Find every digit that can go in the □\square.

Show solution
1 · Understandwhat's really being asked

A number has one digit hidden, in the place that decides rounding to the nearest hundred. Rounding it up must land on the same answer as rounding it the normal half-up way. We need every digit for which that happens.

Givens
  • The number is 357, then the hidden digit, then 7.
  • Rounding up to the nearest hundred and rounding half up must agree.
Unknowns
  • Every digit the box can hold.
Constraints
  • The hidden digit is one of 0 through 9.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #5 Look for a Pattern

Work out what each rounding gives, digit by digit. Rounding up barely depends on the digit; rounding half up depends on it entirely, so comparing them shows exactly where the two agree.

3 · Execute3 carry out the plan

1Find what 'round up' gives

#2 Make a Systematic List 3.NBT.A.1
Rounding up always goes to the next hundred above, no matter which digit is in the box, because the lower digits are never all zero.
35707→35800,35797→3580035707 \to 35800,\quad 35797 \to 35800
That side of the comparison is fixed at 35800.

2See when 'round half up' also gives 35800

#2 Make a Systematic List 5.NBT.A.4
Half up looks at whether the part below the hundred reaches halfway, which is 50. The box digit is what decides that.
35707→35700,35737→35700,35767→35800,35797→3580035707 \to 35700,\quad 35737 \to 35700,\quad 35767 \to 35800,\quad 35797 \to 35800
Half up gives 35800 depending on the digit.

3Match the two results

#5 Look for a Pattern 3.NBT.A.1
Keep the digits where both roundings land on the same hundred: 5, 6, 7, 8, 9. The others (0, 1, 2, 3, 4) disagree.
□∈{5,6,7,8,9}\square \in \{5, 6, 7, 8, 9\}
The answer is 5, 6, 7, 8, 9.
Answer: 5, 6, 7, 8, 9
4 · Reviewdoes it hold up?

Spot-check a boundary: with 5 in the box the number is 35757 and both roundings give 35800; with 0 they part company.

Another way: Mark the halfway point on a number line at 35750 and see which digits put the number on the far side; the same digits survive.

Standardsmin grade 5
  • 3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Rounding up to the nearest hundred for each digit.
  • 5.NBT.A.4 Round decimals to any place — Applying round-half-up and comparing the two results.
💡Takeaway. Rounding half up asks one question: has the part below reached halfway? Find the halfway mark and the digits sort themselves.
Variant 11 hard answer: 0, 1, 2, 3, 4

For the 5-digit number 513□2513\square2, rounding it down to the nearest hundred gives the same result as rounding it to the nearest hundred (round half up).

Find every digit that can go in the □\square.

Show solution
1 · Understandwhat's really being asked

A number has one digit hidden, in the place that decides rounding to the nearest hundred. Rounding it down must land on the same answer as rounding it the normal half-up way. We need every digit for which that happens.

Givens
  • The number is 513, then the hidden digit, then 2.
  • Rounding down to the nearest hundred and rounding half up must agree.
Unknowns
  • Every digit the box can hold.
Constraints
  • The hidden digit is one of 0 through 9.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #5 Look for a Pattern

Work out what each rounding gives, digit by digit. Rounding down barely depends on the digit; rounding half up depends on it entirely, so comparing them shows exactly where the two agree.

3 · Execute3 carry out the plan

1Find what 'round down' gives

#2 Make a Systematic List 3.NBT.A.1
Rounding down always drops to the hundred below, no matter which digit is in the box.
51302→51300,51392→5130051302 \to 51300,\quad 51392 \to 51300
That side of the comparison is fixed at 51300.

2See when 'round half up' also gives 51300

#2 Make a Systematic List 5.NBT.A.4
Half up looks at whether the part below the hundred reaches halfway, which is 50. The box digit is what decides that.
51302→51300,51332→51300,51362→51400,51392→5140051302 \to 51300,\quad 51332 \to 51300,\quad 51362 \to 51400,\quad 51392 \to 51400
Half up gives 51300 depending on the digit.

3Match the two results

#5 Look for a Pattern 3.NBT.A.1
Keep the digits where both roundings land on the same hundred: 0, 1, 2, 3, 4. The others (5, 6, 7, 8, 9) disagree.
□∈{0,1,2,3,4}\square \in \{0, 1, 2, 3, 4\}
The answer is 0, 1, 2, 3, 4.
Answer: 0, 1, 2, 3, 4
4 · Reviewdoes it hold up?

Spot-check a boundary: with 0 in the box the number is 51302 and both roundings give 51300; with 5 they part company.

Another way: Mark the halfway point on a number line at 51350 and see which digits put the number on the far side; the same digits survive.

Standardsmin grade 5
  • 3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Rounding down to the nearest hundred for each digit.
  • 5.NBT.A.4 Round decimals to any place — Applying round-half-up and comparing the two results.
💡Takeaway. Rounding half up asks one question: has the part below reached halfway? Find the halfway mark and the digits sort themselves.
Variant 12 hard answer: 5, 6, 7, 8, 9

For the 5-digit number 704□5704\square5, rounding it up to the nearest hundred gives the same result as rounding it to the nearest hundred (round half up).

Find every digit that can go in the □\square.

Show solution
1 · Understandwhat's really being asked

A number has one digit hidden, in the place that decides rounding to the nearest hundred. Rounding it up must land on the same answer as rounding it the normal half-up way. We need every digit for which that happens.

Givens
  • The number is 704, then the hidden digit, then 5.
  • Rounding up to the nearest hundred and rounding half up must agree.
Unknowns
  • Every digit the box can hold.
Constraints
  • The hidden digit is one of 0 through 9.
2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #5 Look for a Pattern

Work out what each rounding gives, digit by digit. Rounding up barely depends on the digit; rounding half up depends on it entirely, so comparing them shows exactly where the two agree.

3 · Execute3 carry out the plan

1Find what 'round up' gives

#2 Make a Systematic List 3.NBT.A.1
Rounding up always goes to the next hundred above, no matter which digit is in the box, because the lower digits are never all zero.
70405→70500,70495→7050070405 \to 70500,\quad 70495 \to 70500
That side of the comparison is fixed at 70500.

2See when 'round half up' also gives 70500

#2 Make a Systematic List 5.NBT.A.4
Half up looks at whether the part below the hundred reaches halfway, which is 50. The box digit is what decides that.
70405→70400,70435→70400,70465→70500,70495→7050070405 \to 70400,\quad 70435 \to 70400,\quad 70465 \to 70500,\quad 70495 \to 70500
Half up gives 70500 depending on the digit.

3Match the two results

#5 Look for a Pattern 3.NBT.A.1
Keep the digits where both roundings land on the same hundred: 5, 6, 7, 8, 9. The others (0, 1, 2, 3, 4) disagree.
□∈{5,6,7,8,9}\square \in \{5, 6, 7, 8, 9\}
The answer is 5, 6, 7, 8, 9.
Answer: 5, 6, 7, 8, 9
4 · Reviewdoes it hold up?

Spot-check a boundary: with 5 in the box the number is 70455 and both roundings give 70500; with 0 they part company.

Another way: Mark the halfway point on a number line at 70450 and see which digits put the number on the far side; the same digits survive.

Standardsmin grade 5
  • 3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Rounding up to the nearest hundred for each digit.
  • 5.NBT.A.4 Round decimals to any place — Applying round-half-up and comparing the two results.
💡Takeaway. Rounding half up asks one question: has the part below reached halfway? Find the halfway mark and the digits sort themselves.