Practice · Round by looking at lower-place digits · Sensim Math

Variant 1 easy

For the 4-digit number $18\square1$ , 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 $\square$ .

Your answer (check it yourself)

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 \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 \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.

\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

For the 4-digit number $35\square8$ , 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 $\square$ .

Your answer (check it yourself)

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 \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 \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.

\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

For the 5-digit number $29\square47$ , 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 $\square$ .

Your answer (check it yourself)

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 \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 \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.

\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

For the 4-digit number $47\square3$ , 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 $\square$ .

Your answer (check it yourself)

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 \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 \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.

\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

For the 4-digit number $62\square6$ , 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 $\square$ .

Your answer (check it yourself)

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 \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 \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.

\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

For the 5-digit number $68\square25$ , 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 $\square$ .

Your answer (check it yourself)

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 \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 \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.

\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

For the 4-digit number $86\square9$ , 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 $\square$ .

Your answer (check it yourself)

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 \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 \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.

\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

For the 5-digit number $91\square72$ , 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 $\square$ .

Your answer (check it yourself)

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 \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 \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.

\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

For the 5-digit number $240\square4$ , 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 $\square$ .

Your answer (check it yourself)

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 \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 \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.

\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

For the 5-digit number $357\square7$ , 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 $\square$ .

Your answer (check it yourself)

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 \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 \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.

\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

For the 5-digit number $513\square2$ , 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 $\square$ .

Your answer (check it yourself)

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 \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 \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.

\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

For the 5-digit number $704\square5$ , 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 $\square$ .

Your answer (check it yourself)

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 \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 \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.

\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.

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

Generated variants — 12

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