← Choose digits for sum nearest a target · Get Closest to a Target Value

Choose digits for sum nearest a target · 10 practice problems

3.NBT.A.23.NBT.A.13.OA.D.8

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Generated variants — 10

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

Variant 1 easy

$\bigcirc$ is a three-digit number whose tens digit and ones digit are the same. When the sum $155 + \bigcirc$ is as close as possible to $300$ , find the number that $\bigcirc$ stands for.

Your answer

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1 · Understandwhat's really being asked

A three-digit number $\bigcirc$ has matching tens and ones digits. We add it to 155 and want that sum to be as close to 300 as possible. Find the $\bigcirc$ .

Givens

155 + $\bigcirc$ is the sum
The $\bigcirc$ is a three-digit number whose tens digit equals its ones digit
We want the sum as close to 300 as possible

Unknowns

The three-digit number the $\bigcirc$ stands for

Constraints

The tens digit and ones digit of the $\bigcirc$ must be equal
Closeness means the smallest possible gap between the sum and 300

2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

Work backwards from the target: the ideal $\bigcirc$ is 300 - 155 = 145. Then guess-and-check the nearby numbers that have equal tens and ones digits to see which lands closest to 300.

3 · Execute3 carry out the plan

1Find the ideal addend

#11 Work Backwards 3.NBT.A.2

For the sum to equal 300 exactly, the

\bigcirc

would need to be 300 - 155 = 145. But 145 does not have equal tens and ones digits, so 145 itself is not allowed; the real answer must be the nearest allowed number to it.

300 - 155 = 145

Subtracting tells us the perfect addend; the real answer must be the nearest allowed number to it.

2List allowed numbers near 145

#6 Guess and Check 3.NBT.A.2

Numbers with equal tens and ones digits near 145 are 144 (tens 4, ones 4) and 155 (tens 5, ones 5). Check how close each sum gets to 300.

155 + 144 = 299,\quad 155 + 155 = 310

Only a couple of valid candidates sit on either side of the ideal, so checking them is enough.

3Compare the gaps

#6 Guess and Check 3.OA.D.8

299 is 1 away from 300, while 310 is 10 away from 300. Since 1 is smaller, the number 144 gives the sum closest to 300.

|300 - 299| = 1 < 10 = |310 - 300|

The closest sum is the one with the smallest distance to the target, so we pick the smaller gap.

Answer: 144

4 · Reviewdoes it hold up?

144 has equal tens and ones digits, and 155 + 144 = 299, just 1 from 300. No nearby twin-digit number lands closer, so 144 is genuinely closest. The answer is reasonable.

Another way: Round to estimate (tool aligned with 3.NBT.A.1): 300 - 155 rounds toward 145, then nudge to the nearest valid digit-pattern number, again giving 144.

Standardsmin grade 3

3.NBT.A.2 Fluently add and subtract within 1000 — Computing 300 - 155 and the candidate sums near 300
3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Reasoning about which candidate lands nearest the target 300
3.OA.D.8 Solve two-step word problems using four operations within 100 — Comparing the distances to 300 to choose the closest sum

💡Takeaway. Aim for the perfect number first (145), then pick the closest allowed twin-digit number to it!

Variant 2 easy

$\bigcirc$ is a three-digit number whose tens digit and ones digit are the same. When the sum $264 + \bigcirc$ is as close as possible to $400$ , find the number that $\bigcirc$ stands for.

Your answer

Show solution

1 · Understandwhat's really being asked

A three-digit number $\bigcirc$ has matching tens and ones digits. We add it to 264 and want that sum to be as close to 400 as possible. Find the $\bigcirc$ .

Givens

264 + $\bigcirc$ is the sum
The $\bigcirc$ is a three-digit number whose tens digit equals its ones digit
We want the sum as close to 400 as possible

Unknowns

The three-digit number the $\bigcirc$ stands for

Constraints

The tens digit and ones digit of the $\bigcirc$ must be equal
Closeness means the smallest possible gap between the sum and 400

2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

Work backwards from the target: the ideal $\bigcirc$ is 400 - 264 = 136. Then guess-and-check the nearby numbers that have equal tens and ones digits to see which lands closest to 400.

3 · Execute3 carry out the plan

1Find the ideal addend

#11 Work Backwards 3.NBT.A.2

For the sum to equal 400 exactly, the

\bigcirc

would need to be 400 - 264 = 136. But 136 does not have equal tens and ones digits, so 136 itself is not allowed; the real answer must be the nearest allowed number to it.

400 - 264 = 136

Subtracting tells us the perfect addend; the real answer must be the nearest allowed number to it.

2List allowed numbers near 136

#6 Guess and Check 3.NBT.A.2

Numbers with equal tens and ones digits near 136 are 133 (tens 3, ones 3) and 144 (tens 4, ones 4). Check how close each sum gets to 400.

264 + 133 = 397,\quad 264 + 144 = 408

Only a couple of valid candidates sit on either side of the ideal, so checking them is enough.

3Compare the gaps

#6 Guess and Check 3.OA.D.8

397 is 3 away from 400, while 408 is 8 away from 400. Since 3 is smaller, the number 133 gives the sum closest to 400.

|400 - 397| = 3 < 8 = |408 - 400|

The closest sum is the one with the smallest distance to the target, so we pick the smaller gap.

Answer: 133

4 · Reviewdoes it hold up?

133 has equal tens and ones digits, and 264 + 133 = 397, just 3 from 400. No nearby twin-digit number lands closer, so 133 is genuinely closest. The answer is reasonable.

Another way: Round to estimate (tool aligned with 3.NBT.A.1): 400 - 264 rounds toward 136, then nudge to the nearest valid digit-pattern number, again giving 133.

Standardsmin grade 3

3.NBT.A.2 Fluently add and subtract within 1000 — Computing 400 - 264 and the candidate sums near 400
3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Reasoning about which candidate lands nearest the target 400
3.OA.D.8 Solve two-step word problems using four operations within 100 — Comparing the distances to 400 to choose the closest sum

💡Takeaway. Aim for the perfect number first (136), then pick the closest allowed twin-digit number to it!

Variant 3 medium

$\bigcirc$ is a three-digit number whose tens digit and ones digit are the same. When the sum $333 + \bigcirc$ is as close as possible to $500$ , find the number that $\bigcirc$ stands for.

Your answer

Show solution

1 · Understandwhat's really being asked

A three-digit number $\bigcirc$ has matching tens and ones digits. We add it to 333 and want that sum to be as close to 500 as possible. Find the $\bigcirc$ .

Givens

333 + $\bigcirc$ is the sum
The $\bigcirc$ is a three-digit number whose tens digit equals its ones digit
We want the sum as close to 500 as possible

Unknowns

The three-digit number the $\bigcirc$ stands for

Constraints

The tens digit and ones digit of the $\bigcirc$ must be equal
Closeness means the smallest possible gap between the sum and 500

2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

Work backwards from the target: the ideal $\bigcirc$ is 500 - 333 = 167. Then guess-and-check the nearby numbers that have equal tens and ones digits to see which lands closest to 500.

3 · Execute3 carry out the plan

1Find the ideal addend

#11 Work Backwards 3.NBT.A.2

For the sum to equal 500 exactly, the

\bigcirc

would need to be 500 - 333 = 167. But 167 does not have equal tens and ones digits, so 167 itself is not allowed; the real answer must be the nearest allowed number to it.

500 - 333 = 167

Subtracting tells us the perfect addend; the real answer must be the nearest allowed number to it.

2List allowed numbers near 167

#6 Guess and Check 3.NBT.A.2

Numbers with equal tens and ones digits near 167 are 166 (tens 6, ones 6) and 177 (tens 7, ones 7). Check how close each sum gets to 500.

333 + 166 = 499,\quad 333 + 177 = 510

Only a couple of valid candidates sit on either side of the ideal, so checking them is enough.

3Compare the gaps

#6 Guess and Check 3.OA.D.8

499 is 1 away from 500, while 510 is 10 away from 500. Since 1 is smaller, the number 166 gives the sum closest to 500.

|500 - 499| = 1 < 10 = |510 - 500|

The closest sum is the one with the smallest distance to the target, so we pick the smaller gap.

Answer: 166

4 · Reviewdoes it hold up?

166 has equal tens and ones digits, and 333 + 166 = 499, just 1 from 500. No nearby twin-digit number lands closer, so 166 is genuinely closest. The answer is reasonable.

Another way: Round to estimate (tool aligned with 3.NBT.A.1): 500 - 333 rounds toward 167, then nudge to the nearest valid digit-pattern number, again giving 166.

Standardsmin grade 3

3.NBT.A.2 Fluently add and subtract within 1000 — Computing 500 - 333 and the candidate sums near 500
3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Reasoning about which candidate lands nearest the target 500
3.OA.D.8 Solve two-step word problems using four operations within 100 — Comparing the distances to 500 to choose the closest sum

💡Takeaway. Aim for the perfect number first (167), then pick the closest allowed twin-digit number to it!

Variant 4 easy

$\bigcirc$ is a three-digit number whose tens digit and ones digit are the same. When the sum $382 + \bigcirc$ is as close as possible to $500$ , find the number that $\bigcirc$ stands for.

Your answer

Show solution

1 · Understandwhat's really being asked

A three-digit number $\bigcirc$ has matching tens and ones digits. We add it to 382 and want that sum to be as close to 500 as possible. Find the $\bigcirc$ .

Givens

382 + $\bigcirc$ is the sum
The $\bigcirc$ is a three-digit number whose tens digit equals its ones digit
We want the sum as close to 500 as possible

Unknowns

The three-digit number the $\bigcirc$ stands for

Constraints

The tens digit and ones digit of the $\bigcirc$ must be equal
Closeness means the smallest possible gap between the sum and 500

2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

Work backwards from the target: the ideal $\bigcirc$ is 500 - 382 = 118. Then guess-and-check the nearby numbers that have equal tens and ones digits to see which lands closest to 500.

3 · Execute3 carry out the plan

1Find the ideal addend

#11 Work Backwards 3.NBT.A.2

For the sum to equal 500 exactly, the

\bigcirc

would need to be 500 - 382 = 118. But 118 does not have equal tens and ones digits, so 118 itself is not allowed; the real answer must be the nearest allowed number to it.

500 - 382 = 118

Subtracting tells us the perfect addend; the real answer must be the nearest allowed number to it.

2List allowed numbers near 118

#6 Guess and Check 3.NBT.A.2

Numbers with equal tens and ones digits near 118 are 122 (tens 2, ones 2) and 111 (tens 1, ones 1). Check how close each sum gets to 500.

382 + 122 = 504,\quad 382 + 111 = 493

Only a couple of valid candidates sit on either side of the ideal, so checking them is enough.

3Compare the gaps

#6 Guess and Check 3.OA.D.8

504 is 4 away from 500, while 493 is 7 away from 500. Since 4 is smaller, the number 122 gives the sum closest to 500.

|500 - 504| = 4 < 7 = |493 - 500|

The closest sum is the one with the smallest distance to the target, so we pick the smaller gap.

Answer: 122

4 · Reviewdoes it hold up?

122 has equal tens and ones digits, and 382 + 122 = 504, just 4 from 500. No nearby twin-digit number lands closer, so 122 is genuinely closest. The answer is reasonable.

Another way: Round to estimate (tool aligned with 3.NBT.A.1): 500 - 382 rounds toward 118, then nudge to the nearest valid digit-pattern number, again giving 122.

Standardsmin grade 3

3.NBT.A.2 Fluently add and subtract within 1000 — Computing 500 - 382 and the candidate sums near 500
3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Reasoning about which candidate lands nearest the target 500
3.OA.D.8 Solve two-step word problems using four operations within 100 — Comparing the distances to 500 to choose the closest sum

💡Takeaway. Aim for the perfect number first (118), then pick the closest allowed twin-digit number to it!

Variant 5 medium

$\bigcirc$ is a three-digit number whose tens digit and ones digit are the same. When the sum $475 + \bigcirc$ is as close as possible to $600$ , find the number that $\bigcirc$ stands for.

Your answer

Show solution

1 · Understandwhat's really being asked

A three-digit number $\bigcirc$ has matching tens and ones digits. We add it to 475 and want that sum to be as close to 600 as possible. Find the $\bigcirc$ .

Givens

475 + $\bigcirc$ is the sum
The $\bigcirc$ is a three-digit number whose tens digit equals its ones digit
We want the sum as close to 600 as possible

Unknowns

The three-digit number the $\bigcirc$ stands for

Constraints

The tens digit and ones digit of the $\bigcirc$ must be equal
Closeness means the smallest possible gap between the sum and 600

2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

Work backwards from the target: the ideal $\bigcirc$ is 600 - 475 = 125. Then guess-and-check the nearby numbers that have equal tens and ones digits to see which lands closest to 600.

3 · Execute3 carry out the plan

1Find the ideal addend

#11 Work Backwards 3.NBT.A.2

For the sum to equal 600 exactly, the

\bigcirc

would need to be 600 - 475 = 125. But 125 does not have equal tens and ones digits, so 125 itself is not allowed; the real answer must be the nearest allowed number to it.

600 - 475 = 125

Subtracting tells us the perfect addend; the real answer must be the nearest allowed number to it.

2List allowed numbers near 125

#6 Guess and Check 3.NBT.A.2

Numbers with equal tens and ones digits near 125 are 122 (tens 2, ones 2) and 133 (tens 3, ones 3). Check how close each sum gets to 600.

475 + 122 = 597,\quad 475 + 133 = 608

Only a couple of valid candidates sit on either side of the ideal, so checking them is enough.

3Compare the gaps

#6 Guess and Check 3.OA.D.8

597 is 3 away from 600, while 608 is 8 away from 600. Since 3 is smaller, the number 122 gives the sum closest to 600.

|600 - 597| = 3 < 8 = |608 - 600|

The closest sum is the one with the smallest distance to the target, so we pick the smaller gap.

Answer: 122

4 · Reviewdoes it hold up?

122 has equal tens and ones digits, and 475 + 122 = 597, just 3 from 600. No nearby twin-digit number lands closer, so 122 is genuinely closest. The answer is reasonable.

Another way: Round to estimate (tool aligned with 3.NBT.A.1): 600 - 475 rounds toward 125, then nudge to the nearest valid digit-pattern number, again giving 122.

Standardsmin grade 3

3.NBT.A.2 Fluently add and subtract within 1000 — Computing 600 - 475 and the candidate sums near 600
3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Reasoning about which candidate lands nearest the target 600
3.OA.D.8 Solve two-step word problems using four operations within 100 — Comparing the distances to 600 to choose the closest sum

💡Takeaway. Aim for the perfect number first (125), then pick the closest allowed twin-digit number to it!

Variant 6 medium

$\bigcirc$ is a three-digit number whose tens digit and ones digit are the same. When the sum $450 + \bigcirc$ is as close as possible to $600$ , find the number that $\bigcirc$ stands for.

Your answer

Show solution

1 · Understandwhat's really being asked

A three-digit number $\bigcirc$ has matching tens and ones digits. We add it to 450 and want that sum to be as close to 600 as possible. Find the $\bigcirc$ .

Givens

450 + $\bigcirc$ is the sum
The $\bigcirc$ is a three-digit number whose tens digit equals its ones digit
We want the sum as close to 600 as possible

Unknowns

The three-digit number the $\bigcirc$ stands for

Constraints

The tens digit and ones digit of the $\bigcirc$ must be equal
Closeness means the smallest possible gap between the sum and 600

2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

Work backwards from the target: the ideal $\bigcirc$ is 600 - 450 = 150. Then guess-and-check the nearby numbers that have equal tens and ones digits to see which lands closest to 600.

3 · Execute3 carry out the plan

1Find the ideal addend

#11 Work Backwards 3.NBT.A.2

For the sum to equal 600 exactly, the

\bigcirc

would need to be 600 - 450 = 150. But 150 does not have equal tens and ones digits, so 150 itself is not allowed; the real answer must be the nearest allowed number to it.

600 - 450 = 150

Subtracting tells us the perfect addend; the real answer must be the nearest allowed number to it.

2List allowed numbers near 150

#6 Guess and Check 3.NBT.A.2

Numbers with equal tens and ones digits near 150 are 155 (tens 5, ones 5) and 144 (tens 4, ones 4). Check how close each sum gets to 600.

450 + 155 = 605,\quad 450 + 144 = 594

Only a couple of valid candidates sit on either side of the ideal, so checking them is enough.

3Compare the gaps

#6 Guess and Check 3.OA.D.8

605 is 5 away from 600, while 594 is 6 away from 600. Since 5 is smaller, the number 155 gives the sum closest to 600.

|600 - 605| = 5 < 6 = |594 - 600|

The closest sum is the one with the smallest distance to the target, so we pick the smaller gap.

Answer: 155

4 · Reviewdoes it hold up?

155 has equal tens and ones digits, and 450 + 155 = 605, just 5 from 600. No nearby twin-digit number lands closer, so 155 is genuinely closest. The answer is reasonable.

Another way: Round to estimate (tool aligned with 3.NBT.A.1): 600 - 450 rounds toward 150, then nudge to the nearest valid digit-pattern number, again giving 155.

Standardsmin grade 3

3.NBT.A.2 Fluently add and subtract within 1000 — Computing 600 - 450 and the candidate sums near 600
3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Reasoning about which candidate lands nearest the target 600
3.OA.D.8 Solve two-step word problems using four operations within 100 — Comparing the distances to 600 to choose the closest sum

💡Takeaway. Aim for the perfect number first (150), then pick the closest allowed twin-digit number to it!

Variant 7 medium

$\bigcirc$ is a three-digit number whose tens digit and ones digit are the same. When the sum $520 + \bigcirc$ is as close as possible to $700$ , find the number that $\bigcirc$ stands for.

Your answer

Show solution

1 · Understandwhat's really being asked

A three-digit number $\bigcirc$ has matching tens and ones digits. We add it to 520 and want that sum to be as close to 700 as possible. Find the $\bigcirc$ .

Givens

520 + $\bigcirc$ is the sum
The $\bigcirc$ is a three-digit number whose tens digit equals its ones digit
We want the sum as close to 700 as possible

Unknowns

The three-digit number the $\bigcirc$ stands for

Constraints

The tens digit and ones digit of the $\bigcirc$ must be equal
Closeness means the smallest possible gap between the sum and 700

2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

Work backwards from the target: the ideal $\bigcirc$ is 700 - 520 = 180. Then guess-and-check the nearby numbers that have equal tens and ones digits to see which lands closest to 700.

3 · Execute3 carry out the plan

1Find the ideal addend

#11 Work Backwards 3.NBT.A.2

For the sum to equal 700 exactly, the

\bigcirc

would need to be 700 - 520 = 180. But 180 does not have equal tens and ones digits, so 180 itself is not allowed; the real answer must be the nearest allowed number to it.

700 - 520 = 180

Subtracting tells us the perfect addend; the real answer must be the nearest allowed number to it.

2List allowed numbers near 180

#6 Guess and Check 3.NBT.A.2

Numbers with equal tens and ones digits near 180 are 177 (tens 7, ones 7) and 188 (tens 8, ones 8). Check how close each sum gets to 700.

520 + 177 = 697,\quad 520 + 188 = 708

Only a couple of valid candidates sit on either side of the ideal, so checking them is enough.

3Compare the gaps

#6 Guess and Check 3.OA.D.8

697 is 3 away from 700, while 708 is 8 away from 700. Since 3 is smaller, the number 177 gives the sum closest to 700.

|700 - 697| = 3 < 8 = |708 - 700|

The closest sum is the one with the smallest distance to the target, so we pick the smaller gap.

Answer: 177

4 · Reviewdoes it hold up?

177 has equal tens and ones digits, and 520 + 177 = 697, just 3 from 700. No nearby twin-digit number lands closer, so 177 is genuinely closest. The answer is reasonable.

Another way: Round to estimate (tool aligned with 3.NBT.A.1): 700 - 520 rounds toward 180, then nudge to the nearest valid digit-pattern number, again giving 177.

Standardsmin grade 3

3.NBT.A.2 Fluently add and subtract within 1000 — Computing 700 - 520 and the candidate sums near 700
3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Reasoning about which candidate lands nearest the target 700
3.OA.D.8 Solve two-step word problems using four operations within 100 — Comparing the distances to 700 to choose the closest sum

💡Takeaway. Aim for the perfect number first (180), then pick the closest allowed twin-digit number to it!

Variant 8 hard

$\bigcirc$ is a three-digit number whose tens digit and ones digit are the same. When the sum $618 + \bigcirc$ is as close as possible to $800$ , find the number that $\bigcirc$ stands for.

Your answer

Show solution

1 · Understandwhat's really being asked

A three-digit number $\bigcirc$ has matching tens and ones digits. We add it to 618 and want that sum to be as close to 800 as possible. Find the $\bigcirc$ .

Givens

618 + $\bigcirc$ is the sum
The $\bigcirc$ is a three-digit number whose tens digit equals its ones digit
We want the sum as close to 800 as possible

Unknowns

The three-digit number the $\bigcirc$ stands for

Constraints

The tens digit and ones digit of the $\bigcirc$ must be equal
Closeness means the smallest possible gap between the sum and 800

2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

Work backwards from the target: the ideal $\bigcirc$ is 800 - 618 = 182. Then guess-and-check the nearby numbers that have equal tens and ones digits to see which lands closest to 800.

3 · Execute3 carry out the plan

1Find the ideal addend

#11 Work Backwards 3.NBT.A.2

For the sum to equal 800 exactly, the

\bigcirc

would need to be 800 - 618 = 182. But 182 does not have equal tens and ones digits, so 182 itself is not allowed; the real answer must be the nearest allowed number to it.

800 - 618 = 182

Subtracting tells us the perfect addend; the real answer must be the nearest allowed number to it.

2List allowed numbers near 182

#6 Guess and Check 3.NBT.A.2

Numbers with equal tens and ones digits near 182 are 177 (tens 7, ones 7) and 188 (tens 8, ones 8). Check how close each sum gets to 800.

618 + 177 = 795,\quad 618 + 188 = 806

Only a couple of valid candidates sit on either side of the ideal, so checking them is enough.

3Compare the gaps

#6 Guess and Check 3.OA.D.8

795 is 5 away from 800, while 806 is 6 away from 800. Since 5 is smaller, the number 177 gives the sum closest to 800.

|800 - 795| = 5 < 6 = |806 - 800|

The closest sum is the one with the smallest distance to the target, so we pick the smaller gap.

Answer: 177

4 · Reviewdoes it hold up?

177 has equal tens and ones digits, and 618 + 177 = 795, just 5 from 800. No nearby twin-digit number lands closer, so 177 is genuinely closest. The answer is reasonable.

Another way: Round to estimate (tool aligned with 3.NBT.A.1): 800 - 618 rounds toward 182, then nudge to the nearest valid digit-pattern number, again giving 177.

Standardsmin grade 3

3.NBT.A.2 Fluently add and subtract within 1000 — Computing 800 - 618 and the candidate sums near 800
3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Reasoning about which candidate lands nearest the target 800
3.OA.D.8 Solve two-step word problems using four operations within 100 — Comparing the distances to 800 to choose the closest sum

💡Takeaway. Aim for the perfect number first (182), then pick the closest allowed twin-digit number to it!

Variant 9 hard

$\bigcirc$ is a three-digit number whose tens digit and ones digit are the same. When the sum $727 + \bigcirc$ is as close as possible to $900$ , find the number that $\bigcirc$ stands for.

Your answer

Show solution

1 · Understandwhat's really being asked

A three-digit number $\bigcirc$ has matching tens and ones digits. We add it to 727 and want that sum to be as close to 900 as possible. Find the $\bigcirc$ .

Givens

727 + $\bigcirc$ is the sum
The $\bigcirc$ is a three-digit number whose tens digit equals its ones digit
We want the sum as close to 900 as possible

Unknowns

The three-digit number the $\bigcirc$ stands for

Constraints

The tens digit and ones digit of the $\bigcirc$ must be equal
Closeness means the smallest possible gap between the sum and 900

2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

Work backwards from the target: the ideal $\bigcirc$ is 900 - 727 = 173. Then guess-and-check the nearby numbers that have equal tens and ones digits to see which lands closest to 900.

3 · Execute3 carry out the plan

1Find the ideal addend

#11 Work Backwards 3.NBT.A.2

For the sum to equal 900 exactly, the

\bigcirc

would need to be 900 - 727 = 173. But 173 does not have equal tens and ones digits, so 173 itself is not allowed; the real answer must be the nearest allowed number to it.

900 - 727 = 173

Subtracting tells us the perfect addend; the real answer must be the nearest allowed number to it.

2List allowed numbers near 173

#6 Guess and Check 3.NBT.A.2

Numbers with equal tens and ones digits near 173 are 177 (tens 7, ones 7) and 166 (tens 6, ones 6). Check how close each sum gets to 900.

727 + 177 = 904,\quad 727 + 166 = 893

Only a couple of valid candidates sit on either side of the ideal, so checking them is enough.

3Compare the gaps

#6 Guess and Check 3.OA.D.8

904 is 4 away from 900, while 893 is 7 away from 900. Since 4 is smaller, the number 177 gives the sum closest to 900.

|900 - 904| = 4 < 7 = |893 - 900|

The closest sum is the one with the smallest distance to the target, so we pick the smaller gap.

Answer: 177

4 · Reviewdoes it hold up?

177 has equal tens and ones digits, and 727 + 177 = 904, just 4 from 900. No nearby twin-digit number lands closer, so 177 is genuinely closest. The answer is reasonable.

Another way: Round to estimate (tool aligned with 3.NBT.A.1): 900 - 727 rounds toward 173, then nudge to the nearest valid digit-pattern number, again giving 177.

Standardsmin grade 3

3.NBT.A.2 Fluently add and subtract within 1000 — Computing 900 - 727 and the candidate sums near 900
3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Reasoning about which candidate lands nearest the target 900
3.OA.D.8 Solve two-step word problems using four operations within 100 — Comparing the distances to 900 to choose the closest sum

💡Takeaway. Aim for the perfect number first (173), then pick the closest allowed twin-digit number to it!

Variant 10 hard

$\bigcirc$ is a three-digit number whose tens digit and ones digit are the same. When the sum $840 + \bigcirc$ is as close as possible to $1000$ , find the number that $\bigcirc$ stands for.

Your answer

Show solution

1 · Understandwhat's really being asked

A three-digit number $\bigcirc$ has matching tens and ones digits. We add it to 840 and want that sum to be as close to 1000 as possible. Find the $\bigcirc$ .

Givens

840 + $\bigcirc$ is the sum
The $\bigcirc$ is a three-digit number whose tens digit equals its ones digit
We want the sum as close to 1000 as possible

Unknowns

The three-digit number the $\bigcirc$ stands for

Constraints

The tens digit and ones digit of the $\bigcirc$ must be equal
Closeness means the smallest possible gap between the sum and 1000

2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

Work backwards from the target: the ideal $\bigcirc$ is 1000 - 840 = 160. Then guess-and-check the nearby numbers that have equal tens and ones digits to see which lands closest to 1000.

3 · Execute3 carry out the plan

1Find the ideal addend

#11 Work Backwards 3.NBT.A.2

For the sum to equal 1000 exactly, the

\bigcirc

would need to be 1000 - 840 = 160. But 160 does not have equal tens and ones digits, so 160 itself is not allowed; the real answer must be the nearest allowed number to it.

1000 - 840 = 160

Subtracting tells us the perfect addend; the real answer must be the nearest allowed number to it.

2List allowed numbers near 160

#6 Guess and Check 3.NBT.A.2

Numbers with equal tens and ones digits near 160 are 155 (tens 5, ones 5) and 166 (tens 6, ones 6). Check how close each sum gets to 1000.

840 + 155 = 995,\quad 840 + 166 = 1006

Only a couple of valid candidates sit on either side of the ideal, so checking them is enough.

3Compare the gaps

#6 Guess and Check 3.OA.D.8

995 is 5 away from 1000, while 1006 is 6 away from 1000. Since 5 is smaller, the number 155 gives the sum closest to 1000.

|1000 - 995| = 5 < 6 = |1006 - 1000|

The closest sum is the one with the smallest distance to the target, so we pick the smaller gap.

Answer: 155

4 · Reviewdoes it hold up?

155 has equal tens and ones digits, and 840 + 155 = 995, just 5 from 1000. No nearby twin-digit number lands closer, so 155 is genuinely closest. The answer is reasonable.

Another way: Round to estimate (tool aligned with 3.NBT.A.1): 1000 - 840 rounds toward 160, then nudge to the nearest valid digit-pattern number, again giving 155.

Standardsmin grade 3

3.NBT.A.2 Fluently add and subtract within 1000 — Computing 1000 - 840 and the candidate sums near 1000
3.NBT.A.1 Round whole numbers to the nearest 10 or 100 — Reasoning about which candidate lands nearest the target 1000
3.OA.D.8 Solve two-step word problems using four operations within 100 — Comparing the distances to 1000 to choose the closest sum

💡Takeaway. Aim for the perfect number first (160), then pick the closest allowed twin-digit number to it!

Where next?

← Back to the problem: Choose digits for sum nearest a target