Choose digits for sum nearest a target

3.NBT.A.23.NBT.A.13.OA.D.8 · take · grade 3

Archetype: Get Closest to a Target Value · step in a 4-type progression

$\bigcirc$ 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.

Show solution

Understand

A three-digit number (circle) 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 circle.

Givens

475 + (circle) is the sum
The circle 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 circle stands for

Constraints

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

Plan

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

Work backwards from the target: the ideal circle 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.

Execute

#11 Work Backwards 3.NBT.A.2

For the sum to equal 600 exactly, the circle would need to be 600 - 475 = 125. But 125 has tens digit 2 and ones digit 5, which are not equal, so 125 itself is not allowed.

600 - 475 = 125

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

#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 125, so checking both is enough.

#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 than 8, 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

Review

122 has equal tens and ones digits (2 and 2), and 475 + 122 = 597, just 3 below 600. The only nearby alternative, 133, overshoots to 608 (8 away), so 122 is genuinely closest. The answer is reasonable.

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.

Standards · min grade 3

3.NBT.A.2 Fluently add and subtract within 1000 — Computing 600 - 475 and the candidate sums 475 + 122 and 475 + 133
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 two distances to 600 to choose the closest sum

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