Problem
Choose digits for sum nearest a target
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.
Base-ten numbersOperations
Your answer
Watch out It's tempting to answer 125, but its tens digit (2) and ones digit (5) differ, so it doesn't meet the circle's condition.
How to solve
Strategy Work Backwards — 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.
1STEP 1
Find the ideal addend
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.
3.NBT.A.2Work BackwardsFor the sum 475 + (circle) to reach 600 exactly, the circle would have to be 600 - 475 = 125.
Why?
To get from 475 up to 600 you add some amount, so the circle is exactly the amount that fills the gap between 475 and 600.
Why?
That gap-filling amount is found by subtracting 475 from 600, because subtracting undoes the adding and hands back the part that was added.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
Carrying out that subtraction gives 600 - 475 = 125, because you work one place at a time and trade a larger unit for ten smaller ones wherever the top digit is too small to take away from.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
2STEP 2
List allowed numbers near 125
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, 475 + 133 = 608
Only a couple of valid candidates sit on either side of 125, so checking both is enough.
3.NBT.A.2Guess And Check3STEP 3
Compare the gaps
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 less than 8 = |608 - 600|
The closest sum is the one with the smallest distance to the target, so we pick the smaller gap.
3.OA.D.8Guess And CheckAnswer
122
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.
Takeaway
Aim for the perfect number first (125), then pick the closest allowed twin-digit number to it!
- Find the ideal addend
- List allowed numbers near 125
- Compare the gaps
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested