Problem 17 · 2024 Math Kangaroo
Stretch
Number Theory
place-valuesum-constraint
Mia has 3 cards, each showing a three-digit number. When she adds the three numbers she gets 782. Sadly a worm has eaten one digit on each card, so they now read 2 ? 3, 1 ? 4 and 4 1 ?. What do you get when you add the three digits the worm ate?
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Answer: D — 11
Show hints
Hint 1 of 3
First add up only the digits you can still see, putting a 0 in each eaten spot.
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Hint 2 of 3
Compare that total with 782 to see how much the eaten digits must add back.
Still stuck? Show hint 3 →
Hint 3 of 3
Remember: an eaten digit in a tens place is worth that many tens, and an eaten digit in a ones place is worth that many ones.
Show solution
Approach: add the visible digits first, then see how much the eaten digits must add back
- Treat each eaten spot as 0: the cards read 203, 104 and 410, which add to 717.
- But the real total is 782, so the eaten digits must add back 782 − 717 = 65.
- Two of the eaten digits sit in tens places, so together they are worth 60 (meaning those two digits add to 6); the third sits in a ones place worth 5 (so that digit is 5).
- The three eaten digits therefore add to 6 + 5 = 11 (D).
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