Problem 19 · 2021 Math Kangaroo
Medium
Logic & Word Problems
complementary-counting
There were 20 apples and 20 pears in a box. Carl randomly took 20 pieces of fruit from the box and Luca took the rest. Which of the following statements is always true?
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Answer: D — Carl got as many pears as Luca got apples.
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Hint 1 of 2
Carl took exactly 20 fruits, and there are exactly 20 apples in the whole box — the same number.
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Hint 2 of 2
Every apple Carl did NOT take is left for Luca, so think about how Carl filled up his 20 spots.
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Approach: match Carl's missing apples to his pears
- Carl grabbed 20 fruits, and there are 20 apples in all, so the number of apples Carl is missing is exactly the same as the number of pears he picked up to fill his 20 spots.
- All the apples Carl skipped end up with Luca, so the apples Carl missed = the apples Luca got.
- Putting those together: Carl's pears = the apples Carl missed = Luca's apples, so 'Carl got as many pears as Luca got apples' is always true — choice D.
With letters
If Carl took \(a\) apples and \(p\) pears then \(a+p=20\); Luca has \(20-a\) apples, and \(p=20-a\) too, so Carl's pears equal Luca's apples.
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