Problem 15 · 2013 Math Kangaroo
Hard
Counting & Probability
careful-counting
How many different subtractions of two two-digit numbers give the answer 50?
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Answer: A — 40
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Hint 1 of 3
You want two two-digit numbers whose difference is exactly 50.
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Hint 2 of 3
Once you pick the bigger number, the smaller one is forced (it is 50 less).
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Hint 3 of 3
The smaller number must still have two digits (be at least 10), so see which bigger numbers are allowed.
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Approach: count valid values of the larger number
- Pick the bigger number; the smaller one must be 50 less, and it still has to be a two-digit number (at least 10).
- Being 50 less and at least 10 means the bigger number runs from 60 up to 99, and each choice gives exactly one subtraction.
- That is \(99 - 60 + 1 = 40\) different subtractions, which is choice A.
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