Problem 18 · 2022 Math Kangaroo
Hard
Algebra & Patterns
substitutionsum-constraint
Werner writes down some numbers whose sum is 22. Ria subtracts each of Werner’s numbers from 7 and writes down the results; her numbers add up to 34. How many numbers did Werner write down?
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Answer: B — 8
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Hint 1 of 2
Each of Ria's numbers is 7 minus one of Werner's; what does adding them all give in terms of how many numbers there are?
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Hint 2 of 2
If there are n numbers, Ria's total is 7n minus Werner's total; set that equal to 34.
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Approach: sum the transformed numbers
- If Werner wrote n numbers summing to 22, Ria's numbers sum to 7n - 22.
- Set 7n - 22 = 34, so 7n = 56 and n = 8.
- Werner wrote 8 numbers, the answer is B.
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