Problem 15 · 2022 Math Kangaroo
Hard
Algebra & Patterns
substitutionsum-constraint
The sum of two positive integers is three times as big as their difference. The product of the two numbers is four times as big as their sum. How big is the sum of the two numbers?
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Answer: E — 18
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Hint 1 of 2
Translate both sentences into equations in the two numbers.
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Hint 2 of 2
The first equation forces a simple ratio between the numbers.
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Approach: solve the two equations
- a + b = 3(a - b) gives 4b = 2a, so a = 2b.
- a*b = 4(a + b): with a = 2b this is 2b^2 = 12b, so b = 6 and a = 12.
- Their sum is 12 + 6 = 18.
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