Problem 16 · 2018 Math Kangaroo
Hard
Algebra & Patterns
work-backward
Peter wants to buy a book but has no money. His father and his two brothers help. His father gives him half as much as his two brothers give jointly. His older brother gives a third of what the other two give him. The youngest brother gives 10 €. How expensive is the book?
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Answer: A — 24 €
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Hint 1 of 2
Write the total as father + older brother + youngest brother and use each clue as a fraction of the total.
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Hint 2 of 2
Father is one third of the whole; the older brother is one quarter of the whole.
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Approach: express each gift as a fraction of the total
- Father gives half of what the brothers give, so father is \(\tfrac13\) of the total cost.
- The older brother gives a third of what the other two give, so he is \(\tfrac14\) of the total.
- The youngest brother's 10 € is the remaining \(1-\tfrac13-\tfrac14 = \tfrac{5}{12}\) of the total.
- So the total is \(10\div\tfrac{5}{12} = \) 24 €.
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