PK 2018 Q14
14. Peter wanted to buy a book, but he didn't have any money. He bought it with the help of his father and his two brothers. His father gave him half of the amount given by his brothers. His elder brother gave him one third of what the others gave. The younger brother gave him 10 euros. What was the price of the book?
- A 24 euros
- B 26 euros
- C 28 euros
- D 30 euros
- E 32 euros
Pink Kangaroo 2018, Question 14. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: A 24 euros
Let \(D\) be the amount that Peter's dad gave, and \(B\) the amount his elder brother gave.
Then \(D = \tfrac{1}{2}(B + 10) \ldots\) (1) and \(B = \tfrac{1}{3}(D + 10) \ldots\) (2).
Substitute (1) into (2) to get \(B = \tfrac{1}{3}\left(\tfrac{1}{2}(B + 10) + 10\right) = \tfrac{1}{6}B + 5\).
Hence \(\tfrac{5}{6}B = 5\), so \(B = 6\) and \(D = \tfrac{1}{2}(6 + 10) = 8\). This gives a total of 6 + 8 + 10 = 24. So the book cost 24 euros.
Solution © UK Mathematics Trust, from the PK 2018 solutions.