PK 2017 Q13
13. Four brothers have different heights. Tobias is shorter than Victor by the same amount by which he is taller than Peter. Oscar is shorter than Peter by the same amount as well. Tobias is 184 cm tall and the average height of all the four brothers is 178 cm. How tall is Oscar?
- A 160 cm
- B 166 cm
- C 172 cm
- D 184 cm
- E 190 cm
Pink Kangaroo 2017, Question 13. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: A 160 cm
Let \(k\) cm be the difference between the heights.
Then the heights in cm are: Tobias 184, Victor \(184 + k\), Peter \(184 - k\), and Oscar \(184 - 2k\). The mean is 178 so \(\tfrac{1}{4}\left(184 + 184 + k + 184 - k + 184 - 2k\right) = 178\). Hence \(4 \times 184 - 2k = 4 \times 178\), giving \(2k = 4 \times 184 - 4 \times 178 = 4 \times 6 = 24\). Hence \(k = 12\).
Therefore Oscar's height in cm is \(184 - 2 \times 12 = 160\) cm.
Solution © UK Mathematics Trust, from the PK 2017 solutions.