IMC 2016 Q14
14. Tegwen has the same number of brothers as she has sisters. Each one of her brothers has 50% more sisters than brothers.
How many children are in Tegwen's family?
- A 5
- B 7
- C 9
- D 11
- E 13
Intermediate Mathematical Challenge 2016, Question 14. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D 11
Let there be \(g\) girls and \(b\) boys in Tegwen's family. Then, as she has the same number of brothers as she does sisters, \(b = g - 1\). Also, each of her brothers has 50% more sisters than brothers. Therefore \(g = \frac{3}{2}(b - 1)\). So \(b + 1 = \frac{3}{2}(b - 1)\) and hence \(2b + 2 = 3b - 3\). Rearranging this equation gives \(b = 5\). So \(g = 5 + 1 = 6\). Therefore there are 5 + 6 = 11 children in Tegwen's family.
Solution © UK Mathematics Trust, from the IMC 2016 solutions. Longer versions with investigations: Solutions and investigations (PDF).