GK 2016 Q15
15. There are twenty students in a class. Some of them sit in pairs so that exactly one third of the boys sit with a girl, and exactly one half of the girls sit with a boy. How many boys are there in the class?
- A 9
- B 12
- C 15
- D 16
- E 18
Grey Kangaroo 2016, Question 15. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 12
Let the number of boys in the class be \(x\) and let the number of girls be \(y\). Since one third of the boys sit with a girl and one half of the girls sit with a boy, we have \(\tfrac{1}{3}x = \tfrac{1}{2}y\) and hence \(\tfrac{2}{3}x = y\). The total number of students in the class is 20. Therefore \(x + \tfrac{2}{3}x = 20\). Hence \(\tfrac{5}{3}x = 20\), which has solution \(x = 12\). Therefore there are 12 boys in the class.
Solution © UK Mathematics Trust, from the GK 2016 solutions.