GK 2019 Q25
25. A train is made up of 18 carriages. There are 700 passengers travelling on the train. In any block of five adjacent carriages, there are 199 passengers in total. How many passengers in total are in the middle two carriages of the train?
- A 70
- B 77
- C 78
- D 96
- E 103
Grey Kangaroo 2019, Question 25. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D 96
Let \(n\) be the total number of passengers in the middle two carriages. These are the ninth and tenth carriages and so, since there are 199 passengers in any block of five adjacent carriages, the total number of passengers in the sixth, seventh and eighth carriages is \(199 - n\). There are 199 passengers in total in carriages 1 to 5 inclusive, in carriages 9 to 13 inclusive and in carriages 14 to 18 inclusive. Since there are 700 passengers in total on the train, we have \(199 + 199 - n + 199 + 199 = 700\). Therefore \(796 - n = 700\) and hence \(n = 96\). Therefore there are 96 passengers in total in the middle two carriages of the train.
Solution © UK Mathematics Trust, from the GK 2019 solutions.