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## Saturday, July 26, 2008

### Speedy Sums

A salesman drives from Amsterdam to The Hague. The first half of the distance of his journey, he drives at a constant speed of 80 km/h. The second half of the distance of his journey, he drives at a constant speed of 120 km/h.

First Question: What is the salesman's average speed for the complete journey?

A Hint : The solution is not 100 km/h!

Second Question: A race car driver drove, on a 4 km long race course, at an average speed of 120 km/h for the first 2 km. How fast does he have to go the second 2 km to average 240 km/h for the entire course?

Third Question: Makkum and Stavoren are two villages. Michael and Donald want to go from Makkum to Stavoren. They leave at the same time. Michael goes by bicycle. Donald goes by car, which is six times as fast as Michael on his bicycle. Unfortunately, Donald has a car breakdown half-way between Makkum and Stavoren. Fortunately, a passing farmer gives him a lift to Stavoren on his tractor. Unfortunately, the farmer drives only half as fast as Michael drives on his bicycle. Who of the two arrives first in Stavoren?

Fourth Question: Normally, the train between Utrecht and Amersfoort drives at an average speed of 90 km/h. One day, the train was delayed a little. Because of this, the average speed of the train between Utrecht and Amersfoort was only 70 km/h, and the train arrived four minutes late in Amersfoort. What is the distance between the stations of Utrecht and Amersfoort?

First Answer: Let the length of the journey be n km. The first half of the journey takes (1/2×n)/80 = 1/160×n hours, and the second half of the journey takes (1/2×n)/120 = 1/240×n hours. The average speed for the complete journey is n/(1/160×n+1/240×n) = 96 km/h.

Second Answer: At 120 km/h, it took the race car driver 1 minute to cover the first 2 km. To reach an average speed of 240 km/h for 4 km, that distance must be travelled in 1 minute. But that time was already used up in the first half of the course. So the answer is that, no matter how fast the race car drives in the second part of the course, it is impossible to average 240 km/h for the entire course!

Third Answer: Because Donald travels the second part of the journey at half the speed at which Michael travels the whole journey, it takes Donald exactly as much time for the the second part the journey as it takes Michael for the whole journey. So, no matter how fast Donald is in the first part of the journey, Michael will arrive first in Stavoren.

Fourth Answer: Let the distance between the stations of Utrecht and Amersfoort be d km. The four minutes delay is 4/60 part of an hour. Now the following holds:

d / 90 + 4/60 = d / 70.

Solving this equation gives d=21, so the distance between the stations of Utrecht and Amersfoort is 21 km.