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Schools Mathematics Grand Challenge

PROBLEM 7

At a party, John is in charge of pouring cola from bottles into glasses. Each bottle contains a little more cola than is needed to fill a glass. After filling 6 identical glasses using 6 identical bottles (one for each glass), John has enough left in the bottles to fill exactly 1 bottle. If John has 36 full bottles to start with, what is the total number of glasses he can completly fill?

All of the glasses and all of the bottles have the same volume.
HINT: There may be a little cola left at the end, but not enough to fill a glass.

PROBLEM 8

Look at the picture below. It consists of two identical semicircles centred at a and c respectively and one quarter circle with centre at b. What is the ratio between the areas of region I and region II. (If you divide the area of region I by the area of region II, what answer will you get?)

PROBLEM 9

A rally driver recently tested the team's new car on a private track that is closed to other traffic for safety reasons. Part of the track is flat and the rest of it is sloped either uphill or downhill. At all times during the test the rally driver drove according to the following rules:

1. when driving on a flat road, he drove at a constant speed of 126 km/h (kilometres per hour);
2. when driving uphill, he drove at a constant speed of 112 km/h;
3. when driving downhill, he drove at a constant speed of 144 km/h.

In the test the driver drove according to these rules from the start of the track to its end, and then drove back from the end of the track to its start. It took the driver exactly 2 hours to go from the start to the end of the track, and it took exactly 2 hours and 20 minutes to do the return journey from the end to the start.

How many kilometres long is the track?

NOTE: There are uphill, flat and downhill sections on the track.