5. Yesterday at the market, with one hundred tugriks you could buy 9 gingerbreads and 7 pastries (and even get some change), but today this amount is no longer enough. However, with the same one hundred tugriks today, you can buy two gingerbreads and 11 pastries (also with some change), but yesterday this amount would not have been enough. A gingerbread and a pastry cost a whole number of tugriks, and the price of each sweet changed by no more than one tugrik overnight. How much does one gingerbread cost today?
Problem 845
Official solution
Solution. Note that something has become more expensive (otherwise, the situation where we could buy a set yesterday but not today could not have arisen), and another sweet has become cheaper for similar reasons. Nine cookies and seven cakes together have become more expensive, so the cookie has become more expensive, and the cake has become cheaper. This set has become 2 tugriks more expensive overnight, i.e., it cost 99 tugriks yesterday and 101 tugriks today. Thus, today 9 cookies = 101 tugriks - 7 cakes, i.e., by subtracting a multiple of 7 from 101, we should get a number that is a multiple of 9. This is only possible in one case: 101 - 56 = 45. Then a cookie today costs 5 tugriks, and a cake costs 8 tugriks.