PRIZE PUZZLE FOR OCTOBER 2004
THE SMART GOLF CLUBThe Smart Golf Club has less than 200 members. 39.27% of the members are women. How many men are in the club?
In real life, where people throw numbers around, most of the time they are approximations rounded to a few digits. If someone in authority says that 5.7% of the population are out of work it is obvious that this is a rounded figure and that the true value is somewhere between 5.65 and 5.75%. In other words, add and subtract half the last unit value. (Sometimes, in more scientific works, an apparently useless zero is appended to the fraction part of a numerical quantity. But the zero is not redundant: it shows extra precision. For example, if the figure was 5.70% one should assume a range between 5.695 and 5.705%.)
In this problem it soon becomes clear that the percentage of women is not precisely 39.27% because if it were, there would have to be at least 10000 members with 3927 of them women. (That is because there are no common factors of 3927 and 10000.) We are told there are less then 200 members, so we must look for a ratio a/b where b is less than 200 and a/b is in the range 0.39265 and 0.39275.
Finding approximations like this is best done using a technique called continued fractions. What are continued fractions?
If we work out the continued fraction expansion of the two limit values until we get two terms that differ:
0.39265 = /2,1,1,4,1,5.../
and now add 1 to the smaller of 5 and 11 and terminate it there we have:
which evaluates to 75/191.
75/191 = 0.39267... which is within the range we want.
So 75 of 191 members are women. If the other kind of person is restricted to men, that leaves 116 members men.
Most solvers searched for the answer using either a computer program or a spreadsheet or by hard work with a hand calculator. I received correct answers from John Stafford, Peter Greenhalgh, Clem Robertson and Rosemary West. Rosemary won the random draw and a £5 book token. Well done all.