## 1230.42 – Who is Going with Whom?

Three couples solved the mystery of the Senior Bulletin Board and together collected the (lavish!) 5400 reward. They divided the prize amongst themselves unevenly. The women together received 2400 dollars. Sue had 200 dollars more than Jan and Lynn had 200 dollars more than Sue. Lou got half as much as his girlfriend; Bob got the same amount as his girlfriend; and Matt got twice as much as his girlfriend. Who is going with whom? Solution Working with the women first, we have the system of equations: \begin{aligned} Sue &=& 200 + Jan, \\ Lynn &=& 200 + Sue = 200 + 200 + Jan = 400 + Jan, \\ Jan &=& 2400 - Sue - Lynn = 1800 - 2 Jan. \end{aligned} From this we conclude that $Jan = 1800/3 = 600$, and this means that $Sue = 800$ and $Lynn = 1000$. So much for the women. As for the men, one of them will receive exactly what their girlfriend got; another half what his girlfriend got; and in the third case double what she got. These three numbers have to add up to the rest of the prize which is3000. Now, if we are going to get to 3000 using the numbers 600, 800 and 1000, we had better double the largest award, the 1000. Thus, we start by doubling 1000 to 2000 leaving just 1000 more to go using the numbers 600 and 800, one of which is left alone and the other of which is halved. It is easy to see that 600 must be left alone and 800 will be halved.

On this basis, we conclude that Jan goes with Bob, Sue with Lou, and Lynn with Matt.

The mystery must have been extremely serious to prompt such a big reward.