Friday, July 25, 2008

Friday Fun - How Long is the Ring Road around Iceland? - Answer

We should be flying back home from Iceland about this time. Hopefully everyone has had fun with the puzzles while we have been gone. If you haven't had a chance to solve the puzzle about the Iceland Ring Road yet, take a look at last Friday's post and don't read any further. But if you want the answer, read on...
A: Let A be the speed of the first couple and B be the speed of the second couple. After an equivalent amount of time T, one couple has traveled AT miles and the other travels BT miles. For the return, the first couple now travels BT miles in 9 hours, while the other couple travels AT miles in 16 hours.

A = BT/9
B = AT/16

9A = BT
16B = AT

T = 9A/B
T = 16B/A
9A^2 = 16B^2
Take the square root of both sides (which is okay because both are positive)
3A = 4B

This tells us the ratio of their speeds is 4 to 3. In other words, over the same time, the faster couple will travel 4/7 of the ring road, the slower couple will travel 3/7. The difference is 120 miles. And if 1/7 is 120 miles, the whole road is 840 miles.