Girlfriends, Parties ... Math Problems!
You and your girlfriend have decided to sneak out and go for a party! The only problem is that there is nobody to take you, so the both of you will have to either walk or use the one bicycle at your disposal.
The party is 16 km away from home. Both of you agree that you will start on foot and your girlfriend will start on the bicycle. After some time, your girlfriend will leave the bicycle beside the road and continue on foot, so that when you reach the bicycle, you can mount it and cycle the rest of the distance. You walk at 4 km per hour and cycle at 10 km per hour, while your girlfriend walks at 5 km per hour and cycles at 12 km per hour. For what length of time (in minutes) should your girlfriend ride the bicycle, if you are to both arrive at the party at the same time?
Adapted from the South African Mathematics Olympiad (Problem 13 Junior Paper Round 3 2005)
The answer is 30.
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1 solution
Let x be your walking distance, such that
4 x + 1 0 1 6 − x = 1 2 x + 5 1 6 − x ⟹ x = 6 .
The biking time for your girlfriend is 1 2 6 × 6 0 = 3 0 minutes.