Ten Problems

A truck has two front tires that need to be changed every 15 000 km and four back tires that need to be changed every 25 000 km. (The difference is due to the imbalance of the load in the truck.) How can you switch the front and back tires to maximize the distance travelled without getting any new tires? What is that distance?

The Maximum distance is 20454+6/11 km.

Let us call the amount of ‘usable’ rubber in each tire a ‘rubber unit’, so we have six rubber units in total. From the information we are given, for every 1 km travelled, each front tire uses up 1/15000 rubber units and each back tire uses up 1/25000 rubber units. So, for every 1 km travelled, 2/15000 + 4/25000 = 11/37500 rubber units are used. Let us assume that the truck travels x km, so 11x/37500 rubber units are used. Since the maximum number of rubber units we have is 6, therefore 11x/37500 is less than or equal to 6, and x is less than or equal to 20454+6/11 km.

While we are traversing through this distance, we stop every third of the way, and switch two of the back tire with the front. In this case, each of the six tires would be installed at the front for 1/3 of the way and at the back 2/3 of the way. This way, upon arrive at our destination, all six tires will be used up.