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# Rotating Tires

## Question

Tires placed on the rear of your car will wear out after 21000 miles, while tires on the front

of your car will last for 29000 miles.

Suppose you have a new car and five identical new tires (four installed and one spare).

a) What is the maximum distance you can drive, assuming you can easily change the tires any time you want?

b) Describe a rotation schedule that allows you to drive this distance.

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a) 30,450 miles can be driven.

b) Rotate the tires after each mile:
Spare -> Right Rear -> Right Front -> Left Front -> Left Rear -> Spare

Explanation:

For each 5 miles driven using this rotation, each tire will
lose (2/29000)+(2/21000) of its tread. So, suppose we drive
5*N miles like this. Then, each tire will lose
2*N*((1/29000)+(1/21000)) of its tread. We wish to
determine N so that the latter expression gets to be 1,
representing all of each tire's tread. That happens when
N=6090, which is when 5*N (the total miles driven) is
30,450.
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Four tire changes are needed.

Five tires will be worn out when 2n/21 + 2n/29 = 5.
Where n is the trip length in thousands of miles.
100n = 5 x 609, so the trip is 30.45 k miles.

For the tires to wear out simultaneously they must have the same, or equivalent histories.
Let's say s, 2f and 2r thousands of miles, respectively, as a spare, front, and rear tire, and s=r=f.
Thus 5s = 30.45 k miles, so s=r=f = 6090 miles.

Rotate the tires every 6090 miles, four times in all.

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