riddle of the day

Sometimes a question looks like it is about cars, but it is really about conservation. It does not matter how you rotate the wheels: what matters is how much total tire life you consume per kilometer.

The Spare Tire

Curious
Pure logic

Riddle statement

A car has 4 wheels mounted and one spare wheel.

Each tire lasts exactly 20,000 km before becoming unusable.

You may change the wheels' positions whenever you want, including using the spare.

What is the maximum distance the car can travel before all 5 tires are worn out?

Show solution

Solution

Each tire has a useful life of:

$ 20,000 $

kilometers.

Since there are 5 tires, the total available tire life is:

$ 5 \times 20,000 = 100,000 $

tire-kilometers.

But the car uses 4 tires at a time.

Each kilometer traveled consumes:

$ 4 $

tire-kilometers.

Therefore, the maximum possible distance is:

$ \frac{100,000}{4}=25,000 $

kilometers.

To achieve this, the tires must be rotated so that they all wear equally. Each tire must rest one fifth of the time and roll during the other four fifths.

Thus, after 25,000 km, each tire will have rolled:

$ 25,000 \times \frac{4}{5} = 20,000 $

kilometers.

Exactly its useful life.

Answer: the car can travel at most 25,000 km.