## Thursday, January 6, 2011

### New Tires: When Do You Need Them and Tricks You Should or Shouldn't Do With Them

If on a recent winter family vacation you car was screeching, barking and jumping like a scared small puppy (as our car) instead of smoothly gliding through little hills of the freshly pressed snow, you may need new tires.

We learned that a simple penny test can tell whether tires have been so brushed by use that they lost necessary friction. Ridges on the tire should be at least 1.6mm deep for a safe driving on a slippery surface. This means that if you place a penny in the tire ridges, you should not be able to see the tip of the Abraham Lincolns’ hair on the penny. The expected tire use is 6-10 years. It is usually recommended to replace all 4 tires at once.

Now, a puzzle:
Assume that you just bought four new tires for your car. Your car manual says that your front tires could last you 25,000 miles. Your back tires 15,000 miles. You are trying to be creative and come up with an idea of switching your front tires with your back tires after driving 10,000 miles. What tires will go bad now? When?

Answers accepted all day long on Friday, on our Family Puzzle Marathon. They will be hidden until Saturday morning (EST) and everyone who contributed something reasonable will get a puzzle point.

Dennis and Katrina said...

Facts -
1. Front tires last for 25,000 miles
2. Rear tires last for 15,000 miles
3. Switch at 10,000 miles
4. Typical tread depth of new passenger car tires - 10/32" or 11/32" (7.94 mm or 8.73 mm)

Assumptions -
A. Tires are bad @ 1.6 mm
B. Wear on the tires is linear with miles travelled
C. Use new tread depth of 7.94 mm to be conservative

Calculate wear rates for Front and Rear:

Front Tires
Wear = (miles travelled) * (Front wear rate)
(7.94 mm - 1.6 mm) = (25,000) * (FWR)
6.34 mm = (25,000) * FWR
FWR = 6.34 / 25,000 = 0.0002536, or 0.2536 mm per 1,000 miles

Rear Tires
Wear = (miles travelled) * (Rear wear rate)
(7.94 mm - 1.6 mm) = (15,000) * (RWR)
6.34 mm = (15,000) * RWR
RWR = 6.34 / 15,000 = 0.0004227, or 0.4227 mm per 1,000 miles

At 10,000 miles, the front tires have lost (10) * (0.2536), or 2.536 mm
At 10,000 miles, the rear tires have lost (10) * (0.4227), or 4.227 mm

And we switch!

NEW rear tires have 7.94 - 2.536, or 5.404 mm of tread left, with 5.404 - 1.6, or 3.804 mm of that usable.

If we divide 3.804 mm by the RWR, we get 3.804/0.0004227, or 8,999 miles

The NEW front tires have 7.94 - 4.227, or 3.713 mm of tread left, with 3.713 - 1.6, or 2.113 mm of that usable.

If we divide 2.113 by the FWR, we get 2.113/0.0002536, or 8,332 miles.

Therefore, we can expect the tires that started on the front and were moved to the rear to become bad first, at 8,332 miles after the switch, or 18,332 miles after they were purchased.

As an aside, switching them like this makes sense if you are going to replace all four at the same time - the other set only has another 667 miles on it! It would make more sense to switch them at about 9374.5 miles; you'd need to replace all four tires in another 9374.5 miles!

SteveGoodman18 said...

The tires that started in the back and have now been moved to the front will go bad first, after another 8,333 and one-third miles.

After 10,000 miles, the tires that were in front still have 15/25, or 3/5 of their life remaining. When they move to the rear, 3/5 of 15,000 miles gives those tires another 9,000 miles of life.

The tires that started in the rear have 5/15, or 1/3 of their life remaining. When they move to the front, 1/3 of 25,000 miles gives those tires just 8,333 and 1/3 miles of life remaining.

An interesting extension is finding the time when the tires should be switched so that they have equal life left. If we instead had switched the tires after 9,375 miles, then each set would last another 9,375, thus maximizing their simultaneous usage. 9,375 happens to be half of the harmonic mean of 15,000 and 25,000.

Lars said...

the front tires would be bad after 18,950 miles and the back tires after 18,250 miles (however my solution is not very elegant)

kj said...

The tires that started on the back will wear out first.

What we must do is look at how much of each tire's life is left when we rotate them at 10,000 miles.

The front tires have 25,000 miles of life, and we drove 10,000 miles, meaning we used up 10/25 = 2/5 of each tire's life, leaving each of the front tires with 3/5 of its life. We put them on the rear, and now they have only (3/5)*15000 = 9000 miles of life remaining.

The rear tires wear out after 15,000 miles and we drove 10,000 miles on them, meaning we used up 10/15 = 2/3 of each tire's life, leaving each tire with 1/3 of its life. We put them on the front, and now have (1/3)*25000 = 8333.3 miles of life remaining.

So the front tires (formerly the rear tires) will wear out first, after we drive an additional 8,333 miles (18,333 total miles). The rear tires (originally on the front) will wear out shortly after, at 9,000 miles (19,000 total miles).

If we put the tires on today, and average 1,000 miles of driving per month, then we should check the tread in early September (8 1/3 months would be about September 17). The other tires wear out 2/3 of a month later, about October 7.

Donna said...

Switching the tires after 10,000 miles is a pretty good plan if you want the tires to wear out about the same time. At 10,000 miles, the ones that were on the back would have lost 2/3 of their tread (10,000/15,000), so would have 1/3 of their tread left, which if moved to the front would last for about 8,333 miles. (1/3x25,000) The tires that were on the front for the first 10,000 miles would have lost 2/5 of their tread (10,000/25,000) and would have 3/5 of it left. It would take these tires 9000 miles (3/5x15,000) to lose all their tread.

anne-marie said...

The front tires have 60 percent of useful life left.
The back tires have 33.3 percent of useful life left.
After switching the tires,
The back tires could go 9000 more miles and the front tires, 8333.3 miles.
The front tires will go bad first.

Maria said...

You all are amazing.
Dennis and Katrina - you even included thread depth into the puzzle solution!
But it looks like we can also use just a ratio of tire wear.

It seems that everyone agrees that:

1)if we don't switch we'll need to replace rear tires at 15,000 miles, front tires at 20,000 miles.

2)if we switch at 10,000 miles, rear-front tires will last 18,333 miles and front-rear tires 19,000 miles. This gives us 3,333 extra miles but does require significant roadside work.

3)a few mavericks suggested switching tires at 9,375 miles. This will allow for both sets of tires to last 19,375 miles.

Congratulations on another puzzle point to Dennis 7 Katrina, SteveGoodman18, Lars, kj, Donna, anne-marie!