Australian Intermediate Mathematics Olympiad - Question 1?

Question

Trevor's trailer has 3 tyres, two in use and one spare tyre. The three tyres are worn 25000km, 28000km, and 31000km. Find the distance that Trevor's trailer traveled in thousand kilometres.

Answer 1

Here is the correct (not over-complicated or half-baked) solution. First, let's reduce it to

a = 25
b = 28
c = 31

since the extra zeros are going to bug me.

There are three distinct states the trailer could be in, depending on which tyres are in use:

A = a not in use (b and c in use)
B = b not in use (a and c in use)
C = c not in use (a and b in use)

Total = T = A + B + C

a = T - A
b = T - B
c = T - C

a + b + c = 3T - A - B - C = 2T

2T = a + b + c = 25+28+31 = 84

Total = 42 000 km (adding back in those nasty zeros).

Here's an example:

Drive 11 on a and b
Drive 14 on a and c
Drive 17 on b and c

And you get:
a = 11 + 14 = 25
b = 11 + 17 = 28
c = 14 + 17 = 31

But you drove 11+14+17 = 42.

Notice that user "Pi R Squared" initially made the trademark mistake on this question, saying that it was (2/3)(a+b+c). He changed his answer, of course, when I pointed out that he was wrong.

Please also note that the conclusion:
2T = a+b+c
is not "logic" (as one answerer contends) it is a result of basic algebra. You can't wave your hands at a simple piece of algebra and say "no, logic makes this work." It's algebra that makes this work. The fact that logic underlies algebra notwithstanding.

Answer 2

Australian Intermediate Mathematics Olympiad

Answer 3

The thing to keep in mind is that at any given time, he had two tires (please forgive my American spelling of that word) on the trailer. The tire with the longest wear is tire #3, the one with 31,000 km, so we know he at least went THAT far. When tire #3 wasn't on the trailer, the other two were on. So we now have to find the distance traveled when tires #1 and #2 were paired together, and add this to the 31,000.

Let's say tire #1 went x kilometers when it was paired up with tire #3. That means tire #1 went (25,000 - x) when it was paired up with tire #2. So out of the 28,000 km that tire #2 rode, 25000-x of it was with #1 and 28000-(25000-x) was with #3. Also, we know that:

(the wear on #3 when it was paired with #1) +
(the wear on #3 when it was paired with #2) =
the total wear on #3

So:
x + 28000-(25000-x) = 31000, which means
2x + 3000 = 31000
2x = 28000
x = 14000

We said tire #1 went (25,000 - x) when it was paired up with tire #2, so the total distance was:
31,000 + (25000 - 14000) = 42,000 km

Answer 4

Hi,

The total distance on the 3 tires was 25000 + 28000 + 31000 = 84000 km. The total distance traveled is 42000 km because the total of those distances divided between the 2 tires is 42000 km.

I hope that helps!! :-)

Answer 5

Not knowing which tire is the spare or if the spare has been used on this particular trailer or if he bought the tires used the farthest he could have traveled is 25000km

Answer 6

Give logic a chance
instead of crunching numbers.

One bike had two wheels that did 84000.
One bike did 42000.

Could have had ten spares.

Answer 7

i think no because last year i wrote olympaid all my information i wrote with gel pen its was written to write with ball pen my sheet was nt rejected so dont worry all ovals we should mark with pencil.

Answer 8

are you an esquimal, or a native american?