This question is from the 1990 competition and only 8% of students got
this one right. So the idea is that we buy four tyres for a car and the specs
of the tyres state that they will last 40,000 kilometres on the front wheels
and 60,000 kilometres on the back wheels. So that's because when the tyres are on
the front wheels they'll wear out faster. So what we want to do is at some point
swap the tyres around, so move the ones that are on the back wheels to the front
and the front ones to the back, because that way they'll all wear out evenly, and
all wear out at the same time. So the question is: what distance in kilometres
can we go on these four tyres before they all wear out? So you might like to
pause the video here and have a go at this question before you come back. Okay
how did you go? So a lot of people will answer 50,000 kilometres to this
question with the idea that you go twenty-five thousand kilometres and then
swap the back and front tyres, so I just want to start out by seeing if that
works. So the way that I would think about this is like for example if you
went 20,000 kilometres then the front tyres would be worn out half way because
20 thousand is half of 40,000 and the back tyres would be worn out a third of
the way because that's a third of 60,000, so I want to like extend that idea to
25,000. So if we go 25,000 kilometres what sort of fraction is the front tyres
worn down and the back tyres worn down? So to work that out you would do 25,000
over 40,000 - that'll give you the fraction of like how much the tyres are
worn down. So you can simplify that. Let's take out the thousands so like dividing
top and bottom by a thousand. 25 over 40. And then we can simplify that
even further by dividing top and bottom by 5, so that'll give us 5/8, so
if you travel 25,000 kilometres then five-eighths of the front tyres are worn
down. For the back tyres it's going to be 25,000 over 60,000 which we can simplify
to 25 over 60 and then divide top and bottom by 5 again so that makes 5/12 so
5 twelfths of the back tyres will be worn out. So if you then swap the front and
back tyres, each tyre is going to be worn out by like 5/8 plus 5/12
because like either it's on... either it starts on the front and it's worn out
this much and then this much or it starts on the back and it's worn out this much
and then that much. So the question is: is 5/8 plus 5/12... does that add up to being
a hundred percent worn out? So let's test that. We can add these two fractions by
making a common denominator. Now sometimes like if you don't know how to
work out the common denominator you can just make it 8 times 12, but it's gonna
be simpler if you can think of a number that both of those divide into, so that...
the smallest one would be 24, because 12 goes into 24 twice and 8 goes into
it 3 times, so both of those work. So what we want to
do is multiply top and bottom of this fraction by 3; that'll give us 15.
And top and bottom of this one by 2; that'll give us 10. So that actually adds up to
25 over 24, so if you were to do 50,000 kilometres all of the tyres would
actually be more than a hundred cent worn out, you know, because this is more
than 1. So that's not the answer. So if you answered 50,000 you might like to
pause the video here and have another go at the question. Ok
so 25,000 didn't work because it gave us the 25 on 24 instead of 1. So what we can
do is just take this and turn it into an equation by saying, "ok it's not 25000; we
don't know what should go there so let's just call it x, for the unknown." And on
the right we want to have 1. So now this is an equation that we can solve to
work out what x is. So what we can do is add up these two fractions by making a
common denominator, so 120,000 would be the lowest
common multiple of these two numbers and you'd get that by multiplying this one
on top and bottom by 3 so you'd get 3x on 120,000 and multiply this one on
top and bottom by 2 so 2x on 120,000 and that... that's unchanged
on the right. Ok and then we just add these up so you get 5x on 120,000
and then we want to solve for x so we want to start moving stuff away from the left
hand side. We could start with the 120,000, so to move that
across you multiply both sides by 120,000, so that gives
you 5x = 120,000 and then divide both sides by five so that we can
get the x by itself. Now a good trick for dividing by 5 is if we've got
120,000 divided by 5, what you can do is multiply top and bottom of
this by two which actually makes it much easier because now we've got 240,000
on 10 and dividing by 10 is super easy because you can just get rid
of a zero on the end. So that gives us 24,000. Okay, meaning that we would swap the
tyres over after 24,000 kilometres and then drive another 24,000
so the total distance is twice that which is 48,000, so that is the
answer to that question. Don't worry if you couldn't get this question, because
every time you tackle a question that's too hard for you, your brain actually grows
stronger. So give yourself a pat on the back. And there's lots more coming up
with The How and Why of Mathematics so consider subscribing if you want to see
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