Hardest maths questions – inheritance

The question we’ll be working through
today is from the 1983 competition and only 5% of students got this one right.
So the question is: in his will a father leaves all his money to his children in
the following manner: the first child gets a thousand dollars and a tenth of
the rest, the next child gets $2,000 and a tenth of the rest, and the third child gets
$3,000 and a tenth of the rest, and so on. Now once this is done, all the children
end up with the same amount of money. So we have to work out how many children
there are. So pause the video here, have a go, and then come back and see my
solution. Okay how did you go? I think this question is really probably a bit
too hard for students at this level but it is such a great one for people who
are just learning algebra because first of all it’s gonna give you an
appreciation for what algebra’s for because you can do this question without
formal algebra but it’s really hard; it’s much easier with algebra. So that’s
going to show you what algebra is for, and also it’s going to give you an
overview of all the different parts of algebra: expanding, simplifying, solving
equations and it’s going to show you what all those things are for. So don’t
worry if you didn’t get very far with this question. I know it is a really hard
one for kids your age. So what I would do first and what’s really useful for all
sorts of questions is to try try to draw a picture from the information you have.
So what you could do for this one is draw like a stack of money. So if you
imagine like this is all the money, like a stack of bills, that the father is
leaving to his kids. Now the first child takes a thousand
dollars so let’s just sort of take a bit off the top and say that’s a thousand.
And then also a tenth of the rest, so however much money is left at this point,
we want to take off a tenth of that, so I might just cut a bit off and say that’s
a tenth. Okay so all of that bit is what the first child gets.
Now the second child takes off $2,000 so I’ll draw that in and then a tenth
of the rest, so however much money is left at this point – the second child also
takes a tenth of that. So let’s draw that in as well. Okay so all of that is
what the second child gets. So I’ll show you the non-algebra solution first and
I really think that this was much harder to come up with than the algebra
solution. Once you get used to algebra, algebra’s quite easy. So what you could
do here is you could say: okay this amount of money has to be the same as
this amount of money, right? Because each child gets the same amount. And because
this part is a thousand more than this part, that means that this part has to be
a thousand less than this part. I hope that makes sense to you. Don’t worry too
much if this doesn’t make sense because that just demonstrates how useful
algebra is because you don’t have to do any of that kind of thinking once you
do the algebra solution. But okay so I’m saying this bit has to be a thousand less
than that bit. Now because this is a tenth of whatever amount of money it was here,
and this is a tenth of whatever amount of money it was here, that means that
this has to be ten thousand less than that. Again I know this is really
confusing; this is why we really need algebra for this question. So…
hopefully that did make some sense to you though, that this is a thousand less
than this and because this is a tenth of there and that’s a tenth of
there, then the difference between these two points has to be ten thousand.
Okay now if that amount of money is ten thousand and this is two thousand, that
means this bit has to be eight thousand. I’ll just write that in there. So that
bit there is eight thousand, and that is a tenth of the amount of money here.
Therefore this amount of money has to be 80,000. So eighty thousand is like that
whole bit of the stack and therefore the original amount of money, like the whole
stack up to here, that has to be eighty one thousand. So then from there, so we’ve
got eighty one thousand is the total… let’s work out how much the first child
gets. So that was eight thousand there and you’ve got the one thousand there, so
nine thousand dollars is what the first child gets. And then the second child
will also get nine thousand if we’ve done things correctly. So we can work out
how many children there are by doing eighty one thousand divided by nine
thousand and that will give you nine. So nine children is the answer. Now I want
to go through the solution using algebra which may look confusing at first if you
if you haven’t done much algebra but it is actually easier with algebra. So
what you can do is maybe we call the amount of money at this point x. So we
don’t know that amount of money… yet. You could also call the total from there x
but it actually comes out more complicated if you do that, so I decided
to call that amount of money x. So if that’s x and this is a tenth of that
amount, this is actually x divided by ten. So I’ll just write that in, so that’s x
over 10. You can also say something like it’s one over ten times x or
x divided by 10, those are all the same thing. Okay so that’s x over
10, that’s 2,000, therefore the amount of money that we have there is x, so that
level minus that and minus that. And then this, which is a tenth of this amount of money –
it’s just going to be like that divided by ten. So now we can actually use that
to make an equation. And this is the most important part of algebra, like actually
solving equations is something that computers can do for you, so in the real
world being able to turn all these words into an equation is a much more useful
skill than being able to solve equations, so that’s what you should really
concentrate on the most, and I think it’s it’s probably the most interesting part
of algebra. So what we want to do is just take this information that each child
gets the same amount of money and turn that into an equation. So that’s just like
this bit equals this bit. So that’s 1000 + x/10, so that’s the
amount of money that the first child gets, that has to equal this amount of
money which is 2000 plus a tenth of that, so you can write that as like
one over ten times that or you can write it as like a fraction as like x minus x/10
minus 2000 all over ten. All right so now we’ve got the equation and actually
if you wanted to use a computer to solve it you would pretty much be done now. You
can just type that into… I’ll just show you how to do it with Wolfram Alpha. If
you go to wolframalpha.com and just type in “solve” and then the
equation that you want to solve and hit enter that will give you the solution. So
a computer can actually do all of the equation solving for you which is why
that isn’t the most important part of algebra. Like you need to know how to
solve equations to pass your tests in high school but in the real world the
being able to write down the equation in the first place – that’s the important
part. But of course I’ll show you how to solve this by hand as well. And I just
wanted to mention that if you did the equation in terms of the amount of money
at the start, instead of the amount of money here, you would have x – 1000
wherever I have x there. So just in case you wanted to check your
equation if you wrote it down that way. Alright now to solve this… the basic idea
with solving equations is that you want to do the same thing to both sides of the
equation. So in this case the first thing I would do is move the thousand over,
like I want to get rid of that and shift it over to the right, and the way that
you do that is to subtract a thousand from this and because we’re doing that to
the left-hand side, we also have to subtract a thousand from the right. So when you
subtract a thousand from this you get x on 10 and then over to the right, we
subtract a thousand there as well. And so this just becomes one thousand. So that’s cleaned up the equation quite
a bit already and interestingly that corresponds to the first thing that I
did with the non algebra solution, because the first thing I said was that
this one would have to be a thousand less than this, and that’s exactly what
this equation is saying: it’s saying that the x/10 which is that bit is a
thousand more than this bit. So what I would do next, because we’ve got these
fractions, it would really clean things up if we multiply both sides by 10
because that’s gonna like get rid of these fractions. Because if I multiply
this by 10 on the left, we’re just going to get x. So because I’ve done that on
the left, you also have to multiply the right-hand side by 10. Now some people,
when they’re starting out with algebra, they find this a bit confusing, how to
multiply this whole thing by 10. So I’m going to write it out like in two steps
because I think it becomes easier if you do an intermediate line like this first. So now what we need to do is expand this,
so that means we need to do the ten times the thousand and then also have
ten times this. So ten times a thousand is just ten thousand and then when we do
10 times this, the tens are just going to like cancel out, because we’re
multiplying this by ten and then dividing it by ten, so we’re just gonna
get what’s on the top here: x minus x/10 minus 2000. Okay now that actually
corresponds to the next thing I did because if you remember what I said
after that was that the amount of money here has to be ten thousand less than
the amount of money here, so what I’m doing in the algebra is like perfectly
corresponding to what I was doing in the non algebra solution. Alright what I
would do next is combine the 10,000 and the negative 2000, so that’s called
collecting like terms I guess, or just simplifying. So 10,000 minus 2000,
that makes eight thousand. Okay and then what I would do, because on
the left we’ve got just x… what you want to do when you’re solving an equation and
there’s like a bunch of x’s everywhere, you want to move all the x’s over one
side of the equation. So what I want to do is take this x and this x/10 and
shift them over to the left. Now the way you do that… I’ll do that in two steps,
I’ll shift the x/10 first. So what I want to do is add x/10 to both
sides because if I add x/10 to this side then that’s just going to cancel
with the minus x/10 and that’s gonna disappear. So I’ll add x/10 to this
side as well, so we’ve got x + x/10 and then on the right we’ve got 8,000 plus x,
we’ve got like minus x/10 plus x/10 so that’s just nothing. And then
I’ll move the x over to the left as well. Now because that’s just going to cancel
with this x if I subtract x from both sides – that’s just going to get rid of
that and that, so what you can actually do is just put a line through that and
that to say that they’re both gone. Okay so now we’ve got x/10 equals 8,000
and that was actually the next bit that I did here I said that this is 10,000
and this is 2,000 therefore that bit has to be 8,000. Okay now what I would do
next with this to work out what x is, is just multiply both sides of the equation
by 10, so on the left that gives you x and on the right we multiply that by 10
as well and that gives you 80,000. Okay so then that tells you the amount of
money that was there and then again you just add a thousand to that to get the
eighty one thousand as the total, and then once you’ve done that you can work
out that that’s nine thousand therefore there are nine children. Let me
know in the comments if you are still confused about anything in this question
or about maths in general and give this video a like if you want me to make more
like this.