# 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.

Good video. I like your style. Clear, coherent, concise overall.

Here is an easier way to have the number of children, without having to calculate the amout for each child nor the total amount (those came as a bonus, but they are not asked for):

(nor the amount after having removed the fisrt 1000$ as in your video)

T = total amount

M = money for each child

n = number of child

First child : M = 1000 + (T-1000)/10 (A)

Second child : M = 2000 + (T – M – 2000)/10 (B)

…

n-1 child : M = (n-1)1000 + (T-(n-2)M – (n-1)1000)/10 (only for info)

Last child : M = n*1000 (C) (no tenth of anything left, else it means that there is still some money left, so another child…)

(You come to that by going to the last child on your graphic)

(B)-(A) (to remove T)

0 = 1000 -(1000+M)/10

(1000+M)/10 = 1000

use (C) M = n*1000 (to remove M)

(1000+n*1000) =1000*10

(n+1)1000 = 1000*10

n+1 = 10

n = 9

So the father have 9 children

Each of the child get M = 9*1000 = 9000$

from the total of T = 9 * 9000 = 81000$

Remark : for (C), if you doubt that there is no money left after removing n*1000 for the last child, then the full (C) equation should have been M = n*1000 +(T – (n-1)M-n*1000)/10

And we can check that the money left is null : T – (n-1)M-n*1000 = 0?

T – (n-1)M-n*1000 = 81000 – 8*9000 -9*1000 = 81000 – 9*9000 = 81000-81000 = 0

An alternative solution is that there's 1 child and 1000 dollars.

Very good video

Although I am doing the AMC tomorrow and it it quite unlikely you will rspond in the next two hours, I was just wondering how in the non algebraic solution how you determined that the second point of the total is $10,000 less than the first cut off, I understand how you got it through the algebraic solution (which was explained very nicely by the way) but just was wondering. Also you are super gorgeous and your hair is super pretty haha. You're very intelligent and amazing at explanation. Thank you, I will now continue watching the rest of the videos in this playlist. <33