Sherpa In this video, we look at some GCSE maths number problems. We'll be looking at how to solve them using BIDMAS (Brackets, Indices, Division, Multiplication, Addition and Subtraction), standard form and factorisation.

We run through some practice exam questions and answers to help you put your knowledge into practice. Feel free to pause and attempt to answer the questions before we go through our workings out to test yourself.

This video is perfect for anyone studying GCSE maths or wanting to brush up on their problem-solving skills!

So I know what some of you might be thinking when you see number of problems. Considering that most of maths is number of problems, I definitely agree with you. Number of problems appear literally everywhere throughout the entirety of your math education. And consequently learning this stuff is probably more important than the others because it's nice it is to have a good bit of geometry or trigonometry or something similar. It all comes back to using these numbers correctly.

Now you can actually split number problems into multiple different categories as I've done here. The main one that I'm interested in here is Bid Mass, which is kind of our top one, which just shows how we use general operators with regular numbers. And then everything else kind of comes underneath it. Bid Mass is the king up here and everyone else underneath is just kind of working with the things that Bid Mass sets out. So we'll begin with bid mass.

This is probably the most important. You might hear this sometimes called mathematical operators, or alternatively the four operators. But this shows the order in which you need to do mathematical problems. So brackets, indices, division, multiplication, addition, subtraction, Bid Mass. You might also see this sometimes as Bod Mass, but it's the same sort of principle and it shows the order in which you need to do things.

And also encoded in this is all the things you'd be expected to do as a mathematician in GCSE. So for example, you might be expected to do twelve x 36 just as something that you could do without a calculator. So it's useful to know all the different techniques you're going to need when you're doing things like divisions and multiplications. In the case of additions, we mostly need to put things into the correct columns and have various different units which we'll talk about more in standard form for multiplication. Instead, you might be expected to use something like a multiplication grid.

So in this case we do something like ten and two, three and six, and just split everything according to its various different properties. Okay, we will multiply everything together, then add it all up. Okay. So in this case we'll get something like 360 60, which is a bit of a coincidence there, and then twelve, which then gives our overall result of \$432. Okay, so that gives you an idea about the kind of skills you might be needed to do this.

But more generally in Bid Mass as well is taking simpler formula like this and then doing more operations to it over and over again. Okay, so for example, taking twelve times 36 and then subtract eight and the order in which you do it. So if we just take that as an example here, if we move our entire grid across here and I instead add a minus eight at the end, then you might be confused as to do. Well, do you do twelve x 36 first and then subtract eight, or do you do 36, subtract eight and multiply by twelve? In this case, Bid Mask gives us the answer where we do okay, there are any brackets, no indices, no division, no multiplication.

Yes, we do that first and then we move on. Let's say subtraction. So in this case it will be that minus eight, which will then give us a quite satisfying four, two four. Okay, so one thing which this is illustrate though is if any of you are confused about to remember the order of this, generally it's good Practise whenever doing maths and science exams to bracket everything. This is much more clear.

You do everything in a bracket first and everything out of the bracket. And everything from here follows along from Bid mass in no particular order. Then I'm going to pick standard form. So standard form is another very common number problem where what we do is we take numbers and we turn them into, well, a standard. So for example, if we had the number like say 3000, that's three with then three zeros at the end.

Okay? Now this is the equivalent of doing three times ten times ten times ten. Okay. Seems obvious, right? That's no problem there.

This is where we introduce these indices that we were just mentioning. So an indices is like saying multiply by something a certain number of times. The way you write an indices is you would do in this case, if we're multiplying by ten three times, we'd multiply by ten and then you'd write a three here, it's got a power of three. Now, some of you might recognise this already. This is what it looks like when you're squaring something, you put a number at the top and that's what an indices multiplication is like saying do an addition this many times.

And indices are like saying do a multiplication this many times. So some of you might be wondering, well, why do we bother putting something in standard form? And it's not necessarily something that you do as often in math, as much as this is a skill that you see vastly more in things like physics and engineering. So for example, if you're dealing with very big numbers, the mass of the earth is going to be on the order of times ten to the power of 24 plus 240 is at the end of the mass of the earth that's measured in kilogrammes. That gets very tedious, right over and over again.

One thing you're going to learn plenty is the fact that scientists are quite lazy. And so we invent standard form as an easier way to do these kinds of operations. In addition to that, it also means you can treat this sum a lot more easily by doing other things. But I mean, if we have three times ten to the three and then say I want to multiply that by, we'll go for nine times ten to the minus two we get this interesting effect where the three and the nine can multiply and that's just one digit multiplication, which is pretty straightforward. In this case it's 27 that's multiply by and then basically we're saying multiply by 1000.

And then we're also going to be multiplying by ten to the minus two. So we think about what minus two means. So if it's a positive number that tends to the power of three, that means multiply three times minus two would mean divide twice. So in this case we get 27 multiplied by well, if you're multiplying by ten three times, then dividing by two divided by ten twice, you get times ten to the power of one, which just means 10. So 270 standard form is very useful for doing things like that.

Fractions is our kind of next topic here, which I'm not going to spend too long on. Fractions just a way of writing numbers. Rather than doing on one line something like say, multiplication, like say doing two divided by three, we instead write it as two over three like that. Now some of you may have noticed this for actually look at the divide by symbol. You may notice that it's two dots with a line underneath with a line between them.

And that's all that division sign means. It's saying, okay, just take this number and put it in this dot, that number and put it in this dot and then we get two divided by three. Okay, we're not too fussed about fractions at the minute. We'll show that in an easier example. Next.

But fractions are just a way of representing this data. In general it's something which is quite useful because it means that you can add different quantities together without worrying about decimals too much, which is something which in math is a lot more important. The last sort of skill I wanted to talk about today will be factorization, which is taking a number and breaking it down to something more similar. We're already seeing how there's a lot of dependence on different skills here because factorization actually relies quite heavily on the use of standard form. So for example, factorization would mean if we take, say, a number like twelve, how many ways could we write twelve?

factorising, it means breaking it down to simpler parts. So in this case we could break it down into two and then that would be multiplied by six. That would be one way we could factorise it. Then we could also break that six down into two times three, which means that we can actually simplify this down to two times two times three, which we can then simplify even further down to. Well, if we're doing two twice, that's two squared, two twice and times by three.

And if you want to, you can put a little one here saying just multiply by three one. But that's a way that we can break down twelve. And if we were to then put that into something, that's a fraction we might discover, they can simplify further. One thing which we're not going to have plans to talk about today is third. But this is another important skill when doing number theory.

It's just something to keep in mind going forward. Okay, so if we do some examples of this, we'll begin with bid math. So these are mostly exam questions. This year is not an exam question. This actually comes from a math challenge.

So strictly speaking, it's actually even harder. Quite a few math problems. So Lotty has a bag of apples. She gives half of them to Fred. Freddie's two and has four left.

How many apples that Lottie have at the start? So although it doesn't look it, this is actually a question of how do we represent this information as best we can. So in this case, we're going to represent the number of apples as A. Okay, so let's have a bag of apples. She gives half to Fred, so we're having them.

And the equivalent way of saying having a math is dividing by two. And then Fred eats two. So subtract two and then he has four left. So we know that the end result is going to be four. We want to find out what this original A was.

And again, it's not immediately obvious what the first steps that we should do here. So we're going to refer to bid mass. Okay. So there's no brackets, there's no indices. Division.

Okay. So the division happens first. So we can rewrite this as A over two and that would be the way that we'd write it to avoid confusion. Okay. Minus two equals four and then we go along.

Is there any multiplication addition? Okay, not really. Subtraction. There are these minus two. So to balance this equation, we put the two to the other side.

So we get A over two equals six, and then multiply by two on both sides gives us A equals twelve. The original number of apples was twelve. Next one up. We have standard form here and this is simply the same before work on the value of 1.8 times ten to the five divided by nine times ten to the two. So in this case we're doing a very similar problem to before.

So just like before, we're going to rewrite rather than write divide here as one, eight times ten to the five, all divided by nine times ten to the two. So it's like saying 1.8 and then multiply it by ten. Five times divided by two by one, nine multiplied by ten twice. So in this case we could do 1.8 divided by nine. You can do this in the calculator.

Turns out that's 0.5. And then we're doing five lots of ten divided by two lots of ten. That's going to be three lots of ten in the end. And try this in standard form. We don't actually include decimal points in this case.

Instead, what we would do is say, okay, that's the equivalent of doing five divided by ten. So that'd be five times ten to the two. We're taking out three and we're knocking it down by one. Okay, there's an idea about how to do some of these standard form issues. Okay, last one we're going to do today is fractions.

And then I'll leave these two here as a bit of a challenge for you guys to do. So looking at these fractions, a water container is one 8th full, 45 litres of water and airport into the the container. The container is now three quarters full. When the water container is full, how much water does it hold? So the way we might write that is that we've got one eight of the full amount.

So we can call that full amount of this container w for water, then add 45 litres and now it is three quarters full. So three divided by four of this water. Okay, to make everything look the same as best we can, it's actually going to be a lot easier if we take this one eight here and there are three quarters here and try and turn these to have the same denominator, make this number the same on both sides. So in this case, we could say that one eight of the water plus 45 litres is going to be equal to six over eight w. That's the exact same thing.

That's like saying how many sixes can go into eight. It's the exact same way of saying how many threes can go into four. Okay, we can then subtract this one eight from this side. So we end up getting that 45 litres is equal to five eight of the total water. And then we're going to need to multiply across and divide.

So we multiply by eight on both sides, we get 45 times eight equals five w divided by five, which means that we then get 45 times eight divided by five equals the amount of water. 45 of five is nine times eight equals 72 litres. Okay, that gives you a very quick idea of the kind of problems that you'd end up getting with fractions and the kind of skills that you need to do, making sure to multiply and give them the same denominator. Lastly, your factorization, I believe it's a challenge for you guys, write 36 is a product of prime factors. So break it down as much as possible and give it in the same form as we just discussed.

And this little indices challenge here, which is a ball is dropped and bounced up to height. There's 75% of the height which is dropped, then bounces 75% again. How many bounces until it's less than 25%? That's quite a tricky problem, this one. So look forward to seeing some of the answers.

I'd recommend that you guys put them into the chat below. If you've got any ideas. And if not, feel free to leave any comments or questions and I'll do my best to answer them. So best of luck with your studies and hopefully you enjoy number Theory you.

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