Sherpa In this video, we take a look at the topic of AQA GCSE Maths Decimals. Decimal points are used in numbers to separate the whole number from parts of the whole. Like whole numbers, numbers written as decimals can be either positive or negative, for example, 2.6 or -2.6. Decimals are just one way of expressing numbers that are parts of wholes. These numbers can also be written as fractions or percentages. The number 1.5 (decimal) could also be written as (fraction) or 150% (percentage). They are all exactly the same number. Knowledge of converting between decimals, fractions and percentages is important.

We are looking at decimals as well. First do a brief intro on what we mean by place value. This often comes up when we deal with decimals. And then we'll just focus on the four main operators that we can use when we're for example, adding decimals together. So what is place value?

What do we mean by place value? All this means is that each column has its own kind of multiplier, right? So this is 100 column, 10th column, my units column, our 10th column and 100th column. So an example, let's say we've got say 432.7. Well in this case because for example, the three is actually in my 10th column, the three actually has a value of 30 because the seven is in my 100 column, the seven actually has a value of 700.

So that's what we mean by place value. Every digit has its own actual value in a number. So let's now focus on the operations. So let's do adding. First of all, adding decimals together is very much the same as what we used to with regular whole numbers.

So for example, we could do say let's say we want to do 32.7 plus 15.8. Well in this case I actually might just simplify slightly. Let's do 5.8. So in this case, what we're doing is as we normally do, we just add the columns together, right? So seven plus eight gives me 15.

I carry the one from the 15, I then do five plus two, I get seven plus the one, I get eight and then I do three plus nothing and I get three. And so that's how we can add together decimals. We just going need to keep our decimal point in that same column. We don't want to move our decimal point around at all. Minus thing is also the exact same as we used to know because I could keep the same numbers together, we could do 32.7 -5.8.

In this case, I can't do seven I get a negative number. So I steal from the column over, I turn that to into one and I take the one from this column and I give it to this column next to it. So that anything in this column is ten times bigger than in the column next to it. So one here actually becomes a ten here. So this seven becomes a 17.

And then we can do 17 minus eight, that gives us nine and put the point in there. As we always see, we can then do one minus five. But we can't do that because we get a negative number again. So we can steal from the three, turn them into a two and we can do eleven minus five. Or eleven minus five gives us six and we can then do two minus nothing and we get 26.9%.

And dividing are slightly different beasts. So let's say we wanted to times two decimals together. Let's keep it the same way. So let's say we wanted to do 32.7 times 5.8. Well, this is easy to slip up here because during zero seven times zero eight, we actually have to put our answer not in this column, not in the column over, but in the column over from that.

The fact that we've got seven, sorry, seven in the 10th rather than in the units column means we've got a shift over by one. And the fact that this eight is in the 10th column rather than in the usual units column means we've got to shift down again because our actual answer is going to be 100 times smaller than what we'd normally expect if we did seven times eight. So we can do this method if you want to. It's much easier in a way to kind of artificially turn this into a much nicer question. So we can actually turn this into a nice question, but just times in the top by ten and times in the bottom by ten as well.

So I end up with 327 times 58. And at the very end I know that my answer is going to be 100 times too big, right? Because I've times this one by ten and I've times this one by ten. So I can do this standard application. Now, I could go through all these steps.

Eight times seven is 56. I carry the five for the six there. Eight times two is going to give me 16 plus the 521. I can go for this, I won't do it all now. But in times three is 24 plus the two is 26 and I could do the same thing for all of my numbers and I could add them all up together.

I get another line here, I'd add that up together and I'd get some answer and at the very end I'd take that answer and I would divide it by 100 because I know that this number is ten times too big and this number is ten times too big. That's how we can do timing. We'll go through an extra exam question in a second. Dividing again, it's very similar to what you're used to. Let's say we wanted to do 32.7 divided by 5.8.

Well, again, in this case we can either think of this as 32.7 divided by 5.8 and we could set up our bus stop method that we're probably fairly familiar with 32.8. You may have seen this kind of a layout before.

We could do 5.8 into three doesn't go 5.8 into 32. We can put five one eight to 32 probably five times and then we can work out the remainder and we could put it in front of the seven. Or what we can do is it's actually easier to manipulate this rather than the time thing. I can take this number and again, I can times it by ten and I can take this number and I can times it by ten and actually that won't change the final result, right? We know that.

For example, I don't know, like 40 divided by 20 is two, as is four divided by T being equal the same thing. So it doesn't really matter if it's ten times bigger or ten times smaller. I get the same answer either way when I do a divide. So in this case, I could probably make this a bit nicer for myself by just turning these into regular whole numbers. And I could do that kind of standard bus stop method and eventually I'd get down to my answer again.

I won't go through this. Now. We're going to do a Practise exam question in a second. Let's go on now and do some actual questions. So we're going to do an adding timing and dividing.

Obviously adding and monitoring are almost the exact same. So we'll just do one of these, right? So if we're doing zero 89 plus zeroteen, well, we're going to lay this out as we did in the previous page. So we've got zero 89 plus I might just put a zero there just to make it easier. So it is zero plus three, we get three.

One plus nine, we get ten, we carry the one. Eight plus one plus one we get ten. So we carry the one again and zero plus zero is just zero. Add the one, you get one. And our decimal point again stays in that same column as it always does.

So our final answer is going to be 1.3. Timing. Again, it's tempting to try and do this. I think it's easy to slip up. If we just try to dive straight in to doing two times 43.

It's easy to slip up. We're not quite sure exactly how many columns over we have to shift until we hit the right column, right? This is actually 2000 times 3000. What we can do to make it easier for ourselves is a bit of a tricky one. We can actually times both sides, top on the bottom by 1000.

Once they're going to do is if we zoom in, what time is this? By 1000. It's going to move my decimal place across three places. And same here, this decimal point is going to shift three columns and so this is going to become two times 43, which we can do, right? We can know what two times 43 is.

Two times three is six, two times four is eight. And so we end up with 86. But this isn't the final answer. We know that this is 1000 times too big and this one is also 1000 times too big. And so our final answer is going to be 1000 times 1000 times too big.

So you've got to take this 86 and we've got to divide it by six by 1 million because that's going to be equal to 1000, right? This is taking account of the fact that we times it by a thousand again, we terms it by thousand a second time. So any six divided by million. What we do here is we take our decimal point and we just shift it up six places in this, in this case. So it's going to be 122-3456.

And so that is going to be my final answer, right, the final question dividing. So let's look at this. We've got 7.48 divided by one point T. And again, it's tempting to have a go this, we could just have a go at this and keep this one point in. I want to make it slightly nicer and I'm going to times this by ten and times this by ten.

So our final answer isn't going to change at all. It's the same as doing either 40 over 20, like I said before, or four over T, right? It doesn't matter if both are ten times bigger. I could equally do 400 over 200. I get the exact same answer every time.

I just get T in this example. So I'm actually going to change this and I'm going to do twelve into 74.88.

I just need to figure out how many times or twelve times five is 6012. Times six gives me 72 so I can put it in six full times with the remainder of T. I know that two times twelve is 24 and that's the maximum I can make. So I can fit two more twelve into 28. So 212 gives me 24.

I've got a remainder of four and I carry that over. And twelve into 48 goes very nicely. It goes four times. And again, I need to make sure that I keep my decimal point in that same place. And so our answer in this case is going to be 6.24. And that is, the end of decimals!

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