In this AQA GCSE maths video, we are looking at approximation. This is where we round a number up or down to the nearest whole number, decimal or fraction. We also look at truncation and estimation. Truncation is when we cut off a number after a certain point, and estimation is when we guess a value for a number. Finally, we look at bounds which are upper and lower limits that a value falls between.
Sometimes it is a good idea to estimate the value of a calculation rather than work it out exactly. In this situation, round the numbers in the question before performing the calculation. Usually, numbers are rounded to one significant figure. The 'approximately equal to' sign, ≈, is used to show that values have been rounded.
So approximation can actually be broken down into four sub topics. We've got rounding, Truncation, estimation and bounds. So I'll do a brief intro to each of these subtopics and then we'll go through one or two exam questions for both. So, first of all, let's do rounding. So rounding basically tells us we have to shorten the numbers.
So let's say we've got 31.4582 and I want to round this to two decimal places.
Well, that means that that column there is going to be my final column in my new number, right? That four is in my first decimal place and the five is in my second decimal place. What we do is we look at the number directly next to what will end up being my final column. And because that's an eight, well, because it's five or above, we therefore round up and we end up with 31.46 kwh. And that's one new number rounded to two decimal places.
If the eight was, say, a three, then it would be less than five and therefore we would round down. And that five in my final column would stay as a five, rather than changing to a six. Next up is Truncation. Truncation is almost like a lazier version of rounding. It's what calculators and computers do.
We can look at the same number again, right, 31.45, eight, two. And again, we're going to truncate this to two decimal places. Well, all Truncation does is it chops off the end of a number. And so that's my second decimal place column. What all I do now is I just chop off everything else, I just get rid of it.
And so all I'm left with is 31.45. There's actually no math at all involved in Truncation, and what we can see is if we compare these two values, they're different, right? We end up with two different numbers. So we've got to be careful which method we use. Ideally, we use rounding.
We need to understand what Truncation is, because that's what calculators and computers use. They have a limited precision and anything beyond that precision will just get chopped off when they're doing that. Calculations estimation is aware of very quickly doing some lazy math, right? So we could say, for example, 21.8 is roughly equal to 20. If we were doing a calculation.
If we were doing, say, 21.8 times 1.97, well, we'd say this is basically equal to 20 times, because, again, we don't want to be messing around with 1.97 kwh. So we can say this is roughly equal to 40. And if we were to test this on a calculator, we'd end up with very close to that true calculation. Finally, bounce bounds are what we have to deal with when we round things. So, for example, if we think of a ruler, if we were to zoom into a ruler, it would look something like this, right?
We'd end up with 31 centimetres, zero four millimetres. 31 centimetres, zero, five millimetres, and down here we might have thirty one cents three millimetres and if I told you that I measured the length of plaque or wood to 31.4 centimer well in reality that plyka wood could be anywhere down to halfway between these two numbers, right? And it could be anywhere up to halfway between these two numbers and so my actual range of possible values is going to be that interval there and so you'll see in a second when we do an example that's what we have to deal with when we're doing bounce so let's jump into some past paper questions first of all we'll look at branding so we want to write this number correct to three decimal places three DP well we look at the number in the third decimal place column which is that seven there and we then look at the number directly next to the seven because that's five and above it's a six we know we have to round that seven up to an eight so we get 3.848 and we'll do one more we'll look at a significant figure version of rounding so it's one significant figure well one big figure is just my very first digit.
Right? It's that three there and again I look at the number right next to it it's five and above it's seven so I know I need to run that three up to a four and everything else I just ignore so obviously that three is in my 10th column it's actually a 30 and so that 30 gets rounded up to 40 truncation halfway there Sally uses a calculator to work out the value of a number why the mistakens begins 8.3 so it's going to be 8.3 something something we don't know what these numbers are after that three digit well if we look at when we're looking at the error interval we're looking at the maximum and the minimum there could be so we're trying to find the min and the max that this number could actually be what's the minimum that could go?
What's the smallest digit that could go in this box? Well the smallest digit is zero right? And in fact all these boxes could contain zeros so 8300, that's just 8.3 and that's going to be our smallest value we can do the same experiment again but look at what's the largest value that could go in each of these boxes? Well the largest value that the calculator could display in each position would be a nine right? We could have a nine in every single position going on forever well that's just equal to 8.39 recurring right?
So if we have 8.39 recurring that's going to be our largest value that we could possibly have we've got to be careful though because we've got this in a quality sign that says we actually want to find the number that we have to be less than and because we know we could go up to and include 8.39 recurring. The number that we therefore have to be less than is going to be the number directly next to that number, which is going to be 8.4. So that's going to be our maximum number. Finally, last two estimation and bounds, we're a workout estimate for this fairly horrific looking fraction. So we're going to estimate each of these three terms.
So this is going to be roughly 20. This is going to be roughly the square root of 100 and this is going to be roughly 0.2 kwh. So if we then do this calculation well this is just 20 plus square of 100 is going to be ten or divided by 0.2. Well that gives us 30 over zero, which is the same as 30 times five, which is going to give us 150. So that's going to be finally the very last Saturday bounds x is relative to three SIG fig.
The result is 2.15. So we know our number line is going to be counting up in terms of the third significant figure column. So 2.7 is going to go here. Directly below it is going to be 2.15 and directly above it is going to be 2.18. If we then look at what the lowest that this could be is, well, it's going to be directly halfway between 2.16.
Right, anything to the right of this line will round up to 2.17. Anything to the left we round down to 2.16. And likewise directly between zero seven and 00:18. Well to the left we drown down and to the right we drowned up. And so we can see this is our full range of possible values that the true value could actually be.
So now I'm just going to find out what are those maximum minimum values? Well, the minimum it could be is between zero six and 00:17. So that's going to be 2.16 and the maximum or that limiting line there is going to be between zero seven and 18, that's going to be 2.75. And so they are going to be our minimum and our maximum limits. And you can say why is it a 2.75 wouldn't be round up?
How can that be our final answer? Well, it's because we've got this in a quality sign here which says we have to be less than whatever number is in this position. So it's going to be less than 2.75. Anything less than that, we're going to run down to that 2.7. And that includes our lesson on approximation.
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