Overview of Key Mathematical Concepts

to one significant figure. So 6.32 is going to round to 6, 6.51 rounds to 7, and then 0.503 to one significant figure on the bottom is 0.5. Whenever we've got these we've got to work out what's on the top and what's on the bottom. So on the top there, 6 times 7 is 42, and it's 42 divided by 0.5. Obviously if that was a nice little division there we could do that straight away, but dividing by a half does something in particular. So dividing by a half actually doubles your answer. If you think half fits into... 1 twice, so it'll fit into 42, 84 times. So there we go, dividing by 0.5 doubles our number. Obviously if we had something like 3 on the bottom or something like that, that'd be a little easier, but dividing by 0.5 doubles our answer. So that's estimating, just remember to round to one significant figure and then to complete your sums that way whenever you see that word estimate. So write 180 as a product of prime factors and give your answer in index form. So we can do our nice little prime factor tree and again thinking of any two numbers that times to make 180. So I'm going to go for 18 and 10. 10 becomes 2 and 5, locking off those numbers when we see that they're prime. 18 is 2 and 9, and 9 is 3 and 3, and again locking those all off when they're prime. Remember, you can write down some of the prime numbers if it helps. Now writing these in size order, we get 2 times 2 times 3 times 3 times 5, and then we can simplify these. So 2 times 2 becomes 2 squared, 3 times 3 becomes times 3 squared, and then times 5 at the end, and that's an index form there. See? using indices. Okay so there are 15 pencils in a pack and 40 pens in a box. How many packs of pencils and boxes of pens will we need to buy to have the same amount of each? So we're looking at lowest common multiple, so 15 and 40. I'm going to do a list of their times tables, so 15, 30, 45, 60, 75, 90, 105, 120 and there it is. I know that 40, 80, 120, that's going to be our lowest common multiple there. So we're going to do that. So the question said how many packs of each? So that is one on the top we've got one, two, three, four, five, six, seven, eight. That was the pencils so that's eight packs of pencils, eight packs of pencils and below there we have one, two, three, three boxes of pens. Okay so just remember to read the question there what's it actually asking for in the final step even though we were looking at lowest common multiple for that one. Let's have a look at the next one. So find the highest common factor of 72 and 90. So we've got different methods of doing this. We can write down 72, we can write down 90 and start finding all their factors. We can of course use the prime factor method but let's have a look. Let's start with 90. 1 and 90. 2 and 90. and 45, 3 and 30. Let's see what else goes in. 4 doesn't go in, 5 does. 5 goes in 18 times. 6 goes in, let's have a think, 15 times. 7 doesn't, 8 doesn't, 9 does, 9 and 10, now the numbers are right next to each other so there's no more to look at. So starting with the biggest, let's have a look which one goes in. 90 doesn't, 45 doesn't, 30 doesn't go into 72, does 18. 18 actually goes in 4 times. So 4 times 18, don't have to do all the factors of 72 there because we've spotted that it's 18. Of course we could use the prime factor method to find the highest common factor which I'd recommend for numbers like this. So 72 and 90. do their prime factors so 8 and 9, 3 and 3, 2 and 4, 2 and 2, 90 is 9 and 10 and then we have 3, 3, 2 and 5 and we can have a look at what's in both so they both have a 2, they both have a 3 so times 3, they both have another 3 and that's all that they share. 2 times 3 is 6, times 3 again is 18. so we can get our highest common factor using our prime factor method as well. OK, so on the number line, show this inequality. So when we're drawing these sorts of things on the number line, we look at what numbers we've got. We're going to put our little circles above those, so negative 2 and 4, and we're going to join them up, and then we just need to have a look at these symbols. So the 1 here, which is less than or equal to, we're pointing towards that less than or equal to, it has more ink on it, so we just remember to colour that one in, make sure the circle has more ink in there as well, a nice little way of... of remembering it. There we go. So that is between minus 2 and 4. Remember, That is equal to minus 2, so the numbers that this could be is we could have negative 2, negative 1, 0, 1, 2 and 3, but not including the 4 there because it says less than 4. So you can have an inequality like this as well where it just says x is greater than... or equal to 1. So again, putting our circle above the 1, the greater than or equal to has the extra ink on it, so we'll colour that one in. It's equal to that as well, and greater than 1, so we'll just point an arrow to the right saying it's got to be greater than 1. And that's that for that type of inequality. quality. Alright so a washing machine costs £640 plus 20% VAT. James Payne's a £68 deposit and the rest in 10 equal monthly instalments, work out the cost of each monthly payment. So it's a percentage of an amount question so we need to work out 20% of £640. 10% is the most important percentage we ever have to work out so we can work out everything from there. So we'll work out 10%, 10% is £64. 20% will be double that, so double that is £128. So that's the extra amount he's got to pay for. VAT. So if we add these together, 640 add 128, we get 768 pounds. Now it says he's going to pay a 68 pound deposit, so let's take that 68 pounds off because that's what he pays right at the start, and that leaves us with 700 pounds. Now it says in the question he's going to pay that in 10 equal monthly instalments, so if we finish this off, if we divide this by 10, so 700 divided by 10 tells us that he's going to pay 70 pounds. per month. There you go. So the main part here was the working out the percentage. So always work out 10% and remember from there you can figure out any percentage. We could have halved 10%. We could have got 5% as 32. Or we could have actually divided it by 10. We could have got 1%. Dividing 64 by 10, 1% would have been £6.40. And we could have built that up to any percentage that we wanted from there. So £6.40. Let's have a look at something else. Okay, so a mobile phone is reduced by 20% in a sale. sale price is 500, work out the original price. So this is a reverse percentage. It's asking us to work out the price in the past. Now I like to draw this in a slightly different way. I want to identify what percentage is this 500. So if it's been reduced by 20% in a sale, well that means that we have the 80% of the cost and this 20% reduction. If I think of this in terms of two little bars. So I've just got to think which percentage actually is this 500. Is it the 80% or is it the 20%? While the sale price is the amount that we're paying. so not the little 20% sale. So that £500 equals 80%. And I always like to write this down. Now we don't want to know what 80% is, we want to know what 100% is. So I just need to think, how do I actually turn this back into 100%? So if we've got a calculator, this is nice and easy because we can divide that by 80, which would give us 1%. And if we do that on a calculator, 500 divided by 80 gives us £6.25. Okay, so 500 divided by 80. So if we have a calculator, we can do that. we can do 500, divide it by 80, that tells us 1% is £6.25, and then we can times that by 100 to get 100%, and that would be 100%, what we're looking for, equals £625. That's not a 6, £625. Okay, if we don't have a calculator though, we're going to have to think of a different way of doing this. So what we can do is think about actually just dividing this by something to maybe get 10%, and you can divide 80 by 80. 8 to get 10%. So if we divide 500 by 8, we'll also get 10%. And we can do that using a bit of a stop. So off to the side, if we didn't have a calculator, 8 into 500, 8 into 50 goes 40, 48. So it would go in 6 times up to 48 with a remainder of 2. It would go into 20 twice up to 16 with a remainder of 4. So we'd have to carry that over. And then 8 goes into 45 times. So we get that 62.5 is 10%. percent so 10% is 62.5 and then we can times that by 10 to get back to 100% and 100% there would be 625 pounds so we get there either way but that's two different ways of looking at a reverse percentage where we're looking for something in the past when there's a sale going on. Okay so the other way around a house rises in value by three percent the value of the house increases by 7,500 work out the value of the house before the increase so when it's been increased in value again we can visualize this we've got the original price which is our 100% and then it's rising in value by this extra little 3% so again identifying what is this 7500 is it the original 100% or is it the 3% now it says it rises it increases by 7500 so that 7500 is my 3% there you go and again we want to know 100% I can't straightaway times 3 to get to 100 but I can divide both of these by 3 to find out what 1% is And again, we can do that on a couch. We're using better bus stop, but 7,500 divided by 3 is 2,500. So 2,500 equals 1%. And again, we can times that by 100. So over here, if 1% equals 2,500, we can times that by 100. to get our 100% so 100% will equal two five zero zero with another two zeros and that's 250,000 so our answer there will be 250,000 pounds. So 800 pounds is his best for two years offering a 1.5% simple interest rate per annum per year. How much is in the account after two years? Now simple interest means it doesn't keep accumulating so if we first work out what 1.5% is, again if you haven't got a calculator we can do this non-calculator so 10% of 800 would be 80, 1% would therefore be 8 and half a percent which we're going to need there is 4. So joining those two together the 1 and the 1.5 gives us 12 so we get 12 pounds there for our 1.5 percent. Now we can do this using a calculator as well which I'll discuss in a sec but if we're going to get 12 pound per year simple interest just means that's what we're going to get per year so if it's It's two years, so we'll times that by two, and we will get £24. So that's the amount of interest we'll earn. It does say how much is in the account, though. So if we add that back into the account, we'd have £824 as our final answer. Now, if you've got a calculator here, you can always use your multiplier method. So we could do 800, and we could times it by 1.015, and that gives us the answer 812, telling us that in that first year there, we've earned £12 in. interest. So adding in another 12, so add another 12 would give us 824, the same answer. Okay. Or you could times it by 0.015 and that would have just straight up given you the 12 and you can add that on for however many years for simple interest. When it comes to this sort of question, Emily invests £150,000 in a savings account offering a compound interest rate of 2.4% per annum per year for four years. Work out the amount of interest that she earns. So I'm just going to underline that, the amount of interest. So a slightly different part of the question here. So compound interest does keep accumulating. So she earns more in the second year, more in the third year and the fourth year. But we can take a very similar approach. We do 150,000 times it by 1.024 for 2.4%. And rather than finding out what that is and repeating myself, I'm going to do it to the power of four. And that will just keep timesing it by 1.024, adding on that 2.4% every year. Okay. And if we type that into a calculator, 150,000 times 1.024 to the power of 4, we get the answer 164,926 pounds and 74p to the nearest penny. Now, just being careful on this question, because it does say how much interest has this person earned, has Emily earned. So we're going to want to subtract that original 150,000. So take away the 150,000 for this particular one. And that gives us £14,926.74p again to the nearest penny. So there's a bit of compound interest. Just remembering this multiplier here and how that works with compound interest. zero there and putting because if we were to put 1.24 that would increase it by 24% so we don't want that one there. So Mary buys a car for £4,000 each year it depreciates by 20% work out the value of the car in three years so that word depreciates by 20% means it falls in value. Now again with depreciation it's similar to compound interest, it falls in value but it falls in value based on its new price. So if we do this without a calculator we can do it both ways. Year one, the £4,000 is what it's worth, 20% of that we can work out, 10% is £400, 20% is £800 and if we take away that £800 there, so £4,000, take away £800, we end up with £3,200. So we don't keep losing £800 a year. Now in the second year, it's worth £3,200. 10% of that is £320. 20% of that is double that, £640. And we can subtract £640. So £3,200 take away £640 will leave us with our answer for the final year. Take away £640 there gives us £2,560. and we can finish this off, I should have said year 2, we can finish this off on year 3. So our value is £2,560, 10% of that is £256. So 20% will be 512, and then we can subtract 512. So 2,560 take away 512 leaves us with a final value there of 2,000. £2,048 and that would be our final answer. Again, we could do this with a calculator. If something loses 20% in value, then it's only actually worth 80% of its cost the previous year. So if it's worth... 80%, the decimal that I can use for 80% is 0.8. If it's losing value, we won't want a one point number there, we've got 0.8. So we could just times it by 0.8 three times, or times it by 0.8 to the power of three. So I could just type type into my calculator, 4,000 times 0.8 to the power of 3, and I'll get that answer there, 2,048 is my final answer. And let's test that out, 4,000 times 0.8 to the power of 3, and we get 2,048. So that's depreciation. Okay, moving on to fractions, multiplying and dividing fractions. Multiplying fractions is the easiest one, isn't it? So we can times the top numbers, 3 times 2 is 6, and times the bottom numbers, 4 times 3 is 12. And we always want to simplify our answer in a question. will normally ask us to write it in its simplest form. So here we can divide the top and bottom by 6 and we get 1 over 2 as our final answer there. So always looking to simplify. On to the next one, we keep the first fraction the same, we flip the second one over and we multiply them together. So times in the top numbers just like before, 2 times 5 is 10, 1 times 9 is 9 and again we'd normally be asked to write our answer in its simplest form or as a mixed number and here it is an improper fraction, 9 fits into 10. once with a remainder of 1 9th left over. So 1 and 1 9th would be my final answer there. And that's dividing fractions. When it comes to adding and subtracting, the process is the same. I'm just going to go for this one addition. So when we're doing this, we need to obviously look at the fact that there is a mixed number here. And we're going to make that into an improper fraction first. And the same goes for multiplying and dividing. So we're going to do 1 times the 4, the big number times the bottom, and then add the top. So 1 times 4, add the 3, is 7 quarters. And we're going to add to that list. two thirds and again if that was a mixed number we'd also convert that into an improper fraction. Now we need a common denominator when writing and subtracting so that's going to be 12 on the bottom so we can times this left fraction top and bottom by 3 and we can times this right fraction on the top and bottom by 4 and that's going to give us two fractions out of 12 so 7 on the left times 3 is 21 over 12 and on the right 2 times 4 is 8 over 12 and now we can add these together and again if we were subtracting them we'd just subtract those numerators now but we'll add them together and we get 12. 29 over 12. And again, it says to give your answer as a mixed number. So we are going to convert this now into a mixed number. 12 fits into 29 twice, up to 24. And there'd be a remainder of 5, 5 twelfths. And again, there's my final answer. Just be very, very careful at this point. Just to have a look, does this fraction simplify here, the little one? It doesn't. But let's imagine it was 4 twelfths there instead of 5 twelfths. That would actually simplify, wouldn't it? Because if it was 4 twelfths, we could divide the top and bottom by 4. And that would end up being 1. third. So just have a look in that situation. Does it simplify? This one obviously doesn't. It's two and five twelfths. Let's have a look at another one. So in a bag of counters, two-fifths are blue and it says 18 are blue. Work out the total number of counters in the bag. So it's not asking us to work out two-fifths of a number. It's saying that two-fifths are 18. Remember, it could have said work out two-fifths of, let's say, 20. And you divide by the bottom to get one-fifth. You'd say that one-fifth is four. And then times by the top to get two-fifths. fifths and we'd get two fifths would be eight. That's a different question so it's nothing to do with that but it says two fifths are blue and 18 are blue so two fifths equals 18. So if I want to work out one fifth in this scenario I need to divide by the top this time because that'd give me one fifth half of that so one fifth would be nine and then to work out five fifths the original amount we could times that by five so times it by five and we would get five fifths equal Equaling 45 counters. There you go, so that's our scenario there. So slightly different, the other way around obviously to working out a fraction of an amount. It says that that fraction is 18. So dividing it by 2 to get 1 fifth and then times it by 5 to get the original amount. Okay, so writing some numbers in standard form. We're going to make this a number between 1 and 10. Times 10 to the power of however many jumps we have to do in whatever particular direction. So this number here, 5600, I'm going to hop the decimal in between the 5 and the 6. So 1, 2, 3. So that becomes 5.6 times 10 to the power of, it's a big number there, it's not a nought point number, so 10 to the power of 3. Final answer. The next one is a small number, so we're going to have a negative power, because we hopped the decimal the other way this time. So 1, 2, 3 would make it 3.4. And again, this time it would be times 10 to the negative 3, because it's a nought point number, that indicates that it's a small number. When it comes to the other way around, writing them as ordinary numbers, I like to rewrite this little bit at the start to start with. So 2 and 3, and I like to just imagine where that decimal is between the 2 and the 3. So times 10 to the power of 5 makes it a big number, so we need to hop it five places. 1, 2, 3. 4, 5 to the right, and then filling in all those zeros underneath. And if it is messy, just remember to rewrite that. But there's my answer, 230,000. The next one, 8.04 times 10 to the negative 3. So this is going to be a 0 point number. And again, I'm just going to write... write these digits out at the start, these bits here, and imagine where the decimal is at the top, 8.04. So negative 3 means it's going to be a small number, so we're going to jump it left 3, 1, 1, 2, 3. So it's going to go there, fill in all the zeros, and tidy it up at the start. So 0.00804, final answer. Okay, so write this as a power of 3. So when we've got the same base numbers, which we do, these are all 3s, we can add the powers when multiplying and subtract the power. when dividing. So if we tidy up the top to start with, which I always do first, we add the powers there because they're getting divided, sorry multiplied. So 3 to the power of 7 and on the bottom we've got 3 to the power of 3. Now these are being divided so we can subtract the powers, so 7 take away 3 is 4, so we get 3 to the power of 4. And again it doesn't say to work out the value of that, it says to leave it as a power of 3. If it said work out the value of it I'd actually have to work that out and that means 3 times It wouldn't be very nice without a calculator, that one, but we could work it out. But it says to write as a power of 3. OK, so negative and fractional powers. Remembering the negative symbol there just means to do the reciprocal to flip it over. So in this first one, if that's 5 over 1 at the moment, it'll become 1 over 5. Now, if it was just a negative 1 in the power, we'd leave it as it is because it'd be a power of 1. But that 2 is just a normal power, which means we're going to have to square both of these. It's a normal squared power. So 1 squared on the top is 1, and 5 squared on the bottom is 20. 75, and that's our final answer there. Onto a fractional power. Remembering that number on the bottom of the fraction represents a root. So in the case of a 2, that is a square root. So we're going to do the square root of 64, and the square root of 64 is 8. Now the 1 on the top is a normal power, so a normal power of 1 doesn't do anything. But if it was a 3, for example, we'd have to cube it, but it is just a 1. There's no negative in that one, so we don't have to flip it over or anything. It's just the square root, which gives us the answer 8. Okay, I'll... on some combinations. It says in a bike store there are eight different bikes and ten different choices of paint colour. How many different combinations are there? Well we have eight choices of bikes to pick from so that's eight different combinations and all we do is we multiply it by however many choices we have on the second option. So the second option is the paint colour and there are ten so we times it by ten as our second option. So eight times ten gives us eighty options as our final answer. So you just take how many choices you have on each pick and multiply them together and that's combinations. OK, so this one's slightly different. A three-digit padlock uses the numbers from 0 to 9. How many of the possible three-digit number combinations use three different digits, meaning we can't have any repeats here? So let's imagine on the first one, on the first digit of this padlock, let's imagine we've got three numbers. And on that first digit, let's say we pick the number 2. Now, in that first pick there, we had the numbers from 0 to 9, including 0, so that would be 10 options that I had. But now that I've picked the number 2, I can't pick the number 2 again. is no repeats. So I don't have 10 options now, I only have 9 options. Let's imagine we pick the number 3. So on this option now, on this third pick, I can't pick 2, I can't pick 3 out of these 10 numbers, so I only have 8 to choose from. So there are my 3 digit padlocks, let's just imagine we pick the number 4, 2, 3, 4, and we'd have to multiply these all together. So 10 times 9 gives us 90, I thought we've got to times by 8, and 8 times 9 is 72, that's not a 7. 8 times 9 is 72, so 720. And that's how many different combinations we'd have for this particular padlock. So 10 times 9 times 8, perfect, 720. OK, some error intervals. 6.4 has been rounded to one decimal place, and write the error interval. So the error interval, we always tend to put either 6.4 or a letter in the middle, depending on what the question is, and we have our arrows always pointing to the left. So we're going to find the upper and lower bound. Now 6.4, if it's been rounded to one decimal place, it couldn't have... been 6.45. As soon as that 5 goes after that 4, it would have rounded up to 6.5. So we put that on the right hand side as our biggest possible number that it couldn't have been 6.45. So it can't be equal to that number, so we leave that symbol as it is. It's got to be less than 6.45. Now what we did there is we added on this x to the x-axis. So we Extra 0.05 and all we're going to do for the smaller one is take that away. So it would be 6. But it can actually be equal to 6.35, so I'm going to add my little equal to symbol on my inequality just there. So it can always be equal to that lower bound because that would in fact round up to 6.4 if we rounded it to one decimal place. On to the next one. This one's slightly different in the wording. So this one says it's been truncated to one decimal place. And truncated basically just means it's been chopped off after this first decimal, regardless of how it rounded. So if we have a look at writing this is... as an error interval. Again, we write these in the same way. The impossible number that it couldn't have been, it couldn't have been 6.5. So my top one here is 6.5. It had to be 6.4 something if it was just chopped off. Let's imagine just a random how this works. So it could have been 6.4932, but it was just chopped here and we just wrote what was at the start. Okay, but it couldn't have possibly been 6.5 because otherwise we'd have written 6.5. And the smallest it could have been is just 6.5. 6.4 with no digits after it. It couldn't have been 6.3 though because again if we truncate that it would have just been 6.3. So two different scenarios there. One where you've got to add on that extra 5 at the end and take the same size 5 off the start and truncating there where we've just got to think about the fact that it was just chopped off and no rounding at all. And the final little one we're going to have a look at here is using your calculator. So it's important to know where lots of digits are. I've picked quite a complicated one here but you just need to have a play around with your calculator to make sure you can find your your square root, your cube root, your squared, your cubed, your fraction button, your sin, your cos, your tan, and I'm figuring these out. So I've picked quite a complicated one here. So to get your cube root button, depending on what calculator you have, but the majority of them, you still have to press the same things. We press shift and then we go for, let's get a different colour here, we go for the square root button and that allows us to get a cube root. So on your calculator, a cube root will pop up and then we're going to use the fraction button. So the fraction button. button is just next to it in this diagram here, the fraction button, and an empty fraction is going to pop up. So on the top, we're going to just type in on the top of that fraction there, 4.3 multiplied by, press your tan button, and then press 39. And normally, this will go in a bracket, so remember to close the bracket off there. But you don't put this little degrees symbol in, you just put 39, the calculator knows that you're talking about degrees. So let's get that up on my calculator, we've got 4.3 multiplied by 39. So we've got multiplied by tan, the brackets opens for me, 39, close the brackets, press your down button on the calculator to go into the bottom of the fraction, and type this sum in here, 23.4, take away 6.06, so 23.4, take away 6.06, and make sure at this point you write all the numbers down, that equals 0.5855934. 1, 2, 3, 3. Now some calculators may go beyond this, depending on what the answer is on the calculator, but it tends to be that you never really have to go beyond 9 digits, because not all calculators will go that far, so you are limited really to the amount that some people will be able to write, depending on what calculator they use. Now the question will probably usually say, then to round your answer to 1 or 2 decimal places. So if we just have a quick think, if we were to round this to 1 dp, to 1 decimal place, and we could chop that after the 5 there, So one decimal place would round up to 0.6. And we would round that to 0.6. If we had this to two decimal places or beyond, let's have a think. After two decimal places, let's pick a different colour. It would chop it after the 8. And it would be 0.59 because it's a 5 after that one. OK, so you could have different ways of rounding this. But just make sure you write down all the digits on your calculator display before actually rounding anything. Okay, so kicking this off with fractional and negative indices. So you've just got to remember, looking at this one on the left, if there is an index, the negative does the reciprocal, which I always just write as a flip. The bottom is the number underneath, which does the root. And then the number on the top is just a normal power. Okay, so I'm just going to put power with that, so normal power. So for this particular one, we've got all three going on. I always start with flipping it over doing the reciprocal and remembering that 8 is just a number so we could write that as 8 over 1. So when we flip that over it becomes 1 over 8. Then we've got to do either the root or the power. I always like to do the root first, make the number smaller before trying to then do the reciprocal. do the power. So a 3 on the bottom would represent a cube root, so we need to do the cube root of both of these numbers. The cube root of 1 is 1, and the cube root of 8 is 2. 2 times 2 times 2 is 8, so it's 1 half. Then we've got to finish it off, we've still got to do that normal power there. If it was a 1 it wouldn't affect it, but a 2 as a normal power is squaring, so if we square them both we get 1 over 4 as a final answer. And just remember to follow all those rules. Now looking at the one on the left, we've not got a negative, so we don't have to flip it over, but we have got 2. two on the bottom which is the root again and a two is a square root and then we've got a three on the top which is going to cube the numbers. So if I start with the roots to start with if I do the square root of both of these and it's always a little hint there they are square numbers the square root of nine is three the square root of 100 is 10 and then we've got to actually cube them as well so if I cube these as well let's write that in three cubed is 27 and 10 cubed is a thousand so it'll be 27 over a thousand just being very careful to get those right. So that's negative and fractional indices. Okay so when you see this sort of question you see these words here about bounds, there's always something we need to write down straight away. So we've been given two numbers here, so s is correct to two decimal places and t is correct to one decimal place. So the first thing I'm going to write down is the error intervals for them before I move any further. So writing our error interval, arrows pointing to the left and an equal to symbol on our lower bound there. So I always start with the upper bound and I'll just stick a five on the end so eight point two four. with a 5 on the end and then we just need to take that 5 off for the lower bound so it'll be 8.2 not 4 but 3 with a 5 on the end and then do the same for the other one as well so let's write down the upper and lower bound for 3.5 so 3.5 stick a 5 on the end and then for the lower one 3 point drop it down to a 4 and again just stick a 5 on the end now this question here asks us to find the upper bound so when it comes to a fraction and you could have anything here but when it comes to a fraction if we want to get the biggest possible answer we want to to do either, we've got to decide here, either we're going to do the upper bound divided by the lower bound, or we're going to do the lower bound divided by the upper bound. And we've just got to think which one there is going to give us our biggest possible answer. And that is the upper bound divided by the lower bound. So we want the biggest number on top divided by the smallest number, and that'll give us our biggest possible answer. So we want the upper bound of S, S sitting on top, so the 8.245, and the lower bound of T, 3.45 sitting on the bottom. So this is a calculator question. So we just need to type that in on the calculator. calculator 8.245 divided by 3.45 and see what we get as our answer there so if we type that in 8.245 divided by 3.45 and we get the answer it's quite a lot of decimals here 2.38985507 always just write down all the numbers and then round it however you've been asked in the question this one says three decimal places so let's chop it after the nine and that would go to two 2.3, and this is a bit of a dodgy one because the 8 rounds the 9 up, so it'll go to 2.390, keeping the 0 in there, it does say three decimal places, so 2.390 would be my answer there. Obviously it might say lower bound as well, in which case you'd use the other side of the formula, we'd use this one instead. You might even be asked to do both of them and then round it depending on what they both look like, so if we imagine that the lower bound comes out, and if we just make something up here, let's imagine the lower bound comes out. out as 2.295 or something like that. Now it might say considering bounds give your answer to an appropriate degree of accuracy. And if you look at both these numbers, if we just imagine this is the lower bound, it's probably not, but if we imagine it was, these two numbers would not round the same to two decimal places, not to three decimal places, not even to one decimal place because the one on the left there would round to 2.4 and the one on the right would round to 2.3. have to consider bounds to give an answer for this. I just have to say 2 and I would say the answer is 2 and that would be to one significant figure. So I could be asked that as well and I just look at the upper and the lower bound and I round according to how they both actually round. In this case, my little example here, they only round the same to one significant figure. So recurring decimals as fractions. So a little rule, we are going to use a bit of algebra for this. But if there's two recurring decimals, we're going to times it by 100. And if there's one recurring decimal, we times it by 10 and so on. There's lots of different ways of doing this. This is just the way that I do them. So I write what the decimal is. We've got x equals 0.54. And remember, if there's a dot above the 5 and the 4, that means that that decimal there is 0.5454. And continues like that. It's the 5, 4 that's recurring. It's always good just to write that down, just to remind yourself what that decimal actually represents. So if we've got two recurring decimals, we can times it by 100. times that by 100. You'll see why if you've not seen this before. Hopefully it's just a reminder. You get 54.54. Now we can take them away from each other and eliminate these recurring decimals here. If we take them away from each other, we have 99 x's equals 54. And make sure that you actually do use your algebra to show this. Now we can divide both sides by 99. So x equals 54 over 99. And there it is as a fraction. You do just need to check. just to see if your fraction simplifies. So do check to see if this one simplifies. This one actually does. The top and the bottom both divide by 9. And you'd normally be asked to show that it equals this next one, but if it doesn't, you know, you can look to simplify if it says to give it in its simplest form. And if you divide them both by 9, you get 6 over 11. So the question might have said show that it equals 6 over 11, and that's how you would show it. You'd show how it simplifies. Now on to the one below. It's only got one recurring decimal, but it's a bit of a nasty one. So if we think about what this decimal actually looks like, 0.23. with a dot above the 3. That's 0.2 and then a recurring set of 3s. It's not a full repeating pattern of 2, 3, 2, 3, 2, 3. So if we have a look at this one, and again, there are different ways of doing this type of question here, but I just follow the same process for all of them. So x equals 0.23 with a dot above the 3. And there's one recurring decimal, so let's times it by 10. So we get 10x equals 2.33, and then the 3 keeps going. I'll just balance it out so there's the same amount of decimal places in both. Now if we take these away from each other, something kind of interesting happens here. We get 9x equals, and 2.33 take away 0.23 is 2.1. Now if we take the same approach as before and turn that into a fraction, so dividing both sides by 9, we get x equals 2.1 over 9. And obviously you can't have decimals in fractions, so we need to remove this decimal from the top. So we can times the top and bottom by 10, and that would give us 21 over 90. There you go, and there's your fraction and again You just need to check to make sure that's if see it let to see if that simplifies or not and actually does the top and bottom both divide by 3 so you can divide the top and bottom by 3 and You would get 7 over 30 Again, so it might have said show that this fraction equals 7 over 30 But always just checks it at the end see if it simplifies obviously if it just said to write it as a fraction 21 over 90 would be fine, but if it said to give it a simplest form we would have to look to see simplify there. Okay, so some compound interest. James invests £2,500 for three years in a savings account and gets 3% per annum compound interest in the first year and then X% for two years. He has this much at the end of the three years work out the value of it. Well just like normal compound interest here we can actually do the first year. So we can work out, I'm just going to label this up, so after the first year let's see how much money he has. So £2,500 for 3% we would times that by 1.03. Obviously we don't need to add any powers into there because it is just the first year that he's getting this. So if we do this on the calculator times 1.03 and we get an answer of £2,575. Now we've got a bit of a reverse compound interest scenario and we're going to have to create a little formula for this. the second and third year he gets x percent. Now his starting number, and let's just write down second and third, his starting number now is 2,575. Now if we knew the percentage we'd multiply it by that percentage which we'll call x in this scenario because we don't know what it is and we'd do that to the power of two as it's two years and what it's saying in the question is that gives us the answer 2,705 pounds and 36p. Obviously at this point here we could just get guess some numbers we could just guess an x value and if it's too high go lower if it's too low go higher and that's a fine way of doing it as well but you can save yourself a bit of time here just guessing lots of numbers by actually just rearranging this so we can divide both sides by 2575 and it will give us the value of x squared. So x squared is going to equal, and again let's just do this on the calculator, 270536 divided by that gives us this answer here, 1.050625243. That's obviously the value of x squared, so we need to square root both sides. So if we square root that answer, we get the answer 1.02. five zero zero zero one one eight. So that gives us our little percentage here. So it's hidden within this decimal. Now our actual percentage here is not 1.025 because that one point is what we increased by for the percentage. So it's just this 0.25 that we're interested in and 0.25 if you think back to what we did at the start here with 0.3 being 3% well 0.25 would be 2.5%. There we go so that's 2.5%. So just be careful that you do read that off there and don't put 1.025 in this type of question. But again, you could have guessed. If you'd have guessed 1.02, it's normally going to be lower than that first one. That's normally what banks do. They give you a better one in the first year and then not so good following that. OK, so there's just two ways that you can approach that. You can guess some numbers or you could obviously just rearrange it and find it there. Just be very careful that you actually read the percentage from that one point number. OK, so we've got some SIRDS questions. So 3 plus root 5 squared in the form a plus b root 5. where a and b are integers. Okay so we need to have a look at this so that obviously is a double bracket there so if I rewrite this as a double bracket we've got 3 plus root 5 and 3 plus root 5 and we just need to expand that now and see what we get so lots of little surd rules going on in this one. So 3 times 3 gives us 9, 3 times root 5 is 3 root 5 5 root 5 times 3 is another 3 root 5 and then root 5 times root 5 is root 25 and it just becomes 5 remembering similar serves when all these exact same serves when you multiply them together become the whole number and if we simplify all this down 9 and 5 makes 14 and then we can add together these thirds in the middle because they both have a root 5 so we've got 3 lots of root 5 and 3 lots of root 5 which is 6 lots of root 5 and there's your answer just there so a would be 14 and B would be 6 and that is in the form a plus b root 5 and there's our answer with a bit of thirds. Let's have a look at another thirds question. So we've got to show that this can be written in this form. Now when we've got thirds on the bottom of a fraction we're always going to look to rationalise it. Now with some of the easier thirds you just times the top and bottom by whatever this root is just here, so by root 3. Unfortunately that's not going to work for this one because we've got 2 plus root 3 and if we did times the bottom by root 3 there we'd get 2 root 3 plus 3 and we'd still have a third on the bottom. So what you do when you've got this on the bottom is you times it by the denominator still but we change. this little sign here so whatever that sign is we're going to flip it to the opposite so we're going to times it by 2 minus root 3 so always write that down okay because you do get marks for these these elements of you working out so I'm just going to rewrite the fraction 2 plus root 3 and I'm going to show next to it what I'm timesing the top and bottom by so I'm going to times by 2 minus root 3 and I'll put that in a bracket because I'm going to do a double bracket here and I'm also going to times the top by 2 minus root 3 because we're creating an equivalent fraction here so it's up to you if you want to put these in brackets as well just remind yourself you're doing a double bracket and then you just need to multiply both of them out and see what we get. So on the top and I'm not going to draw all over it I'm going to times them both by the 5 first and then times them both by the 2 root 3. So 5 times 2 on the top gives us 10 5 times negative root 3 gives us negative 5 root 3. On to the 2 root 3 so 2 root 3 times 2 is positive 4 root 3 and then 2 root 3 times negative root 3 and this is the multiplication you always just got to be careful of. gives you two, I'm going to do it to the side, we get two lots of root 9. Two lots of root 9 is 2 times 3, which is 6. So that number at the end there, and careful because one's positive, one's negative, so we get negative 6. I'll tidy that all up in a sec, let's have a look at what's on the bottom. Now the whole reason we're doing this on the bottom is to eliminate the thirds in the middle, so you can do a bit of a trick when you're doing these. You can just do the first two, which is 2 times 2, which is 4, and the last two. So root 3 times root 3 is 3, and it's minus 3. There we go, so that's why our fraction's disappearing, because it's 4 minus 3 on the bottom. Let's tidy up what's on the top. So on the top there we have 6 and the 10, so we've got 10 minus 6, which is 4. We've got negative, let's do this in a different colour, we've got negative 5 root 3 add 4 root 3. Negative 5 add 4 is negative 1, so minus 1 root 3, or minus root 3, and that's all over 4 take away 3, so it's over 1. So when we're dividing by 1, we can get rid of the fraction there, and we get 4 minus. root 3. So there's some surd elements to be thinking about. Obviously there are so many different questions on surds that you can actually have, but there's a couple of things there to be thinking about. The other thing is obviously to remember that surds can only be collected together like in here, when the number underneath the surd is the same. Okay, so you will have to know how to simplify surds as well. I've got plenty of videos on that to check out. Okay, so we've got some standard forms, we've got some multiplication and some division. I'm going to start with this multiplication. Now I'm primarily thinking this is like a non-calculator style. question because obviously if this was on a calculator you could just type it in nice and easy and write it in standard form. I have to do a bit of a trick here. So 3.2, I can times that by 4 to figure out what my starting numbers would be. So 3.2 times 4, again, you've just got to work that out, whatever method you're used to using. I'm going to do 32 times 4 and hop the decimal back in. times 2 is 8 and 3 times 4 is 12 and then hop the decimal back in so it's 12.8. So the answer I get there is 12.8, it's going to be times by 10 and when we're multiplying we can add together the powers here so 3 plus 4 gives us 7. Obviously though it says to write your answer in standard form and that number here, 12.8, is not between 1 and 10. So if I hop the decimal in making the number 1 place value smaller, I'd get 1.28. And if I make the number smaller, I have to make the power 1 bigger, so it'd be 10 to the power of 8. Okay, so don't be put off by these sorts of questions. Do your multiplication, follow your power rules and then balance it out. Just remember if you make the number smaller, you make the power bigger. We'll take exactly the same approach for this one except it's a divide. So I need to do 3.6 divide. divided by 4. And again, it's up to you. You can hop the decimal out and hop it back in, but I'm just going to do 3.6 divided by 4 using a bit of a stop. So 4 doesn't go into 3, so carry the 3. And then 4 goes into 36 nine times. There we go. So it's 0.9. So when we divide that, we get 0.9 times 10 to the power of, and when we're dividing, we can subtract the powers. So the first power there is 2. The second power is negative 3. So just write that down to the side. What have we got? We've got 2 take away negative 3, which is to add 3. which gives us a power of 5. There we go, so the power is 5. And then again we just need to balance that out because obviously this is not between 1 and 10. So I'm going to hop the decimal this way making it 9. So this time I've made the number bigger so I need to make the power smaller so it goes down to a 4. And there's our final answer 9 times 10 to the 4. And our last question on this section. So we've got in 2008 Alice bought a house in 2000, sorry in 2008 Alice bought a house. In 2014 she sold it for 20% profit and in 2019... it was sold at a 5% loss and it gives us the value that she sold it for. Work out the cost of the house in 2008, so right back at the start. So we had 2008, she bought a house and she sold it for a 20% profit. Then in 2014, it was sold for a 5% loss. And then in 2019, the value of it was £182,400. Right, so at this point in 2019, this cost here to whoever she sold the house to in 2019, 2014, this was sold at a 5% loss. So to her, that was only 95% of the value that she bought it for. So if we've got a calculator here, in order to turn that back into 100%, quite nice and simply we can divide it by 95. to get 1%. So if that there is 95%, we can divide that by 95. So divide by 95, which would give us 1%. So 1% would equal, and we'll just do this on the calculator, 182,400 divided by 95 gives us 1,920. And then to get back to 100%, we just times that by 100. So we can just add two zeros on for that. So 1920 with extra two zeros. There we go, so 192,000. So that's the price that it was sold for. Sorry, she did actually paid for it before. So before that, it was 192,000. That's not a zero. 192,000. And then we've got to think about the original sale in the first place. So that means that that 192,000 there was a 20% profit. So if that was a profit to Alice in the first place, then to her this amount of money was 120%. Okay, it was 20% more than she originally paid for it. So we can follow exactly the same approach now. If that's 120%, we would divide that by 120 to give it us 1%. So 1% here would equal, and we'll just work that out. Divide that by 120 and 1% is 1,600. And again, you can times that by 100 to get 100%. And our original cost up here would be 160. Zero with two zeros on the end, so 160,000 pounds. There we go, so 160,000. There we go. So that's it using a calculator. That's moving through two reverse percentages there. One with a decrease, one with an increase. You just got to remember if something's been decreased, then we get this 95%, whatever was left over. And then there was also the scenario where at one point it was increased. making it 120%, that extra 20% there. So this can be done in calculator and it can be done in non-calculator. The 95% one wouldn't be very nice, but thinking about how you could have done this 120%, which is quite common when VAT is added on because they often have these VAT questions. or they say VAT was added on at 20%. There is a way of doing that without a calculator and what you could do when that happens is you can divide by 12 and then times by 10. So you could do that without a calculator. You could do a bit of a stop there and divide 192,000 by 12 and then just add on a zero when you times by 10.

So on the top there, 6 times 7 is 42, and it's 42 divided by 0.5. Obviously if that was a nice little division there we could do that straight away, but dividing by a half does something in particular. So dividing by a half actually doubles your answer.

If you think half fits into... 1 twice, so it'll fit into 42, 84 times. So there we go, dividing by 0.5 doubles our number. Obviously if we had something like 3 on the bottom or something like that, that'd be a little easier, but dividing by 0.5 doubles our answer. So that's estimating, just remember to round to one significant figure and then to complete your sums that way whenever you see that word estimate.

So write 180 as a product of prime factors and give your answer in index form. So we can do our nice little prime factor tree and again thinking of any two numbers that times to make 180. So I'm going to go for 18 and 10. 10 becomes 2 and 5, locking off those numbers when we see that they're prime. 18 is 2 and 9, and 9 is 3 and 3, and again locking those all off when they're prime. Remember, you can write down some of the prime numbers if it helps.

Now writing these in size order, we get 2 times 2 times 3 times 3 times 5, and then we can simplify these. So 2 times 2 becomes 2 squared, 3 times 3 becomes times 3 squared, and then times 5 at the end, and that's an index form there. See?

using indices. Okay so there are 15 pencils in a pack and 40 pens in a box. How many packs of pencils and boxes of pens will we need to buy to have the same amount of each? So we're looking at lowest common multiple, so 15 and 40. I'm going to do a list of their times tables, so 15, 30, 45, 60, 75, 90, 105, 120 and there it is.

I know that 40, 80, 120, that's going to be our lowest common multiple there. So we're going to do that. So the question said how many packs of each?

So that is one on the top we've got one, two, three, four, five, six, seven, eight. That was the pencils so that's eight packs of pencils, eight packs of pencils and below there we have one, two, three, three boxes of pens. Okay so just remember to read the question there what's it actually asking for in the final step even though we were looking at lowest common multiple for that one. Let's have a look at the next one. So find the highest common factor of 72 and 90. So we've got different methods of doing this.

We can write down 72, we can write down 90 and start finding all their factors. We can of course use the prime factor method but let's have a look. Let's start with 90. 1 and 90. 2 and 90. and 45, 3 and 30. Let's see what else goes in. 4 doesn't go in, 5 does.

5 goes in 18 times. 6 goes in, let's have a think, 15 times. 7 doesn't, 8 doesn't, 9 does, 9 and 10, now the numbers are right next to each other so there's no more to look at. So starting with the biggest, let's have a look which one goes in. 90 doesn't, 45 doesn't, 30 doesn't go into 72, does 18. 18 actually goes in 4 times.

So 4 times 18, don't have to do all the factors of 72 there because we've spotted that it's 18. Of course we could use the prime factor method to find the highest common factor which I'd recommend for numbers like this. So 72 and 90. do their prime factors so 8 and 9, 3 and 3, 2 and 4, 2 and 2, 90 is 9 and 10 and then we have 3, 3, 2 and 5 and we can have a look at what's in both so they both have a 2, they both have a 3 so times 3, they both have another 3 and that's all that they share. 2 times 3 is 6, times 3 again is 18. so we can get our highest common factor using our prime factor method as well. OK, so on the number line, show this inequality. So when we're drawing these sorts of things on the number line, we look at what numbers we've got.

We're going to put our little circles above those, so negative 2 and 4, and we're going to join them up, and then we just need to have a look at these symbols. So the 1 here, which is less than or equal to, we're pointing towards that less than or equal to, it has more ink on it, so we just remember to colour that one in, make sure the circle has more ink in there as well, a nice little way of... of remembering it. There we go.

So that is between minus 2 and 4. Remember, That is equal to minus 2, so the numbers that this could be is we could have negative 2, negative 1, 0, 1, 2 and 3, but not including the 4 there because it says less than 4. So you can have an inequality like this as well where it just says x is greater than... or equal to 1. So again, putting our circle above the 1, the greater than or equal to has the extra ink on it, so we'll colour that one in. It's equal to that as well, and greater than 1, so we'll just point an arrow to the right saying it's got to be greater than 1. And that's that for that type of inequality. quality.

Alright so a washing machine costs £640 plus 20% VAT. James Payne's a £68 deposit and the rest in 10 equal monthly instalments, work out the cost of each monthly payment. So it's a percentage of an amount question so we need to work out 20% of £640.

10% is the most important percentage we ever have to work out so we can work out everything from there. So we'll work out 10%, 10% is £64. 20% will be double that, so double that is £128. So that's the extra amount he's got to pay for. VAT.

So if we add these together, 640 add 128, we get 768 pounds. Now it says he's going to pay a 68 pound deposit, so let's take that 68 pounds off because that's what he pays right at the start, and that leaves us with 700 pounds. Now it says in the question he's going to pay that in 10 equal monthly instalments, so if we finish this off, if we divide this by 10, so 700 divided by 10 tells us that he's going to pay 70 pounds.

per month. There you go. So the main part here was the working out the percentage.

So always work out 10% and remember from there you can figure out any percentage. We could have halved 10%. We could have got 5% as 32. Or we could have actually divided it by 10. We could have got 1%. Dividing 64 by 10, 1% would have been £6.40.

And we could have built that up to any percentage that we wanted from there. So £6.40. Let's have a look at something else.

Okay, so a mobile phone is reduced by 20% in a sale. sale price is 500, work out the original price. So this is a reverse percentage.

It's asking us to work out the price in the past. Now I like to draw this in a slightly different way. I want to identify what percentage is this 500. So if it's been reduced by 20% in a sale, well that means that we have the 80% of the cost and this 20% reduction. If I think of this in terms of two little bars.

So I've just got to think which percentage actually is this 500. Is it the 80% or is it the 20%? While the sale price is the amount that we're paying. so not the little 20% sale.

So that £500 equals 80%. And I always like to write this down. Now we don't want to know what 80% is, we want to know what 100% is.

So I just need to think, how do I actually turn this back into 100%? So if we've got a calculator, this is nice and easy because we can divide that by 80, which would give us 1%. And if we do that on a calculator, 500 divided by 80 gives us £6.25.

Okay, so 500 divided by 80. So if we have a calculator, we can do that. we can do 500, divide it by 80, that tells us 1% is £6.25, and then we can times that by 100 to get 100%, and that would be 100%, what we're looking for, equals £625. That's not a 6, £625. Okay, if we don't have a calculator though, we're going to have to think of a different way of doing this.

So what we can do is think about actually just dividing this by something to maybe get 10%, and you can divide 80 by 80. 8 to get 10%. So if we divide 500 by 8, we'll also get 10%. And we can do that using a bit of a stop. So off to the side, if we didn't have a calculator, 8 into 500, 8 into 50 goes 40, 48. So it would go in 6 times up to 48 with a remainder of 2. It would go into 20 twice up to 16 with a remainder of 4. So we'd have to carry that over.

And then 8 goes into 45 times. So we get that 62.5 is 10%. percent so 10% is 62.5 and then we can times that by 10 to get back to 100% and 100% there would be 625 pounds so we get there either way but that's two different ways of looking at a reverse percentage where we're looking for something in the past when there's a sale going on. Okay so the other way around a house rises in value by three percent the value of the house increases by 7,500 work out the value of the house before the increase so when it's been increased in value again we can visualize this we've got the original price which is our 100% and then it's rising in value by this extra little 3% so again identifying what is this 7500 is it the original 100% or is it the 3% now it says it rises it increases by 7500 so that 7500 is my 3% there you go and again we want to know 100% I can't straightaway times 3 to get to 100 but I can divide both of these by 3 to find out what 1% is And again, we can do that on a couch. We're using better bus stop, but 7,500 divided by 3 is 2,500.

So 2,500 equals 1%. And again, we can times that by 100. So over here, if 1% equals 2,500, we can times that by 100. to get our 100% so 100% will equal two five zero zero with another two zeros and that's 250,000 so our answer there will be 250,000 pounds. So 800 pounds is his best for two years offering a 1.5% simple interest rate per annum per year.

How much is in the account after two years? Now simple interest means it doesn't keep accumulating so if we first work out what 1.5% is, again if you haven't got a calculator we can do this non-calculator so 10% of 800 would be 80, 1% would therefore be 8 and half a percent which we're going to need there is 4. So joining those two together the 1 and the 1.5 gives us 12 so we get 12 pounds there for our 1.5 percent. Now we can do this using a calculator as well which I'll discuss in a sec but if we're going to get 12 pound per year simple interest just means that's what we're going to get per year so if it's It's two years, so we'll times that by two, and we will get £24.

So that's the amount of interest we'll earn. It does say how much is in the account, though. So if we add that back into the account, we'd have £824 as our final answer.

Now, if you've got a calculator here, you can always use your multiplier method. So we could do 800, and we could times it by 1.015, and that gives us the answer 812, telling us that in that first year there, we've earned £12 in. interest. So adding in another 12, so add another 12 would give us 824, the same answer. Okay.

Or you could times it by 0.015 and that would have just straight up given you the 12 and you can add that on for however many years for simple interest. When it comes to this sort of question, Emily invests £150,000 in a savings account offering a compound interest rate of 2.4% per annum per year for four years. Work out the amount of interest that she earns.

So I'm just going to underline that, the amount of interest. So a slightly different part of the question here. So compound interest does keep accumulating.

So she earns more in the second year, more in the third year and the fourth year. But we can take a very similar approach. We do 150,000 times it by 1.024 for 2.4%.

And rather than finding out what that is and repeating myself, I'm going to do it to the power of four. And that will just keep timesing it by 1.024, adding on that 2.4% every year. Okay. And if we type that into a calculator, 150,000 times 1.024 to the power of 4, we get the answer 164,926 pounds and 74p to the nearest penny.

Now, just being careful on this question, because it does say how much interest has this person earned, has Emily earned. So we're going to want to subtract that original 150,000. So take away the 150,000 for this particular one. And that gives us £14,926.74p again to the nearest penny. So there's a bit of compound interest.

Just remembering this multiplier here and how that works with compound interest. zero there and putting because if we were to put 1.24 that would increase it by 24% so we don't want that one there. So Mary buys a car for £4,000 each year it depreciates by 20% work out the value of the car in three years so that word depreciates by 20% means it falls in value. Now again with depreciation it's similar to compound interest, it falls in value but it falls in value based on its new price. So if we do this without a calculator we can do it both ways.

Year one, the £4,000 is what it's worth, 20% of that we can work out, 10% is £400, 20% is £800 and if we take away that £800 there, so £4,000, take away £800, we end up with £3,200. So we don't keep losing £800 a year. Now in the second year, it's worth £3,200.

10% of that is £320. 20% of that is double that, £640. And we can subtract £640.

So £3,200 take away £640 will leave us with our answer for the final year. Take away £640 there gives us £2,560. and we can finish this off, I should have said year 2, we can finish this off on year 3. So our value is £2,560, 10% of that is £256. So 20% will be 512, and then we can subtract 512. So 2,560 take away 512 leaves us with a final value there of 2,000.

£2,048 and that would be our final answer. Again, we could do this with a calculator. If something loses 20% in value, then it's only actually worth 80% of its cost the previous year. So if it's worth...

80%, the decimal that I can use for 80% is 0.8. If it's losing value, we won't want a one point number there, we've got 0.8. So we could just times it by 0.8 three times, or times it by 0.8 to the power of three.

So I could just type type into my calculator, 4,000 times 0.8 to the power of 3, and I'll get that answer there, 2,048 is my final answer. And let's test that out, 4,000 times 0.8 to the power of 3, and we get 2,048. So that's depreciation. Okay, moving on to fractions, multiplying and dividing fractions. Multiplying fractions is the easiest one, isn't it?

So we can times the top numbers, 3 times 2 is 6, and times the bottom numbers, 4 times 3 is 12. And we always want to simplify our answer in a question. will normally ask us to write it in its simplest form. So here we can divide the top and bottom by 6 and we get 1 over 2 as our final answer there. So always looking to simplify.

On to the next one, we keep the first fraction the same, we flip the second one over and we multiply them together. So times in the top numbers just like before, 2 times 5 is 10, 1 times 9 is 9 and again we'd normally be asked to write our answer in its simplest form or as a mixed number and here it is an improper fraction, 9 fits into 10. once with a remainder of 1 9th left over. So 1 and 1 9th would be my final answer there.

And that's dividing fractions. When it comes to adding and subtracting, the process is the same. I'm just going to go for this one addition.

So when we're doing this, we need to obviously look at the fact that there is a mixed number here. And we're going to make that into an improper fraction first. And the same goes for multiplying and dividing.

So we're going to do 1 times the 4, the big number times the bottom, and then add the top. So 1 times 4, add the 3, is 7 quarters. And we're going to add to that list.

two thirds and again if that was a mixed number we'd also convert that into an improper fraction. Now we need a common denominator when writing and subtracting so that's going to be 12 on the bottom so we can times this left fraction top and bottom by 3 and we can times this right fraction on the top and bottom by 4 and that's going to give us two fractions out of 12 so 7 on the left times 3 is 21 over 12 and on the right 2 times 4 is 8 over 12 and now we can add these together and again if we were subtracting them we'd just subtract those numerators now but we'll add them together and we get 12. 29 over 12. And again, it says to give your answer as a mixed number. So we are going to convert this now into a mixed number.

12 fits into 29 twice, up to 24. And there'd be a remainder of 5, 5 twelfths. And again, there's my final answer. Just be very, very careful at this point.

Just to have a look, does this fraction simplify here, the little one? It doesn't. But let's imagine it was 4 twelfths there instead of 5 twelfths. That would actually simplify, wouldn't it?

Because if it was 4 twelfths, we could divide the top and bottom by 4. And that would end up being 1. third. So just have a look in that situation. Does it simplify?

This one obviously doesn't. It's two and five twelfths. Let's have a look at another one. So in a bag of counters, two-fifths are blue and it says 18 are blue. Work out the total number of counters in the bag.

So it's not asking us to work out two-fifths of a number. It's saying that two-fifths are 18. Remember, it could have said work out two-fifths of, let's say, 20. And you divide by the bottom to get one-fifth. You'd say that one-fifth is four.

And then times by the top to get two-fifths. fifths and we'd get two fifths would be eight. That's a different question so it's nothing to do with that but it says two fifths are blue and 18 are blue so two fifths equals 18. So if I want to work out one fifth in this scenario I need to divide by the top this time because that'd give me one fifth half of that so one fifth would be nine and then to work out five fifths the original amount we could times that by five so times it by five and we would get five fifths equal Equaling 45 counters. There you go, so that's our scenario there.

So slightly different, the other way around obviously to working out a fraction of an amount. It says that that fraction is 18. So dividing it by 2 to get 1 fifth and then times it by 5 to get the original amount. Okay, so writing some numbers in standard form.

We're going to make this a number between 1 and 10. Times 10 to the power of however many jumps we have to do in whatever particular direction. So this number here, 5600, I'm going to hop the decimal in between the 5 and the 6. So 1, 2, 3. So that becomes 5.6 times 10 to the power of, it's a big number there, it's not a nought point number, so 10 to the power of 3. Final answer. The next one is a small number, so we're going to have a negative power, because we hopped the decimal the other way this time.

So 1, 2, 3 would make it 3.4. And again, this time it would be times 10 to the negative 3, because it's a nought point number, that indicates that it's a small number. When it comes to the other way around, writing them as ordinary numbers, I like to rewrite this little bit at the start to start with. So 2 and 3, and I like to just imagine where that decimal is between the 2 and the 3. So times 10 to the power of 5 makes it a big number, so we need to hop it five places.

1, 2, 3. 4, 5 to the right, and then filling in all those zeros underneath. And if it is messy, just remember to rewrite that. But there's my answer, 230,000. The next one, 8.04 times 10 to the negative 3. So this is going to be a 0 point number.

And again, I'm just going to write... write these digits out at the start, these bits here, and imagine where the decimal is at the top, 8.04. So negative 3 means it's going to be a small number, so we're going to jump it left 3, 1, 1, 2, 3. So it's going to go there, fill in all the zeros, and tidy it up at the start.

So 0.00804, final answer. Okay, so write this as a power of 3. So when we've got the same base numbers, which we do, these are all 3s, we can add the powers when multiplying and subtract the power. when dividing.

So if we tidy up the top to start with, which I always do first, we add the powers there because they're getting divided, sorry multiplied. So 3 to the power of 7 and on the bottom we've got 3 to the power of 3. Now these are being divided so we can subtract the powers, so 7 take away 3 is 4, so we get 3 to the power of 4. And again it doesn't say to work out the value of that, it says to leave it as a power of 3. If it said work out the value of it I'd actually have to work that out and that means 3 times It wouldn't be very nice without a calculator, that one, but we could work it out. But it says to write as a power of 3. OK, so negative and fractional powers.

Remembering the negative symbol there just means to do the reciprocal to flip it over. So in this first one, if that's 5 over 1 at the moment, it'll become 1 over 5. Now, if it was just a negative 1 in the power, we'd leave it as it is because it'd be a power of 1. But that 2 is just a normal power, which means we're going to have to square both of these. It's a normal squared power. So 1 squared on the top is 1, and 5 squared on the bottom is 20. 75, and that's our final answer there.

Onto a fractional power. Remembering that number on the bottom of the fraction represents a root. So in the case of a 2, that is a square root.

So we're going to do the square root of 64, and the square root of 64 is 8. Now the 1 on the top is a normal power, so a normal power of 1 doesn't do anything. But if it was a 3, for example, we'd have to cube it, but it is just a 1. There's no negative in that one, so we don't have to flip it over or anything. It's just the square root, which gives us the answer 8. Okay, I'll...

on some combinations. It says in a bike store there are eight different bikes and ten different choices of paint colour. How many different combinations are there?

Well we have eight choices of bikes to pick from so that's eight different combinations and all we do is we multiply it by however many choices we have on the second option. So the second option is the paint colour and there are ten so we times it by ten as our second option. So eight times ten gives us eighty options as our final answer. So you just take how many choices you have on each pick and multiply them together and that's combinations. OK, so this one's slightly different.

A three-digit padlock uses the numbers from 0 to 9. How many of the possible three-digit number combinations use three different digits, meaning we can't have any repeats here? So let's imagine on the first one, on the first digit of this padlock, let's imagine we've got three numbers. And on that first digit, let's say we pick the number 2. Now, in that first pick there, we had the numbers from 0 to 9, including 0, so that would be 10 options that I had.

But now that I've picked the number 2, I can't pick the number 2 again. is no repeats. So I don't have 10 options now, I only have 9 options. Let's imagine we pick the number 3. So on this option now, on this third pick, I can't pick 2, I can't pick 3 out of these 10 numbers, so I only have 8 to choose from.

So there are my 3 digit padlocks, let's just imagine we pick the number 4, 2, 3, 4, and we'd have to multiply these all together. So 10 times 9 gives us 90, I thought we've got to times by 8, and 8 times 9 is 72, that's not a 7. 8 times 9 is 72, so 720. And that's how many different combinations we'd have for this particular padlock. So 10 times 9 times 8, perfect, 720. OK, some error intervals.

6.4 has been rounded to one decimal place, and write the error interval. So the error interval, we always tend to put either 6.4 or a letter in the middle, depending on what the question is, and we have our arrows always pointing to the left. So we're going to find the upper and lower bound.

Now 6.4, if it's been rounded to one decimal place, it couldn't have... been 6.45. As soon as that 5 goes after that 4, it would have rounded up to 6.5.

So we put that on the right hand side as our biggest possible number that it couldn't have been 6.45. So it can't be equal to that number, so we leave that symbol as it is. It's got to be less than 6.45. Now what we did there is we added on this x to the x-axis.

So we Extra 0.05 and all we're going to do for the smaller one is take that away. So it would be 6. But it can actually be equal to 6.35, so I'm going to add my little equal to symbol on my inequality just there. So it can always be equal to that lower bound because that would in fact round up to 6.4 if we rounded it to one decimal place. On to the next one. This one's slightly different in the wording.

So this one says it's been truncated to one decimal place. And truncated basically just means it's been chopped off after this first decimal, regardless of how it rounded. So if we have a look at writing this is... as an error interval. Again, we write these in the same way.

The impossible number that it couldn't have been, it couldn't have been 6.5. So my top one here is 6.5. It had to be 6.4 something if it was just chopped off. Let's imagine just a random how this works.

So it could have been 6.4932, but it was just chopped here and we just wrote what was at the start. Okay, but it couldn't have possibly been 6.5 because otherwise we'd have written 6.5. And the smallest it could have been is just 6.5.

6.4 with no digits after it. It couldn't have been 6.3 though because again if we truncate that it would have just been 6.3. So two different scenarios there.

One where you've got to add on that extra 5 at the end and take the same size 5 off the start and truncating there where we've just got to think about the fact that it was just chopped off and no rounding at all. And the final little one we're going to have a look at here is using your calculator. So it's important to know where lots of digits are.

I've picked quite a complicated one here but you just need to have a play around with your calculator to make sure you can find your your square root, your cube root, your squared, your cubed, your fraction button, your sin, your cos, your tan, and I'm figuring these out. So I've picked quite a complicated one here. So to get your cube root button, depending on what calculator you have, but the majority of them, you still have to press the same things. We press shift and then we go for, let's get a different colour here, we go for the square root button and that allows us to get a cube root. So on your calculator, a cube root will pop up and then we're going to use the fraction button.

So the fraction button. button is just next to it in this diagram here, the fraction button, and an empty fraction is going to pop up. So on the top, we're going to just type in on the top of that fraction there, 4.3 multiplied by, press your tan button, and then press 39. And normally, this will go in a bracket, so remember to close the bracket off there.

But you don't put this little degrees symbol in, you just put 39, the calculator knows that you're talking about degrees. So let's get that up on my calculator, we've got 4.3 multiplied by 39. So we've got multiplied by tan, the brackets opens for me, 39, close the brackets, press your down button on the calculator to go into the bottom of the fraction, and type this sum in here, 23.4, take away 6.06, so 23.4, take away 6.06, and make sure at this point you write all the numbers down, that equals 0.5855934. 1, 2, 3, 3. Now some calculators may go beyond this, depending on what the answer is on the calculator, but it tends to be that you never really have to go beyond 9 digits, because not all calculators will go that far, so you are limited really to the amount that some people will be able to write, depending on what calculator they use. Now the question will probably usually say, then to round your answer to 1 or 2 decimal places.

So if we just have a quick think, if we were to round this to 1 dp, to 1 decimal place, and we could chop that after the 5 there, So one decimal place would round up to 0.6. And we would round that to 0.6. If we had this to two decimal places or beyond, let's have a think. After two decimal places, let's pick a different colour. It would chop it after the 8. And it would be 0.59 because it's a 5 after that one.

OK, so you could have different ways of rounding this. But just make sure you write down all the digits on your calculator display before actually rounding anything. Okay, so kicking this off with fractional and negative indices.

So you've just got to remember, looking at this one on the left, if there is an index, the negative does the reciprocal, which I always just write as a flip. The bottom is the number underneath, which does the root. And then the number on the top is just a normal power. Okay, so I'm just going to put power with that, so normal power. So for this particular one, we've got all three going on.

I always start with flipping it over doing the reciprocal and remembering that 8 is just a number so we could write that as 8 over 1. So when we flip that over it becomes 1 over 8. Then we've got to do either the root or the power. I always like to do the root first, make the number smaller before trying to then do the reciprocal. do the power. So a 3 on the bottom would represent a cube root, so we need to do the cube root of both of these numbers.

The cube root of 1 is 1, and the cube root of 8 is 2. 2 times 2 times 2 is 8, so it's 1 half. Then we've got to finish it off, we've still got to do that normal power there. If it was a 1 it wouldn't affect it, but a 2 as a normal power is squaring, so if we square them both we get 1 over 4 as a final answer.

And just remember to follow all those rules. Now looking at the one on the left, we've not got a negative, so we don't have to flip it over, but we have got 2. two on the bottom which is the root again and a two is a square root and then we've got a three on the top which is going to cube the numbers. So if I start with the roots to start with if I do the square root of both of these and it's always a little hint there they are square numbers the square root of nine is three the square root of 100 is 10 and then we've got to actually cube them as well so if I cube these as well let's write that in three cubed is 27 and 10 cubed is a thousand so it'll be 27 over a thousand just being very careful to get those right. So that's negative and fractional indices.

Okay so when you see this sort of question you see these words here about bounds, there's always something we need to write down straight away. So we've been given two numbers here, so s is correct to two decimal places and t is correct to one decimal place. So the first thing I'm going to write down is the error intervals for them before I move any further. So writing our error interval, arrows pointing to the left and an equal to symbol on our lower bound there.

So I always start with the upper bound and I'll just stick a five on the end so eight point two four. with a 5 on the end and then we just need to take that 5 off for the lower bound so it'll be 8.2 not 4 but 3 with a 5 on the end and then do the same for the other one as well so let's write down the upper and lower bound for 3.5 so 3.5 stick a 5 on the end and then for the lower one 3 point drop it down to a 4 and again just stick a 5 on the end now this question here asks us to find the upper bound so when it comes to a fraction and you could have anything here but when it comes to a fraction if we want to get the biggest possible answer we want to to do either, we've got to decide here, either we're going to do the upper bound divided by the lower bound, or we're going to do the lower bound divided by the upper bound. And we've just got to think which one there is going to give us our biggest possible answer.

And that is the upper bound divided by the lower bound. So we want the biggest number on top divided by the smallest number, and that'll give us our biggest possible answer. So we want the upper bound of S, S sitting on top, so the 8.245, and the lower bound of T, 3.45 sitting on the bottom.

So this is a calculator question. So we just need to type that in on the calculator. calculator 8.245 divided by 3.45 and see what we get as our answer there so if we type that in 8.245 divided by 3.45 and we get the answer it's quite a lot of decimals here 2.38985507 always just write down all the numbers and then round it however you've been asked in the question this one says three decimal places so let's chop it after the nine and that would go to two 2.3, and this is a bit of a dodgy one because the 8 rounds the 9 up, so it'll go to 2.390, keeping the 0 in there, it does say three decimal places, so 2.390 would be my answer there. Obviously it might say lower bound as well, in which case you'd use the other side of the formula, we'd use this one instead.

You might even be asked to do both of them and then round it depending on what they both look like, so if we imagine that the lower bound comes out, and if we just make something up here, let's imagine the lower bound comes out. out as 2.295 or something like that. Now it might say considering bounds give your answer to an appropriate degree of accuracy.

And if you look at both these numbers, if we just imagine this is the lower bound, it's probably not, but if we imagine it was, these two numbers would not round the same to two decimal places, not to three decimal places, not even to one decimal place because the one on the left there would round to 2.4 and the one on the right would round to 2.3. have to consider bounds to give an answer for this. I just have to say 2 and I would say the answer is 2 and that would be to one significant figure.

So I could be asked that as well and I just look at the upper and the lower bound and I round according to how they both actually round. In this case, my little example here, they only round the same to one significant figure. So recurring decimals as fractions. So a little rule, we are going to use a bit of algebra for this.

But if there's two recurring decimals, we're going to times it by 100. And if there's one recurring decimal, we times it by 10 and so on. There's lots of different ways of doing this. This is just the way that I do them.

So I write what the decimal is. We've got x equals 0.54. And remember, if there's a dot above the 5 and the 4, that means that that decimal there is 0.5454. And continues like that. It's the 5, 4 that's recurring.

It's always good just to write that down, just to remind yourself what that decimal actually represents. So if we've got two recurring decimals, we can times it by 100. times that by 100. You'll see why if you've not seen this before. Hopefully it's just a reminder. You get 54.54. Now we can take them away from each other and eliminate these recurring decimals here.

If we take them away from each other, we have 99 x's equals 54. And make sure that you actually do use your algebra to show this. Now we can divide both sides by 99. So x equals 54 over 99. And there it is as a fraction. You do just need to check. just to see if your fraction simplifies. So do check to see if this one simplifies.

This one actually does. The top and the bottom both divide by 9. And you'd normally be asked to show that it equals this next one, but if it doesn't, you know, you can look to simplify if it says to give it in its simplest form. And if you divide them both by 9, you get 6 over 11. So the question might have said show that it equals 6 over 11, and that's how you would show it.

You'd show how it simplifies. Now on to the one below. It's only got one recurring decimal, but it's a bit of a nasty one. So if we think about what this decimal actually looks like, 0.23. with a dot above the 3. That's 0.2 and then a recurring set of 3s.

It's not a full repeating pattern of 2, 3, 2, 3, 2, 3. So if we have a look at this one, and again, there are different ways of doing this type of question here, but I just follow the same process for all of them. So x equals 0.23 with a dot above the 3. And there's one recurring decimal, so let's times it by 10. So we get 10x equals 2.33, and then the 3 keeps going. I'll just balance it out so there's the same amount of decimal places in both.

Now if we take these away from each other, something kind of interesting happens here. We get 9x equals, and 2.33 take away 0.23 is 2.1. Now if we take the same approach as before and turn that into a fraction, so dividing both sides by 9, we get x equals 2.1 over 9. And obviously you can't have decimals in fractions, so we need to remove this decimal from the top.

So we can times the top and bottom by 10, and that would give us 21 over 90. There you go, and there's your fraction and again You just need to check to make sure that's if see it let to see if that simplifies or not and actually does the top and bottom both divide by 3 so you can divide the top and bottom by 3 and You would get 7 over 30 Again, so it might have said show that this fraction equals 7 over 30 But always just checks it at the end see if it simplifies obviously if it just said to write it as a fraction 21 over 90 would be fine, but if it said to give it a simplest form we would have to look to see simplify there. Okay, so some compound interest. James invests £2,500 for three years in a savings account and gets 3% per annum compound interest in the first year and then X% for two years.

He has this much at the end of the three years work out the value of it. Well just like normal compound interest here we can actually do the first year. So we can work out, I'm just going to label this up, so after the first year let's see how much money he has.

So £2,500 for 3% we would times that by 1.03. Obviously we don't need to add any powers into there because it is just the first year that he's getting this. So if we do this on the calculator times 1.03 and we get an answer of £2,575. Now we've got a bit of a reverse compound interest scenario and we're going to have to create a little formula for this.

the second and third year he gets x percent. Now his starting number, and let's just write down second and third, his starting number now is 2,575. Now if we knew the percentage we'd multiply it by that percentage which we'll call x in this scenario because we don't know what it is and we'd do that to the power of two as it's two years and what it's saying in the question is that gives us the answer 2,705 pounds and 36p. Obviously at this point here we could just get guess some numbers we could just guess an x value and if it's too high go lower if it's too low go higher and that's a fine way of doing it as well but you can save yourself a bit of time here just guessing lots of numbers by actually just rearranging this so we can divide both sides by 2575 and it will give us the value of x squared. So x squared is going to equal, and again let's just do this on the calculator, 270536 divided by that gives us this answer here, 1.050625243.

That's obviously the value of x squared, so we need to square root both sides. So if we square root that answer, we get the answer 1.02. five zero zero zero one one eight. So that gives us our little percentage here.

So it's hidden within this decimal. Now our actual percentage here is not 1.025 because that one point is what we increased by for the percentage. So it's just this 0.25 that we're interested in and 0.25 if you think back to what we did at the start here with 0.3 being 3% well 0.25 would be 2.5%.

There we go so that's 2.5%. So just be careful that you do read that off there and don't put 1.025 in this type of question. But again, you could have guessed.

If you'd have guessed 1.02, it's normally going to be lower than that first one. That's normally what banks do. They give you a better one in the first year and then not so good following that.

OK, so there's just two ways that you can approach that. You can guess some numbers or you could obviously just rearrange it and find it there. Just be very careful that you actually read the percentage from that one point number.

OK, so we've got some SIRDS questions. So 3 plus root 5 squared in the form a plus b root 5. where a and b are integers. Okay so we need to have a look at this so that obviously is a double bracket there so if I rewrite this as a double bracket we've got 3 plus root 5 and 3 plus root 5 and we just need to expand that now and see what we get so lots of little surd rules going on in this one. So 3 times 3 gives us 9, 3 times root 5 is 3 root 5 5 root 5 times 3 is another 3 root 5 and then root 5 times root 5 is root 25 and it just becomes 5 remembering similar serves when all these exact same serves when you multiply them together become the whole number and if we simplify all this down 9 and 5 makes 14 and then we can add together these thirds in the middle because they both have a root 5 so we've got 3 lots of root 5 and 3 lots of root 5 which is 6 lots of root 5 and there's your answer just there so a would be 14 and B would be 6 and that is in the form a plus b root 5 and there's our answer with a bit of thirds.

Let's have a look at another thirds question. So we've got to show that this can be written in this form. Now when we've got thirds on the bottom of a fraction we're always going to look to rationalise it. Now with some of the easier thirds you just times the top and bottom by whatever this root is just here, so by root 3. Unfortunately that's not going to work for this one because we've got 2 plus root 3 and if we did times the bottom by root 3 there we'd get 2 root 3 plus 3 and we'd still have a third on the bottom.

So what you do when you've got this on the bottom is you times it by the denominator still but we change. this little sign here so whatever that sign is we're going to flip it to the opposite so we're going to times it by 2 minus root 3 so always write that down okay because you do get marks for these these elements of you working out so I'm just going to rewrite the fraction 2 plus root 3 and I'm going to show next to it what I'm timesing the top and bottom by so I'm going to times by 2 minus root 3 and I'll put that in a bracket because I'm going to do a double bracket here and I'm also going to times the top by 2 minus root 3 because we're creating an equivalent fraction here so it's up to you if you want to put these in brackets as well just remind yourself you're doing a double bracket and then you just need to multiply both of them out and see what we get. So on the top and I'm not going to draw all over it I'm going to times them both by the 5 first and then times them both by the 2 root 3. So 5 times 2 on the top gives us 10 5 times negative root 3 gives us negative 5 root 3. On to the 2 root 3 so 2 root 3 times 2 is positive 4 root 3 and then 2 root 3 times negative root 3 and this is the multiplication you always just got to be careful of. gives you two, I'm going to do it to the side, we get two lots of root 9. Two lots of root 9 is 2 times 3, which is 6. So that number at the end there, and careful because one's positive, one's negative, so we get negative 6. I'll tidy that all up in a sec, let's have a look at what's on the bottom. Now the whole reason we're doing this on the bottom is to eliminate the thirds in the middle, so you can do a bit of a trick when you're doing these.

You can just do the first two, which is 2 times 2, which is 4, and the last two. So root 3 times root 3 is 3, and it's minus 3. There we go, so that's why our fraction's disappearing, because it's 4 minus 3 on the bottom. Let's tidy up what's on the top.

So on the top there we have 6 and the 10, so we've got 10 minus 6, which is 4. We've got negative, let's do this in a different colour, we've got negative 5 root 3 add 4 root 3. Negative 5 add 4 is negative 1, so minus 1 root 3, or minus root 3, and that's all over 4 take away 3, so it's over 1. So when we're dividing by 1, we can get rid of the fraction there, and we get 4 minus. root 3. So there's some surd elements to be thinking about. Obviously there are so many different questions on surds that you can actually have, but there's a couple of things there to be thinking about. The other thing is obviously to remember that surds can only be collected together like in here, when the number underneath the surd is the same.

Okay, so you will have to know how to simplify surds as well. I've got plenty of videos on that to check out. Okay, so we've got some standard forms, we've got some multiplication and some division. I'm going to start with this multiplication.

Now I'm primarily thinking this is like a non-calculator style. question because obviously if this was on a calculator you could just type it in nice and easy and write it in standard form. I have to do a bit of a trick here. So 3.2, I can times that by 4 to figure out what my starting numbers would be.

So 3.2 times 4, again, you've just got to work that out, whatever method you're used to using. I'm going to do 32 times 4 and hop the decimal back in. times 2 is 8 and 3 times 4 is 12 and then hop the decimal back in so it's 12.8.

So the answer I get there is 12.8, it's going to be times by 10 and when we're multiplying we can add together the powers here so 3 plus 4 gives us 7. Obviously though it says to write your answer in standard form and that number here, 12.8, is not between 1 and 10. So if I hop the decimal in making the number 1 place value smaller, I'd get 1.28. And if I make the number smaller, I have to make the power 1 bigger, so it'd be 10 to the power of 8. Okay, so don't be put off by these sorts of questions. Do your multiplication, follow your power rules and then balance it out. Just remember if you make the number smaller, you make the power bigger.

We'll take exactly the same approach for this one except it's a divide. So I need to do 3.6 divide. divided by 4. And again, it's up to you. You can hop the decimal out and hop it back in, but I'm just going to do 3.6 divided by 4 using a bit of a stop.

So 4 doesn't go into 3, so carry the 3. And then 4 goes into 36 nine times. There we go. So it's 0.9.

So when we divide that, we get 0.9 times 10 to the power of, and when we're dividing, we can subtract the powers. So the first power there is 2. The second power is negative 3. So just write that down to the side. What have we got?

We've got 2 take away negative 3, which is to add 3. which gives us a power of 5. There we go, so the power is 5. And then again we just need to balance that out because obviously this is not between 1 and 10. So I'm going to hop the decimal this way making it 9. So this time I've made the number bigger so I need to make the power smaller so it goes down to a 4. And there's our final answer 9 times 10 to the 4. And our last question on this section. So we've got in 2008 Alice bought a house in 2000, sorry in 2008 Alice bought a house. In 2014 she sold it for 20% profit and in 2019...

it was sold at a 5% loss and it gives us the value that she sold it for. Work out the cost of the house in 2008, so right back at the start. So we had 2008, she bought a house and she sold it for a 20% profit. Then in 2014, it was sold for a 5% loss.

And then in 2019, the value of it was £182,400. Right, so at this point in 2019, this cost here to whoever she sold the house to in 2019, 2014, this was sold at a 5% loss. So to her, that was only 95% of the value that she bought it for. So if we've got a calculator here, in order to turn that back into 100%, quite nice and simply we can divide it by 95. to get 1%. So if that there is 95%, we can divide that by 95. So divide by 95, which would give us 1%.

So 1% would equal, and we'll just do this on the calculator, 182,400 divided by 95 gives us 1,920. And then to get back to 100%, we just times that by 100. So we can just add two zeros on for that. So 1920 with extra two zeros.

There we go, so 192,000. So that's the price that it was sold for. Sorry, she did actually paid for it before.

So before that, it was 192,000. That's not a zero. 192,000.

And then we've got to think about the original sale in the first place. So that means that that 192,000 there was a 20% profit. So if that was a profit to Alice in the first place, then to her this amount of money was 120%. Okay, it was 20% more than she originally paid for it.

So we can follow exactly the same approach now. If that's 120%, we would divide that by 120 to give it us 1%. So 1% here would equal, and we'll just work that out. Divide that by 120 and 1% is 1,600.

And again, you can times that by 100 to get 100%. And our original cost up here would be 160. Zero with two zeros on the end, so 160,000 pounds. There we go, so 160,000.

There we go. So that's it using a calculator. That's moving through two reverse percentages there. One with a decrease, one with an increase.

You just got to remember if something's been decreased, then we get this 95%, whatever was left over. And then there was also the scenario where at one point it was increased. making it 120%, that extra 20% there.

So this can be done in calculator and it can be done in non-calculator. The 95% one wouldn't be very nice, but thinking about how you could have done this 120%, which is quite common when VAT is added on because they often have these VAT questions. or they say VAT was added on at 20%.

There is a way of doing that without a calculator and what you could do when that happens is you can divide by 12 and then times by 10. So you could do that without a calculator. You could do a bit of a stop there and divide 192,000 by 12 and then just add on a zero when you times by 10.

Transcript for:Overview of Key Mathematical Concepts

Transcript for:
Overview of Key Mathematical Concepts