Sherpa

A percentage is a proportion that shows a number as parts per hundred. The % symbol represents the percentage. An example; 12% can be displayed as 12/100 or 12 out of every 100.

Percentages are just one way of expressing numbers that are part of a whole. These numbers can also be written as fractions or decimals. 50% can also be written as a fraction, 1/2, or a decimal, 0.5. They are all exactly the same amount.

We are looking at percentages. So we'll first focus on what are they and what's the point of them. We'll do some subtopics and then we'll finish off with some example questions at the end. So percentage will percent dollars just means per 100. And if we're talking about maybe 42% of the population has blond hair, well, this is saying 42 out of every 100 people has blonde hair out of 100.

And as a fraction, 42 out of 100 is just 42 over 100, right? So this is how we can see we can convert from percentage to fractions. Also 42 over 100 or 42 divided by 100. This just gives us four point.

And so we can actually also convert from percentages to decimals. We can see we just divide by 100 to convert from percentages to decimals. Percentages. Take a step back. They're very useful for talking about percentage changing things at the moment.

You may have seen news articles about the inflation rate 10.2% or something, where it's very useful to be able to talk about how things are costing more relative to what they cost last year, right? The easiest way to do that is with percentages. Or if you ever go shopping for clothes or whatever, you might see it's a 15% discount. But it's very useful to be able to appreciate what actually means. If you see the discounted price is £25, we can always work out what the original price was.

Or if you own a shop, you need to know what the new price is going to be if you take off 15%, say. Unfortunately, they're kind of pervasive everywhere. We kind of can't get away from percentages. So finally, percentage of something. So let's say we want to discount we own a shop and want to discount a jumper by 45% or something.

Let's say we want to work out as 30%. We want to do a 30% discount. We want 30% of the original price, which was £48. But in this case, there are two ways to do this. We can either work out 10% is going to be 10th of this £4.8, and then we could times it by three to get up to 30%, say 30% is going to be 4.8 times three.

Or what we could do is we could just dive straight in, right? We could convert this percentage into a decimal. We know we just divide it by 130% of the decimal is going to be not .3. And again, we just take our original amount and we multiply it by not .3. So we're doing 48 times.

I shouldn't say that on something better times 0.3. Well, you can see in both cases, in both ways of thinking we did the same thing, right? We needed 4.8 times three or 48 times, not .3. Both of these would actually go as the exact same answer, right? This is we've made the top number ten times bigger and the bottom number ten times smaller.

So the answer will be the same. Just two different ways of thinking about it, finding something as a percentage.

Let's say we decide that we want a dress in our shop to go from costing £64 down to costing just £24. We want to put a price tag on it and say the percentage drop in price actually is. The customers can see this. Well, in this case, we want to work out what is 24 as a percentage of 64 or what is the difference between these two as a percentage, if we're looking at the percentage drop, so ideally we can look at what the discount price actually gone down by going from 64 down to 24. Well, we've gone down by £40, right?

So we've gone from 64 to 24. So we're actually looking at the discounted amount, how much we discounted by as a percentage of the original amount of \$64. And this is going to tell customers how much we've locked off the price. So what is 40 as a percentage of 64? Well, in this case, we're just doing 40 relatives of 64, right?

It's 40 out of a total of 64. We know in this case, we're literally just paying 40 over 64. This is just a fraction. We now have to deal with fractions. We can simplify them down if you know times tables.

We've got a common factor of eight in the top and the bottom. We can divide them both by eight and we get five over eight. That's not our final answer, right? We want it as a percentage. This is our aim, to get a percentage.

Well, we know one eight if you don't know one eight as a percentage. A quarter is 25%, an 8th is just half a quarter. So it's half of 25%, which is going to be 12.5%. It's going to be five times 12.5%. That's going to give us five times ten and a half.

It's going to give us twelve and a half. Add on five times ten, which is 50, that's going to give us 60%. And that is our financer. So we've taken off 62.5% of our original price percentage change. Again, this is fairly similar to what we just did, right?

We could be given a percentage or we can be asked to work out a percentage and apply that to get the change of something. It's really what we just did in the last two topics. I did a very quick example. Let's say we want to work out the status change. Let's say we've gone from charging, I don't know, £540 for a TV, and now we only charge £480 for a TV.

But what's the percentage change? How much have we taken off to go from 540 to 480? Well, it's very similar to this one, right? We've taken off £60. So we want to work out 60 out of 540.

And we can just go through the same stages as we did to get down to that result, let's go on to something more interesting. Percentage interest. So this is breaking down into two kind of subtopics. We've got simple interest and we've got compound interest.

Simple interest is simple. It's very easy to calculate. Let's say we put £500 into the bank and the bank gives us 4% simple interest per year. This is a very nice bank. And 4% of 501% is £5.

4% will be £20. And so every year the bank is going to put £20 into our bank. So after the first year, after year one, we'll have £520. After year two, we have £540. After year three, we'll have £560.

Every year we just add on the same amount. It's 4% of the original amount that we add on. Compound is slightly different. Compound looks at just the previous year. It doesn't care about what we started with.

So, again, let's stick with the same example. But in this case, we'll be adding on compound interest. So the first year is the exact same as the above case. After one year, we just add on 4% on the previous year. So in this case, it's 4% of 500.

So we add on £20 to get 520. After year two, we actually no longer look at what we had at the start. We only care about the previous year. And so we look at the previous year, 520 and we calculate 4% of 520. And then we add that on to 520.

So in this case of 520 will be £5.20, times four will be 20 pound 80. And so we end up with £540, \$80. You can see these two versions are already divergent. We're already at richer with the compound interest than we are with the simple interest. If you keep doing this, what we end up actually seeing is our bank account will actually do something like that.

It will kind of exponentially increase in value. The more we have in our account, the more we then put into our account next year and so on. That's why it's always good to be rich, right? That's why people say money, people with money get more money. It's a simple fact of compound interest.

The more you have, the more you get. When we're talking about interest, whereas simple interest, we just add on the same amount each year. Every single year we just add on a bit extra, right? And we add on the extra and so on. We don't care about how much you've got, we just care about the initial amount that we had in our bank.

So common interest is generally good if you're paying money to account. It's bad if you're borrowing money, right? Because then the lender is making all this lovely money and you are having to pay them that money, right? Let's now look at a couple of example questions. So, Anne buys a dress in a sale.

Normal price is reduced by 20%. The normal price is 36 pound 80. What is the sale price? And so if we take off 20%, we're left with 80%. And so, really, we're calculating 80% of 36 pound 80.

Well, again, we could do this one or two ways. We could say, well, 10% is going to give us three pound 68. And so 80% is just going to be equal to eight times this 10%, right? Eight times three pound 68. Alternatively, we could just dive straight in and we could convert this into a decimal.

We could say this is as a decimal to 0.8, and we can do zero eight times 36, 80. Both methods, I'm not going to do the calculations, but both methods are the exact same again, right? 0.8 is ten times smaller than 836, 80 is ten times bigger than 368. And so both of these will go with the exact same answer. Again, it's just two different ways of thinking about this problem.

Question three. Mario invest two grand for three years at 5% per annum, compound interest, calculating the value of interest of the investment rate at the end of the three years. And so we're going to take £2000 if we're adding on an extra 5%. Well, we've already got like we're taking 100% of our initial investment, adding on another 5%, we end up with a total of 105% after the first year. And so we just take this, times it by 1.5%, that's 105% of the decimal, and that's the end of our first year.

We then take this end of year amount and we timed it again by 105% to get the end of year two amount. And then that whole thing will give us year two. We take this whole thing and we timed it again by 1.5 to calculate year three. And what we're seeing here, right, is we're taking our initial amount, £2000, and we're timing it by 1.5 times one, five times 1.5. Well, this thing 1.5 times itself, three times, it's just 1.5 to the power of three.

And so we can put that into a calculator and that will be our answer. In general, we take our initial starting amount, we call it N, and we timed it by the percentage change as a decimal. We call this zero C, and we do it to the power of how many years or how many periods that we're looking at here. We could call this Why in this case? And that would be our kind of general formula that we can apply to these kind of questions.

Book a lesson with this tutor

Oli W

A young, gifted tutor who uses his engineering experience to ignite your curiosity in Maths. A straight 'A' student, he knows exactly how you need to learn to help you ace your exams.