So, in this lesson, what we’re going to look at is the applications of percentages. So, percentages, we’ve seen many times before. And we use it with things like percentage increase or decrease or percentage of amounts. And also, we use in everyday life. And we talk about it in things like newspapers where they talk about interest rates. We also see it on the things we eat, so the percentage of recommended daily amounts like fat intake or carbohydrates. Also, we see it at the shops.
But what does percent mean? Well, percent is from the Latin phrase per centum, which means by the 100. Okay, so great, we’ve talked about what percentages are, how we might use them, so now let’s get on and see what we would do with percentages.
So, before we can apply our percentage knowledge, we need to see what skills we need, so what percentage skill will we need for this lesson. Well, we’re gonna need to know how to calculate percentages of amounts, for example, 25 percent of 300 dollars. And within this, we’re gonna have to look at things like multipliers, so how we’d use a multiplier to help us find a percentage of amount.
And we’re also gonna have a look at percentage increase and decrease. For example, if we’re looking at percentage decrease, we could have things like a sale. So, 30 percent off, and we’ve gotta work out the value after our sale amount. And if we’re looking at percentage increase, we could look at things like interest, for example, five percent PA, which means per annum, which is every year.
And then, finally, we can look at something called reverse percentages. And this is where we work backwards. So, for example, we know that something’s 15 percent off in a sale. And it’s now 200 dollars. Then, we can work out how much it was before the sale began. And yes, these are all gonna be really useful cause they can help us to answer different questions in the maths that we’re doing.
But also, as you can see, they’re useful in everyday life because we know that we can go into a shop, use percentages to make sure we’re getting the best deal or to make sure we’re not being ripped off. So, we now know which skills we’re going to need. Let’s take a look at some questions that’s gonna utilize these skills.
If the tax rate is seven percent, what will the sales tax be for a truck that costs 16000 dollars?
So, in this question, we have two key bits of information. We’re told the tax rate, which is seven percent. And we’re told the amount that the truck was originally. And that is 16000 dollars. And what we’re trying to do is work out how much the sales tax is going to be on this amount. And what’s always good practice with this type of question, so an applications of percentage question, is to reframe it into what we’re actually looking for. And that is, what is seven percent of 16000 dollars?
Now, because this is quite a straightforward question because we’re looking to just find the percentage of an amount, what I’m gonna do is use this as an opportunity to show a couple of methods. So, first of all, there’s the mental method. And then, we’re gonna show the multiplier method, which is what we’d use if we have a calculator.
So, in the mental method, what we want to do is find seven percent of 16000. The first thing we’re gonna do is find one percent. And one percent is gonna be equal to 160. And the way we find this is by dividing our 16000 by 100. And that’s cause if we think about percent, this means by the 100, or out of 100. So therefore, if we divide by 100, that would give us what one percent is. So therefore, if we want to work out what seven percent is, this is gonna be equal to 160 multiplied by seven.
And probably, the quickest way to work this out would be to do 16 multiplied by seven. So, we could do that using the column method. And this’d give us 112. And because it was 160 multiplied by seven, we just add on a zero. So therefore, we could say that seven percent is gonna be equal to 1120. Okay, so that was our mental method. But how could we work it out using the multiplier method?
Well, when we’re dealing with percentage problems, if we say seven percent of an amount, this means seven percent multiplied by that amount. So, we’ve got seven percent means seven by the 100, or seven out of 100. And we could work this out as a decimal because all we do is we divide seven by 100. Which will be equal to 0.07. And we can usually work this out quickly in our head because we know that if we divide by 100, then all we do is we move each of our digits two place values to the right.
So, now, as I said, all we need to do to work this out is multiply our percentage, which is 0.07, by our amount, which is 16000, which, once again, would give us 1120. So therefore, we can say that if the tax rate is seven percent, then the sales tax for a truck that costs 16000 dollars is gonna be 1120 dollars. And we’ve shown a couple ways of working this out.
So, as I said, in this lesson, what we’re gonna do is build up the skills that we use to solve problems that apply percentages. So, in this problem, we found the percentages of amount. And what we’ve also done is we’ve broken it down because we’ve used the mental method and showed how that could be used. But also, we’ve looked at the multiplier method, which has introduced us to multipliers, which can be very useful in our other questions. So, now, what we’re gonna move on to are a couple of questions that look at percentage increase and percentage decrease. So, we’re gonna start with percentage increase.
Determine, to the nearest cent, the total price of an inflatable dolphin that costs five dollars and nine if the sales tax is 4.25 percent.
So, what we do first is rewrite the question so we can see exactly what we’re asked to do. And that is increase five dollars and nine by 4.25 percent. So, if we want to work out the total price of our inflatable dolphin, what we want to do is add on 4.25 percent. And we can do this using the multiplier method. And the multiplier for an increase of 4.25 percent is gonna be 1.0425. But how do we get this?
Well, there’re a couple ways of thinking about it. The first is if we think about the whole amount, that’s 100 percent. And if we want to increase it by 4.25 percent, we’re gonna have 100 percent plus 4.25 percent, which is 104.25 percent. And to have a multiplier, what we want to do is convert it into a decimal. So, as percent means out of 100, or by the 100, we can divide this by 100, which is gonna give us 1.0425. So, that’s one way of finding out what the multiplier would be.
The other way of considering the multiplier is to think that one is gonna be the whole of something. And then, we’re gonna add 4.25 percent to it, which is gonna be one add 4.25 over 100. And that’s because 4.25 over 100 will give us it as a decimal, which would give us one plus 0.425, which, once again, would give us our multiplier of 1.0425. Okay, great, so now we’ve got our multiplier, we can calculate the value of this question.
So, what we do to calculate the total price after the sales tax is multiply 5.09 by 1.0425, which is gonna be equal to 5.306325 dollars. Well, have we finished there? Well, not quite because the question asked us to leave it to the nearest cent. Well, if we round to the nearest cent, we’re gonna look at the second decimal place. So, then, we can see the digit that’s to the right of this, which is a six. Because it’s five or above, we round the zero to a one. So, we get five dollars and 31 cents.
So, now, what we’ve done is we’ve added one more skill to the skills that we’ve used in these real-life applications of percentage questions. And that is percentage increase. So, now, let’s quickly move on and see a question where we can use percentage decrease.
A clothing store is having a sale of 16 percent. Determine, to the nearest cent, the cost of four pairs of pants that regularly cost 23 dollars and 99 cents.
So, the first thing we do is look at the information in the question. So, we can see that we’re having a sale of 16 percent. So, we know, as it’s a sale, it’s gonna be a percentage-decrease question. We’re also told that there’re four pairs of pants and that their regular cost is 23 dollars and 99 cents.
Well, there are, in fact, a couple of ways that we could attempt this question. First of all, we could work out the sale price of each of our items and then multiply it by four because there are four pairs. Or we could find the total cost of the four pairs of pants first by multiplying four by 23 dollars and 99 and then work out the sale price afterwards. And that’s the method we’re going to use.
So, first of all, we’re gonna work out the total cost before the sale. And we’re gonna do that by multiplying four by 23 dollars and 99 cents. And we could do that using a calculator. However, it is worth noting, just a quick tip if you wanted to use a mental method, you could do four multiplied by 24 dollars, which would give us 96 dollars. And then, just remove four cents. And that’s because there’d four cents left over because each of our pants was only 23 dollars and 99 cents. And again, that would give us 95 dollars and 96 cents.
Okay, great, so now what we need to do is to work out the price after the sale. So, this is now where we like to make sure that we’re clear about what we need to do. And that is decrease our 95 dollars and 96 cents by 16 percent. And to do this, what we’re gonna do is use the multiplier method.
I’m gonna quickly look at a couple ways that we could get this multiplier. First of all, if we’re decreasing by 16 percent, well, the whole amount is 100 percent. And we’re gonna subtract 16 percent from it. So, this is gonna give us 84 percent of the original, which is gonna be equal to 84 over 100. And that’s because percent means out of or by the 100. Which is gonna give us our multiplier of 0.84. Okay, great, so that’s how we’d find our multiplier.
Well, the other way we could think for our multiplier is one, because that’s all of something, minus 16 percent, which would be one minus 0.16. And that’s because 16 percent is the same as 0.16, which, once again, would give us our 0.84. Okay, great, we’ve got our multiplier. So, all we need to do is multiply our 95 dollars and 96 cents by our 0.84.
So, when we multiply 95 dollars and 96 cents by 0.84, well, this would give us 80 dollars and 61 cents. Now, have we answered the question fully? Well, yes because it asked us to give our answer to the nearest cent. So therefore, we can say that the cost of the four pairs of pants after the sale is 80 dollars and 61 cents.
So, great, what we’ve done now is we’ve added another skill. Because we’ve looked at percentage decrease and shown how we could use this in a real-life context. So, now, what’s next? What skills are we going to apply now?
Well, now, what we’re gonna do is we’re gonna look at working backwards. But what do I mean by this, working backwards? How’s this gonna help us? Well, eventually, we’re gonna take a look at reverse percentages, which is where we work out the original amount from a given amount. But before we get onto that, what we’re gonna quickly look at is how to calculate what is the percentage increase or decrease of an amount.
Emma works at a clothing store where she gets an employee discount. If she paid 44 dollars and 87 for a pair of pants that costs 64 dollars and 10 cents, determine the percentage discount.
Well, this question is a percentage-change question cause we’re looking to see what the percentage change is between our two values. And we have a formula to help us. And that is that the percentage change is the difference between the values divided by the original amount and then multiplied by 100 to put it into a percentage. So, first of all, what we can do is work out our difference. And it’s gonna be 64.10 minus 44.87 cause it’s 64 dollars 10 cents minus 44 dollars and 87 cents. Which is gonna give us 19.23, or 19 dollars and 23 cents.
Okay, great, so now we can put this into our formula. So, when we do this, we’re gonna get our percentage discount is equal to 19.23 because that’s our difference over and then our original amount. And that’s the price before the sale, which is 64.10, or 64 dollars 10 cents. Then, we multiply the whole lot by 100, which is gonna give us 30 percent. So therefore, we could say that the percentage discount is 30 percent.
Now, what we can do is do a quick check to check this is the correct answer. So then, before our check, what we can do is do 64 dollars 10 because that was the original price. Then, multiply it by 0.7 because this would be our multiplier if we reduced by 30 percent. And this would give us 44 dollars and 87 cents, which is exactly what Emma paid. So, yes, this is correct.
So, great, we’ve now used another skill. And that was calculating percentage change. So, let’s move on to our final skill that we’re gonna look at. And that is reverse percentages. And we’ve got a good question here to show us how to do these.
A shoe store is having a sale in which all items are discounted 25 percent. Ethan paid 39 dollars, tax included, for a pair of sneakers. If the sales tax is four percent, how much did the pair of sneakers originally cost?
So, there are three key bits of information here. The first is that the items are all in a sale and they’re discounted 25 percent. The next is that Ethan paid 39 dollars. And then, finally, we’re told the sales tax is four percent. The key thing to notice here is that sales tax is put on at the end. So, it’s put on whatever value or amount Ethan paid for the sneakers after the discount. So therefore, our first calculation is to work out how much Ethan paid before the sales tax.
And what this is is something called a reverse percentage because we have the final amount and we want to find the original amount. And to solve reverse percentages, we have a couple of methods we can use. The first method is to consider the original amount and the final amount. Now, if we were to get to the final amount from the original amount, and we’re looking at a percentage increase or decrease, then all we would do is multiply by our multiplier.
So, therefore, if we wanted to go the other way, so from our final amount to our original amount, then what we would do is the converse of this. And we’d, in fact, divide by our multiplier. So, that’s one method, or model, that we can use. But there is another way of thinking about reverse percentage. And this is the method, first of all, that I’ll use to solve this problem. And that’s because it’s easier to demonstrate when we have values.
Well, if we start by thinking about the sales-tax part of the question, then our 39 dollars paid by Ethan is equal to the total amount of the shoes before sales tax, which is 100 percent of that amount, plus the four percent that the sales tax is. Well, therefore, we could think of 39 dollars as 104 percent of the original amount that was paid after discount but before sales tax.
So therefore, if we want to find out what one percent is gonna be, we could divide this by 104, which will be equal to 0.375 dollars. And then, to work out how much the price after discount but before sales tax was, all we do is multiply this by 100 to get the 100 percent. Which will give us 37 dollars and 50 cents. Okay, great, so this is the first part of the question answered.
Well, now, for the second part of the question so we can answer the question fully, what we’re gonna use is the original method that we mentioned. And that is the multipliers. Because if we got our original amount, and that’s the amount after discount but before tax, then if we multiply this by 0.75, we’ll get 37 dollars and 50. And we got that cause one minus 0.25 is 0.75. And that’s cause we’ve got a discount of 25 percent. Then, if we wanna work backwards, then what we’ll do is divide by 0.75. And when we divide 37 dollars and 50 cents by 0.75, we get 50 dollars. So therefore, the original cost is 50 dollars.
So, now, we’ve used the final skill we wanted to. And we’ve calculated reverse percentages. So, now, all that’s left is talk about the key points of today’s lesson. Well, the first key point, that is if we want a percentage as a decimal, then all we do is divide the percentage by 100. Next, if you wanna find the multiplier of a percentage increase or decrease, it’s one plus or minus the percentage as a decimal.
To calculate percentage change, this is equal to the difference over the original multiplied by 100. And finally, to calculate reverse percentage, all we do is we have the final amount and we divide by the multiplier to get to the original amount.