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Video: Increasing and Decreasing by a Percentage

Tim Burnham

Through a series of examples, we show you how to increase or decrease a number by a given percentage. First, using a two-step process of finding a percentage and adding it, and then a one-step process of finding the resultant percentage directly.

14:02

Video Transcript

In this video, we’re gonna be going through some questions which ask us to increase or decrease numbers by a certain percentage, and we’re gonna look at a couple of different ways of answering those sorts of questions.

Add eleven percent to twenty-five dollars. Okay, we’re gonna go through this entire question two different ways. So first method is gonna be we’re gonna find eleven percent of twenty-five and then we’re going to add that to twenty-five. Now, to find eleven percent of twenty-five, eleven percent means eleven out of a hundred. So, it’s eleven hundredths of twenty-five. I’m gonna write that out. I could say eleven over a hundred times twenty-five over one. So twenty-five over one is the fraction equivalent of twenty-five. And now we’ve got fraction multiplication, which means I can do a bit of cancelling. If I divide twenty-five by twenty-five, I get one. If I divide a hundred by twenty-five, I get four. So the answer is eleven over four. Now I could change that to two point seven five if I wanted to or two and three-quarters, but I’m just gonna leave it in its completely accurate form, eleven over four. So that’s eleven percent of twenty-five. Now we have to add that eleven percent to the original twenty-five. And when we see that calculation, I can hear some of you screaming at me, “We should’ve changed eleven over four into two point seven five. That would’ve been much easier. Twenty-five plus two point seven five is twenty-seven point seven five.” And I’ve got a lot of sympathy for that. It makes a lot of sense in this case. But sometimes, the fractional answer that you get doesn’t translate itself into a nice easy accurate decimal like that one did. So I’m gonna go through it in fraction format so that you’re practiced in the technique should you encounter one of these less friendly sets of number shall we say.

Well, twenty-five plus eleven over four. Well, we can change that into its fraction format by saying twenty-five over one plus eleven over four. And when we’re adding fractions, we need to have a common denominator. So what I need to do is multiply that first term by one, well four over four, that version of one, because then we’ll have an equivalent fraction to twenty-five, which has a denominator of four. So four times twenty-five is a hundred and four times one is four. So twenty-five over one is the same as a hundred over four. So the calculation becomes a hundred over four plus eleven over four, which is obviously a hundred and eleven over four. And that is twenty-seven and three-quarters, or twenty-seven dollars and seventy-five cents. Now I do grant you that in this particular case if we’d have worked in decimals from the start, that we’d have got our answer sort of more easily and just as accurately. But just bear in mind that sometimes, as I say, those fractions don’t turn out to be nice simple accurate decimals and you will need to work in fractions to get the most accurate possible answer.

Right. Let’s have a look at another method then. This method uses the fact that adding eleven percent of something is the same as finding a hundred percent plus eleven percent. So we’ve got the original hundred percent of the number and then we’re adding another eleven percent of that. So we’re actually finding a hundred and eleven percent of the number. And to find a hundred and eleven percent of something, you do a hundred and eleven divided by a hundred times the number. So in this calculation, times-ing by a hundred and eleven over a hundred is the bit that’s doing the calculation a hundred and eleven percent or a hundred and eleven out of a hundred times twenty-five. We’re gonna convert the twenty-five into a fraction: twenty-five over one. And then we can do some cancelling: twenty-five divided by twenty-five is one and a hundred divided by twenty-five is four. So again, we’ve got the same answer. That’s good. A hundred and eleven over four.

So both methods give you the same answer at the end of the day and it’s entirely up to you which method that you use. Method one’s a bit longer and you’ve got the risk of having to round something and carry it forward. And there are two set of calculations that you have to do. So there are two opportunities for you to type the wrong number in on the calculator or to make a silly little mistake writing something down. So maybe method two is more accurate and more efficient. You do just have to make sure that you add these two numbers together correctly to find the correct percentage at the end of the day. Easy on the adding ones, perhaps slightly less easy if you’re subtracting a percentage from a number.

So let’s look at a subtract question. Subtract fifteen percent from three hundred ninety-seven dollars. So using method one first, we’ll find fifteen percent of three hundred and ninety-seven. So that’s fifteen percent; it’s fifteen out of a hundred. That’s the proportion times three hundred and ninety-seven. Well that does come out as a nice simple decimal, so this time I’m just gonna work in decimals. And it gives us fifty-nine dollars and fifty-five cents. So we found fifteen percent. We now need to take that away from three hundred and ninety-seven. And when we take fifty-nine point five five away from three hundred and ninety-seven, we get three hundred and thirty-seven dollars and forty-five cents. Moving onto method two then. Subtracting fifteen percent is the same as starting out with a hundred percent and taking away fifteen percent. Well, that means you’re left with eighty-five percent. So we’re gonna find eighty-five percent of three hundred and ninety-seven. And when you do that calculation, you get straight to the final answer: three hundred and thirty-seven dollars and forty-five cents.

And that’s basically it for increasing, decreasing by a given percentage. The only thing that they can do in questions is make them more complicated by putting lots of words around them. So let’s look at a couple of wordy questions.

A woman deposits two hundred and fifty thousand dollars into a savings account that gives an interest rate of five point two five percent per year. She doesn’t touch the account for a whole year. How much money will be in the account after a year, and how much interest was added?

Let’s pick out the key points from this question. We’re depositing two hundred and fifty thousand dollars. So that’s our starting amount. The interest rate is five point two five percent per year. So that is basically the percentage change that we’re applying. How much interest was added? So that’s saying, what is five point two five percent of two hundred and fifty thousand, and how much money will be in the account after a year? And that means, add the five point two five percent to the two hundred and fifty thousand dollars. Now in this question, we’re asked how much interest was added and how much money will be in the account after a year. So we’ve gotta do both of those steps that were in method one. So we may as well use method one. Find five point two five percent of two hundred and fifty thousand. And to find five point two five percent of something, that’s five point two five divided by a hundred times that thing. And when we do that calculation, we get thirteen thousand one hundred and twenty-five. So, that’s the amount of interest that was added. Now we’ve gotta add that to the original two hundred and fifty thousand, and that gives us two hundred and sixty-three thousand one hundred and twenty-five.

So the final stage is to write your answer out nice and clearly and make it obvious which number means which thing. Two hundred and sixty-three thousand one hundred and twenty-five dollars is in the account at the end of the year, and thirteen thousand one hundred and twenty-five dollars were added in interest.

Let’s look at this question then. A watch costs two hundred and seventy-five dollars, not including sales tax. Sales tax of four point seven five percent will be added to the price in this state. Today the store is offering a discount of thirty-nine point five percent on the watch. If sales tax is added after the discount is deducted, how much would I pay for the watch today?

So the starting price for the watch is two hundred and seventy-five dollars, and today the store is offering a discount of thirty-nine point five percent. So we’ve got to reduce that two hundred and seventy-five dollars by thirty-nine point five percent. And then on top of that price, we’ve got to add four point seven five percent sales tax to get our final price.

So let’s just write down the milestones in our strategy here. First of all, we’re aiming to reduce two hundred and seventy-five by thirty-nine point five percent. Then, we’re gonna take that result and add four point seven five percent to it. Now, we could use either of our methods before. I’m gonna use the first method first, and then I’m gonna do this question again using the second method. So let’s have a look at that.

Well the discount is thirty-nine point five percent. That’s thirty-nine point five over a hundred times the two hundred and seventy-five. And that works out as an exact decimal. So I’m gonna leave it as a decimal for now cause we are dealing with money, a hundred and eight point six two five. So, a hundred and eight dollars sixty-two and a half cents. Now to work out the sale price of the watch then, we need to take that discount off of the original price of two hundred and seventy-five dollars. So two hundred and seventy-five minus the hundred and eight point six two five gives us a hundred and sixty-six dollars thirty-seven and a half cents. And sales tax is four point seven five percent of this amount. So four point seven five divided by a hundred times a hundred and sixty-six point three seven five. Well it gives us an answer in decimals that has got seven decimal places, but it is an accurate answer. So, seven point nine o two eight one two five. So the total price is gonna be the sale price plus the sales tax. And that comes out to a hundred and seventy-four point two seven seven eight one two five. So we’re gonna have to round that to two decimal places for dollars and cents. And a hundred and seventy-four point two seven seven, so that two seven is gonna round up to a two eight. So today’s price including sales tax is one hundred and seventy-four dollars and twenty-eight cents.

I just wanna draw your attention to the fact we’ve labelled each of our lines of our calculation. We haven’t just randomly written numbers on the page. We’ve written down what it is that we’re calculating in each case. The first case, we’ve worked out what the discount is, then we worked out what the sale price of the watch is before sales tax, then we worked out what the sales tax would be on that, and then we’ve got the total price, sale price, including sales tax. And then we’ve clearly written our answer as well at the end and put it in a nice box. So it’s really easy for whoever’s marking our question to find the logic of your working out and your final answer.

Now we’re going through the question again using the other method and just see what the differences are between the two approaches. My first strategy milestone was to reduce the two hundred and seventy-five by thirty-nine point five percent. Now that’s the same as starting up with a hundred percent and taking away thirty-nine point five percent. Well that’s the same as finding sixty point five percent of the number. And the second milestone was to take that result and add four point seven five percent to it. So to add four point seven five percent to something, you start off with a hundred percent and then you add four point seven five percent. That’s the same as finding a hundred and four point seven five percent of the number. So milestone one is to find sixty point five percent of the original price of two hundred and seventy-five dollars. So that’s this calculation: sixty point five over a hundred times two hundred and seventy-five. And then we’re gonna take that result and find a hundred and four point seven five percent of it. So we’re gonna take that result and multiply it by a hundred and four point seven five over a hundred. So we’re doing the whole calculation in one big step. And luckily that gives us the same answer: a hundred and seventy-four dollars and twenty-eight cents, including sales tax.

So the second method involved a bit less writing, although we have still documented our thinking. So we’ve documented our logic here, and we’ve annotated the calculation here so that it’s clear what steps we’re doing in each phase. But something else is revealed when we do the calculation this way. Let me just rub out some of this working out. Put this way, you can see the we’ve just got three numbers multiplied together. Now putting those parentheses around the second two forces us to do those two things first. But because multiplication is associative, it actually doesn’t matter if we have those parentheses there. So it doesn’t matter whether you do the first two terms multiplied together first, and then multiplied by two hundred and seventy-five; or you do the second two terms first and multiply by a hundred and four point seven five over a hundred. So, we can remove those parentheses. Now multiplication is also commutative. And that means if I multiply the first two terms together, it doesn’t matter whether I do the first times the second or the second times the first; I’ll still get the same answer.

And if we take a closer look at that calculation, having turned those two terms round, you can see that what we’re doing here is adding the sales tax to the original price of two hundred and seventy-five before we’re taking off the discount. But we’re still getting the same answer. Now in the question it says, if sales tax is added after the discount is deducted, how much would I pay for the watch today? Well what we’ve just seen here by rearranging that equation is: it doesn’t matter whether we apply the sales tax before we deduct the discount or after we deduct the discount; we’ll still get the same answer. Obviously the question didn’t ask us to do that, but it just goes to show that if you try doing different methods of working out, you can get deeper insights into what’s actually going on within the calculations.

So all you need to do now is decide which method you’re going to use for increasing and decreasing values by percentage. Or better still, you need to get lots of practice for both.