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Percent Increases


We'll talk now about percent increases. First thing I'll say is that the test loves to ask about percent increases and percent decreases. So these are some examples of things that the test might ask. Remember what we said about multipliers in the video, working with percents. There we said that the decimal form of a percent was its multiplier.

Here, we'll change that a little bit. The decimal form of P percent is the multiplier for finding P percent of something, and so if I wanna find 40% of something, I'd use the multiplier 0.4, just change the percent to a decimal. So 0.3 is the multiplier to find 30% of something, but we will use different multipliers for a 30% increase or a 30% decrease.

We'll talk about multipliers for an increase in this video, multipliers for a decrease to the next video. Percent increases. This might be phrased as Y increase by 30%, or X is greater than 30% of Y, is 30% greater than Y. So either one of those is a statement of a percent increase.

And either way, Y was increased. Let's think about this. If we knew, say, 30% of Y, how would we increase Y by 30%? Think about what it means to say that Y is increased by 30%? It means that we still have all of Y, 100% of the original plus 30% more. So a 30% increase of Y equals all of Y plus 30% more of Y.

That's what a 30% increase means. All across all of Y, that's just Y. 33% of Y, we use to multiply that we learned about already, just change the percent to a decimal and multiply 0.3. And notice that we can factor out of Y from that. We have 1 times Y + 0.3 times Y, so we can factor out (1 + 0.3) times Y, or just add those together, 1.3 times Y.

If we were simply dealing with numbers, we could do the calculation directly. So for example, if we wanted to increase some something by 60%, we could figure out, first of all, 60%of something and just add that to the original. So suppose we had to increase 60% by 800, this is probably a little bit easier than the what you'd actually see on the text.

But if we had to increase 60% by 800, first of all, we'd find 60% of 800. So, of course, 10% of 800 is just 80. So we'll multiply that by 6. 60% of 800 is 480, so that's 60% of 800, and now that get's added to the original. So we add that to the original, 800 + 480 is 1,280. And so when we increase 800 by 60%, we get 1,280.

We get all the original 800 plus an additional 60% of that original. Note that with numbers, it's relatively straightforward to do that forward calculation, start with the starting value and then get the percent increase, that's a relatively straightforward calculation. Going backwards though is much trickier, in other words, if we're given the result of the percent increase and we want the starting value, that's a little bit harder.

For example, if the problem told us that our price was increased by 60% and the result was 1,280, what was the starting value? That would be a hard question. We actually know that it's 800 cuz we just did the forward calculation, but the point is if we didn't know that, that would be a harder thing to figure out just like playing with the numbers.

And this is where multipliers would help us a lot. Furthermore, with numbers, we can do direct calculations sometimes but with variables, we need to use multipliers. If I want something that 60% is greater than K, well, I can't figure out 60% of K as number because it's a variable. So I just have to think of it this way, 60% greater than K, that's all of K plus 60% of K, so that's gonna be K + 0.6K.

Again, factor out the K, we're gonna get 1 + 0.6 which is 1.6. And that is the multipier for 60% increase. So 0.6 is the multiplier for 60% of something. And 1.6 is the multiplier for a 60% increase. Notice we could build that multiplier directly.

In general, the problem asks about a P% increase. All we have to do is take the percent, change it to a decimal, and add one. And that's what we follow. So for example, if we wanted a 46% increase, well, the multiplier for that would be 1+0.46, which is 1.46. The multiplier form for a percent increase is quite useful.

We need either when the quantities are variables, where when we have the numerical value of the increase of the final result and we want the starting value. And again, the recipe for building this is very simple. Change the percent to a decimal, add 1. That's all you have to do to build the multiplier for our percent increase.

So here's a practice problem, pause the video and then we'll talk about it. Okay, after a 30% increase, the price of something is $78. What was the orgininal price? All right, a tricky question. So to find the multiplier, what we're gonna do to 30%, we're gonna change it to a decimal, 0.3.

And then we're gonna add 1, so 1.3 is the multiplier for a 30% increase and then the unknown originals multiplied by this. So we're just gonna create a variable for that unknown original just called x, so x times 1.3 = 78. So x = 78 divided by 1.3, we can multiply numerator down there by 10, 78 divided by 13 and we use a calculator to figure that out, that turns out to be 60.

So that is the answer, that was the original price. If we increase 60 by 30% we get 78. Here's another practice problem. Pause the video and then we'll talk about this. Okay, at the beginning of last year, Item X was at some fixed original price. At the end of last year, it's price was increased by 45%, and since that increase, its price has been T.

And we want to express the original in terms of T. So we're definitely gonna have to introduce a variable for that starting price, the price that we want is the answer. So let's just call that P, starting price. What's the multiplier? The multiplier, we wanna take 45%, change that to a decimal, 0.45, and then add 1, 1.45.

That's the multiplier for our percentage increase. So we multiply that by the original price and we get T, the final price, because the original price increases 45%. Now we wanna just solve this for P, so we divide by 0.45. And that actually right there is as far as we can go. That expresses the original prize in terms of T.

In summary, if N increases by some percent, that's all of N plus the additional percentage of N. And if something increases by 35%, that is all of it plus 35% more of it. The multiplier for a P% increase, all we have to do is change P to a decimal and add one. In using multipliers to find the percent change, especially when you have the result, the need starting value or when everything is in variables.

Those are the two times that multipliers are particularly helpful in thinking about percent increases.

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