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Percent Increases and Decreases
Percent increases and decreases. The test absolutely loves to ask about percent increase and decrease problems. This is perhaps one of their favorite kinds of problems. So here are a couple examples of things that you might see. We’re going to be talking about this in the next couple of videos. Remember what we said first of all, about multipliers in the video, Working with Percents.
There, we said that the decimal form of a percent was its multiplier. Here we will change that a little bit. The decimal form of P percent is the multiplier for finding P percent of something. So if I wanted for example, 30% of something, I would use the multiplier, use the decimal form.
If 0.30 is the multiplier to find 30% of something but we would use different multipliers for a 30% increase or a 30% decrease, and that’s what we will be talking about in this particular video. First of all let’s talk about percent increases. So this may be phrased as Y increased by 30% or X is 30% greater than Y, either way, this is a percent increase for Y.
Let’s think about this. If Y increases by 30%, the whole original part of Y is still there, plus 30% more. Therefore, the multiplier for a 30% increase is 1 plus 0.3 which gives us 1.3. The one represents the whole that is there, that remains there, and then the 0.3 is the 30% that is added, and so it combines to our single multiplier 1.3. In general, if the problem talks about a P% increase, the multiplier for a P percent increase is one plus P percent as a decimal.
This is very, very powerful. Thus, the multiplier for 46% increase is 1 plus 0.46, which is 1.46. So let’s talk about how to use this in problems. An item originally cost $800. The price increased by 20%. What is the new price?
So all we have to do is take 800 and multiply it by the multiplier for 20% increase, 20% is a decimal 0.2, so the multiplier would be 1.2. So this would be 800 times 1.2, and of course we can just multiply that out and we get 960. Now, I might do it this way with a multiplier, or I might do it just with some number sense.
I know that 10% is 80, 20% is 160, so in other words, we’re adding 160, 800 plus 160 is 960. That’s another perfectly good way to do this. After a 30% increase, the price of something is $78. What was the original price? Hm, that’s a bit trickier.
So that means we had some original x, we increased it by 30%, which would be a 1.3 multiplier, and that should equal 78. So x times 1.3 equals 78. We’ll divide 78 by 1.3, slide the decimal place over, 780 divided by 13 that turns out to be 60 and that’s the answer. Percent decreases.
This might be phrased as Y decreased by 30 percent or X is 30 percent less than Y. Either way, this is a percent decrease for y. Let’s think about this. If Y decreases by 30%, most of the original whole of Y is still there, except for 30% that’s now missing. Okay, and one way to say it really is that if 30% leaves, 70% of it is still there.
Therefore the multiplier for a 30% decrease is 1 minus 0.3, which gives us 0.7, and this reiterates what I said a moment ago. If 30% leaves, 70% is still there. In general, if the problem talks about a P% decrease, the multiplier for a P% decrease is 1 minus P% as a decimal. So for example, if I want the multiplier for a 28% decrease, that would be 1 minus 0.28, which gives us 0.72.
That is the multiplier for a 20% decrease. A $170 item is discounted 30%. What is the new price? So the multiplier way to do this, I formed the multiplier doing 1 minus 0.3, I get 0.7. That’s the multiplier for a 30% decrease.
Then I multiply the price times that number, multiply out, and I get $119. Now a totally different way to do it is, I might notice, again this is number sense. Notice that 10% of $17, the thirty percent is three times that, $51 dollars. So in other words a hundred and seventy goes down by $51, and here’s how I think about this, price goes down by $50 and then it goes down by $1, goes down by $50 by 170 to 120, and then by $1 from 120 to 119.
That makes it very easy to do in my head. Here’s another problem. After an item was discounted 80%, the new price is $150. What was the original price? You might wanna take a moment just to work on this, and then I’ll show the solution. So here’s how we’ll solve it.
It was discounted 80%. So the multiplier for that is 1 minus 0.8, which is point 0.2. And so it means that the original x, which we don’t know, is multiplied by 0.2 and winds up equaling 150. So we divide that by 0.2, move the decimal and we get 750. Now a number sense way of approaching this, I might notice that if 80% is gone, 20% is left, that 20% is 150.
Therefore 10% is half of that, 75, therefore the whole is 750. That’s kind of a number sense way of approaching it. Finally, finding the percent. Some problems give us the starting and ending values, and ask us to find the percent of increase or decrease. Since the new equals the multiplier times the old, we could say that the multiplier equals the ratio of new price divided by old price.
We have to remember that this ratio gives us a multiplier, and we have to change from that, back to a percent. What was the, the price of an item increases from $60 to 102. What was the percent increase? So again, doing this a straight multiplier way, the multiplier equals new divided by old.
So that equals 102 divided by 60. Divide both the numerator and the denominator by 6, I get 17 over 10, which is 1.7. That’s the multiplier for a 70% increase. For this same problem I might notice that the difference is $42, and of course 10% of 60 is 6.
And there’s 70 of those in 42, so that is another way to see that it’s a 70% increase, 70% greater. The price of an item decreases from $250 to $200. What was the percent decrease? Well again, we can use this ratio. Plug in 200 over 250 divide by 50, you get 20 over 25.
We can also write this as 80 over 100, and of course, this is 0.8. That’s the multiplier for 20% decrease. I might also notice that the price drops $50 and $50 is one fifth of 250. That’s a drop of one fifth or 20%. That’s another way to see that it’s a 20% decrease. If the price, price of an item increases from $200 to $800, what was the percent increase?
Now this is a really tricky one, and there’s a trap here. Pause the video and think about this for a minute, and then we’ll talk about this. . So, obviously the price was multiplied by 4. The trap answer is to believe that this is a 400% increase. That’s a trap.
Now why is that a problem? Four is the multiplier, so it’s the multiplier for 4 minus 1, 3 or a 300% increase. Now this is really subtle. We have to be careful here. 800 is 400% of 200.
So that’s one kind of question we can ask 800 is what percent of 200? It is 400% of 200, but 800% is three, 800 is 300% greater than 200. Now, that’s a very different question. Not a question of what percent of, but what percent greater than. Those are two very different questions, and they have two different answers here.
So, in this video we talked about percent increases and decreases. We talked about the multiplier for a P% increase, 1 plus P as a decimal, the multiplier for a P% decrease, 1 minus P as a decimal, and we talked about using multipliers to find the percent change.
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