Sequential Percent Changes

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Sequential percent changes. So this is about when we have percent increases and decreases following each other. When you combine percent increases or percent decreases. There are a few common mistakes involving percent increases and decreases. And the test loves to exploit these common oversights.

They love to write problems where people make very predictable mistakes. These mistakes center on scenarios in which two or more percent changes follow in a sequence. For example, you might have a certain percent increase in a price and then an employee will buy with a certain percent employee discount. And so you have a percent up or percent down.

How do they combine? So for example, here's a typical problem, an item initially cost $100, the beginning of the year, the price increased by 30%. After the increase, an employee purchased it with a 30% decrease, a 30% discount. What price did the employee pay? So pause the video right now, work on this on your own, and then restart the video when you're ready.

The first thing I'll say is that the answer is not $100. That is the trap answer, that is the predictable mistake answer. More than half the people who take the test will guess that and they will be wrong. That is the most common mistake in this whole subject. The same percent up and then down, or down and then up, does not put you back in the same place.

People think, 30% up, 30% down and they cancel. They absolutely do not, you do not wind up back in the same place. We answered this using a product of multipliers. Though a 30% increase, that's a multiplier of 1.3. A 30% decrease is a multiplier of 0.7 and so we just multiply 100 by each 1 of those multipliers.

So 100 times 1.3 times 0.7 gives us 91 and that's the actual price that the employee paid. Now this might be antituitive for some people. Let's think about this. Starts out at 100, a 30% increase means it goes up to 130. Well, then, the employee comes along with a 30% discount, but they're not getting a 30% discount on the price of 100.

They're getting a 30% discount on a price of 130. And 30% of 130 is bigger than 30% of 100, which is why the amount it goes down is larger than the amount it goes up, which is why the employee winds up paying a price less than $100. Here's another problem. At the beginning of the year, the price of an item increased 30%.

After the increase, an employee purchased it with a 40% discount. The price the employee paid was what percent below the original price? In other words, what percent below the price before the increase? So again, pause the video, take a moment to work this out on your own. The first thing I'll talk about here is the mistake. The common mistake people are gonna say, well, up 30% then down 40%, 30- 40 is -10%, must be a 10% decrease.

Guaranteed more than half, maybe even three-quarters of the people who take the test will fall into this trap. It's as if the test writer just set up a huge butterfly net, and people just run into it in hordes, it's absolutely, 100%, predictable how many people make this mistake. That's why it's so important to recognize this and understand not to make it.

Whenever you have two or more percent changes in a row, never add or subtract the percents. That will always be wrong. You never want to add or subtract percent increases and decreases. Instead, what do you do, of course, we use multipliers, we always use multiplier for a percent increases and decreases.

So a 30% increase, that's a 1.3 multiplier, a 40% decrease, that's a 0.6 multiplier. We're just gonna multiply those two multipliers. They multiply to 0.78, 0.78 is the multiplier for a 22% decrease. And so this means that what we have going on here is a 22% decrease. The price of the employee paid was 22% less than the original price.

Here's another, the price of a stock increase 20% in January, dropped 50% in February and increased 40% in March. So in other words, that's the first quarter of the year, find the percent change for these three-month period. The percent change for the first quarter. So again, pause the video here and see if you can work this out.

The first thing I'll say is that, of course, the mistake answer, the very predictable mistake that more than half the people who take the test will make, they're just gonna do plus 30, minus 50, plus 40, that it gives us positive 10, so it must be a 10% increase. That's the mistake that people are gonna make. Again, a very predictable mistake.

The test writers absolutely love it when they can write a question that has such a predictable mistake. That's why it's so very important to understand the nature of this mistake so you don't fall into this trap. Of course, again, we're gonna use multipliers. The multiplietr for 20% increase, 1.2, the multiplier for 50% decline, that's 0.5, the multiplier for 40% increase, 1.4 we're just gonna multiply those 3.

To make things simple, we're first gonna multiply the last 2, 0.5 or one-half times 1.4 will give me 0.7, and then 0.7 times 1.2 gives me 0.84. That is the multiplier for a 16% decrease. And so that's what's happened here. For the first quarter of the year, the stock decreased 16%.

So in this topic, it's very important to understand both the nature of the mistakes, the very tempting mistakes, as well as to understand what the right thing is to do. Mistake number 1, an increase and a decrease by the same percent do not get us back to the same original starting point. Mistake number 2, in a series of percent changes, never add or subtract the individual percent, that would be wrong 100% of the time.

Instead, what we're always gonna do is figure out multipliers and multiply all the individual multipliers together.

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