## Percent Decreases

### Transcript

Percent decreases. In the previous video, we discussed how useful the multipliers for a percent increase can be. Here, we'll discuss the parallel process for percent decreases. So, let's talk about percent decreases. This might be phrased as Y is decreased by 30% or X is 30% less than Y.

Either one of those is a 30% decrease of Y. Let's think about this. If we decrease Y by 30%, that means that we start with all of Y, and we subtract 30% of it. Once we get rid of that 30%, the difference that's left is the result of the percent decrease.

So that's a big idea right there, a percent decrease is a difference. It is, at its very essence, something that results from subtraction. For example, if we decrease T by 70%, we start with all of T and then we subtract 70% of T. So the percent decrease = (all of T)- (70% of T), all of T, that's just T. To find 70% of T, we change 70% to a decimal, 0.7, and multiply it times T.

So this is gonna be T- 0.7*T. We could factor out the T, so we have 1*T- 0.7*T, that's (1- 0.7)T and 1- 0.7 is 0.3. So that's the multiplier for 70% decrease, 0.3. Think about that, this same number 0.3 is the multiplier for 30% of something or the multiplier for 70% decrease.

How can that be? This makes sense, because if we start with 100% of anything and we take away 70% of it, then we always will be left with 30% of it. So that make sense. In fact, that's one way to think about percent decreases. Think about the percent that is gonna be left over and just calculate that percent.

As you might expect, we also could build the multiplier for a percent decrease using a method very much like the one we used to build the percent increase multiplier. So for here, step 1 is the same. We change the percent to a decimal but then we subtract because it's percent decrease we subtract that decimal from 1.

In general, the problem talks about a P% decrease. We can think of this formula, the multiplier for a P% decreases 1 minus the percent as a decimal. So, for example, the multiplier for a 28% decrease is 1- 0.28, and that happens to be 0.72. And another way to say that is if we removed 28% of anything, we're gonna be left with 72% of it.

As with percent increases, if we are given numerical starting values, we often can work directly with the numbers to calculate a percent decrease. As in the case a percent increases, the multiplier is more useful when we have the numerical value of the decrease, in other words, the final result. And we want the starting value, in other words, we wanna go backwards or when everything is in variables.

Those are the cases that are much more likely to show up on the test. And those are the cases where we need the multiplier. Here is a practice problem. Pause the video and then we will talk about this. So an item was discounted by 80%, the new price is \$150, what was the original price?

So one way to talk about this is with the multiplier, we'll build the multiplier. So 80% as a decimal is 0.8, 1- 0.8 = 0.2. So that's the multiplier for 80% decrease. We'll just say that the original unknown price, call that P. So P*0.20 = 150. Then we can divide 150 by 0.20, multiply numerator and denominator by 10 and then it's just 1,500 divided by 2 which is 750, and so that's the original price.

So that's the way to find with the multiplier. Another way to think about it, I might notice that if 80% is gone only 20% is left. So whatever the original price was, that 150 is 20% of that original price. So that's 20%, divide that by 2. That means that 10% is 75.

Well, if 10% is 75, all I have to do is multiply that by 10. And, therefore, the original price had to be 750. That's another way to think about this problem. Here's another practice problem. Pause the video, read this, think about this, and then we'll talk about it. Okay, on September 1, the amount of water in a reservoir was at its maximum capacity.

During the month, the amount of water decreased by 32%, and on September 30, there were D cubic meters remaining in the reservoir. Express the maximum capacity in terms of D. So again, we have to introduce a variable for the starting amount. Well, first we have to create the multiplier, so 1- 0.32 for a percentage decrease, that turns out to be 0.68.

The maximum capacity, I'm just gonna introduce the variable M and so that M was decreased by 32%. So it was multiplied by 0.68 and that results in the D at the end of the month. And all we wanna do is solve for M. So divide by 0.68 and that's the answer. Fundamentally, a percent decrease is a difference, the starting value minus the percent by which it decreases.

We can find the multiplier, and the rule for this is just change the percent to a decimal and subtract that from 1. And the multiplier form is most useful when we have the decreased value and we want the starting value, or when everything is in variables.