シャドーイング練習: Math Antics - Calculating Percent Change - 動画で英語スピーキングを学ぶ

So, with a customer acquisition cost of $35 ,000, and a weighted sales pipeline of $1 .2 ,000, and a monthly recurring revenue of $2 .4 million,

we had net sales go from 4 .9 million APR to approximately 5 .1 million per capita.

But what about the percent change?

Ah, yes, percent change.

It's always good to know percent change.

I'll explain that all to you right now.

Oh, sorry, I'm getting a phone call.

I got to take this, but then I'll explain all that percent change to you.

Hi, I'm Rob.

Welcome to Math Antics.

In this lesson, we're going to learn how to calculate percent increase and decrease, known collectively as percent change.

If you're not very familiar with percents, I'd highly recommend watching some of our other videos about them before continuing on.

Lots of times, when you have a change in value, you just say how much something goes up or down in absolute terms,

like: "The population of this city increased by a thousand people." or the cost of this shirt decreased by $15.

But you can also express those sorts of changes in relative terms using percentages.

Unlike an absolute change, a percent change always relates the amount of change to the number 100.

The term 'percent' literally means 'per 100'.

So 'percent change' means 'per 100 change' or the change 'per 100'.

So let's start by imagining that you have 100 of something.

Like 100 bucks.

Oh yeah!

If you start out with 100, but then you get 20 more, that would be a 20 % increase, because the amount went up by 20 per the original 100.

Likewise, if you start out with exactly 100 bucks but then you lose 15, that would be a 15 % decrease because it went down by 15 per the original 100.

So as you can see, it's pretty easy to figure out the percent change when the original amount is exactly 100.

But you don't have to start with 100 to express change as a percentage.

Almost any original value and any amount of change can be represented as a percent change thanks to equivalent fractions.

For example, instead of $100, suppose that you start out with $750.

Then imagine that you get $150 more.

What percent increase is that?

To figure that out, let's use a simple diagram.

This blue bar represents the original $750.

And this green bar represents the $150 increase.

Now let's use our imagination and ask, "What if that original amount was only $100?" what would the equivalent change in value be?

Basically, we're asking: If you had the fraction 150 over 750,

what would an equivalent fraction be that has 100 as the bottom number?

Put another way, if you have 750 and get 150 more, it's equivalent to having 100 and getting 'x' more.

We're using the letter 'x' to temporarily represent the missing value.

The top number of the original fraction is the absolute change.

and the top number of the equivalent fraction is the percent change.

So let's figure out what the missing value is in two different ways.

First, visually using our diagram, and second, using simple arithmetic.

By definition, if you divide any amount up into 10 equal parts, then each one of those parts will be 10 % of the original amount.

So, if you divided the original $750 up into 10 equal amounts, Each of those amounts would be $75.

That means that a $75 increase would be equivalent to a 10 % increase.

Of course, we had an increase of $150, not $75.

150 is exactly 75 plus 75, so that would be another 10 % of the original amount.

As you can see from the diagram, if you start with 750 and then you get 150 more, that's equivalent to starting with 100 and getting 20 more.

In other words, it's a 20 % increase.

Now let's see how we could get that same answer without using a diagram.

Using a little basic algebra, we can solve for the unknown value x, All we need to do is multiply both sides of the equation by 100.

Doing that gives us x all by itself on this side of the equation because the 100 over 100 cancels out.

And on the other side, we have the change in value divided by the original value all times 100.

Using a calculator, 150 divided by 750 equals 0 .2 and 0 .2 times 100 equals 20,

or 20 percent, which is the exact same answer we got from our diagram.

So the formula for calculating percent change is simple:

All you have to do is take the absolute change and divide that by the original amount, and then multiply the result by 100.

This formula may look even more intuitive to you if we put it back in the equivalent fraction form.

These are just two different ways of writing the exact same relationship.

Now that we have a formula for calculating percent change, let's try using it in a couple quick examples.

Suppose a doggy daycare takes care of 25 dogs on Friday, but on Saturday, three more dogs join the group.

What percent increase is that?

Well, the original amount of dogs is 25 and the change in dogs is +3.

According to our formula, we just need to divide the change by the original

and multiply it by 100 to get the percent change.

Using a calculator, we get 3 divided by 25 equals 0 .12.

and then 0 .12 times 100 equals 12.

That means the number of dogs at the daycare increased by 12 % from Friday to Saturday.

That was pretty easy, but what about this example?

Suppose you want to buy a pair of shoes that cost $65.

but you have a discount coupon that will reduce the price by $15.

What would the percent decrease in price be if you use your coupon?

Well, the original price is $65 and the change in price will be -15.

It's negative because it's a decrease.

So let's plug those numbers into our formula.

That gives us percent change equals negative 15 divided by 65 times 100.

Again, using a calculator, negative 15 divided by 65 equals negative 0 .23 rounded off to two decimal places.

And negative 0 .23 times 100 equals negative 23.

so the coupon will decrease the price of the shoes by 23%.

Okay, so if you're given an original amount and told how much that amount changes, it's really easy to calculate the percent change using this simple formula.

But sometimes math problems don't tell you what the absolute change in a value is.

Instead, they just give you an original value and a new value.

In that case, you need to calculate the change yourself.

Here's how you do that: Suppose you're given a problem that says last year your school had 420 students, but this year it has 441 students.

What's the percent change in student population?

This problem doesn't directly say what the absolute change in student population was.

It just tells us what the value was originally and what it is now.

We know that there was a change because of the difference in the numbers.

And in math, what does the word "difference" make you think of?

Yep, subtraction.

We can figure out the absolute change just by subtracting.

But order matters in subtraction.

So should we subtract the original amount from the new amount or the new amount from the original amount?

Well, the standard way of doing it is to start with the new amount and subtract the original amount from it.

If the new amount is bigger than the original, the answer you get will be a positive number.

which means that you have a percent increase.

But if the new amount is smaller than the original, the answer you get will be a negative number, which means you have a percent decrease.

So if we do that, we have 441 minus 420, which is positive 21.

So we have an increase of 21 students.

Positive 21 divided by the original amount, 420, equals positive 0 .05.

100

and 0 .05 times 100 equals 5.

101

Since that's positive, we have a 5 % increase in students.

102

But what if you subtracted in the wrong order and got negative 21 instead?

103

If you plug that into the formula for percent change, you'll get -21 divided by 420 which equals -0 .05.

104

and then multiplying by 100 gives you -5, which suggests a 5 % decrease because the sign is negative.

105

But, since you're paying attention, you'll realize that you couldn't possibly have a 5 % decrease in students since the number got bigger over time.

106

The problem tells us that it was 420 last year and this year it's 441.

107

So you must really have a 5 % increase.

108

The point here is that in math, it's always important to use your intuition and ask yourself if an answer makes sense,

109

rather than simply trying to memorize a formula without thinking about what it really means.

110

And speaking of intuition, before we wrap up, I want to explore just a few more situations that will hopefully give you a better intuition about percent increase and decrease.

111

First, let's consider the case where you start with 1 of something and end up with 2.

112

What would the percent increase be?

113

Well, the original amount is 1, and the change is also 1.

114

Plugging those numbers into the formula gives 1 over 1 times 100, which simplifies to 100.

115

So the percent increase is 100%.

116

That may seem kind of odd, but it makes total sense if you think about it.

117

If you have 1 and then you get 1 more, you're gaining 100 % of what you started with.

118

And that's true any time the original amount doubles.

119

If you start with 2 and get 2 more, for a total of 4, the increase is 100 % because 2 divided by 2 times 100 is 100.

120

And if you start with 5 and then get 5 more, for a total of 10, The increase is 100%.

121

because 5 divided by 5 times 100 is also 100.

122

So anytime the original amount you have doubles, it's an increase of 100%.

123

But what if you start with 2 and then end up with 1?

124

Considering what we just learned, you might be tempted to think that that's a decrease of 100%.

125

But if we use our formula, we'll see that that's not the case.

126

Since the original amount is 2, we put a 2 on the bottom of the fraction.

127

and the change is -1 since we decreased from 2 to 1.

128

So a negative 1 goes up on top.

129

Now, if we simplify, we get negative 1 divided by 2, which is negative 0 .5.

130

and negative 0 .5 times 100 is negative 50 or a 50 % decrease.

131

The reason that the percent changes are different in these two cases, doubling the amount versus cutting it in half, is that the percent change always compares the change to the original amounts,

132

which are different in these two cases.

133

Finally, let's determine what the percent increase would be if you start with 1 and end up with 3.

134

And conversely, what would the percent decrease be if you start with 3 and end up with 1?

135

In the first case, the change is positive 2 and in the second case, it's negative 2.

136

Let's plug those values into our formula for percent change, along with the original values in each case, and see what answers we get.

137

Going from 1 to 3, positive 2 divided by 1 times 100 equals 200, or a 200 % increase.

138

And, going from 3 to 1, -2 divided by 3 times 100 equals -67, rounded to the nearest whole number,

139

or a 67 % decrease.

140

Again, even though the magnitude of the change was the same, the percent changes are different because we started out with different original amounts.

141

And this example also shows that you can get a percent change that's greater than 100%.

142

Alright, so now you know what percent change is and how to calculate it.

143

The formula for calculating it is pretty simple, so you should be able to remember it after you've used it on several problems.

144

And that's the key to learning math.

145

You can't just watch videos about it.

146

You need to actually use it to solve problems.

147

So, be sure to practice what you've learned in this video!

148

As always, thanks for watching Math Antics, and I'll see you next time!

149

That's what my calculator says.

150

Learn more at MathAntics .com.

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