So, with a customer acquisition cost of 35…
and a weighted sales pipeline of 1.2…
and a monthly reoccurring 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… Ah yes… percent change…
It’s always good to know percent change…
I’ll… I’ll explain that all to you right now…
Oh! Sorry… I’m getting a phone call… I gotta take this
but then I’ll explain all that percent change to ya.
Hi I’m Rob. Welcome to Math Antics!
In this lesson, we’re gonna 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 1,000 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 dollars?”
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 (which is currently missing) is the percent change.
So let’s figure out what the missing value is in two different way;
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 PERCENT 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 (150) divided by the original value (750) all times 100.
Using a calculator, 150 divided by 750 equals 0.2
And 0.2 times 100 equals 20 or 20%
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 (or how much the amount has increased or decreased)
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 example.
Suppose a Doggie Day Care takes care of 25 dogs on Friday,
but on Saturday, 3 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 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 day care increased by 12% from Friday to Saturday.
That way 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 -15 divided by 65 times 100.
Again, using a calculator: -15 divided by 65 equals -0.23 (rounded off to two decimal places)
and -0.23 times 100 equals -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.
Is 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.
+21 divided by the original amount 420 equals +0.05
And 0.05 times 100 equals 5.
Since that’s positive, we have a 5% increase in students.
But what if you subtracted in the wrong order and got -21 instead.
If you plug that into the formula for percent change,
you’ll get -21 divided by 420 which equals -0.05
and then multiplying by 100 gives you -5
which suggests a 5% decrease because the sign is negative.
But, since you're paying attention,
you’ll realize that you couldn’t possible have a 5% decrease in students since the number got bigger over time.
The problem tells us that it was 420 last year and this year it’s 441.
So you must really have a 5% increase.
The point here is that in math,
it’s always important to use your intuition and ask yourself if an answer makes sense,
rather than simply trying to memorize a formula without thinking about what it really means.
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.
First let’s consider the case where you start with 1 of something and end up with 2.
What would the percent increase be?
Well, the original amount is 1 and the change is also 1.
Plugging those numbers into the formula gives 1 over 1 times 100 which simplifies to 100.
So the percent increase is 100%.
That may seem kind of odd, but it makes total sense if you think about it.
If you have 1 and then you get 1 more, you’re gaining 100% of what you started with.
And that’s true any time the original amount doubles.
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.
And if you start with 5 and then get 5 more (for a total of 10)
the increase is 100% because 5 divided by 5 times 100 is also 100.
So any time the original amount you have doubles, it’s an increase of 100%
But what if you start with 2 and then end up with 1?
Considering what we just learned, you might be temped to think that’s a decrease of 100%,
but if we use our formula, we’ll see that’s not the case.
Since the original amount is 2,
we put a 2 on the bottom of the fraction,
and the change is -1 (since we decreased from 2 to 1) so a -1 goes up on top.
Now if we simplify, we get -1 divided by 2 which is -0.5
and -0.5 times 100 is -50 or a 50% decrease.
The reason that the percent changes are different in these two cases
(doubling the amount vs. cutting it in half)
is that the percent change always compares the change to the original amounts,
which are different in these two cases.
Finally, let’s determine what the percent increase would be if you start with 1 and end up with 3
and conversely, what would the percent decrease be if you start with 3 and end up with 1.
In the first case, the change is +2 and in the second case it’s -2.
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.
Going from 1 to 3:
+2 divided by 1 times 100 equals 200,
or a 200% increase.
And going from 3 to 1:
-2 divided by 3 times 100 equals -67 (rounded to the nearest whole number)
or a 67 % decrease.
Again, even though the magnitude of the change was the same,
the percent changes are different because we started out with different original amounts.
And this example also shows that you can get a percent change that’s greater than 100%.
Alright, so now you know what percent change is and how to calculate it.
The formula for calculating it is pretty simple,
so you should be able to remember it after you’ve used it on several problems.
And that’s the key to learning math!
You can’t just watch videos about it… you need to actually use it to solve problems,
so be sure to practice what you’ve learned in this video.
As always, thanks for watching Math Antics and I’ll see ya next time.
Ah… Yes… Percent Change.
So percent change in this case is…
negative 1,000 percent.
So I guess you’re all fired.
That’s what my calculator says.
Learn more at www.mathantics.com