Main content

### Course: Algebra (all content) > Unit 2

Lesson 12: Old school equations with Sal# Growing by a percentage

In this example we grow a whole number by a percentage of itself. Growing by percentage is a common skill often used when figuring how much is owed or earned with interest. Created by Sal Khan.

## Want to join the conversation?

- What is portfolio??(119 votes)
- Kartikeye,

In the Banking and Investment field, (which is part of Sal's life before Khan Academy), a "portfolio" means an "investment portfolio".

The "My Portfolio" to which Sal refers in the video is an "Investment Portflio" which is an abstract concept refferring to all the various investments of his money added together.

When he says "My Portfolio grows by 25%", he means the total value of all of his investments grew by 25%.

Investments means things of value which are purchased with money you saved and are purchased for the purpose of the thing becoming more valuable so that later when you need to spend your saved money, it will be worth more than you originally saved.

Investments can be a savings account in a bank, a purchase of land or a building, a purchase of a business, a purchase of part of a business which is usually in the form of a stock, a loan to someone who pays you back the money borrowed plus more in interest, a part of a loan to a government which is usually called a bond.

There is a section in Khan Academy on Finance and Capital Markets that provides some fascinating information on investments. http://www.khanacademy.org/science/core-finance(68 votes)

- 7:05How do we know which number is the numerator and which number is the denominator?(1 vote)
- Haha this is going to sound silly, but I remember that the denominator is on the bottom because as I kid, I initially read the word as "DEMON"inator. I always viewed demons or monsters or whatever as staying down below in a dungeon or something, so the "denominator" always went below in a fraction. Silly, I know, but I always remembered it!(28 votes)

- for this problem coudn't you just say 25% =1/4 and 1/4 of 100 =25 and then do 100 - 25 = 75 .

I thought the answer was $75.(10 votes)- u forgot that it was 150 and u have to MULTIPLY it by 4 because it says 150 is 25 percent OF what. so 150 x 4 = 600(1 vote)

- if 9/20 of females students are at a college and there are 2160 of them how many male students are there at the college?(7 votes)
- If you mean 9/20 of students at a college are female, then you can solve this question this way:

9/20 x=2160, xbeing the total number of students (male and female) at the college

x=4800

Then the number of male students is 11/20*4800=2640 male students(10 votes)

- Is there a need to remember that 25%is 1/4?(6 votes)
- You will use this frequently. So, it is worth memorizing.

Others that you should know: 50% = 1/2

75% = 3/4

100% = 1(8 votes)

- wouldn't that answer be 75 because 25 is a quarter and then you grew to 100 so that would be 75 plus 25=100(4 votes)
- In this case, he is actually finding that $80 plus
**25% of $80**(which is $20) is $100, instead of 25% of 100, because if his portfolio was $80 and THAT amount increases by 25%, it means that the portfolio increased by $20. Hope this helps :)(3 votes)

- Couldn't you use 100% +
*p_% * _n*(where _p_ is the percent in question, in this video 15 and n is the number, in this case 95) as the formula for finding the growth of a number? I have seen this in my math textbook. But I guess this would almost be the same as the method that Sal used, because (in this case) if 95 = 100%, then 100% + 15% * 95 = 95 + 15% * 95, right?

Thanks in advance,

Parth, a.k.a. allies4ever(5 votes) - I don't get when Sal added the 1x+.25x to get 1.25x and then

1.25x=100

Why does it go from 1x+.25x to 1.25x. Isn't there two 'x'? Why do we end up with one 'x'? It seems like one of the 'x' just disappears. I'm a little confused. Can someone please explain? thank you.(2 votes)- Sal adds one x with .25x (a quarter of x). So in total he has one and a quarter x's.

1x + .25x = 1.25x

'x' can stand for anything...

let x = a whole pizza

1 pizza plus a quarter of a pizza is one and a quarter pizzas

let x = 4

1*(4) + .25*(4) = 1.25*(4)

4 + 1 = 5

5 = 5

I'm going to work backwards and see if that helps too. I'm going to start with 1.25x and prove that it equals 1x + .25x

1.25x ----> (1 + .25)x

All I did was replace 1.25 with 1 + .25 because they are the same thing

(1 + .25)x ----> 1*x + .25*x ----> 1x + .25x

For the second step I just used the distributive property.

I hope this helps in some way!(6 votes)

- He did the equation wrong .15 times 95 is 14.25?(3 votes)
- He immediately finds and fixes his error. So, what's the issue? His final result is 14.25.(4 votes)

- I don't understand how he got 80 as the starting number. I tried to find out 25% of 100 and that was 25. Then I subtracted it by 100 and got 75 as my starting number. :I

Also, wouldn't 80 be 4/5 of 100? So to get 80, shouldn't the rate grow by 20% instead?(3 votes)- You are so close to being correct, the only error in your work is that 80 is 80% of 100, and 100 is 125% of 80. Because you need an extra 20, and 20 is 25% of 80, therefore to get to 100, you must multiply by 125%. But, you would decrease by 20% if you were talking about 100 to 80.

So close though!

-AD(3 votes)

## Video transcript

Let's do some more
percentage problems. Let's say that I start
this year in my stock portfolio with $95.00. And I say that my portfolio
grows by, let's say, 15%. How much do I have now? OK. I think you might be able to
figure this out on your own, but of course we'll do some
example problems, just in case it's a little confusing. So I'm starting with $95.00,
and I'll get rid of the dollar sign. We know we're working
with dollars. 95 dollars, right? And I'm going to earn, or I'm
going to grow just because I was an excellent stock
investor, that 95 dollars is going to grow by 15%. So to that 95 dollars, I'm
going to add another 15% of 95. So we know we write 15% as a
decimal, as 0.15, so 95 plus 0.15 of 95, so this is
times 95-- that dot is just a times sign. It's not a decimal, it's a
times, it's a little higher than a decimal-- So 95 plus
0.15 times 95 is what we have now, right? Because we started with 95
dollars, and then we made another 15% times what
we started with. Hopefully that make sense. Another way to say it, the 95
dollars has grown by 15%. So let's just work this out. This is the same thing as 95
plus-- what's 0.15 times 95? Let's see. So let me do this, hopefully
I'll have enough space here. 95 times 0.15-- I don't
want to run out of space. Actually, let me do it up here,
I think I'm about to run out of space-- 95 times 0.15. 5 times 5 is 25, 9 times 5 is
45 plus 2 is 47, 1 times 95 is 95, bring down the 5,
12, carry the 1, 15. And how many decimals
do we have? 1, 2. 15.25. Actually, is that right? I think I made a mistake here. See 5 times 5 is 25. 5 times 9 is 45, plus 2 is 47. And we bring the 0 here, it's
95, 1 times 5, 1 times 9, then we add 5 plus 0 is 5,
7 plus 5 is 12-- oh. See? I made a mistake. It's 14.25, not 15.25. So I'll ask you an
interesting question? How did I know that
15.25 was a mistake? Well, I did a reality check. I said, well, I know in my head
that 15% of 100 is 15, so if 15% of 100 is 15, how can
15% of 95 be more than 15? I think that might
have made sense. The bottom line is 95
is less than 100. So 15% of 95 had to be less
than 15, so I knew my answer of 15.25 was wrong. And so it turns out that I
actually made an addition error, and the answer is 14.25. So the answer is going to be 95
plus 15% of 95, which is the same thing as 95 plus 14.25,
well, that equals what? 109.25. Notice how easy I made
this for you to read, especially this 2 here. 109.25. So if I start off with $95.00
and my portfolio grows-- or the amount of money I have-- grows
by 15%, I'll end up with $109.25. Let's do another problem. Let's say I start off with some
amount of money, and after a year, let's says my portfolio
grows 25%, and after growing 25%, I now have $100. How much did I originally have? Notice I'm not saying that
the $100 is growing by 25%. I'm saying that I start with
some amount of money, it grows by 25%, and I end up with
$100 after it grew by 25%. To solve this one, we
might have to break out a little bit of algebra. So let x equal what
I start with. So just like the last problem,
I start with x and it grows by 25%, so x plus 25% of x is
equal to 100, and we know this 25% of x we can just rewrite as
x plus 0.25 of x is equal to 100, and now actually we have a
level-- actually this might be level 3 system, level 3 linear
equation-- but the bottom line, we can just add the
coefficients on the x. x is the same thing
as 1x, right? So 1x plus 0.25x, well that's
just the same thing as 1 plus 0.25, plus x-- we're just doing
the distributive property in reverse-- equals 100. And what's 1 plus 0.25? That's easy, it's 1.25. So we say 1.25x
is equal to 100. Not too hard. And after you do a lot of these
problems, you're going to intuitively say, oh, if some
number grows by 25%, and it becomes 100, that means that
1.25 times that number is equal to 100. And if this doesn't make sense,
sit and think about it a little bit, maybe rewatch the video,
and hopefully it'll, over time, start to make a lot
of sense to you. This type of math is
very very useful. I actually work at a hedge
fund, and I'm doing this type of math in my
head day and night. So 1.25 times x is equal
to 100, so x would equal 100 divided by 1.25. I just realized you
probably don't know what a hedge fund is. I invest in stocks
for a living. Anyway, back to the math. So x is equal to 100
divided by 1.25. So let me make some space
here, just because I used up too much space. Let me get rid of my
little let x statement. Actually I think we know
what x is and we know how we got to there. If you forgot how we got
there, you can I guess rewatch the video. Let's see. Let me make the pen thin
again, and go back to the orange color, OK. X equals 100 divided by 1.25,
so we say 1.25 goes into 100.00-- I'm going to add a
couple of 0's, I don't know how many I'm going to need,
probably added too many-- if I move this decimal over two to
the right, I need to move this one over two to the right. And I say how many times does
100 go into 100-- how many times does 125 go into 100? None. How many times does
it go into 1000? It goes into it eight times. I happen to know that in my
head, but you could do trial and error and think about it. 8 times-- if you want to think
about it, 8 times 100 is 800, and then 8 times 25 is
200, so it becomes 1000. You could work out if you like,
but I think I'm running out of time, so I'm going
to do this fast. 8 times 125 is 1000. Remember this thing isn't here. 1000, so 1000 minus 1000 is 0,
so you can bring down the 0. 125 goes into 0 zero times,
and we just keep getting 0's. This is just a decimal
division problem. So it turns out that if your
portfolio grew by 25% and you ended up with $100.00
you started with $80.00. And that makes sense, because
25% is roughly 1/4, right? So if I started with $80.00 and
I grow by 1/4, that means I grew by $20, because
25% of 80 is 20. So if I start with 80
and I grow by 20, that gets me to 100. Makes sense. So remember, all you have to
say is, well, some number times 1.25-- because I'm growing
it by 25%-- is equal to 100. Don't worry, if you're still
confused, I'm going to add at least one more presentation
on a couple of more examples like this.