Growing by a percentage Growing a quantity by a percentage
Growing by a percentage
- 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, fifteen percent.
- How much do I have now?
- 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.
- ninety-five dollars, right?
- And I'm going to earn, or I'm going to grow just because
- I was an excellent stock investor, that ninety-five dollars
- is going to grow by fifteen percent.
- So to that ninety-five dollars, I'm going to add another fifteen percent of ninety-five.
- 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 ninety-five dollars, and then we made
- another fifteen percent times what we started with.
- Hopefully that make sense.
- Another way to say it, the ninety-five dollars has grown by fifteen percent.
- 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.
- five times five is twenty-five, nine times five is forty-five plus two is forty-seven, one times ninety-five is
- ninety-five, bring down the five, twelve, carry the one, fifteen.
- And how many decimals do we have?
- one, two.
- Actually, is that right?
- I think I made a mistake here.
- See five times five is twenty-five.
- five times nine is forty-five, plus two is forty-seven.
- And we bring the zero here, it's ninety-five, one times five, one times nine, then
- we add five plus zero is five, seven plus five is twelve-- oh.
- 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 fifteen percent of one hundred is fifteen, so if
- fifteen percent of one hundred is fifteen, how can fifteen percent of ninety-five be more than fifteen?
- I think that might have made sense.
- The bottom line is ninety-five is less than one hundred.
- So fifteen percent of ninety-five had to be less than fifteen, 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 ninety-five plus fifteen percent of ninety-five, which is the
- same thing as 95 plus 14.25, well, that equals what?
- Notice how easy I made this for you to read,
- especially this two here.
- So if I start off with $95.00 and my portfolio grows-- or the
- amount of money I have-- grows by fifteen percent, 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 twenty-five percent, and after growing
- twenty-five percent, I now have $one hundred.
- How much did I originally have?
- Notice I'm not saying that the $one hundred is growing by twenty-five percent.
- I'm saying that I start with some amount of money, it grows
- by twenty-five percent, and I end up with $one hundred after it grew by twenty-five percent.
- 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
- twenty-five percent, so x plus twenty-five percent of x is equal to one hundred, and we know this
- 25% of x we can just rewrite as x plus 0.25 of x is equal to
- one hundred, and now actually we have a level-- actually this might be
- level three system, level three linear equation-- but the bottom
- line, we can just add the coefficients on the x.
- x is the same thing as onex, 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 one hundred.
- 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 twenty-five percent, and it
- becomes 100, that means that 1.25 times that number
- is equal to one hundred.
- 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 one hundred go into one hundred-- how many
- times does one hundred and twenty-five go into one hundred?
- How many times does it go into one thousand?
- 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.
- eight times-- if you want to think about it, eight times one hundred is
- eight hundred, and then eight times twenty-five is two hundred, so it becomes one thousand.
- You could work out if you like, but I think I'm running out of
- time, so I'm going to do this fast.
- eight times one hundred and twenty-five is one thousand.
- Remember this thing isn't here.
- one thousand, so one thousand minus one thousand is zero, so you can bring down the zero.
- one hundred and twenty-five goes into zero zero times, and we just keep getting zero's.
- This is just a decimal division problem.
- So it turns out that if your portfolio grew by twenty-five percent and
- you ended up with $100.00 you started with $80.00.
- And that makes sense, because twenty-five percent is roughly one / four, right?
- So if I started with $80.00 and I grow by 1/4, that means I
- grew by $twenty, because twenty-five percent of eighty is twenty.
- So if I start with eighty and I grow by twenty,
- that gets me to one hundred.
- 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.
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