More percent problems Slightly harder percent problems
More percent problems
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- Let's say I go to a store and I have fifty dollars in my pocket.
- fifty dollars in my wallet.
- And at the store that day they say it is a twenty-five percent
- off marked price sale.
- So twenty-five percent off marked price means that if the marked price is
- one hundred dollars the price I'm going to pay is going to be
- twenty-five percent less than $one hundred.
- So my question to you is if I have $fifty, what is the highest
- marked price I can afford?
- Because I need to know that before I go finding something
- that I might like.
- So let's do a little bit of algebra.
- So let x be the highest marked price that I can afford.
- So if the sale is twenty-five percent off of x, we could say that the new
- price, the sale price will be x minus twenty-five percent of x is equal
- to the sale price.
- And I'm assuming that I'm in a state without sales tax.
- Whatever the sale price is, is what I have to pay in cash.
- So x minus twenty-five percent x is equal to the sale price.
- The discount is going to be twenty-five percent of x.
- But we know that this is the same thing as x minus 0.25x.
- And we know that that's the same thing as-- well, because
- we know this is onex, x is the same thing is onex.
- 1x minus 0.25x.
- Well, that means that 0.75x is equal to the sale price, right?
- All I did is I rewrote x minus 25% of x as 1x minus 0.25x.
- And that's the same thing as 0.75x.
- Because 1 minus 0.25 is 0.75.
- So 0.75x is going to be the sale price.
- Well, what's the sale price that I can afford?
- Well, the sale price I can afford is $fifty.
- So 0.75x is going to equal $50.
- If x is any larger number than the number I'm solving for,
- then the sale price is going to be more than $fifty and I
- won't be able to afford it.
- So that's how we set the-- the highest I can pay is $fifty
- and that's the sale price.
- So going back to how we did these problems before.
- We just divide both sides by 0.75.
- And we say that the highest marked price that I can afford
- is $50 divided by 0.75.
- And let's figure out what that is.
- 0.75 goes into 50-- let's add some 0's in the back.
- If I take this decimal two to the right.
- Take this decimal, move it two to the right, goes right there.
- So 0.75 goes into 50 the same number of times
- that seventy-five goes into five thousand.
- So let's do this.
- seventy-five goes into fifty zero times.
- seventy-five goes into five hundred-- so let me think about that.
- I think it goes into it six times.
- Because seven times is going to be too much.
- So it goes into it six times.
- six times five is thirty.
- six times seven is forty-two.
- Plus three is forty-five.
- So the remainder is fifty.
- I see a pattern.
- Bring down the zero.
- Well, same thing again.
- seventy-five goes into five hundred six times.
- six times seventy-five is going to be four hundred and fifty again.
- We're going to keep having that same pattern over
- and over and over again.
- It's actually 66.666-- I hope you don't think I'm an evil
- person because of this number that happened to show up.
- But anyway, so the highest sale price that I can afford or the
- highest marked price I can afford is $sixty-six dollars.
- And if I were to around up, and $0.67 if I were to
- round to the nearest penny.
- If I were to write this kind of as a repeating decimal, I could
- write this as 66.66 repeating.
- Or I also know that 0.6666 going on forever is
- the same thing as two / three.
- So it's sixty-six and two / three.
- But since we're working with money and we're working with
- dollars, we should just round to the nearest penny.
- So the highest marked price that I can afford is $66.67.
- So if I go and I see a nice pair of shoes for
- $fifty-five, I can afford it.
- If I see a nice tie for $seventy, I can't afford it with
- the $fifty in my pocket.
- So hopefully not only will this teach you a little bit of math,
- but it'll help you do a little bit of shopping.
- So let me ask you another problem, a very
- interesting problem.
- Let's say I start with an arbitrary-- let's put
- a fixed number on it.
- Let's say I start with $one hundred.
- And after one year it grows by twenty-five percent.
- And then the next year, let's call that year
- two, it shrinks by twenty-five percent.
- So this could have happened in the stock market.
- The first year I have a good year, my portfolio
- grows by twenty-five percent.
- The second year I have a bad year and my portfolio
- shrinks by twenty-five percent.
- So my question is how much money do I have at the
- end of the two years?
- Well a lot of people might say, oh, this is easy, Sal.
- If I grow by twenty-five percent and then I shrink by twenty-five percent I'll end up with
- the same amount of money.
- But I'll show you it's actually not that simple because the twenty-five percent
- in either case or in both cases is actually a different
- amount of money.
- So let's figure this out.
- If I start with $one hundred and I grow it by twenty-five percent-- twenty-five percent of $one hundred is $twenty-five.
- So I grew it by $twenty-five.
- So I go to $one hundred and twenty-five.
- So after one year of growing by twenty-five percent I end up with $one hundred and twenty-five.
- And now this $one hundred and twenty-five is going to shrink by twenty-five$.
- So if something shrinks by twenty-five percent, that means it's just going to
- be 0.75 or 75% of what it was before, right?
- one minus twenty-five percent.
- 0.75 times $125.
- So let's work that out here.
- $125 times 0.75.
- And just in case you're confused, I don't want to
- repeat it too much, but if something shrinks by twenty-five percent it is
- now seventy-five percent of its original value.
- So if $125 shrinks by 25% it's now 75% of $125 or 0.75.
- Let's do the math.
- five times five is twenty-five.
- two times five is ten plus two is twelve.
- one times five-- seven.
- seven times five is thirty-five.
- seven times two is fourteen.
- Plus three is seventeen.
- seven times one is seven.
- Plus one is eight.
- So it's five, seven, and then this is seven actually.
- 94.75, right?
- Two decimal points.
- So it's interesting.
- If I start with $one hundred and it grows by twenty-five percent, and then it
- shrinks by twenty-five percent I end up with less than I started with.
- And I want you to think about why that happens.
- Because twenty-five percent on $one hundred is the amount that I'm gaining.
- That's a smaller number than the amount that I'm losing.
- I'm losing twenty-five percent on $one hundred and twenty-five.
- That's pretty interesting, don't you think?
- That's actually very interesting when a lot of
- people compare-- well, actually I won't go into stock
- returns and things.
- But I think that should be a pretty interesting thing.
- You should try that out with other examples.
- Another interesting thing is for any percentage gain, you
- should think about how much you would have to lose-- what
- percentage you would have to lose to end
- up where you started.
- That's another interesting project.
- Maybe I'll do that in a future presentation.
- Anyway, I think you're now ready to do some of those
- percent madness problems.
- Hope you have fun.
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