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# 百分比問題 (英): ck12.org 代數 I 課本第三章的題目

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- Let's do a couple of warm up problems converting fractions
- to percentages and then converting
- percentages to fractions.
- And then we can do some actual word problems.
- So on our first warm up, let's convert 5/24 to a percentage.
- And the way I like to do it, I like to convert it to a
- decimal first. And once you have it in the decimal form,
- it's pretty straight forward to convert it to a percentage.
- So the way you convert to a decimal is you divide the
- denominator into the numerator.
- This can, literally, be read as 5 divided by 24.
- So let's do a little bit of division.
- 24 goes into 5.
- And we're going to have to add some extra spaces to the 5
- because, obviously, this is going to be less than 1.
- 24 is a larger number then 5.
- So let's put our decimal right there.
- And we can now do some division.
- 24 goes in to 5 zero times.
- 0 times 24 is 0.
- 5 minus 0 is 5.
- Bring down this 0 right here.
- 24 goes into 50.
- Well it goes into that two times.
- 2 times 24 is-- 2 times 4 is 8.
- 2 times 2 is 4.
- Now we subtract.
- 50 minus 48 is 2.
- Bring down another 0.
- We're going to keep bringing down 0's
- until we have no remainder.
- So bring down another 0.
- 24 goes into 20 zero times.
- 0 times 24 is 0.
- 20 minus 0 is 20.
- Let's bring down another 0.
- Let's bring down another 0.
- 24 goes into 200-- let's see, eight times?
- It'd be nine times?
- It's 180.
- No it'd be eight times, I believe.
- Let's see if eight times works.
- There's always a little bit of an art to this.
- If this is too little, we might have to increase it.
- 8 times 4 is 32.
- 8 times 2 is 16 plus 3 is 19.
- No, that was right.
- Eight times.
- 200 minus 192, that's 8.
- That is 8.
- Bring down another 0.
- 24 goes into 80 three times.
- Can't be four times.
- 3 times 4 is 12.
- Carry the 1.
- 3 times 2 is 6.
- Add 170 and then you have another 80.
- You bring down another 0.
- We have infinite supply of 0's here as we need them.
- And we're going to have another 80.
- Well once again, 24 is going to go into that three times.
- And we're going to start repeating these 3's over and
- over and over again.
- So 5 over 24-- let me write this down.
- 5 over 24-- I want you to understand
- why it keeps repeating.
- Every time we do this now, we're going to get a 3.
- And come down, get a 72.
- 80 from 72, we're going to get another 80, and we're just
- going to have a big string of 3's there.
- So 5 divided by 24 is 0.2083 repeating.
- Now, if we want to write this as a percentage, the word
- percentage-- the word percent.
- Let me split it up.
- Per cent.
- Cent comes from the word for 100.
- This is per 100.
- So what is this?
- This is 0.20.
- What is this per 100, where 100 is a whole?
- And that might confuse you or it might not.
- But one way you can do this is this is the same thing as--
- you could do it this way.
- This is the same thing.
- This decimal 0.2083 repeating is the same thing as 20.83
- repeating over 100.
- This is how many we have per 100%.
- Or you could say that this is equal to 20.83%: per 100.
- These are equivalent.
- A very quick way to think about how to go from decimals
- to percent is you can multiply by 100, and then put the
- percentage sign.
- Or, if you go backwards, you divide by 100 and get rid of
- the percentage sign.
- That was a good warm up.
- Let's do one more.
- Let's convert a percentage into a fraction.
- I'll do it in blue.
- So let's say we have 16%.
- Remember this means 16 per 100.
- So this is the same thing as 16 per 100.
- That's what per cent-- century is 100 years.
- There are 100 cents in a dollar.
- So 16 over 100.
- Put that in the lowest-- let's see, you can divide the
- numerator and the denominator by 4.
- We get 4 over 25.
- So that's our warm up.
- Let's do some actual problems now.
- All right.
- They tell us a TV is advertised on sale.
- It is 35% off, and has a new price of $195.
- What was the pre-sale price?
- So if x is the pre-sale price, when you take 35% off of that,
- it has a new price of $195.
- So x minus 35% of x-- 0.35 is the same thing as 35%.
- So if I take x and I subtract 35% of x from x, I'm
- going to get $195.
- That's what that is telling me.
- So now we just solve for x.
- So you can view this as 1x minus 0.35x.
- That will be 0.65x.
- So this is 0.65x, right?
- If you add 0.65 and 0.35, you get 1.
- 1 minus 0.35 is 0.65x is equal to 195.
- And now you can divide both sides of this equation.
- Actually, I like to do it more as a fraction, so let me write
- it that way.
- So if you have 65 over 100x is equal to 195, now we can
- multiply both sides of this equation by the inverse.
- Actually, even before I do that, let me
- simplify this fraction.
- I can divide the numerator and denominator by 5.
- This becomes 13 over 20.
- So we get 13 over 20x is equal to 195.
- And now we can multiply both sides by the inverse of this.
- So 20 over 13 times that is equal to 195 times 20 over 13.
- These cancel out, and we get x is equal to 195 times 20 over
- 13, which we could figure out, but I'll take the calculator
- out for this one.
- So let's see.
- You get 195 times 20 divided by 13 is equal to $300.
- I should have done that-- so this is equal to-- that right
- there is equal to $300.
- So $195 is actually divisible by 13.
- I should be able to do that without a calculator.
- Anyway, so the original price was $300.
- You take 35% of $300, which is $105, and you will
- be left with $195.
- So that was our original price.
- Let's do another one.
- An employee at a store is currently paid $9.50 per hour.
- If she works a full year, she gets a 12% pay raise.
- What will be her new hourly rate after the raise?
- So today, she makes 9.50.
- If she works for 12 years, she'll make 9.50 plus 12%.
- Let me write this way; plus 12% times 9.50.
- That's how much her raise will be.
- So her new hourly rate will be this entire thing.
- Well, we could view it this way.
- This is the same thing as 9.50 plus 0.12 times 9.50.
- This is the same thing as 1 times 9.50
- times 0.12 times 9.50.
- So this is the same thing as 1.12.
- I'm just adding that.
- Let me do this in a different color.
- I'm just adding that and that to get that, times 9.50.
- We are growing 9.50 by 12%.
- You already have the 9.50 plus another 12%.
- It's 1.12 times 9.50 is going to be equal to our new number.
- Let me multiply this just to make up for me using the
- calculator the last time.
- So 9.50 times 1.12.
- 2 times 0 is 0.
- 2 times 5 is 10.
- 2 times 9 is 18 plus 1 is 19.
- And I'll put a 0 here.
- 1 times 0 is 0.
- 1 times 5 is 5.
- 1 times 9 is 9.
- Let's put two 0's here.
- 1 times 0 is 0.
- 1 times 5 is 5.
- 1 times 9 is 9.
- And now we can add.
- The 0's add up to 0.
- 9 plus 5 is 14.
- 2 plus 9 is 11 plus 5 is 16.
- 1 plus 9 is 10.
- And then we count the number of decimals.
- We have one, two, three, four numbers behind the decimals.
- So in our answer, we have to have one, two, three, four
- numbers behind the decimal.
- So her new salary will be $10.64 per hour.
- $10.64.
- You don't have to write those trailing 0's right there.
- So this is her new hourly rate after the raise.
- All right.
- Let's do one more.
- Store A and store B both sell bikes.
- And both buy bikes from the same
- supplier at the same price.
- Store A has a 40% mark-up.
- That means whatever they buy the bike for, they sell it for
- 40% above that.
- While store B has a 250% markup.
- Store B has a permanent sale, and always will sell at 60%
- off those prices.
- Which store has a better deal?
- So let's say that they're both buying the bikes, so x is
- equal to the price from supplier.
- So that's the price that both bike stores buy
- their bicycles at.
- They both buy bikes from the supplier at the same price.
- That's this x that I'm going to start off with.
- Now lets do the scenario of store A.
- What does store A sell their bike for?
- They sell the bike for x plus 40% of x, which is equal to
- 1.4 times x.
- That's how much store A sells their bike for.
- Now store B is a little bit more interesting.
- Store B-- so they have a 250% markup.
- Well, they claim to sell their bike for x plus 250%.
- That's the same thing as 2.5 times x.
- A 100% markup would literally mean another x.
- This is 250%.
- And as we said, you can just divide by 100.
- You get 2.5.
- Get rid of the percentage sign.
- So they sell it x plus 2.5x, which is equal to 3.5x.
- That's kind of their ticket price.
- But then, they sell at 60% off those prices.
- So then they take 60% off of this price right here.
- So the real selling price is going to be 3.5 times the
- price from the supplier minus 0.6.
- Minus 0.6.
- Minus 60% of this price.
- Of 3.5x.
- And we could view this as 1 times 3.5x
- minus 0.6 times 3.5X.
- And that's the same thing as 1 minus 0.6 times 3.5x.
- This is going to be 0.4.
- This is 0.4.
- So if you're taking 60% off, it's the equivalent of selling
- it at 40% of the ticket price.
- That's what we're doing right there.
- You might, eventually, do this in your head-- immediately
- say, oh, 60% off is the same thing as selling at 40%, or
- selling at 0.4 of the price.
- 3.5x.
- And let's multiply that out.
- So if we take 3.5x times 0.4, 4 times 5 is 20.
- 4 times 3 is 12 plus 2 is 14.
- And we have one, two numbers behind the decimal spot.
- Two numbers.
- One, two.
- 1.4.
- So they're going to sell it at 1.4x.
- So in either store, you actually going to
- pay the same price.
- Let's say they buy the bikes from the supplier at $100.
- Then at store A, you're going to pay $140 for that bike, and
- in store B, you're going to spend $140 for that bike.
- But in store B, you think you're getting 60% off.

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