百分比問題 (英)

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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.