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# Solving percent problems

We'll use algebra to solve this percent problem. Created by Sal Khan and Monterey Institute for Technology and Education.

Video transcript

We're asked to identify the
percent, amount, and base in this problem. And they ask us, 150 is
25% of what number? They don't ask us to solve it,
but it's too tempting. So what I want to do is first
answer this question that they're not even asking
us to solve. But first, I want to answer
this question. And then we can think about what
the percent, the amount, and the base is, because
those are just words. Those are just definitions. The important thing is
to be able to solve a problem like this. So they're saying 150 is
25% of what number? Or another way to view this,
150 is 25% of some number. So let's let x, x is equal
to the number that 150 is 25% of, right? That's what we need
to figure out. 150 is 25% of what number? That number right here
we're seeing is x. So that tells us that if we
start with x, and if we were to take 25% of x, you could
imagine, that's the same thing as multiplying it by 25%, which
is the same thing as multiplying it, if you
view it as a decimal, times 0.25 times x. These two statements
are identical. So if you start with that
number, you take 25% of it, or you multiply it by 0.25, that
is going to be equal to 150. 150 is 25% of this number. And then you can solve for x. So let's just start with
this one over here. Let me just write it separately,
so you understand what I'm doing. 0.25 times some number
is equal to 150. Now there's two ways
we can do this. We can divide both sides of this
equation by 0.25, or if you recognize that four quarters
make a dollar, you could say, let's multiply both
sides of this equation by 4. You could do either one. I'll do the first, because
that's how we normally do algebra problems like this. So let's just multiply
both by 0.25. That will just be an x. And then the right-hand side
will be 150 divided by 0.25. And the reason why I wanted to
is really it's just good practice dividing
by a decimal. So let's do that. So we want to figure out what
150 divided by 0.25 is. And we've done this before. When you divide by a decimal,
what you can do is you can make the number that you're
dividing into the other number, you can turn this into
a whole number by essentially shifting the decimal
two to the right. But if you do that for the
number in the denominator, you also have to do that
to the numerator. So right now you can view
this as 150.00. If you multiply 0.25 times
100, you're shifting the decimal two to the right. Then you'd also have to do
that with 150, so then it becomes 15,000. Shift it two to the right. So our decimal place
becomes like this. So 150 divided by 0.25
is the same thing as 15,000 divided by 25. And let's just work it
out really fast. So 25 doesn't go into 1, doesn't
go into 15, it goes into 150, what is that? Six times, right? If it goes into 100 four
times, then it goes into 150 six times. 6 times 0.25 is-- or actually,
this is now a 25. We've shifted the decimal. This decimal is sitting
right over there. So 6 times 25 is 150. You subtract. You get no remainder. Bring down this 0 right here. 25 goes into 0 zero times. 0 times 25 is 0. Subtract. No remainder. Bring down this last 0. 25 goes into 0 zero times. 0 times 25 is 0. Subtract. No remainder. So 150 divided by 0.25
is equal to 600. And you might have been able
to do that in your head, because when we were at this
point in our equation, 0.25x is equal to 150, you could
have just multiplied both sides of this equation
times 4. 4 times 0.25 is the same
thing as 4 times 1/4, which is a whole. And 4 times 150 is 600. So you would have gotten
it either way. And this makes total sense. If 150 is 25% of some number,
that means 150 should be 1/4 of that number. It should be a lot smaller than
that number, and it is. 150 is 1/4 of 600. Now let's answer their
actual question. Identify the percent. Well, that looks like 25%,
that's the percent. The amount and the base
in this problem. And based on how they're wording
it, I assume amount means when you take the 25% of
the base, so they're saying that the amount-- as my best
sense of it-- is that the amount is equal to the percent
times the base. Let me do the base in green. So the base is the number you're
taking the percent of. The amount is the quantity
that that percentage represents. So here we already saw
the percent is 25%. That's the percent. The number that we're taking
25% of, or the base, is x. The value of it is 600. We figured it out. And the amount is 150. This right here is the amount. The amount is 150. 150 is 25% of the
base, of 600. The important thing is how
you solve this problem. The words themselves, you know,
those are all really just definitions.