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# Percent word problem: 78 is 15% of what number?

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78 is 15% of what number? So there's some unknown number
out there, and if we take 15% of that number, we
will get 78. So let's just call that
unknown number x. And we know that if we take
15% of x, so multiply x by 15%, we will get 78. And now we just literally
have to solve for x. Now, 15% mathematically, you
can deal directly with percentages, but it's
much easier if it's written as a decimal. And we know that 15% is the
same thing as 15 per 100. That's literally per cent. Cent means 100, which is
the same thing as 0.15. This is literally
15 hundredths. So we could rewrite this as
0.15 times some unknown number, times x,
is equal to 78. And now we can divide both sides
of this equation by 0.15 to solve for x. So you divide the left side
by 0.15, and I'm literally picking 0.15 to divide both
sides because that's what I have out here in
front of the x. So if I'm multiplying something
by 0.15 and then I divided by 0.15, I'll just
be left with an x here. That's the whole motivation. If I do it to the left-hand
side, I have to do it to the right-hand side. These cancel out, and
I get x is equal to 78 divided by 0.15. Now, we have to figure
out what that is. If we had a calculator, pretty
straightforward, but let's actually work it out. So we have 78 divided by,
and it's going to be some decimal number. It's going to be
larger than 78. But let's figure out what it
ends up being, so let's throw some zeroes out there. It's not going to be
a whole number. And we're dividing it by 0.15. Now, to simplify things, let's
multiply both this numerator and this denominator by 100,
and that's so that 0.15 becomes 15. So 0.15 times 100 is 15. We're just moving the decimal
to the right. Let me put that in
a new color. Right there, that's where
our decimals goes. Let me erase the other one,
so we don't get confused. If we did that for the 15,
we also have to do that for the 78. So if you move the decimal two
to the right, one, two, it becomes 7,800. So one way to think about it, 78
divided by 0.15 is the same thing as 7,800 divided by 15,
multiplying the numerator and the denominator by 100. So let's figure out
what this is. 15 does not go into 7, So you
could do it zero times and you can do all that, or you can just
say, OK, that's not going to give us anything. So then how many times
does 15 go into 78? So let's think about it. 15 goes into 60 four times. 15 times 5 is 75. That looks about right,
so we say five times. 5 times 15. 5 times 5 is 25. Put the 2 up there. 5 times 1 is 5, plus 2 is 7. 75, you subtract. 78 minus 75 five is 3. Bring down a zero. 15 goes into 30 exactly
two times. 2 times 15 is 30. Subtract. No remainder. Bring down the next zero. We're still to the left
of the decimal point. The decimal point is
right over here. If we write it up here, which
we should, it's right over there, so we have one
more place to go. So we bring down
this next zero. 15 goes into 0 zero times. 0 times 15 is 0. Subtract. No remainder. So 78 divided by 0.15
is exactly 520. So x is equal to 520. So 78 is 15% of 520. And if we want to use some of
the terminology that you might see in a math class, the 15%
is obviously the percent. 520, or what number before we
figured out it was 520, that's what we're taking the
percentage of. This is sometimes referred
to as the base. And then when you take some
percentage of the base, you get what's sometimes referred
to as the amount. So in this circumstance,
78 would be the amount. You could view it as the amount
is the percentage of the base, but we were able
to figure that out. It's nice to know those, if
that's the terminology you use in your class. But the important thing
is to be able just answer this question. And it makes sense, because
15% is a very small percentage. If 78 is a small percentage of
some number, that means that number has to be pretty big, and
our answer gels with that. This looks about right. 78 is exactly 15% of 520.