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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.