Scientific notation word problem: U.S. national debt

On February 2, 2010 the U.S. Treasury estimated the national debt at 1.2278 times 10 to the 13th power. And just to get a sense of things, 1 times 10 to the sixth is a million, 1 times 10 to the ninth is a billion, 1 times 10 to the 12th is a trillion. So we're talking on the order of magnitude of 10 trillion dollars. So this is about 12 trillion dollars. Then they tell us that the U.S. Census Bureau's estimate for the U.S. population was about 3.086 times 10 to the eighth power. So this is a little over 300 million people. So that's an interesting number right there, it's the population. And then they say, using these estimates calculate the per-person share of the national debt. So essentially, we want to take the entire debt and divide by the number of people. That'll give us the per-person share of the national debt. Use scientific notation to make your calculations and express your answer in both scientific and decimal notation. Which means just as a regular number. Round to four decimal places while making calculations. So we want the per-person debt. So we want to take the total debt and divide by the number of people. So the total debt is 1.2278 times 10 to the 13th. And we want to divide that by the total number of people, which is 3.086 times 10 to the eighth. And we could separate this into two division problems. We could say that this is equal to the division right here, 1.2278 divided by 3.086. And then times 10 to the 13th divided by 10 to the eighth. Now, what's 10 to the 13th divided by 10 to the eighth? Let me do it over here. The way I think about it, this is the exact same thing as 10 to the 13th times 10 to the negative eight. This is an eight right here. If you have a 10 to the eighth in the denominator, that's like multiplying by 10 to the negative eight. So you have 13, you the same base 10, so 10 to the 13th times 10 to the negative eight is going to be 10 to the 13 minus 8. Which is 10 to the fifth. Or another way to think about it: If you have the base in the denominator, you subtract the exponents. So it's 13 minus 8. 10 to the fifth. So it's this blue expression times 10 to the fifth. And let's get a calculator out to calculate this right here. And they say round everything to four decimal places, so I'll keep that in mind. Let me turn my calculator on. 1.2278 divided by 3.086 is equal to 0.3979. Because we want to round right there. Let me remember that. Or let me just put it on the side so I can still look at it. So this this little dividing decimals problem results in 0.3979. And of course, times 10 to the fifth dollars per person. Once again, you might be tempted to say, hey this is in scientific notation. I have some number times a power of ten. But notice, this number is not greater than or equal to 1. Remember, this number, if you want to be formal about scientific notation, has to be greater than or equal to 1, or less than 10. So what we can do here is we can multiply. If we don't want to change the number, we can multiply this number by 10 and divide this number by 10. Or another way you can think about it is, this whole thing can be rewritten as 0.3979 times 10 times 10 to the fourth. What I did was just now was I broke up the 10 of the fifth into a 10 and a 10 to the fourth. And I did that because I want to multiply this by 10 so I can get a 3 out front instead of a 0.3. So let's do that. So essentially, I took a 10 out of the 10 to the fifth. I divided it by 10, I multiplied this other guy by 10, not changing the whole number. So then this right here will become 3.979 and then times 10 to the fourth. So that's how much debt there is per-person in scientific notation. So this is debt per person in scientific notation. Now, in the problem they also wanted us to express it in decimal notation. Which is just kind of standard writing it as a number with our standard numeric decimal system. So what is 3.979 times 10 to the fourth? Let's think about it. We have 3.979 times ten to the fourth. Well let me just do it this way. Let's just move the decimal space. If we multiply it by 10, we're going to get 39.79. If we multiply it by 10 squared, we're going to get 397.9. If we multiply it by 10 to the third, we're going to get 3,979. If we multiply it by 10 to the fourth, we're going to get one more zero right there. So we're essentially going to move the decimal four to the right. So I could write it like this. This is equal to $39,790. So if you think about the national debt per person. Every man, woman, and child in the United States essentially owes $39,790.