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Scientific notation word problem: U.S. national debt

CCSS.Math:

Video transcript

On February 2nd 2010 the US Treasury estimated the national debt at one point to two seven eight times 10 to the 13th power and just to get a sense of things 10 to the 6th you're talking of 1 times 10 to the 6 is a million 1 times 10 to the 9th is a billion 1 times 10 to the 12th is a trillion so we're talking about on the order of magnitude of 10 trillion dollar so this is about 12 trillion dollars then they tell us that the US Census Bureau estimate for the US population was about three point zero eight six times 10 to the 8th power so this is a little over 300 million people so that's an interesting number right there it's a 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 notations all right which means just as a regular number round to four decimal places while making calculations so we want the per person debt so we wanna take the total debt divided by the number of people so the total debt is 1.2 to 7 8 times 10 to the 13th power and we want to divide that by the total number of people the total number of people which is three point zero eight six times 10 to the 8th and we could separate this into two division problems we could say that this is equal to the division right here these guys 1.2 2 7 8 / 3.06 so 1.2 - 7 8 divided by three point zero eight six and then x times 10 to the 13th divided by 10 to the 8th now what's 10 to the 13th divided by 10 to the 8th let me do it over here 10 to the 13th over 10 to the 8th the way I think about it this is the exact same thing this is equal to 10 to the 13 times 10 to the negative 8 right 1 over this is an 8 right here if you have a 10 to the 8 in the denominator that's like multiplying by 10 to the negative 8 so you have 13 you have the same base 10 so 10 to the 13th times 10 to the negative 8 is going to be 10 to the 13 minus 8 which is 10 to the 5th or another way to think about it use if you have if you have the base in the denominator you subtract the exponent so it's 13 minus 8 10 to the fifth so it's this blue expression times 10 to the 5th 10 to the 13th divided by 10 to the 8th is 10 to the 5th 10 to the 5th power and let's get a calculator out to calculate this right here let's get the calculator out and they say round everything to 4 decimal places so I'll keep that in mind turn my calculator on one point two two seven eight divided by three point zero no three point zero eight six is equal to three let's see how many three nine seven nine because we want to round right there four digits so three nine seven nine let me remember that let me just put it on the side you put on the eye side so I can still look at it so this this little dividing decimals problem results in this is zero zero point three nine seven nine and of course times 10 to the fifth times 10 to the fifth dollars per person now once again you might be tempted to say hey this is in scientific notation I have some number times a power of 10 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 lets 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 0.397 9 times 10 times 10 to the 4th times 10 to the 4th what I did just now is I broke up the 10 to the 5th into a 10 and a 10 to the 4 and 10 to the 4th 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 if you multiply this out so essentially I took a 10 out of the 10 to the 5th I divided it by 10 and I multiplied this other guy by 10 not changing the whole number so then this right here will become 3 point 9 7 9 and then times 10 to the 4th times times 10 to the 4th power 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 to wanted us to express it in decimal notation decimal notation which is just kind of standard you know writing it as a number with our standard numeric decimal system so what is 3 point 9 7 9 times 10 to the 4 so let's think about it we have 3 point 9 7 9 times 10 to the 4th well let me just do it this way times 10 to the 4th so if we multiply it let's just move the decimal space if we multiply it by 10 we're going to get thirty nine point seven nine if we multiply it by 10 squared we're going to three hundred ninety seven point nine if we multiply 10 to the third we're going to get 3979 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 and we get it's 39 so I could write it like this this is equal to 39 thousand seven hundred and ninety dollars so if you think about the national debt per person every man woman and child in the United States essentially owes thirty nine thousand dollars seven hundred or thirty nine thousand seven hundred and ninety dollars