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## 8th grade (Illustrative Mathematics)

### Unit 7: Lesson 7

Lesson 14: Multiplying, dividing, and estimating with scientific notation

# Scientific notation word problem: U.S. national debt

Ever wonder what your part of the national debt is? It might surprise you. What isn't surprising is that you can use scientific notation and division to figure out the answer. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• At around , what would you do if you had to divide 10 to the power of 6 by 10 to the negative first?
(24 votes)
• Since you subtract exponents when dividing with exponents, you would do 10^6/10^-1=
10^6-(-1), which is equivalent to 10^6+1, which equals 10^7.
(4 votes)
• I often wonder about when is the correct time to round a number during a calculation. At you rounded to the fourth decimal place. Should you wait to round until you have moved the decimal for the last time? Making it 3.9786*10 to the 4th?
I have gotten the answer wrong by not rounding during the problem, but it is less accurate too.
Am I correct in rounding after, or should it occur during the problem calculation?
(13 votes)
• You should estimate (round) an answer only at the very end, not before that.

Although, for most of the calculations, rounding up to the 6th or 7th place is enough to still get a precise answer at the end. I'd just like to make it clear that this is NOT a fact, this is only my personal opinion based on thousands of exercises I've done.
(14 votes)
• In he changes 10^8 into 10^-8. How does this work. Wont that change the outcome? Please explain.
(11 votes)
• So, look carefully between and - in first case you have a fraction (10^13)/(10^8), where 10^8 is under the fraction; afterwards, Sal transformed the division into multiplication, where the number under the fraction is with a negative exponent, tha's how 10^13 divided by 10^8 becomes 10^13 multiplied by 10^-8;
(4 votes)
• I don't understand, where did you get the number 9 when the number said .3978 (in the calculator)
(6 votes)
• When he divided 1.2278 by 3.086, The calculator gave him: .397861 The question said to round to four decimal places. The number in the fourth place is 8, so when you want to round it, look at the number to the right of it ( which is the lesser number) That number is 6. If the number is 4 or below, keep the number the same, if it's 5 or above, round the number up. Since 6 is above 5, you would round the 8 to a 9.
(13 votes)
• why did the eight turn into a negative 8 in ?
(4 votes)
• Because 1/10⁸ is the same as 10⁻⁸.

When you move a number with an exponent from the numerator to the denominator (or from the denominator to the numerator), you must change the sign of the exponent.
(3 votes)
• I don't understand why it's 10^4? He moved it one to the right. Wouldn't it go from 10 ^5 to 10^6? I"m confused, 0.3979, you move it 1 to the right (multiply by 1) to make it 3.979. So wouldn't you add 10^5 + 10^1=10^6. But the exponent went from 10^5 to 10^4. Doesn't that only happen when you divide by 1? The decimal point was moved 1 to the right, not to the left...?
(4 votes)
• When you move the decimal point one to the right, you multiply the decimal by 10. So, you have to adjust the exponent to compensate for that. Hence, you divide 10^5 by 10 and get 10^4.

Essentially you multiply the number by 10 and divide it by 10. So you didn't change the value, only the expression.
(3 votes)
• What is the u.s census bureau?
(3 votes)
• "The United States Census Bureau (or USCB) is a principal agency of the U.S. Federal Statistical System responsible for producing data about the American people and economy." - Wikipedia
(4 votes)
• Around Sal said 10^13/10^8 = 10^5 , What if it was 10^-13/10^8 or 10^13/10^-8.. Do I subtract regularly like 13-(-8) or -13-8?
(4 votes)
• 10^-13/10^8 = 10^-13 * 10^-8 = 10^-21
10^13/10^-8 = 10^13 * 10^8 = 10^21
(2 votes)
• At , does everyone really owe that much or do some people owe more than others?
(3 votes)
• Our national debt is not directly allocated to individual people. You will sometimes seem problems like the one in the video that show the average amount every person would owe to pay of the debt. This gives you a sense of how large the debt is.

In reality, our federal income tax is used to pay for federal expenses. Individuals and companies pay income tax. And, the rates are based on income levels. So, the average cost per person does not directly align to how taxes are paid.
(2 votes)
• If you were to solve:

4.82/5.16 times 10 to the first/ 10 to the sixth

what would happen to 4.82/5.16?
(2 votes)
• Hi, evilcupcake6663

First off, 4.82 divided by 5.16, followed by multiplication by 10^1 divided by 10^6, yields 0.934 multiplied by 10^-5 - but the number 0.934 is NOT greater than or equal to 1 AND less than 10. Thus, we have to multiply 0.934 by 10; to make it so that it results in a number that is greater than or equal to 1 and less than 10 - or, in other words, shift the decimal one place to the right - in order to express it in proper scientific notation, which equals 9.34; but because we are multiplying it by 10, we have to take off a power from our 10^-5 yield from before - in other words, 9.34 multiplied by 10^1, followed by multiplication by 10^-5; but remember, when you multiply powers, you are effectively subtracting them away. So, in our case, 10^1 multiplied by 10^-5, yields 10^-4.

Final answer: 9.34 multiplied by 10^-4.

I hope this helps. Feel free to ask further questions if you seek clarification.

Happy math-ing!
(3 votes)

## Video transcript

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.