Solving problems using powers of 10 in context
The narrator reads the total as 9 x 10^9 packages per year at the start, then works with 8 x 10^9 (which gives the stated answer 1 x 10^4). Interventions use 8 x 10^9.
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The video solves a rate problem using numbers written as products with powers of 10. It divides total packages by the number of workers and then by hours per worker, using the rule that dividing powers of 10 means subtracting exponents, to get about 1 x 10^4 packages per worker per hour.
You'll be able to
- rewrite a number like 400 as a single digit times a power of 10.
- divide numbers written with powers of 10 by dividing the front numbers and subtracting the exponents.
- use division with powers of 10 to find a rate such as packages per worker per hour.
Key words
- power of 10
- A number like 10^3 that means 10 multiplied by itself a certain number of times, here 10 x 10 x 10 = 1,000.
- exponent
- The small raised number that tells you how many times the base is multiplied by itself.
- per
- For each one, so 'packages per hour' means how many packages in each hour.
Transcript
- [Instructor] A warehouse ships about 9 times 10 to the 9th packages per year. There are 400 workers who prepare the packages. Each worker works about 2 times 10 to the 3rd hours per year. About how many packages does each worker prepare per hour? So pause this video, see if you can have a go at that before we can do it together.
All right, now let's do this together. So the total per year for all of the workers is 8 times 10 to the 9th packages per year in total. And if we divide that by the number of workers, if we divide that by 400, this would give us how many packages per year per worker.
So I'm taking all of the workers, what they do in a year, divide by the number of workers. This will tell us each worker per year. So this is going to be the same thing as 8 times 10 to the 9th divided by... Actually, I'm going to write this as a product of a power of 10 as well.
This is 4 times 100, which is the same thing as 4 times 10 to the 2nd power. And what's useful about that is we can see that if you divide both 8 and 4 by 4 you get a 2, here and you get a 1 here. And then if you divide both 10 to the 9th and 10 to the 2nd by 10 to the 2nd you would get a 1 here.
And then this would be 10 to the 7th. You might remember your exponent properties where if you take 10 to the 9th divided by 10 to the 2nd you would subtract the exponents. You could also imagine if you had 10 times 10... Actually I don't, if you had nine 10s up here...
Ah, let me just do it. 10 times 10, times 10, times 10, this is a lot of them. Times 10. How many is that? 6? Times 10, times 10, times 10. One, two, three, four, five, six, seven, eight, nine. And if you divide that by 10 times 10, you could say these two cancel out with these two.
And then you have, or any of these two, and then you would have seven 10s left. So that's 10 to the 7th. So what we have left is the dominator, we just have 1. So we're just left with 2 times 10 to the 7th. Now we're not done. This is packages per year per worker, this number right over there.
So if we want to know packages per year, or if we want to know packages per hour per worker, we would divide this by the number of hours. So we would get 2 times 10 to the 7th divided by the number of hours in a year. So that's two times 10 to the 3rd. And this is equal to, the 2s cancel out.
And 10 to the 7th divided by 10 to the 3rd, we can subtract the exponents just like we talked about there, which is going to be 10 to the 4th. So this is 10 to the 4th, or 1 times 10 to the 4th which is choice C, right over there.
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