Back To CourseCollege Algebra: Help and Review
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Amy has a master's degree in secondary education and has taught math at a public charter high school.
Division is the equal sharing of something into groups. So, for smaller numbers, we can apply this definition to figure out our answers. For example, if we wanted to divide 8 by 4, we could picture having 8 candy bars and then sharing them equally among a group of 4 friends. We could start by handing each friend one candy. We would keep this going as long as we can hand out an equal number of candy bars to each person.
After doing this, we see that 8 divided by 4 is 2 because each person would end up with 2 candy bars. But, what if we wanted to divide larger numbers such as 96 by 3? What would we do then? It wouldn't be particularly useful for us to picture 96 candy bars being distributed to 3 people. That would take quite a bit of time and a lot of candy bars! What we can do is follow a procedure called long division, a special procedure to divide larger numbers.
How does this procedure work? Let me give you a quick overview before we go into some examples. To put it simply, what we do is take our number that we want to divide and break it up into little pieces, and we divide these little pieces by our divisor, the number we are dividing by. This makes our job of dividing larger numbers easier. Because we can easily divide smaller numbers, this process allows us to break down our larger division into manageable pieces. So, now let's see some examples of how it is done.
Why don't we try to divide 96 by 3 right now? To set up our problem for long division, we will first write our divisor, the number we are dividing by. Yes, our numbers are switched for long division - our 3 comes first. Then we have our long division symbol covering our dividend, the number that we want to divide. We end up writing our answer on top of our long division symbol.
So, now that we have our problem all set up, what we do now is to try to divide the first digit of our dividend by our divisor. The first digit we have is 9. So, we will try to divide 9 by 3. We want to see how many 3's we can take out of our 9. These are small numbers and we can easily do this. 9 divided by 3 is 3. So, we write this part of our answer on top of our 9.
Now, what we want to do is we multiply the number we just wrote down on top by our divisor, our 3. What is 3 * 3? It is 9. We write this 9 underneath our 9. We draw a line underneath to show that we are going to be performing an operation here. What we do now is we subtract this 9 from the number directly on top of it. So, we write a minus sign in front of our second 9. 9 - 9 is 0. Yes, we are ignoring all the digits to our right; we are only concerned about the digits we are currently on and the digits to our left. We write this 0 underneath the line we just drew.
Since we have more digits to work with in our dividend, we check to see if we can divide our result here by our divisor. Does 3 go into 0? No, it doesn't. So, that means we need to pull down our next digit in our dividend to work with. Our next digit is 6. So, we pull that down and write it next to our 0. Now we ask ourselves the same question. Does 3 go into 6? Yes, it does! How many times? 3 goes into 6 two times. So, 6 divided by 3 is 2. We write this 2 on the very top next to our 3.
Next, we multiply our 2 by our divisor. We multiply 3 by 2. We get 6; write this number underneath our 6. We draw a line underneath this and put a minus sign out front. We are now repeating the steps we took with our first digit. We subtract: 6 - 6 is 0. Are there more numbers to pull down? No. Is my last line a 0? Yes. That means we are done with dividing. Our answer is 32.
Notice how we treat each digit like we did the last digit. Once we are done with one digit, we move ourselves over to the right one place and we repeat what we did before. This is the process of long division. Yes, there is quite a bit of repetition. But it is an easy to follow procedure that works with all numbers. To help us divide our smaller numbers, we can always go back to thinking about sharing candy bars or whatever else you want to think about.
Let's look at another number. How about dividing 123 by 12? We set up our problem by writing the 12 first, followed by our long division symbol covering 123. Good. Next, we look at our first digit of our dividend, the 1. Can I divide 1 by 12? No. So, I can write 0 on top of my 1 in the answer part. That means I need to look at the next digit of the dividend, the 2.
So, I'm now looking at 12. Can I divide 12 by 12? Yes, I can. It is 1. So, I write this 1 on my answer line next to the 0. I now multiply this 1 with my divisor, my 12. 1 multiplied by 12 is 12. I write this 12 underneath the 12 I already have. I draw a line underneath and add a minus sign. What is 12 - 12? It's 0.
Okay, so far, so good. Does 12 go into 0? No. So, that means I need to pull down my next digit. So, I pull down my 3. Now the number I am looking at is 3. Can I divide 3 by 12? No, so I write a 0 on my answer line next to my 1. Now, because my last line is not 0 even though I've reached the end of my dividend, I need to pull down another digit. But wait, I have no more digits to pull down. What do I do?
At this point, we realize that all numbers have a decimal point here followed by an infinite number of zeroes. So, I put down the decimal point here after the 3. I also write this decimal point in my answer line directly above. Now I can pull down a 0. The number that I am now looking at on my very last line of my current problem is 30. Does 12 go into 30? Yes. How many times? It can go into 30 two times because I can pull out two 12s from my 30.
There isn't enough to take out another 12. So, I write 2 on my answer line after the decimal point. I multiply this 2 by my 12. I get 24. I write this 24 on a new line at the way bottom lining up the numbers with the 30. I draw a line underneath plus a minus sign. 30 - 24 is 6. Does 12 go into 6? No. So, that means I need to pull down another digit.
So, I pull down another 0. That makes my number 60. Does 12 go into 60? Yes. How many times? 5! I write 5 on my answer line. What is 5 multiplied by 12? It's 60. I write this on a new line at the way bottom lining it up with the 60. I draw a line underneath and add a minus sign. What is 60 - 60? It's 0. Since we don't have any more numbers to pull and my answer is a 0, we are done dividing. So, my answer is 10.25.
Let's review what we've learned. We reviewed that division is about equally sharing something with a group. Long division is the procedure for dividing larger numbers. The process of long division requires you to take the dividend, the number you want to divide, and divide it one digit at a time with the divisor, the number you are dividing by.
Long division is a repetitive procedure, but it's easy once you have the process down. The process involves looking at the first digit of our dividend and checking to see if our divisor can be divided into it. If so, we write how many times our divisor goes into that digit on our answer line. We then multiply this number we just wrote down with our divisor. We subtract to find our remainder. We pull down our next digit and repeat the process.
Once we have no more digits to pull down and our answer is 0, we are done. If we run out of digits to pull down, we remember that all numbers have a decimal point followed by an unending stream of zeroes. We can pull down a zero whenever we need more digits to pull. We just make sure we note the position of the decimal point.
Once you have completed this lesson, you will have learned how to use long division to solve a division problem.
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Back To CourseCollege Algebra: Help and Review
27 chapters | 228 lessons