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Introduction When dividing, we always have two numbers to consider - the divisor and the dividend. Divisiors are the number we are dividing by. A dividend is the number being divided. Long division can be used to solve problems involving all kinds of divisors and dividends. There are certain rules we must follow when we divide decimals by whole numbers and vice versa.  In this lesson, we will look at various situations where we divide with decimals. Dividing Whole Numbers by Decimals When we divide anything by a decimal, our first step is always going to be to shift the decimal point of our divisor until it becomes a whole number. Dividing Whole Numbers by Decimals     Let's take a look at dividing a whole number by a decimal. In this case, we are dividing 4 by 0.25.     We need to shift the decimal point of both the numbers involved in the division. Since 4 is a whole number, as we shift the decimal point to the right, we will add zeroes at the end of our number.     Shift the decimal points as many times as it takes until we are dividing by a whole number.     Our expression can now be solved as a regular long division problem. 25 goes into 40 one time. Be sure to record our answer above the second unit place since we are dividing a two digit number. Subtract 1 times 25 from 40.     Bring down the next digit of our dividend. How many times does 25 go into 150? 25 will go into 150 six times.     Record the number 6 with the answer above.  Subtract 6 times 25 from 150. Since our result is zero, we are done with this problem. This tells us that 4 divided by 0.25 is equal to 16.     To recap, when we divide a whole number by a decimal, we must shift the decimal points of both numbers until we are dividing by a whole number. After that, we solve our expression like a regular long division problem. Dividing Decimals by Whole Numbers When we divide a decimal by a whole number, things are a little bit different. There are a couple of situations to watch out for when performing this type of division. Occasionally, you may have to annex a zero or your answer might have a recurring decimal. First, let's take a look at solving a regular decimal-whole number division problem. Dividing Decimals by Whole Numbers     We are dividing 36.32 by 8.     First, record the decimal point directly above its location in the dividend.     With the decimal point recorded, proceed as you would for a normal long division problem. 8 does not go into 3 so we will move on to the next digit. 8 goes into 36 four times. Record the 4 above the second digit, then subtract 8 times 4 from 36.     Bring down the next digit. How many times does 8 go into 43?  8 will go into 43 five times. Record the 5 above the third digit, then subtract 5 times 8 from 43.     Bring down the next digit. How many times does 8 go into 32?  8 will go into 32 four times. Record the 4 above the last digit, then subtract 4 times 8 from 32. Since we are left with a zero, we are done solving this problem. 36.32 divided by 8 is equal to 4.54.     Occasionally, we will need to add zeroes to the end of our dividend to continue dividing. This process of adding zeroes is called annexing. Sometimes, our numbers have a repeating decimal, meaning the digits repeat forever. We call these numbers recurring decimals. Dividing Decimals by Whole Numbers: Special Cases     Let's take a look at dividing a whole number by a decimal. In this case, we are dividing 1.1 by 9.     First, record the decimal point directly above its location in the dividend.     With the decimal point recorded, proceed as you would a normal long division problem. 9 does not go into 1 so we will move on to the next digit. 9 goes into 11 one time. Record the 1 above the second digit, then subtract 1 times 9 from 11.     Since we are out of digits, we will have to annex a zero. Add a zero to the end of the dividend and bring it down. This process of adding a zero to the end of our dividend is called annexing. You'll notice we added a zero in front of our answer in red as well. Since we passed up the first units place and recorded our answer after the decimal point we will have a zero before the decimal point.     How many times does 9 go into 20? 9 will go into 20 two times. Record the two above the zero we annexed. Subtract 9 times 2 from 20.     We will have to annex another zero. You'll notice that we are stuck in a repeating cycle of adding a zero and then subtracting 18 from 20. We will never be really done with this problem since our decimal will keep repeating.     Our last two digits will keep repeating forever. At this point, we can record our answer as 0.122 with a bar on top of the repeating digits. We call this a recurring decimal.     In this example, we saw two special case scenarios at work. The first was that we had to add zeroes to the end of our dividend to keep dividing it. The second was the situation when our decimals keep repeating. These techniques may be used together or separately to solve many of these types of division problems. Dividing Decimals by Decimals Finally, we have the situation when we are dividing a decimal by another decimal. When performing this type of division we always start by shifting the decimal points of our numbers until we are dividing by a whole number. Furthermore, you may be required to annex zeroes or your answer might have a recurring decimal. Let's take a look at dividing a decimal by a decimal.   Dividing Decimals by Decimals     We are dividing 6.46 by 0.2.     Shift the decimal point of both the numbers involved in the division until our divisor is a whole number.     Next, record the decimal point directly above where it is in the dividend.     Proceed with solving as you would a regular long division problem. 2 goes into 6 three times. Record the 3 above the first digit. Subtract 2 times 3 from 6.     Since our 6's cancelled each other out, we can ignore the zero that resulted and bring down our next digit. How many times does 2 go into 4? 2 will go into 4 twice. Record the 2 above the second digit and subtract two times two from four.     Since our 4s cancelled each other out, we can ignore the zero that resulted and bring down our next digit. How many times does 2 go into 6? 2 will go into 6 three times. Record the 3 above the second digit and subtract 2 times 3 from 6. Since we are left with a zero, we are finished with the problem. This tells us that 6.46 divided by 0.2 is equal to 32.3.     We have learned how to deal with all types of division problems involving decimals and whole numbers. To recap, always ensure that our divisior is a whole number. Occasionally, you will have to add a zero to the end of our dividend to finish up the problem. Sometimes, your answer may have repeating digits. Show Edit Destroy

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