  Mathematics is a part of our lives in many ways. We use math to do everything from buying stuff to navigating across town to making pizzas. We even have to use math at work. If you had to measure out some raw materials for a project or quoted a price to a customer, you did math. Mathematics is so prevalent in our lives that we create tools called calculators to help us with it. However, there are times when we cannot use a calculator. We may have forgotten to bring one, or security restrictions may have prohibited them. Either way, we have to do math on our own. This would be fine if we only had to deal with whole numbers, but we live in a world that has fractional quantities. While we can reduce some of the hassle by using those fractions as decimal numbers, that leaves us having to do long division with decimals.

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# Understanding Decimals

Decimal numbers let us save space when we write fractions. They are formed by a sequence of digits following a period serving as a decimal point. This sequence of digits represents our fraction as the numerator of a faction with a power of ten as the denominator. Thus, 0.23 is the fraction 23/100. We write a 0 in front of the decimal when there is no whole number associated with it. If there is a whole number, we write the whole number in front of the decimal point such as 5345.23. Here, 5345 is a whole number while .23 is our decimal. Multiplying this construct by a power of ten, we just shift the location of the decimal point. For example, 5345.23 multiplied by 1000 is the number 5345230.  In this way, we can use whole numbers as fractions a long with our decimals. For instance, 5345.23 is the fraction 534523/100.

Because decimals are fractions, the same rules we use for fractions apply to decimals. Since decimals are really fractions, you are really dividing a fraction by a fraction when we divide a number with decimal with a number with a decimal. You should also note that we can view division, including long division, as a fraction itself. When we say “4 divided by 2”, we are describing the fraction 4/2. The beauty of fractions is that we can multiply them with any other fraction that has the value of one to create a new fraction. For instance, we can multiply 4/2 by 2/2 to get 2/1 which we can easily see evaluates to 2. Dividing with decimals requires us to do the same to create a much easier problem.

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# Long Division using Decimals

In long division, the numerator is located inside the enclosure while the divisor is located outside to the left. We write the quotient on top of the enclosure. This format lets us take a step by step approach to division. While we can use approach with other formats, long division streamlines the process by including our steps under the enclosure.

Long division proceeds by inspecting each digit of the numerator and figuring out what number, if multiplied by our divisor, will produce that digit or at least a number close to it. If the number is 0, we attach the digit to the next digit in sequence and try again. If we find a number that works, we multiply the number by the divisor and place the result under the numerator placing the last digit under our current numerator digit. Subtracting the result from the numerator digits above gives us the remainder. If the numerator still has digits, we move to the next one and bring it down to the remainder. We then use this remainder to repeat the process until we have no more digits in the numerator.

If we still have a remainder after all the numerator digits are used up, we then move on to decimals. In this case, we add a decimal point to our quotient and numerator, and continue the process as if the decimal point didn’t exist. We insert 0s to the numerator as needed until the remainder vanishes. The quotient we come up with is the true solution to our long division problem.

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# Long Division with Decimals

If there is a decimal point is in the divisor, we have to do one other step before we can do our long division.  Here is where our “division is a faction” concept from earlier comes into play. As I said earlier, we can change a decimal number by multiplying it by a power of ten. This has the effect of moving the decimal by the number of places indicated by the exponent. We can use this procedure to get rid of the decimal point in our divisor. However, we must also do the same procedure to the numerator to keep the end result the same. If the numerator does not have enough digits to support such as move, we have to insert 0s to make up the difference.

Once the decimal in the divisor is gone, we can proceed with our long division.  Fortunately, we don’t have to do anything to the decimal point in the numerator. We just have to remember to include it in the quotient when we come to it. Otherwise, the decimal point doesn’t affect our long division in any way. For an example, let’s long divide 6.85 by 0.5.

We multiply both numbers by 10 to get rid of the decimal point in the divisor.

We proceed with the long vision, adding a decimal point to the quotient with we come to the decimal point in the numerator.

With this quick an easy step, you can perform long division with any decimal in the world. While we usually use calculators these days to perform these calculations, I hope this procedure will come in handy the next time you have to perform a long division with decimals.

Page Last Updated: February 2020

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