The Secrets to Dividing Decimals by Decimals

dividing decimals by decimalsWe use math every day without realizing it. From buying stuff to navigating across town, math is everywhere. This would be fine if we only have to work with whole numbers, but fractional numbers fill our world. We have to know how to deal with these decimal numbers in order to live and play. We have to work with decimals to divide up the check at a restaurant or bar. We have to divide decimal numbers to know when we must fill up the gasoline tank before we go on a trip. Dividing decimals by decimals isn’t rocket science. Anyone can do it. You just have to understand what decimal numbers are.

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Understanding Decimals

Decimal numbers are a way to save space while writing fractions. Instead of writing the fractions as two numbers arranged vertically with a bar between them, you can write any fraction as a single number followed by a period. These numbers are technically just the top numbers from fractions with a power of 10 on the bottom. 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.

This short review of decimal numbers exists to show you the reasoning behind the techniques I am about to show you for dividing decimals by decimals. Since decimals are really fractions, you are really dividing a fraction by a fraction, and the techniques will reflect this. You should also note that we can view division as a fraction itself. When we say “4 divided by 2”, we are describing the fraction. 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  by to get  which we can easily see evaluates to 2. Dividing decimals by decimals requires us to do the same to create a much easier problem.

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Understanding Division

Division is the opposite operation to multiplication. It depicts a partitioning of something. We write it by places two numbers on either side of a operator sign The first number is called the dividend or numerator and it is the number we are dividing from. The second number is called the divisor or denomination, and it is the number we are dividing the dividend by. The operator’s form is often set by convenience. We can write the operation as a fraction as we did in the last section with either a horizontal bar or a / symbol. We can also write it on a single line using the ÷ or | symbol though | has a special meaning I will say later.

When decimals are involved, you must identify which decimal number is the dividend and the divisor, as the techniques for dividing then differ between the two. You can modify the operation to get rid of the decimal divisor, but not the dividend.

Any time you see a decimal divisor, you can get rid of it just by creating a new problem with the same answer. You can do this by multiplying the dividend and divisor by the same number. Generally, you want to use the power of ten associated with the decimal in fraction form. For instance, when dividing 37.6 by 2.5 you can multiply both numbers by 10 since .5 is  in faction form as I do in the following example.

As you can see, we have nothing to fear about a decimal divisor. We can change the problem to get rid of it. You just have to know the power of ten associated with its digits and multiply both it and the dividend by that number.

For the decimal dividend, we can only modify it as much as we can modify the divisor. Once we remove the decimal from the divisor, we have to live with how the dividend looks from that point on. Fortunately, the decimal point does not play a role in the mathematics. Since we divide numbers by inspecting the dividend’s digits from left to right, you can deal with the decimal point when you come to it. Plus, you just have to add a decimal point in the quotient at the time. The dividend’s decimal point does nothing else. You just carry it over into the quotient and leave it there. Otherwise, you ignore it. You then continue to complete the division as if the decimal point wasn’t there. Once done, the decimal point should be in the right position in the quotient.

That is how easy it is to divide decimals by decimals. You first identify the dividend and divisor. You then multiply both numbers by the power of ten that will get rid of the decimal point in the divisor. You then proceed with the division as if the dividend did not have a decimal point until you come to it. Then, you just add a decimal point to the quotient. You then continue the division as if the decimal point doesn’t exist. The resulting quotient will be the solution to your initial problem with the decimal point in the right position.

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The | Division Operator

When dividing decimals by decimals, the above tricks are all you need to know. However, it would be rude if I did not explain the | division operator. Before, I said it has special meaning, but you may never come across it outside of computer programming and math classes. The | symbol denotes integer or whole number division which is the same thing as regular division but you discard the remainder. The key thing to remember is that integer division requires both the dividend and divisor to be whole numbers. Otherwise, it has no solution. In context with our discussion on dividing decimals by decimals, if you see something like 4.6|0.23, you immediately know the solution is DNE (“does not exist”). However, like I said before, you may never have to use this type of division. I just included it as a joke or conversation starter you can use with your friends.