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Dividing Decimals

There are $3$ cases when dividing decimals.

1. Dividing a decimal by a whole number
2. Dividing a whole number by a decimal
3. Dividing a decimal by another decimal.

They all follow the same basic process as dividing integers, but they need a little bit of manipulation to solve. Often we use fractions to represent division, but there's no need to this time. We'll use long division symbols instead.

Decimal Divided by Whole Number

Dividing a decimal by a whole number is almost exactly the same as dividing two whole numbers. Let's look at an example by dividing $4.26$ by $6$.

First, we're going to set up our division. We need to move the decimal point up to the top of our division. This guarantees that we won't forget where it goes later in the process. Since the number under the division sign is less than the number we are dividing it by, we will place a $0$ to the left of the decimal point. This indicates our final answer will be between $0$ and $1$ on the number line.

The rest of the process is the same as dividing two whole numbers. We're going to treat $4.26$ as $426$ for our division. $4.26$ divided by $6$ is $0.71$.

Whole Number Divided by Decimal

Dividing a whole number by a decimal has an extra step. Let's look at dividing $16$ by $0.8$. We need to move the decimal point in $0.8$ to the right in order to make it a whole number. In addition, we need to move the decimal point in $16$ to the right the same amount of places. This makes it so we are dividing numbers with the same ratio, but with easier numbers to work with. We're going to move this newly placed decimal point on top. Now, we solve this same way we would solve $160$ divided by $8$. $16$ divided by $0.8$ is $20$.

Decimal Divided by Decimal

The final case is when we need to divide a decimal by another decimal. This case follows the exact same process as dividing a whole number by a decimal, but you might not get clean answers. In particular you will sometimes get repeating decimals.

Let's look at dividing $8.25$ by $1.2$.

First, we're going to set up the division. We need to move the decimal point in $1.2$ to the right one place, so we are going to do the same with $8.25$. This means our division will show $82.5$ divided by $12$. Now we just perform the same long division that we already know!

Wrap-Up

The key difference between dividing decimals and dividing integers is that we need to manipulate our decimals a little bit before we can divide. We need to change our divisor into a whole number and mirror that change in our dividend. We divide the numbers the same way we divide integers once we have made the numbers easier to work with.

Dividing Decimals

Here is an example of how to divide one decimal by another: $$13.75 \div 5.5=2.5$$

In the step where you "move both decimal points until the divisor is an integer," make sure you move the decimal point in the divisor and the decimal point in the dividend to the right an equal amount of times until the divisor is an integer!

In the step where you perform "usual long division," don't worry about dropping down any decimal points as you divide. Here is a refresher on dividing positive integers.

Let's do another example to practice.

First things first—if we divide a negative number by a positive number, we will get a negative result. Since we are dividing $-5.175,$ a negative number, by $1.5,$ a positive number, we know the result will be negative. For now, let's do $5.175 \div 1.5,$ and at the end we will put a negative sign in front of that answer to get the actual final answer to $-5.175 \div 1.5.$

Step One: Set Up Division

Step Two: Move Decimal Points

Remember, here we are going to move the the decimal point in the divisor and the decimal point in the dividend to the right an equal amount of times until the divisor is an integer!

In this example, we move the decimal point in $1.5$ one place to the right so that the new divisor is $15.$ We also have to move the decimal place in $5.175$ one place to the right so that the new dividend is $51.75.$ Here's what our new division looks like after this process: We can get rid of the decimal point after the $15$ to make it look a little nicer if we want. The number is still the same.

Step Three: Move Up Decimal Point

Step Four: Perform Normal Long Division

Step Five: Include a Negative Sign

So, $5.175 \div 1.5 = 3.45.$ Remember, the problem was asking us to find $-5.175 \div 1.5,$ so all that's left to do is include a negative sign in our final answer: $$-5.175 \div 1.5=-3.45$$

(Videos) Dividing Decimal Numbers

by mathman1024

Here's a video by mathman1024 that shows you how to divide decimals. There are more videos with examples later in the explanation.

$\large{\text{Summary}}$

There are two types of problems with decimal division: dividing by whole numbers and dividing by decimal numbers. This video deals with the easier of the two, dividing by whole numbers.

Let's work through the example $49.91\div7$. You set up the long division as you do normally.

Next, you place the decimal up top in the same position.

Now, you do the division like you would any other problem. Ignore the decimal when dividing.

Try this next one on your own and then watch the video to see the answer!

1. $2.76\div3$

Here's a quick video with another example. Try this one on your own and then watch the video to see the answer!

1. $23.3\div8$

Here's another video with an example where the solution is an irrational number. Try this on your own and then watch the video to see the solution!

1. $5.2\div3$