4.6: Dividing by Decimals
Introduction
The Sand Experiment
Most students love to participate in handson projects, and the students in Mrs. Andersen’s class aren’t any exception. At the science museum there is a whole section that is a Discovery Center. In the Discovery Center, students can use real objects to work on experiments.
Mrs. Andersen has asked her students to bring a notebook and a pencil into the Discovery Center. The students need to keep track of the experiments that they work on. They will each have an opportunity to share their discoveries when they return to the classroom.
When Miles enters the Discovery Center he is immediately overwhelmed with all of the options. After looking around, he finally decides to work on an experiment that involves an hour glass. To complete the experiment, Miles needs to figure out how long it takes 1.25 pounds of sand to go through the hour glass. There is bucket of sand that is 6.25 pounds in front of Miles. He has a scale and another bucket to hold the sand he needs for his experiment.
Miles needs to complete the experiment as many times as he can with the 6.25 pound bucket of sand. Miles picks up the scoop and begins to sort out the sand. Remember he needs 1.25 pounds of sand each time he does the experiment.
If Miles needs 1.25 pounds of sand, how many times can he complete the experiment if he has a 6.25 pound bucket?
Pretend you are Miles. If you were completing this experiment, how many times could you do it given the amount of sand you have been given and the amount of sand that you need?
In this lesson, you will find all of the information that you need. Dividing decimals by other decimals will help you with this experiment.
What You Will Learn
In this lesson you will learn to:
 Divide decimals by decimals by rewriting divisors as whole numbers.
 Find quotients of decimals by using additional zero placeholders.
 Solve realworld problems involving division by decimals.
Time to Practice
I. Divide Decimals by Decimals by Rewriting Divisors as Whole Numbers
In our introductory problem, Miles is working on dividing up sand. If you were going to complete this problem yourself, you would need to know how to divide decimals by decimals.
How can we divide a decimal by a decimal?
To divide a decimal by a decimal, we have to rewrite the divisor. Remember that the divisor is the number that is outside of the division box. The dividend is the number that is inside the division box.
Let’s look at an example.
Example
\begin{align*}2.6 \overline{)10.4 \;}\end{align*}
In this problem, 2.6 is our divisor and 10.4 is our dividend. We have a decimal being divided into a decimal. Whew! This seems pretty complicated. We can make our work simpler by rewriting the divisor as a whole number.
How can we do this?
Think back to the work we did in the last section when we multiplied by a power of ten. When we multiply a decimal by a power of ten we move the decimal point one place to the right.
We can do the same thing with our divisor. We can multiply 2.6 times 10 and make it a whole number. It will be a lot easier to divide by a whole number.
2.6 \begin{align*}\times\end{align*}
What about the dividend?
Because we multiplied the divisor by 10, we also need to multiply the dividend by 10. This is the only way that it works to rewrite a divisor.
10.4 \begin{align*}\times\end{align*}
Now we have a new problem to work with.
\begin{align*}& \overset{ \qquad 4}{26\overline{ ) 104 \;}}\end{align*}
Our answer is 4.
What about if we have two decimal places in the divisor?
Example
\begin{align*}.45 \overline{)1.35 \;}\end{align*}
In this example, we want to make our divisor .45 into a whole number by multiplying it by a power of ten. We can multiply it by 100 to make it a whole number. Then we can do the same thing to the dividend.
Here is our new problem and quotient.
\begin{align*}& \overset{ \qquad 3}{45\overline{ ) 135 \;}}\end{align*}
Now it is time for you to practice a few. Rewrite each divisor and dividend by multiplying them by a power of ten. Then find the quotient.

\begin{align*}1.2 \overline{)4.8 \;}\end{align*}
1.2)4.8¯¯¯¯¯¯¯¯¯¯ 
\begin{align*}5.67 \overline{)11.34 \;}\end{align*}
5.67)11.34¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ 
\begin{align*}6.98 \overline{)13.96 \;}\end{align*}
6.98)13.96¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Take a minute to check your rewrite and quotient with a peer. Is your work accurate?
II. Find Quotients of Decimals by Using Additional Zero Placeholders
The decimals that we divided in the last section were all evenly divisible. This means that we had whole number quotients. We didn’t have any decimal quotients.
What can we do if a decimal is not evenly divisible by another decimal?
If you think back, we worked on some of these when we divided decimals by whole numbers. When a decimal was not evenly divisible by a whole number, we had to use a zero placeholder to complete the division.
Here is a blast from the past problem.
Example
\begin{align*}5 \overline{)13.6 \;}\end{align*}
When we divided 13.6 by 5, we ended up with a 1 at the end of the division. Then we were able to add a zero placeholder and finish finding a decimal quotient. Here is what this looked like.
\begin{align*}& \overset{ \quad 2.72}{5 \overline{ ) {13.60 \;}}}\\
& \underline{10 \;\;}\\
& \quad \ 36\\
& \ \ \underline{35\;\;}\\
& \qquad 1  \ \text{here is where we added the zero placeholder}\\
& \qquad 10\\
& \quad \ \underline{10}\\
& \qquad \ \ 0\end{align*}
We add zero placeholders when we divide decimals by decimals too.
Example
\begin{align*}1.2 \overline{)2.79 \;}\end{align*}
The first thing that we need to do is to multiply the divisor and the dividend by a multiple of ten to make the divisor a whole number. We can multiply both by 10 to accomplish this goal.
\begin{align*}12 \overline{)27.9 \;}\end{align*}
Now we can divide.
\begin{align*}& \overset{ \quad \ \ 2.3}{12 \overline{ ) {27.9 \;}}}\\
& \ \underline{24 \;\;}\\
& \quad \ \ 39\\
& \quad \underline{36}\\
& \qquad \ 3\end{align*}
Here is where we have a problem. We have a remainder of 3. We don’t want to have a remainder, so we have to add a zero placeholder to the problem so that we can divide it evenly.
\begin{align*}& \overset{ \quad \ \ 2.32}{12 \overline{ ) {27.90 \;}}}\\
& \ \underline{24\;\;}\\
& \quad \ \ 39\\
& \quad \underline{36\;\;}\\
& \qquad \ 30\\
& \quad \ \ \underline{24}\\
& \qquad \quad 6\end{align*}
Uh Oh! We still have a remainder, so we can add another zero placeholder.
\begin{align*}& \overset{ \quad \ 2.325}{12 \overline{ ) {27.900 \;}}}\\
& \ \ \underline{24\;\;}\\
& \ \quad \ \ 39\\
& \ \quad \underline{36\;\;}\\
& \ \qquad \ 30\\
& \ \quad \ \ \underline{24\;\;}\\
& \ \qquad \quad 60\\
& \ \qquad \ \underline{60\;\;}\\
& \ \qquad \quad \ \ 0\end{align*}
Sometimes, you will need to add more than one zero. The key is to use the zero placeholders to find a quotient that is even without a remainder.
Real Life Example Completed
The Sand Experiment
Congratulations you have finished the lesson! Now you are ready for the experiment.
Here is the problem once again.
Most students love to participate in handson projects, and the students in Mrs. Andersen’s class aren’t any exception. At the science museum there is a whole section that is a Discovery Center. In the Discovery Center, students can use real objects to work on experiments.
Mrs. Andersen has asked her students to bring a notebook and a pencil into the Discovery Center. The students need to keep track of the experiments that they work on. They will each have an opportunity to share their discoveries when they return to the classroom.
When Miles enters the Discovery Center he is immediately overwhelmed with all of the options. After looking around, he finally decides to work on an experiment that involves an hour glass. To complete the experiment, Miles needs to figure out how long it takes 1.25 pounds of sand to go through the hour glass. There is bucket of sand that is 6.25 pounds in front of Miles. He has a scale and another bucket to put the sand he needs for his experiment.
Miles needs to complete the experiment as many times as he can with the 6.25 pound bucket of sand. Miles picks up the scoop and begins to sort out the sand. Remember he needs 1.25 pounds of sand each time he does the experiment.
If Miles needs 1.25 pounds of sand, how many times can he complete the experiment if he has a 6.25 pound bucket?
First, underline the important information.
Next, write a division problem.
\begin{align*}1.25 \overline{)6.25 \;}\end{align*}
You can start by multiplying the divisor by a power of ten to rewrite it as a whole number. Do this to the dividend too. Since there are two places in the divisor, we can multiply it by 100 to make it a power of ten.
\begin{align*}125 \overline{)625 \;}\end{align*}
Next, we divide. Our answer will tell us how many times Miles can complete the hourglass experiment.
\begin{align*}& \overset{ \qquad \ \ 5}{125 \overline{ ) {625 \;}}}\\
& \quad \underline{625}\\
& \qquad \ \ 0\end{align*}
Miles can complete the experiment 5 times using 1.25 pounds of sand from his 6.25 pound bucket.
Vocabulary
Here are the vocabulary words that are found in this lesson.
 Divisor
 the number doing the dividing, it is found outside of the division box.
 Dividend
 the number being divided. It is found inside the division box.
 Quotient
 the answer in a division problem
Technology Integration
Khan Academy Dividing Decimals
James Sousa Example of Dividing Decimals
James Sousa Another Example of Dividing Decimals
Other Videos:
http://www.mathplayground.com/howto_dividedecimals.html – Good basic video on dividing decimals
Time to Practice
Directions: Divide the following decimals. Use zero placeholders when necessary.

\begin{align*}1.3 \overline{)5.2 \;}\end{align*}
1.3)5.2¯¯¯¯¯¯¯¯¯¯ 
\begin{align*}6.8 \overline{)13.6 \;}\end{align*}
6.8)13.6¯¯¯¯¯¯¯¯¯¯¯¯ 
\begin{align*}4.5 \overline{)13.5 \;}\end{align*}
4.5)13.5¯¯¯¯¯¯¯¯¯¯¯¯ 
\begin{align*}2.5 \overline{)10 \;}\end{align*}
2.5)10¯¯¯¯¯¯¯¯ 
\begin{align*}3.3 \overline{)19.8 \;}\end{align*}
3.3)19.8¯¯¯¯¯¯¯¯¯¯¯¯ 
\begin{align*}8.5 \overline{)17 \;}\end{align*}
8.5)17¯¯¯¯¯¯¯¯ 
\begin{align*}9.3 \overline{)27.9 \;}\end{align*}
9.3)27.9¯¯¯¯¯¯¯¯¯¯¯¯ 
\begin{align*}1.2 \overline{)7.2 \;}\end{align*}
1.2)7.2¯¯¯¯¯¯¯¯¯¯ 
\begin{align*}5.3 \overline{)26.5 \;}\end{align*}
5.3)26.5¯¯¯¯¯¯¯¯¯¯¯¯ 
\begin{align*}6.5 \overline{)13 \;}\end{align*}
6.5)13¯¯¯¯¯¯¯¯  \begin{align*}1.25 \overline{)7.5 \;}\end{align*}
 \begin{align*}3.36 \overline{)20.16 \;}\end{align*}
 \begin{align*}5.87 \overline{)52.83 \;}\end{align*}
 \begin{align*}2.5 \overline{)3 \;}\end{align*}
 \begin{align*}3.2 \overline{)8 \;}\end{align*}
 \begin{align*}4.6 \overline{)10.58 \;}\end{align*}
 \begin{align*}8.1 \overline{)17.82 \;}\end{align*}
 \begin{align*}3.1 \overline{)28.52 \;}\end{align*}
 \begin{align*}8.7 \overline{)53.94 \;}\end{align*}
 \begin{align*}5.4 \overline{)18.9 \;}\end{align*}
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