4.4: Dividing by Whole Numbers
Introduction
The Discount Dilemma
When the students in Mrs. Andersen’s class came out of the dinosaur exhibit, Sara, one of the people who works at the museum, came rushing up to her.
“Hello Mrs. Andersen, we have some change for you. You gave us too much money, because today we have a discount for all students. Here is $35.20 for your change,” Sara handed Mrs. Andersen the money and walked away.
Mrs. Andersen looked at the change in her hand.
Each student is due to receive some change given the student discount. Mrs. Andersen tells Kyle about the change. Kyle takes out a piece of paper and begins to work.
If 22 students are on the trip, how much change should each student receive?
In this lesson you will learn about dividing decimals by whole numbers. When finished with this lesson, you will know how much change each student should receive.
What You Will Learn
In this lesson you will learn how to:
- Divide decimals by whole numbers.
- Find decimal quotients of whole numbers using additional zero placeholders.
- Divide decimals by whole numbers and round to a given place.
- Solve real-world problems involving the division of decimals by whole numbers.
Teaching Time
I. Divide Decimals By Whole Numbers
To divide means to split up into equal parts. You have learned how to divide whole numbers in an earlier lesson. Now we are going to learn how to divide decimals by whole numbers.
When we divide a decimal by a whole number, we are looking at taking that decimal and splitting it up into sections.
Let’s look at an example.
Example
4.64 \begin{align*}\div\end{align*} 2 \begin{align*}=\end{align*} ______
The first thing that we need to figure out when working with a problem like this is which number is being divided by which number. In this problem, the two is the divisor. Remember that the divisor goes outside of the division box. The dividend is the value that goes inside the division box. It is the number that you are actually dividing.
\begin{align*}2 \overline{)4.64 \;}\end{align*}
We want to divide this decimal into two parts. We can complete this division by thinking of this problem as whole number division.
We divide the two into each number and then we will insert the decimal point when finished. Here is our problem.
\begin{align*}& \overset{232}{2\overline{ ) 4.64 \;}}\end{align*}
Finally, we can insert the decimal point into the quotient. We do this by bringing up the decimal point from its place in the division box right into the quotient. See the arrow in this example to understand it better, and here are the numbers for each step of the division.
\begin{align*}& \overset{\overset{ \ 2.32} {\uparrow}}{2 \overline{ ) 4.64 \;}}\\ & \quad \underline{4 \quad }\\ & \quad \ 0 6\\ & \quad \ \underline{ \ \ 6 \ }\\ & \qquad 04 \end{align*}
Our answer is 2.32.
As long as you think of dividing decimals by whole numbers as the same thing as dividing by whole numbers it becomes a lot less complicated.
Always remember to notice the position of the decimal point in the dividend and bring it up into the quotient.
Here are a few for you to try.
- 36.48 \begin{align*}\div\end{align*} 12
- 2.46 \begin{align*}\div\end{align*} 3
- 11.5 \begin{align*}\div\end{align*} 5
Take a minute to check your work with a peer. Did you put the decimal point in the correct spot?
II. Find Decimal Quotients of Whole Numbers Using Additional Zero Placeholders
In our last lesson, you learned to divide a decimal by a whole number. Remember here that the divisor is the whole number which goes outside of the division box and the dividend is the decimal that goes inside of the division box.
The examples in the last section were evenly divisible by their divisors. This means that at the end there wasn’t a remainder.
How do we divide decimals by whole numbers when there is a remainder?
Let’s look at an example.
Example
14.9 \begin{align*}\div\end{align*} 5 \begin{align*}=\end{align*} ______
The first thing that we can do is to set up this problem in a division box. The five is the divisor and the 14.9 is the dividend.
\begin{align*}5 \overline{)14.9 \;}\end{align*}
Next we start our division. Five goes into fourteen twice, with four left over. Then we bring down the 9. Five goes into 49, 9 times with four left over. Before you learned about decimals, that 4 would just be a remainder.
\begin{align*}& \overset{2.9 \ \ } { \ 5 \overline{ ) {14.9}} \ {r \ 4} \;}\\ & \underline{- \ 10 \ \; \;}\\ & \quad \ 49\\ & \ \underline{- \ 45 \; \;}\\ & \quad \ \ \ 4 \end{align*}
However, when we work with decimals, we don’t want to have a remainder. We can use a zero as a placeholder. In this example, we can add a zero to the dividend and then see if we can finish the division. We add a zero and combine that with the four so we have 40. Five divides into forty eight times.
Here is what that would look like.
\begin{align*}& \overset{ \quad 2.98}{5 \overline{ ) {14.90 \;}}}\\ & \underline{-10 \ \ }\\ & \quad \ 49\\ & \ \ \underline{-45 \ }\\ & \qquad 40 \\ & \ \ \ \ \underline{-40 \ }\\ & \qquad \ \ 0 \end{align*}
Our final answer is 2.98.
When working with decimals, you always want to add zeros as placeholders so that you can be sure that the decimal is as accurate as it can be. Remember that a decimal shows a part of a whole. We can make that part as specific as necessary.
Try a few of these on your own. Be sure to add zero placeholders as needed.
- 13.95 \begin{align*}\div\end{align*} 6 \begin{align*}=\end{align*} _____
- 2.5 \begin{align*}\div\end{align*} 2 \begin{align*}=\end{align*} _____
- 1.66 \begin{align*}\div\end{align*} 4 \begin{align*}=\end{align*} _____
Take a minute to check your work with a neighbor.
III. Divide Decimals by Whole Numbers and Round to a Given Place
You have learned how to divide decimals by whole numbers and how to use zero placeholders to find the most accurate decimal quotient. We can also take a decimal quotient and round it to a specific place.
Let’s say we have a decimal like this one.
Example
.3456210
Wow! That is a mighty long decimal. It is so long that it is difficult to decipher the value of the decimal.
If we were to round the decimal to the thousandths place, that would make the size of the decimal a lot easier to understand.
.3456210 Five is in the thousandths place. The number after it is a six, so we round up.
.346
Our answer is .346.
Now let’s try it with an example. Divide and round this decimal quotient to the nearest ten-thousandth.
Example
1.26484 \begin{align*}\div\end{align*} 4 = ______
Use a piece of paper to complete this division.
Our answer is .31621.
Now we want to round to the nearest ten-thousandth.
.31621 Two is in the ten-thousandths place. The number after this is a one so our two does not round up.
Our answer is .3162.
Divide these decimals and whole numbers and then round each to the nearest thousandth.
- .51296 \begin{align*}\div\end{align*} 2 \begin{align*}=\end{align*} _____
- 10.0767 \begin{align*}\div\end{align*} 3 \begin{align*}=\end{align*} _____
Check your work with a peer. Did you round the quotient to the correct place?
Real Life Example Completed
The Discount Dilemma
Now that you have learned about dividing decimals by whole numbers, we are ready to help Kyle figure out the change from the science museum.
When the students in Mrs. Andersen’s class came out of the dinosaur exhibit, Sara, one of the people who works at the museum, came rushing up to her.
“Hello Mrs. Andersen, we have some change for you. You gave us too much money because today we have a discount for all students. Here is $35.20 for your change,” Sara handed Mrs. Andersen the money and walked away.
Mrs. Andersen looked at the change in her hand.
Each student is due to receive some change given the student discount. Mrs. Andersen tells Kyle about the change. Kyle takes out a piece of paper and begins to work.
If 22 students are on the trip, how much change should each student receive?
First, let’s go back and underline the important information.
Now that we know about dividing decimals and whole numbers, this problem becomes a lot easier to solve.
Our divisor is the number of students, that is 22.
Our dividend is the amount of change = 35.20
\begin{align*}& \overset{ \quad \ 1.60}{22 \overline{ ) {35.20 \;}}}\\ & \ \ \underline{-22 \ \ }\\ & \quad \ \ 132\\ & \ \ \ \underline{-132 \ }\\ & \qquad \ \ \ 0 \end{align*}
Our answer is $1.60.
Kyle shows his work to Mrs. Andersen, who then hands out $1.60 to each student.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Divide
- to split up into groups evenly.
- Divisor
- a number that is doing the dividing. It is found outside of the division box.
- Dividend
- the number that is being divided. It is found inside the division box.
- Quotient
- the answer to a division problem
Technology Integration
Khan Academy Dividing Decimals 2
James Sousa Example of Dividing a Decimal by a Whole Number
Other Videos:
http://www.schooltube.com/video/8431c6dd1e794831b100/13-Dividing-Decimals-by-Whole-Numbers-Ex-1 – Blackboard video on dividing decimals by whole numbers
Time to Practice
Directions: Divide each decimal by each whole number. Add zero placeholders when necessary.
- \begin{align*}5 \overline{)17.5 \;}\end{align*}
- \begin{align*}8 \overline{)20.8 \;}\end{align*}
- \begin{align*}4 \overline{)12.8 \;}\end{align*}
- \begin{align*}2 \overline{)11.2 \;}\end{align*}
- \begin{align*}4 \overline{)14.4 \;}\end{align*}
- \begin{align*}5 \overline{)27.5 \;}\end{align*}
- \begin{align*}6 \overline{)13.8 \;}\end{align*}
- \begin{align*}7 \overline{)16.8 \;}\end{align*}
- \begin{align*}7 \overline{)23.1 \;}\end{align*}
- \begin{align*}6 \overline{)54.6 \;}\end{align*}
- \begin{align*}8 \overline{)41.6 \;}\end{align*}
- \begin{align*}9 \overline{)86.4 \;}\end{align*}
- \begin{align*}10 \overline{)52 \;}\end{align*}
- \begin{align*}10 \overline{)67 \;}\end{align*}
- \begin{align*}11 \overline{)57.2 \;}\end{align*}
- \begin{align*}10 \overline{)96 \;}\end{align*}
- \begin{align*}8 \overline{)75.2 \;}\end{align*}
- \begin{align*}9 \overline{)32.4 \;}\end{align*}
- \begin{align*}12 \overline{)38.4 \;}\end{align*}
- \begin{align*}12 \overline{)78 \;}\end{align*}
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