2.1: Adding and Subtracting Decimals
Introduction
The School Store
The student council of F.W. Harris Middle School has decided to open a school store. There always seems to be a need for extra money whether it for school dances or for sporting events or to help with field trip costs. The student council has students on it from sixth, seventh and eighth grade and they have decided that this will be the best way to tackle fundraising in an ongoing way.
“What do you think Mr. Janus?” Kelly asked of their teacher advisor at the meeting.
“I think that it is a good idea. There will be some upfront costs involved however. Have you thought of how you are going to handle that?” Mr. Janus asked.
“Yes,” Tyler responded. “Each grade has some money in their account. We have each decided to use this money to help purchase supplies for the store.”
“Alright kids, it seems to me that you have this under control. Why don’t you begin by figuring out the sum of the money that you have so that you know what we have to work with?” Mr. Janus suggested.
“Okay, let’s start their, Trevor, how much is in the sixth grade budget?” Kelly asked.
Trevor flipped a few pages in his notebook before responding.
“There is $345.67 in the sixth grade budget.”
“Okay, let’s write that down. Mallory how about seventh grade?”
“There is $504.89 in seventh grade,” Mallory answered.
“Great and I know that there is $489.25 in the eighth grade budget,” Kelly responded.
“How much do we have to work with?” Trevor asked. “Let’s start by estimating.”
This is where you come in. This lesson is about adding sums and figuring out differences of decimals. Trevor’s suggestion is a great way to begin tackling the sum, with an estimate. Pay attention to this lesson and you will learn all about estimating and adding sums with decimals. At the end of the lesson, you will have the chance to solve this problem for yourself.
What You Will Learn
By the end of this lesson you will be able to demonstrate the following skills.
 Add and subtract decimals with and without rounding.
 Estimate decimal sums and differences using frontend estimation.
 Identify and apply the properties of addition and subtraction in decimal operations using numerical and variable expressions.
 Model and solve realworld problems using simple equations involving decimal addition and subtraction.
Teaching Time
I. Add and Subtract Decimals with and without Rounding
By this point in your learning of mathematics, you have some experience working with decimals. In this lesson we are going to review some of what you have learned before, but we will also build upon this knowledge and expand what you know to include some new ideas. First, let’s think about identifying decimals.
What is a decimal?
A decimal is a number that uses a decimal point and place value to show tenths, hundredths, thousandths, and so on. The decimal point divides the whole number portion from the fractional portion of the number. For example, look at this number.
35.492
The whole number portion is 35, or 3 tens and 5 ones. The fractional portion is 0.492, or 4 tenths, 9 hundredths, and 2 thousandths. Sometimes there are decimals with both wholes and parts, and sometimes, there are decimals with only parts.
Let's take a look at adding and subtracting decimals.
You can add and subtract decimals by adding according to place value or by rounding the values before adding them.
First, let's take a look at adding according to place value.
Example
Add: \begin{align*}48.08+6.215\end{align*}
We can add decimals like we add whole numbers: by lining up the place values. For decimals, this means lining up the decimal points. This means that we add each place value with its common place value. We don’t add hundredths and tens. We add hundredths and hundredths. If you think about this logically, it makes perfect sense. Here is what a problem looks like when it is lined up according to place value.
\begin{align*}& \ 48.080\\
& \underline{+6.215}\end{align*}
Now add each place value, remembering to carry when necessary.
The sum is 54.295
Next, we can find a sum by estimating. Remember that when you estimate you will find an approximate answer, but it will not be exact.
One way to estimate is by rounding.
We round each value to the nearest whole number. To determine which whole number to round a number to, we look at the decimal portion of the number. If the decimal part is less than .5, then we round down. If the decimal part is .5 or greater we round up. Let’s look at an example.
Example
Round 4.56 to the nearest whole number
In this example, we look at the fraction part to decide whether we will round the 4 up to 5 or leave it alone. In this case the fraction part is .56 which is greater than .5 so we round up to 5.
4.56 rounds up to 5
Example
Round 2.3 to the nearest whole number
In this example, the decimal part of the number is .3. It is less than .5, so we round down and the 2 remains the whole number.
2.3 rounds down to 2.
Rounding decimals is very useful when estimating a sum. We round each decimal in the problem and then add the whole numbers. This will give us our sum by rounding.
Let’s look at the last example.
Example
\begin{align*}48.08+6.215\end{align*}
Let’s start with rounding 48.08. The decimal part of the value is less than .5 in fact it is .08 a very small decimal. Therefore, the number rounds down to 48.
48
Next, we round the second value. We look at the decimal part of the value .215 is less than .5. Therefore we round down to 6.
\begin{align*}48 + 6\end{align*}
Our answer is 54.
You can see that the actual addition and the estimate by rounding are very close. That is how you can tell that your work is accurate.
We can also subtract decimals by using place value or by rounding. Let’s look at an example.
Example
\begin{align*}56.9310.14\end{align*}
First, we line up the values according to place value so that we can subtract one from the other.
The difference is 46.79.
We can also find the difference by rounding to the nearest whole number. We round each number to the nearest whole number and then we find the difference between the two values.
56.93 rounds up to 57
10.14 rounds down to 10
\begin{align*}5710 = 47\end{align*}
Our answer is 47.
Notice once again that the answers are close. This lets us know that our work is accurate.
II. Estimate Decimal Sums and Differences using FrontEnd Estimation
In the last section, you learned to estimate using rounding. Now we are going to use a different method of estimation. It is called frontend estimation.
Frontend estimation is a particular way of rounding numbers to estimate sums and differences. To use frontend estimation, add or subtract only the numbers in the greatest place value.
Example
Estimate the sum using frontend estimation: \begin{align*}4.8+3.2+7.2\end{align*}
First add the digits in the ones places: \begin{align*}4+3+7=14\end{align*}
Now look at the digits in the tenths places: \begin{align*}8+2+2=12\end{align*}
\begin{align*}14+1=15\end{align*}
A good estimate for the sum is 15. This is our answer.
Example
Estimate the difference using frontend estimation: \begin{align*}9.523.39\end{align*}
First subtract the digits in the ones places: \begin{align*}93=6\end{align*}
Now look at the digits in the tenths places. Since the difference of 5 and 3 is 2, it will not affect your first estimate.
A good estimate for the difference is 6. This is our answer.
Now you know how to find sums and differences in three different ways. You can add them according to place value for an exact answer. You can estimate using rounding or you can estimate using frontend estimation.
Take a few minutes to write down some notes about each of the ways of finding decimal sums and differences. Be sure to include a definition so that you can remember the differences between the three ways.
III. Identify and Apply the Properties of Addition and Subtraction in Decimal Operations, using Numerical and Variable Expressions
Do you remember working with properties? A property is a rule that applies to mathematical statements. The great thing about a property is that the rule has been proven so it is always true. Properties help us to understand certain ways of doing things in mathematics.
Here are two properties of addition that you should be familiar with.
Associative Property of Addition
The grouping of addends does not affect the sum: \begin{align*}4.5+(2.1+9.6)=(4.5+2.1)+9.6\end{align*}
Commutative Property of Addition
The order of addends does not change the sum: \begin{align*}6.3+8.7=8.7+6.3\end{align*}
Example
Which of the following shows the Commutative Property?
a. \begin{align*}x+9.5=9.5x\end{align*}
b. \begin{align*}x9.5=9.5x\end{align*}
c. \begin{align*}x+9.5=9.5+x\end{align*}
Consider choice a.
This equation states that a number added to 9.5 is equal to that number multiplied by 9.5. This is not correct.
Consider choice b.
This equation states that the difference of a number and 9.5 is equal to the difference of 9.5 and a number. This is not correct.
Consider choice c.
This equation states that the sum of a number and 9.5 is equal to the sum of 9.5 and a number. The Commutative Property states that the order of addends does not change the sum, so this is the correct equation.
You can also use properties to help you simplify numerical expressions.
That is a great question and the best way to understand it is to look at an example. Let’s do that now.
Example
Simplify: \begin{align*}10.5+(3.2+4.5)\end{align*}
You can use addition properties to reorganize this expression to make it easier to simplify.
First apply the commutative property.
\begin{align*}10.5+(3.2+4.5)=10.5+(4.5+3.2)\end{align*}
Then apply the associate property.
\begin{align*}10.5+(4.5+3.2)=(10.5+4.5)+3.2\end{align*}
Now you can easily use mental math to find the sum.
\begin{align*}(10.5+4.5)+3.2=15+3.2=18.2\end{align*}
The answer is 18.2
IV. Model and Solve Real – World Problems using Simple Equations Involving Decimal Addition and Subtraction
Decimals are all around us in the world. Anytime a number does not divide evenly, you will have to work with a decimal. Think about measurement, items are often not perfect whole measurements. Think about money, we often receive change when we buy something. Because decimals are such a big part of our everyday life, it is important for you to achieve some proficiency and mastery when working with them.
Example
While at the grocery store, Candace bought a box of cereal for $3.65, a carton of juice for $4.78, and fish for $10.28. If all prices include tax, what is a reasonable estimate for the amount Candace spent? What is the exact amount that Candace spent?
There are two parts to the answer for this question. First, we need to find an estimate and then we need to figure out the exact amount that Candace spent.
First, to find an estimate, let’s use rounding.
Round each decimal to a whole number and then find the sum.
3.65 rounds up to 4
4.78 rounds up to 5
10.28 rounds down to 10
\begin{align*}4+5+10=19\end{align*}
Candace spent about $19.
Now let’s work on figuring out the exact amount that Candace spent.
To find the exact amount Candace spent, write a simple equation to represent the problem. Let \begin{align*}x\end{align*}
\begin{align*}x &= 3.65+4.78+10.28\\
&=8.43+10.28\\
&=18.71\end{align*}
Candace spent exactly $18.71.
Example
Nick bought one book for $5.75 and one book for $7.15. He paid with a $20 bill. How much change should he receive?
First find the total amount Nick spent.
\begin{align*}5.75+7.15=12.9\end{align*}
Then subtract to find the change he should receive.
\begin{align*}2012.9=7.1\end{align*}
Nick should receive $7.10 in change.
Now let’s go back to the problem in the introduction and use what we have learned to help us solve that problem.
RealLife Example Completed
The School Store
Now you will have an opportunity to apply what you have learned in this lesson. Reread the problem and then work out a solution in your notebook. You should have an estimate and an accurate sum when you are finished.
The student council of F.W. Harris Middle School has decided to open a school store. There always seems to be a need for extra money whether it for school dances or for sporting events or to help with field trip costs. The student council has students on it from sixth, seventh and eighth grade and they have decided that this will be the best way to tackle fundraising in an ongoing way.
“What do you think Mr. Janus?” Kelly asked of their teacher advisor at the meeting.
“I think that it is a good idea. There will be some upfront costs involved however. Have you thought of how you are going to handle that?” Mr. Janus asked.
“Yes,” Tyler responded. “Each grade has some money in their account. We have each decided to use this money to help purchase supplies for the store.”
“Alright kids, it seems to me that you have this under control. Why don’t you begin by figuring out the sum of the money that you have so that you know what we have to work with?” Mr. Janus suggested.
“Okay, let’s start their, Trevor, how much is in the sixth grade budget?” Kelly asked.
Trevor flipped a few pages in his notebook before responding.
“There is $345.67 in the sixth grade budget.”
“Okay, let’s write that down. Mallory how about seventh grade?”
“There is $504.89 in seventh grade,” Mallory answered.
“Great and I know that there is $489.25 in the eighth grade budget,” Kelly responded.
“How much do we have to work with?” Trevor asked. “Let’s start by estimating.”
Now it is time for you to complete this work in your notebook. Remember that you need to have two answers.
Solution to Real – Life Example
The first thing that the students need to do is to find an estimate.
To find an estimate, we round each number to the nearest whole number.
$345.67 rounds to $346
$504.89 rounds to $505
$489.25 round up to $490
We add \begin{align*}346 + 505 + 390 = \$ 1341\end{align*}
Look at your estimate is it close to this one? Why or why not?
Now we can find the actual sum.
\begin{align*}& \quad \$ 345.67\\
& \quad \$ 504.89\\
& \underline{+ \$ 489.25}\\
& \ \$ 1339.81\end{align*}
Notice that our estimate is reasonable given the actual answer. In fact, our estimate is very close to the actual sum.
Vocabulary
Here are the vocabulary words that are found in this lesson.
 Decimal
 a part of a whole. The numbers to the left of the decimal point represent whole quantities. The numbers to the right of the decimal point represent parts.
 Estimate
 to find an approximate answer that is reasonable or makes sense given the problem.
 FrontEnd Estimation
 a method of estimating where you only add the digits in the greatest place value
 Associative Property of Addition
 states that the grouping of numbers does not impact the sum of those numbers.
 Commutative Property of Addition
 states that the order of the numbers as you add them does not impact the sum of those numbers.
Time to Practice
Directions: Find the exact sum or difference by adding or subtracting the following decimals according to place value.

\begin{align*}16.27 + 3.45 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
16.27+3.45=−−−−− 
\begin{align*}22.34 + 9.21 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
22.34+9.21=−−−−− 
\begin{align*}34.5 + 1.234 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
34.5+1.234=−−−−− 
\begin{align*}5.6 + 8.9 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
5.6+8.9=−−−−− 
\begin{align*}1.02 + 12.34 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
1.02+12.34=−−−−− 
\begin{align*}67.89 + 23.45 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
67.89+23.45=−−−−− 
\begin{align*}123.4 + 7.89 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
123.4+7.89=−−−−− 
\begin{align*}34.05 + 102.10 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
34.05+102.10=−−−−− 
\begin{align*}34.56  11.23 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
34.56−11.23=−−−−− 
\begin{align*}67.09  2.34 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
67.09−2.34=−−−−− 
\begin{align*}88.9  13.24 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
88.9−13.24=−−−−− 
\begin{align*}234.5  16.7 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
234.5−16.7=−−−−−  \begin{align*}708.90  45.67 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
 \begin{align*}27.56  1.20 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
Directions: Find each estimate by rounding or by using frontend estimation.
 \begin{align*}45.67 + 3.04 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
 \begin{align*}55.10 + 5.6 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
 \begin{align*}88.99  2.10 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
 \begin{align*}80.09  12.78 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
 \begin{align*}34.75  3.05 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
 \begin{align*}5.67 + 3.87 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
 \begin{align*}5.6 + 3.2 + 8.9 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
 \begin{align*}7.8 + 2.3 + 5.3 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
 \begin{align*}8.1 + 5.4 + 5.8 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
Directions: Use the associative and commutative properties of addition to solve each problem.
 \begin{align*}(7.2 + 9.1) + 3.2 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
 \begin{align*}5.4 + 2.1 + 5.4 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
 \begin{align*}(1.2 + 6.7) + 1.3 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
 \begin{align*}(4.1 + 9.2) + 9.0 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}