3.6: Adding and Subtracting Decimals
Introduction
The Broken Cash Register
When Julie arrived for her shift at the ice cream stand, she was surprised to find out that the cash register was broken.
“You can just figure out each total and the customer’s change,” Mr. Harris said to Julie with a smile.
Julie grimaced as she got out a pad of paper and pencil. She knew that she was going to need to do some quick addition and subtraction to make this whole day work.
Very soon her first customer arrived. This customer ordered a small cone for $2.25 and gave Julie exact change.
“Maybe this won’t be so tough after all,” Julie thought.
Then her luck ended. A woman arrived and ordered a small cone with sprinkles, caramel, and an extra scoop of ice cream.
Julie quickly jotted the following numbers down on a piece of paper.
While Julie was working to figure out the sum, the woman handed Julie a $10.00 bill and two quarters.
“I am so glad that I have the change,” she said to Julie.
Julie frantically began to work out the math on her piece of paper.
How can Julie add up the decimals?
Is there a way for her to do it mentally?
What about the customer’s change? If the woman gave Julie a ten dollar bill and two quarters, how much change should she get back?
This lesson is going to teach you all about adding and subtracting decimals.
Hold on Julie, help is right around the corner!!
What You Will Learn
In this lesson, you will learn the following skills:
- Adding and Subtracting Decimals by rewriting with additional zero place holders.
- Using mental math to add/subtract decimals
- Identifying the commutative and associative properties of addition in decimal operations, using numerical and variable expressions
- Solving real world problems involving decimal addition and subtraction
Teaching Time
In our last lesson we learned how to estimate the sums and differences of problems with decimals. Remember, an estimate only works when we don’t need an exact answer.
Let’s think about Julie. She can’t use an estimation to solve her problem. She needs to know the exact cost of the ice cream cone with all of the additions and she needs to know the exact change to give back to the customer. Think about how funny it would be if Julie told her what an estimate of the cost would be and if she gave back an estimate of the change.
In problems like Julie’s situation, we need to know how to add and subtract decimals.
Let’s begin by learning how to find an exact sum or an exact difference.
I. Adding and Subtracting Decimals by Rewriting With Additional Zero Place Holders
To add or subtract decimals, we are going to be working with the wholes and parts of the numbers separately.
We want to add or subtract the parts and then add or subtract the wholes.
How can we do this?
The best way to do this is to keep the parts together and keep the wholes together.
To do this, we simply line up the decimal points in each number that we are adding or subtracting.
Let’s look at an example.
Example
Add 3.45 + 2.37 = _____
In this problem we have parts and wholes. Let’s rewrite the problem vertically, lining up the decimal points.
\begin{align*}3.45 \\ \underline{+\ 2.37}\end{align*}Next, we can add the columns vertically and bring the decimal point down into the answer of the problem.
\begin{align*}3.45 \\ \underline{+\ 2.37} \\ 5.82\end{align*}
Our answer is 5.82.
Does this work the same way when finding a difference?
Yes. We can line up the decimals in a subtraction problem and subtract the digits the same way.
Example
6.78 - 2.31 = _____
First, we line up the problem vertically.
\begin{align*}6.78 \\ \underline{-\ 2.31} \end{align*}
Next, we subtract each digit vertically.
\begin{align*}6.78 \\ \underline{-\ 2.31} \\ 4.47\end{align*}
Our answer is 4.47.
These examples both had the same number of digits in them. They each had one whole number and a decimal in the hundredths.
What happens when you have decimals with different numbers of digits in them?
When we have a problem like this, we still line up the decimal points, but we add zeros to help hold places where there aren’t numbers. This helps us to keep our addition and subtraction straight.
Let’s look at an example.
Example
5 + 3.45 + .56 = _____
First, we line up the problem vertically.
\begin{align*}5.00 \\ 3.45 \\ \underline{+\ 0.56} \end{align*}
Notice that we added in zeros to help hold places where we did not have numbers. Now each number in the problem has the same number of digits. We can add them with ease.
\begin{align*}5.00\\ 3.45 \\ \underline{+\ 0.56} \\ 9.01 \end{align*}
Our answer is 9.01.
We can do the same thing with a subtraction problem. We add zeros to help hold places where there are not digits. That way each number has the same number of places.
Example
67.89 - 18.4 = _____
First, we line up the problem vertically with the decimal point.
\begin{align*}67.89 \\ \underline{-\ 18.40} \\ 49.49 \end{align*}
Our answer is 49.49.
Now it is time for you to try a few on your own.
- 4.56 + .89 + 2.31 = _____
- 16 - 12.22 = _____
- 88.92 + .57 + 3.12 = _____
Take a few minutes to check your addition and subtraction with a peer. Did you remember to add in the zeros for place holders?
II. Use Mental Math to Add/Subtract Decimals
Sometimes, you don’t need to go through all of the work of lining up decimal points and filling in the zeros. Sometimes you can use mental math to figure out a sum.
When is mental math most helpful with decimal sums and differences?
When you have a decimal where the decimal parts can easily add up to be one whole, you can use mental math to figure out the sum.
Think about this. If you had .30 + .70, you know that 3 + 7 is 10, therefore you know that .30 + .70 is 1.00.
Let’s apply this information.
Example
5.30 + 6.70 = _____
Here we can start by looking at the decimals, since .30 + .70 is 1. Then we combine the whole numbers and add the total of the decimals to get an answer:
5 + 6 = 11 + 1 = 12
Our answer is 12.
What about subtraction?
We can use mental math to complete subtraction problems too.
We just look for which decimals add up to be wholes and go from there.
Let’s look at an example.
Example
25.00 - 22.50 = _____
We are subtracting 25.00 - 22.50, we can think about this problem in reverse to make the mental math simpler.
“What plus 22.50 will give us 25.00?" Think: 2.50 plus what equals 5.00?
25.00 - 22.50 = 2.50
Our answer is 2.50.
Not all problems will be able to be solved mentally, but when we can mental math makes things a whole lot simpler!!
Here are few for you to work on. Add or subtract using mental math.
- 33.50 + 5.50 = _____
- 10 - 3.75 = _____
- 18.25 + 2.25 = _____
Take a few minutes to check your work with a peer. Do your answers match? If so, move on. If not, recheck your work.
III. Identify and Apply the Commutative and Associative Properties of Addition in Decimal Operations
We have just learned how to add and subtract decimals both by using mental math and by completing the arithmetic on a piece of paper by lining up the decimal points.
We can also apply two properties to our work with decimals.
A property is a rule that remains true when applied to certain situations in mathematics.
We are going to work with two properties in this section, the associative property and the commutative property.
Let’s begin by learning about the commutative property.
The commutative property means that you can switch the order of any of the numbers in an addition or multiplication problem around and you will still receive the same answer.
Here is an example.
Example
4 + 5 + 9 = 18 is the same as 5 + 4 + 9 = 18
The order of the numbers being added does not change the sum of these numbers. This is an example of the commutative property.
How can we apply the commutative property to our work with decimals?
We apply it in the same way. If we switch around the order of the decimals in an addition problem, the sum does not change.
Example
4.5 + 3.2 = 7.7 is the same as 3.2 + 4.5 = 7.7
Now we can look at the associative property.
The associative property means that we can change the groupings of numbers being added (or multiplied) and it does not change the sum. This applies to problems with and without decimals.
Example
(1.3 + 2.8) + 2.7 = 6.8 is the same as 1.3 + (2.8 + 2.7) = 6.8
Notice that we use parentheses to help us with the groupings. When we regroup numbers in a different way the sum does not change.
What about variables and decimals?
Sometimes, we will have a problem with a variable and a decimal in it. We can apply the commutative property and associative property here too.
Example
\begin{align*}x + 4.5\end{align*} is the same as \begin{align*}4.5 + x\end{align*}
\begin{align*}(x + 3.4) + 5.6\end{align*} is the same as \begin{align*}x + (3.4 + 5.6)\end{align*}
The most important thing is that the order of the numbers and the groupings can change but the sum will remain the same.
Look at the following examples and name the property illustrated in the example.
- 3.4 + 7.8 + 1.2 = 7.8 + 1.2 + 3.4
- (1.2 + 5.4) + 3.2 = 1.2 + (5.4 + 3.2)
- \begin{align*}x + 5.6 + 3.1 = 3.1 + x + 5.6\end{align*}
Check your work with a peer. Did you name the correct property?
Real Life Example Completed
The Broken Cash Register
Alright Julie, help is now on the way.
Now that we have learned how to add and subtract decimals, we are ready to help Julie with her customer.
Let’s look at the problem once again.
When Julie arrived for her shift at the ice cream stand, she was surprised to find out that the cash register was broken.
“You can just figure out each total and the customer’s change,” Mr. Harris said to Julie with a smile.
Julie grimaced as she got out a pad of paper and pencil. She knew that she was going to need to do some quick addition and subtraction to make this whole day work.
Very soon her first customer arrived. This customer ordered a small cone for $2.25 and gave Julie exact change.
“Maybe this won’t be so tough after all,” Julie thought.
Then her luck ended. A woman arrived and ordered a small cone with sprinkles, caramel, and an extra scoop of ice cream.
Julie quickly jotted the following numbers down on a piece of paper.
While Julie was working to figure out the sum, the woman handed Julie a $10.00 bill and two quarters.
“I am so glad that I have the change,” she said to Julie.
Julie frantically began to work out the math on her piece of paper.
How can Julie add up the decimals?
Is there a way for her to do it mentally?
What about the customer’s change? If the woman gave Julie a ten dollar bill and two quarters, how much change should she get back?
First, let’s underline all of the important information.
Next, we need to figure out the cost of the ice cream cone.
Here are the numbers that Julie wrote down.
2.25 + .10 + .30 + .85 = _____
Next, we need to line up the numbers vertically.
\begin{align*}2.25 \\ .10 \\ .30 \\ \underline{+\ .85} \\ 3.50\end{align*}
The cost of the ice cream cone is $3.50.
Julie took the ten dollar bill and the two quarters from the customer.
We can use mental math to figure out the customer’s change.
\begin{align*}\$10.50 - 3.50 & = \underline{\;\;\;\;\;\;\;\;\;\;}\\ .50 - .50 & = 0\\ 10 - 3 & = 7\end{align*}
Julie confidently handed the customer $7.00 in change. The customer smiled, thanked Julie and left eating her delicious ice cream cone.
Vocabulary
Here are the vocabulary words in this lesson.
- Properties
- the features of specific mathematical situations.
- Associative Property
- a property that states that changing the grouping in an addition problem does not change the sum.
- Commutative Property
- a property that states that changing the order of the numbers in an addition problem does not change the sum.
Technology Integration
Khan Academy Subtracting Decimals
James Sousa, Adding and Subtracting Decimals
Other Videos:
- http://www.gamequarium.org/cgi-bin/search/linfo.cgi?id=7544 – Blackboard video on how to add decimals.
- http://www.gamequarium.org/cgi-bin/search/linfo.cgi?id=7545 – Blackboard video on how to subtract decimals.
Time to Practice
Directions: Add or subtract the following decimals.
1. 4.5 + 6.7 = _____
2. 3.45 + 2.1 = _____
3. 6.78 + 2.11 = _____
4. 5.56 + 3.02 = _____
5. 7.08 + 11.9 = _____
6. 1.24 + 6.5 = _____
7. 3.45 + .56 = _____
8. 87.6 + 98.76 = _____
9. 76.43 + 12.34 = _____
10. 5 + 17.21 = _____
11. 17.65 - 4 = _____
12. 18.97 - 3.4 = _____
13. 22.50 - .78 = _____
14. 27.99 - 1.99 = _____
15. 33.11 - 3.4 = _____
16. 44.59 - 11.34 = _____
17. 78.89 - 5 = _____
18. 222.56 - 11.2 = _____
19. 567.09 - 23.4 = _____
20. 657.80 - 3.04 = _____
Directions: Use mental math to compute each sum or difference.
21. .50 + 6.25 = _____
22. 1.75 + 2.25 = _____
23. 3.50 + 4.50 = _____
24. 7.25 + 1.25 = _____
25. 8.75 + 3.25 = _____
26. 8.50 - 2.50 = _____
27. 10 - 4.50 = _____
28. 12 - 3.75 = _____
29. 15.50 - 5.25 = _____
30. 20 - 15.50 = _____
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