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## Add and subtract decimals by lining up the decimal points.

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Have you ever been on a relay team? Have you ever run a relay race?

Last year, the girl’s middle school relay team ran the \begin{align*}4 \times 100\end{align*} meters is 53.87 and won the regional championship. This year, there are four new girls on the team and they have set a goal of running the \begin{align*}4 \times 100\end{align*} in a faster time than the girls did the year before.

Jessica is the lead runner on the team. The second runner is Karin. The third runner is Tasha and the fourth runner and fastest runner is Uniqua. All four girls worked and trained hard. They didn’t know each prior to making the team, so it took the girls most of the season to get to know each other’s strengths and weaknesses. Their coach, Ms. Sutter worked hard to help the girls work through their differences while maintaining a good attitude. Ms. Sutter doesn’t put up with a lot of nonsense and all of the runners show respect to her and their team mates.

Because of their hard work, the relay team made it all the way to the Regional championship.

“Just be sure to run your best race,” Ms. Sutter told them prior to the start.

All four girls shook hands and prepared to run. Here are their individual times:

Jessica = 13.00

Karin = 14.10

Tasha = 14.23

Uniqua = 12.40

Did the girls accomplish their goal of running the relay faster than the previous team?

This Concept is all about adding decimals. To figure out the total time of the relay team, you will need to know how to add decimals. Finish this Concept and you will be ready!

### Guidance

Decimals appear in all aspects of life—any time there is a part of a whole number—from money to measurement to percentages. Meghan finished 0.75 of her homework; Leo ran 1.34 kilometers; the class ate 5.6 pizzas.

Previously we worked on how to compare, order, and round decimals.

In this Concept, we will learn how to add them. Once you know how to add decimals, you can apply your knowledge to a whole new dimension of real-world situations, some of which involve algebraic reasoning and equations.

How do we get started when we want to add decimals?

Well, first of all, it all comes back down to place value. To add decimals, we line up the digits according to place value. To keep the place values straight, we always align the decimal points of the decimals.

Once we have the decimal points lined up, we add exactly like we add whole numbers—from right to left—taking care to bring the decimal point down into our answer.

There are a few things to notice about this problem.

First, the numbers you are adding have been lined up vertically. When you were in elementary school, most of your problems were probably lined up vertically. Now that you are in middle school from now on, you will need to do this step yourself.

Second, you can see that a zero was added to help even out the columns to line up the decimal points. That zero is in red to help you to see that it was added.

\begin{align*}4.5 + 2.137 = \underline{\;\;\;\;\;\;\;\;}\end{align*}

To add this problem, let’s remember to line up the decimal points and each digit according to place value. Notice that there is one digit after the decimal point in the first number and three in the second number. You will need to add some zeros to help with the addition. Now let’s line up the problem.

Sometimes, particularly in real-world situations, decimal numbers are just too long to work with. In these cases, to find a quick and workable solution we might round the decimals before we add. For instance, rather than getting bogged down in the ten-millionth place, we can round to the tenth or hundredth place and then add. Look for clues in the problem to tell you whether or not to round before adding. If you are asked to round, make sure you round to the right place!

Let’s look at a problem where rounding would be helpful.

Round each number to the nearest hundredth then find the sum of 9.199, 11.353, 10.648.

This problem asks us to round before adding and it asks us to round to the hundredth place. To make the rounding steps clear, it helps to underline the number you’re rounding to and bolding or circling the number directly to the right of it. Because we’re rounding to the hundredth place, we’re going to round to the two places to the right of the decimal place. The bolded number, the thousandths place, is the one you’ll look at when deciding to round up or down.

9.199 \begin{align*}\rightarrow\end{align*} rounded to the hundredths place \begin{align*}\rightarrow\end{align*} 9.20

11.353 \begin{align*}\rightarrow\end{align*} rounded to the hundredths place \begin{align*}\rightarrow\end{align*} 11.35

10.648 \begin{align*}\rightarrow\end{align*} rounded to the hundredths place \begin{align*}\rightarrow\end{align*} 10.65

Now that the numbers are rounded, we add zeros, line up the decimal places, and add.

Our answer is 31.2.

Now it's time to try a few on your own.

#### Example A

\begin{align*}4.56 + 1.2 + 37.89 = \underline{\;\;\;\;\;\;\;\;}\end{align*}

Solution: \begin{align*}43.65\end{align*}

#### Example B

\begin{align*}14.2 + 56.78 + 123.4 = \underline{\;\;\;\;\;\;\;\;}\end{align*}

Solution: \begin{align*}194.38\end{align*}

#### Example C

\begin{align*}189.34 + 123.5 + 7.2 = \underline{\;\;\;\;\;\;\;\;}\end{align*}

Solution: \begin{align*}320.04\end{align*}

Now back to the relay race.

Here is the original problem once again.

Last year, the girl’s middle school relay team ran the \begin{align*}4 \times 100\end{align*} meters is 53.87 and won the regional championship. This year, there are four new girls on the team and they have set a goal of running the \begin{align*}4 \times 100\end{align*} in a faster time than the girls did the year before.

Jessica is the lead runner on the team. The second runner is Karin. The third runner is Tasha and the fourth runner and fastest runner is Uniqua. All four girls worked and trained hard. They didn’t know each prior to making the team, so it took the girls most of the season to get to know each other’s strengths and weaknesses. Their coach, Ms. Sutter worked hard to help the girls work through their differences while maintaining a good attitude. Ms. Sutter doesn’t put up with a lot of nonsense and all of the runners show respect to her and their team mates.

Because of their hard work, the relay team made it all the way to the Regional championship.

“Just be sure to run your best race,” Ms. Sutter told them prior to the start.

All four girls shook hands and prepared to run. Here are their individual times:

Jessica = 13.00

Karin = 14.10

Tasha = 14.23

Uniqua = 12.40

Did the girls accomplish their goal of running the relay faster than the previous team?

First, we can write an equation to figure out the time of the race.

\begin{align*}j+k+t+u<53.87\end{align*}

The girls want to have a time that is less than 53.87. In this way, we are going to be combining comparing decimals and adding decimals.

Next, we fill in the given times for each girl into the equation.

\begin{align*}13.00 + 14.10 + 14.23 + 12.40 < 53.87\end{align*}

Now, we can line up the decimal point to add the values.

The time of the relay is 53.73.

Now we can compare this with the previous team’s time.

53.73 < 53.87

This is a true statement. The girls have beaten the previous relay team’s time. They also achieved a medal at the Regional Championship!!

### Vocabulary

Decimal
a part of a whole represented by digits to the right of a decimal point.
Estimate
to find an approximate solution to a problem.
Rounding
a method of estimating where you rewrite a decimal or whole number according to the place value that it is closest to.

### Guided Practice

Here is one for you to try on your own.

Round each value to the nearest tenth and then find the sum.

\begin{align*}4.56, 3.47, 2.04,6.21\end{align*}

First, we round each to the nearest tenth.

4.56 = 4.6

3.47 = 3.5

2.04 = 2.0

6.21 = 6.2

Now we add them up to find the sum.

\begin{align*}16.3\end{align*}

This is our answer.

### Practice

Directions: Find each sum.

1. 29.451 + 711.36

2. 0.956 + 0.9885

3. 13.69 + 42.23

4. 58.744 + 12.06 + 8.8

5. 67.90 + 1.2 + 3.456

Directions: Round to the nearest tenth, then find the sum.

6. 88.95 + 23.451

7. 690.026 + 831.163

8. 3.27 + 7.818

9. .99 + .909 + .048

10. 5.67 + 3.45 + 1.23

Directions: Round to the nearest hundredth, then find the sum.

11. 511.51109 + 99.0986

12. .744 + 7.005 + .7071

13. 32.368 + 303.409

14. 1.0262 + 1.0242

15. 1.309 + 3.4590

### Vocabulary Language: English

Decimal

Decimal

In common use, a decimal refers to part of a whole number. The numbers to the left of a decimal point represent whole numbers, and each number to the right of a decimal point represents a fractional part of a power of one-tenth. For instance: The decimal value 1.24 indicates 1 whole unit, 2 tenths, and 4 hundredths (commonly described as 24 hundredths).
Estimate

Estimate

To estimate is to find an approximate answer that is reasonable or makes sense given the problem.
Rounding

Rounding

Rounding is reducing the number of non-zero digits in a number while keeping the overall value of the number similar.