Chapter Notes: Adding Decimals (Tenths)
When you work with money, measurements, or scores, you often use numbers that are not whole numbers. These are called decimals. A decimal has a decimal point (a dot) that separates the whole number part from the fractional part. In this chapter, you will learn how to add decimals when the numbers have tenths. Tenths are the first place to the right of the decimal point. Adding decimals with tenths is very similar to adding whole numbers, but you need to pay careful attention to lining up the decimal points!
Understanding Tenths
Before you can add decimals with tenths, you need to understand what tenths are. A tenth is one part out of ten equal parts. When you divide something into ten equal pieces, each piece is one tenth.
We write one tenth as a decimal like this: 0.1
The digit 1 is in the tenths place, which is the first place to the right of the decimal point.
Here are some more examples of decimals with tenths:
- 0.3 means 3 tenths
- 0.7 means 7 tenths
- 2.5 means 2 ones and 5 tenths
- 4.9 means 4 ones and 9 tenths
Think of a dollar. If you divide one dollar into ten equal parts, each part is one dime, or one tenth of a dollar. So 0.1 dollar equals 1 dime, 0.5 dollar equals 5 dimes, and so on.
Place Value with Tenths
Understanding place value helps you add decimals correctly. Each digit in a number has a place and a value.
The number 123.4 means 1 hundred, 2 tens, 3 ones, and 4 tenths. The decimal point separates the whole number part (123) from the decimal part (4 tenths).
Modeling Decimals with Tenths
Visual models help you understand decimals better. You can use base-ten blocks or grid models to show decimals with tenths.
Using a Grid Model
Imagine a rectangle divided into 10 equal strips. Each strip represents one tenth, or 0.1.
- If you shade 3 strips out of 10, you have shaded 3 tenths, or 0.3
- If you shade 7 strips out of 10, you have shaded 7 tenths, or 0.7
Think of a pizza cut into 10 equal slices. If you eat 4 slices, you have eaten 4 tenths (0.4) of the pizza.
Using a Number Line
A number line can also show decimals with tenths. Between each whole number, you mark 10 equal spaces. Each space represents one tenth.
On a number line from 0 to 1, the marks would be at 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0.
Adding Decimals with Tenths Using Models
Before you learn the standard way to add decimals, it helps to see addition using models. This builds your understanding of what happens when you add decimals.
Example: Add 0.2 + 0.5 using a grid model.
What is 0.2 + 0.5?
Solution:
Draw a rectangle divided into 10 equal strips.
Shade 2 strips to show 0.2.
Shade 5 more strips to show 0.5.
Count the total number of shaded strips = 7 strips.
7 strips out of 10 means 7 tenths = 0.7
The sum of 0.2 + 0.5 is 0.7.
Example: Add 1.3 + 0.4 using base-ten blocks.
What is 1.3 + 0.4?
Solution:
Use 1 flat (representing 1 whole) and 3 rods (representing 3 tenths) to show 1.3.
Use 4 rods (representing 4 tenths) to show 0.4.
Combine all the pieces: 1 flat and 7 rods total.
This represents 1 whole and 7 tenths = 1.7
The sum of 1.3 + 0.4 is 1.7.
Adding Decimals with Tenths Using the Standard Algorithm
The standard way to add decimals is very similar to adding whole numbers. The most important rule is: line up the decimal points.
Steps for Adding Decimals with Tenths
- Write the numbers vertically (one on top of the other).
- Line up the decimal points so they are directly above each other.
- Add the digits in each place value column, starting from the right.
- Bring the decimal point straight down into your answer.
- If needed, regroup (carry) just like with whole numbers.
Adding Without Regrouping
When the sum of the digits in any place is less than 10, you do not need to regroup.
Example: Add 3.2 + 1.5
What is 3.2 + 1.5?
Solution:
Write the numbers vertically and line up the decimal points:
3.2
+ 1.5Add the tenths: 2 tenths + 5 tenths = 7 tenths
Add the ones: 3 ones + 1 one = 4 ones
Bring the decimal point straight down:
3.2
+ 1.5
---
4.7The sum is 4.7.
Example: Add 5.1 + 2.3
What is 5.1 + 2.3?
Solution:
Write the numbers vertically and line up the decimal points:
5.1
+ 2.3Add the tenths: 1 tenth + 3 tenths = 4 tenths
Add the ones: 5 ones + 2 ones = 7 ones
Bring the decimal point straight down:
5.1
+ 2.3
---
7.4The sum is 7.4.
Adding With Regrouping
Sometimes when you add the tenths, the sum is 10 or more. When this happens, you need to regroup (also called carrying). You regroup 10 tenths as 1 whole.
Example: Add 2.7 + 1.5
What is 2.7 + 1.5?
Solution:
Write the numbers vertically and line up the decimal points:
2.7
+ 1.5Add the tenths: 7 tenths + 5 tenths = 12 tenths
Regroup 12 tenths as 1 one and 2 tenths. Write 2 in the tenths place and carry 1 to the ones place.
Add the ones: 2 ones + 1 one + 1 (carried) = 4 ones
Bring the decimal point straight down:
¹
2.7
+ 1.5
---
4.2The sum is 4.2.
Example: Add 3.8 + 2.6
What is 3.8 + 2.6?
Solution:
Write the numbers vertically and line up the decimal points:
3.8
+ 2.6Add the tenths: 8 tenths + 6 tenths = 14 tenths
Regroup 14 tenths as 1 one and 4 tenths. Write 4 in the tenths place and carry 1 to the ones place.
Add the ones: 3 ones + 2 ones + 1 (carried) = 6 ones
Bring the decimal point straight down:
¹
3.8
+ 2.6
---
6.4The sum is 6.4.
Adding a Decimal and a Whole Number
Sometimes you need to add a decimal with tenths to a whole number. Remember that every whole number has a decimal point, even if we do not write it. For example, 5 is the same as 5.0.
When adding a whole number and a decimal, you can write a decimal point and a zero after the whole number to help you line up the place values correctly.
Example: Add 4 + 2.3
What is 4 + 2.3?
Solution:
Write 4 as 4.0 to show the tenths place.
Write the numbers vertically and line up the decimal points:
4.0
+ 2.3Add the tenths: 0 tenths + 3 tenths = 3 tenths
Add the ones: 4 ones + 2 ones = 6 ones
Bring the decimal point straight down:
4.0
+ 2.3
---
6.3The sum is 6.3.
Example: Add 7 + 1.8
What is 7 + 1.8?
Solution:
Write 7 as 7.0.
Write the numbers vertically and line up the decimal points:
7.0
+ 1.8Add the tenths: 0 tenths + 8 tenths = 8 tenths
Add the ones: 7 ones + 1 one = 8 ones
Bring the decimal point straight down:
7.0
+ 1.8
---
8.8The sum is 8.8.
Adding More Than Two Decimals
You can add three or more decimals with tenths using the same method. Just line up all the decimal points and add column by column.
Example: Add 1.2 + 3.4 + 2.1
What is 1.2 + 3.4 + 2.1?
Solution:
Write all three numbers vertically and line up the decimal points:
1.2
3.4
+ 2.1Add the tenths: 2 tenths + 4 tenths + 1 tenth = 7 tenths
Add the ones: 1 one + 3 ones + 2 ones = 6 ones
Bring the decimal point straight down:
1.2
3.4
+ 2.1
---
6.7The sum is 6.7.
Example: Add 0.5 + 1.8 + 2.3
What is 0.5 + 1.8 + 2.3?
Solution:
Write all three numbers vertically and line up the decimal points:
0.5
1.8
+ 2.3Add the tenths: 5 tenths + 8 tenths + 3 tenths = 16 tenths
Regroup 16 tenths as 1 one and 6 tenths. Write 6 in the tenths place and carry 1 to the ones place.
Add the ones: 0 ones + 1 one + 2 ones + 1 (carried) = 4 ones
Bring the decimal point straight down:
¹
0.5
1.8
+ 2.3
---
4.6The sum is 4.6.
Real-World Applications of Adding Decimals with Tenths
Adding decimals with tenths is very useful in everyday life. Here are some situations where you might use this skill.
Money
Money problems often use decimals. When you add amounts of money, you are adding decimals.
Example: Maria bought a pencil for $0.50 and an eraser for $0.30.
How much did she spend in total?How much money did Maria spend?
Solution:
Add the two amounts: $0.50 + $0.30
Write the numbers vertically and line up the decimal points:
$0.50
+ $0.30Add the hundredths: 0 hundredths + 0 hundredths = 0 hundredths
Add the tenths: 5 tenths + 3 tenths = 8 tenths
Bring the decimal point and dollar sign down:
$0.50
+ $0.30
---
$0.80Maria spent $0.80 in total.
Measurement
When you measure length, weight, or capacity, you often get decimal numbers. Adding these measurements uses the same skill.
Example: A ribbon is 2.4 meters long.
Another ribbon is 1.5 meters long.
If you connect them end to end, how long are they together?What is the total length of both ribbons?
Solution:
Add the two lengths: 2.4 + 1.5
Write the numbers vertically and line up the decimal points:
2.4
+ 1.5Add the tenths: 4 tenths + 5 tenths = 9 tenths
Add the ones: 2 ones + 1 one = 3 ones
Bring the decimal point straight down:
2.4
+ 1.5
---
3.9The total length is 3.9 meters.
Sports and Scores
In some sports, scores are given as decimals. Judges in gymnastics or diving give scores with tenths.
Example: A gymnast received scores of 8.7, 9.2, and 8.9 from three judges.
What is the total of all three scores?What is the sum of the three scores?
Solution:
Add the three scores: 8.7 + 9.2 + 8.9
Write the numbers vertically and line up the decimal points:
8.7
9.2
+ 8.9Add the tenths: 7 tenths + 2 tenths + 9 tenths = 18 tenths
Regroup 18 tenths as 1 one and 8 tenths. Write 8 in the tenths place and carry 1 to the ones place.
Add the ones: 8 ones + 9 ones + 8 ones + 1 (carried) = 26 ones
Bring the decimal point straight down:
¹
8.7
9.2
+ 8.9
---
26.8The total score is 26.8.
Common Mistakes and How to Avoid Them
When adding decimals with tenths, students sometimes make mistakes. Here are the most common errors and tips to avoid them.
Not Lining Up the Decimal Points
The most common mistake is not lining up the decimal points. When the decimal points are not aligned, you are adding digits from different place values, which gives the wrong answer.
Wrong way:
2.3
+ 14.5
---
16.8 (This is incorrect!)Correct way:
2.3
+ 14.5
---
16.8 (Line up the decimal points correctly!)
Tip: Always write the numbers so the decimal points form a straight vertical line.
Forgetting to Bring Down the Decimal Point
After you finish adding, you must bring the decimal point straight down into your answer. If you forget, your answer will be wrong.
Tip: As soon as you write the problem, draw a decimal point in the answer row directly below the other decimal points. This helps you remember.
Not Regrouping When Needed
When the sum of the tenths is 10 or more, you must regroup. Forgetting to regroup gives the wrong answer.
Tip: Always check if the sum in any column is 10 or more. If it is, write the ones digit and carry the tens digit to the next column.
Summary
Adding decimals with tenths is an important skill you will use often. Remember these key points:
- A tenth is one part out of ten equal parts, written as 0.1
- The tenths place is the first place to the right of the decimal point
- Always line up the decimal points when adding decimals
- Add from right to left, just like with whole numbers
- Bring the decimal point straight down into your answer
- Regroup (carry) when the sum in any place is 10 or more
- You can write a whole number as a decimal by adding .0 at the end
With practice, adding decimals with tenths will become easy and automatic. You will use this skill when working with money, measurements, and many other real-world situations!
