Decimals are a common occurrence in daily life in addition to whole numbers. Situations related to money, height, weight, and other quantities expressed as fractional parts are examples of decimal application. Computing both the whole number component and the decimal component helps us solve problems precisely when we know how to add and subtract decimals correctly.

## Addition of Decimals

Decimal addition is somewhat like fraction addition in terms of how it works. Remember that you can add the whole numbers and fractional parts separately to get the total when adding mixed fractions. However, there are additional procedures to consider based on the given numbers and whether they need to be simplified. Nevertheless, by examining the fundamental steps in adding fractions, adding decimals uses the same efforts with careful handling of the decimal points because alignment is essential.

**How to Add Decimals**

The steps to add decimals are as follows:

*Step 1**:* Arrange the numbers so that the decimal points are vertically aligned and written one on top of the other.** Step 2: **Insert placeholder zeros (0s) where needed.

**Add the numbers as whole numbers.**

*Step 3:***Align the decimal point in the sum with the decimal points of the addends.**

*Step 4:***Addition of ***Like*** Decimals**

*Like*

The addition of like decimals is simply aligning the decimal points vertically and adding the given as whole numbers. The addition of like decimals has addends with the same number of digits after the decimal points. Examples of like decimals are

0.78 and 0.85, they both have two (2) digits after the decimal point

12.875 and 255. 679, they both have three (3) digits after the decimal point

44.7, 9.5, 17.8, and 16.5 have one (1) digit after the decimal place.

Let us follow the steps in adding 0.23 and 0.45.

*Step 1:** Arrange the numbers so that the decimal points are vertically aligned and written one on top of the other.*

The two numbers provided, 0.23 and 0.45, are *like* decimals with two digits following the decimal point.

*Step 2:*** ***Insert placeholder zeros (0s) where needed.*

Inserting zeros is not applicable since both numbers have the same number of digits after the decimal place.

*Step 3:**Add the numbers as whole numbers.*

The addition must start at the right and move left. Add 3 and 5, 2 and 4, and 0 and 0, respectively.

*Step 4:**Align the decimal point in the sum with the decimal points of the addends.*

Write the decimal point in the sum between 0 and 6 since the decimal points must be vertically aligned.

Thus, the sum of 0.23 and 0. 45 is **0.68.**

**Addition of ***Unlike*** Decimals**

*Unlike*

Adding unlike decimals requires careful inserting of zeros as a placeholder to ensure that the given numbers are vertically aligned. Since, unlike decimals do not have the same digits after the decimal point, inserting zeros allow us to align the addends properly before addition. The examples of unlike decimals are 23.56 and 12.156, and 1.4 and 3.89.

23.56 has two digits after the decimal point, while 12.156 has three digits after the decimal point.

1.4 has one digit after the decimal point, while 3.89 has two digits after the decimal point.

Let us add the following numbers: 20.123, 56.47, and 10.3

*Step 1:** Arrange the numbers so that the decimal points are vertically aligned and written one on top of the other.*

The numbers 20.123, 56.47, and 10.3 are unlike decimals.

*Step 2: **Insert placeholder zeros (0s) where needed.*

Since 20.123 has three decimal digits, the most after the decimal point, we must also write 56.47 and 10.3 with three decimal digits by adding zeros. 10.3 should be written as 10.300, and 56.47 as 56.470.

*Step 3: **Add the numbers as whole numbers.*

The addition must start at the right (thousandth place) and move left.

3 + 0 + 0 = 3

2 + 7 + 0 = 9

1 + 4 + 3 = 8

0 + 6 + 0 = 6

2 + 5 + 1 = 8

*Step 4:** Align the decimal point in the sum with the decimal points of the addends.*

Write the decimal point in the sum between 6 and 8 to align it with the addends.

Therefore, on adding decimals 20.123, 56.47, and 10.3, we get **86.893.**

**Addition of Whole Numbers and Decimals**

Adding whole numbers and decimals follows the same process as adding, unlike decimals. Inserting zero is essential to ensure that the given become aligned vertically with their decimal points.

Let us say, for example: Add 674 and 11.93

*Step 1:** Arrange the numbers so that the decimal points are vertically aligned and written one on top of the other.*

The decimal point of 674 appears after 4. Converting the whole number to a decimal number is necessary to align the addends vertically with their decimal points.

*Step 2: **Insert placeholder zeros (0s) where needed.*

We need to insert two zeros to the whole number 674 and write it as 674.00 since there are two digits after the decimal point of 11.93.

*Step 3: **Add the numbers as whole numbers.*

Perform addition by starting at the right and moving left.

0 + 3 = 3

0 + 9 = 9

4 + 1 = 5

7 + 1 = 8

6 + 0 = 6

*Step 4:** Align the decimal point in the sum with the decimal points of the addends.*

In the sum, the decimal point must appear between 5 and 9.

Hence, the sum is **685.93** if we add 674 and 11.93.

**Addition of Decimals with Regrouping**

The addition of decimals with regrouping is like adding whole numbers with regrouping. When regrouping happens in addition, it means that when we add the addends vertically in each column, from right to left, the sum of the addends is greater than nine, and we need to carry over the extra digit to the following column on the left.

Let us understand the addition of decimals with regrouping using this example:

Add 197. 356 + 18.39

*Step 1:*

*Step 2: **Insert placeholder zeros (0s) where needed.*

Insert one zero to 18.39.

*Step 3: **Add the numbers as whole numbers.*

Perform addition by starting at the right and move left. Only 1 digit can be entered in each column, so in the column 5+9=14, only 4 is written and 1 is carried over to the column to the left. In the column 3+3, since we have carried over one, this will be 3+3+1 which is equal to 7. Regrouping also happened in the columns 7+8 and 9+1. That is,

6 + 0 = 6

5 + 9 = 14

3 + 3 + 1 ( carry over ) = 7

7 + 8 = 15

9 + 1 + 1 ( carry over ) = 11

1 + 1 ( carry over ) = 2

*Step 4:** Align the decimal point in the sum with the decimal points of the addends.*

In the sum, the decimal point must appear between 5 and 7.

Thus, 197. 356 + 18.39 is equal to **215.746**.

## Subtraction of Decimals

Subtraction of decimals involves the usual steps in subtracting whole numbers; however, the alignment of decimal points is critical, allowing place values to fall in place. There may be cases that we will subtract like decimals, unlike decimals, or whole numbers and decimals. Regrouping or borrowing may also take place as needed. Regrouping happens when the number in the minuend is smaller than the number in the subtrahend when we vertically arrange the given decimal numbers.

**How to Subtract Decimals**

The following are the steps to subtract decimals:

*Step 1:** *Arrange the numbers such that the decimal points are vertically aligned, and they are written one on top of the other.** Step 2: **Insert placeholder zeros (0s) where needed.

**Subtract the numbers as whole numbers.**

*Step 3:*

*Step 4:**Align the decimal point in the difference with the decimal points of the given.*

**Subtraction of ***Like*** Decimals**

*Like*

The key to subtracting like decimal is aligning the decimal points vertically. The process of subtracting like decimals is the same as subtracting whole numbers, and the decimal points are aligned correctly.

Let us look at this example.

Subtract: 46.89 – 32.51

*Step 1:** Arrange the numbers such that the decimal points are vertically aligned, and they are written one on top of the other.*

46.89 is the minuend while 32.51 is the subtrahend. The given numbers are like decimals since they have the same number of digits after the decimal point.

*Step 2: **Insert placeholder zeros (0s) where needed.*

Since the given numbers have the same digits after the decimal point, there is no need to insert zeros.

*Step 3: **Subtract the numbers as whole numbers.*

9 – 1 = 8

8 – 5 = 3

6 – 2 = 4

4 – 3 = 1

*Step 4:** Align the decimal point in the difference with the decimal points of the given.*

The decimal point must appear between 4 and 3.

**Subtraction of ***Unlike*** Decimals**

*Unlike*

In subtracting unlike decimals, the steps are like subtracting whole numbers, but it is necessary to convert them to like decimals by inserting placeholder zeros.

Let us subtract these decimals step-by-step.

Subtract the decimals: 24.865 – 12.5

*Step 1:** Arrange the numbers such that the decimal points are vertically aligned, and they are written one on top of the other.*

24.865 is the minuend while 12.5 is the subtrahend. These are unlike decimals since they do not have the same number of digits after the decimal point. 24.865 has three (3) digits after the decimal point while, 12.5 has only one (1) digit after the decimal point.

*Step 2: **Insert placeholder zeros (0s) where needed.*

To make the given unlike decimals to like decimals, we need to add two (2) zeros to 12.5. Thus, 24.865 and 12.5 have three (3) digits after the decimal point.

*Step 3: **Subtract the numbers as whole numbers.*

5 – 0 = 5

6 – 0 = 5

8 – 5 = 3

4 – 2 = 2

2 – 1 = 1

*Step 4:** Align the decimal point in the difference with the decimal points of the given.*

The decimal point must appear between 2 and 3.

Therefore, when we subtract 12.5 from 24.865, the difference is **12.365**.

**Subtraction of Decimals with Regrouping**

Regrouping or borrowing happens when the number in the minuend is smaller than the number in the subtrahend.

Let us subtract 24.87 from 55.46 and follow the steps in subtracting decimals.

*Step 1:** Arrange the numbers such that the decimal points are vertically aligned, and they are written one on top of the other.*

**In this example, 55.46 is the minuend while 24.87 is the subtrahend. **

*Step 2: **Insert placeholder zeros (0s) where needed.*

Inserting zeros is not needed. 24.87 and 55.46 are like decimals, so it is easy to align them vertically since they have the same number of digits after the decimal point.

*Step 3: **Subtract the numbers as whole numbers.*

Regrouping or borrowing must occur since we cannot take seven (7) ones from six (6) ones. Therefore, we will borrow one (1) ten from four (4) of the column in the minuend to the left. In the next column, 3 – 8, borrowing must also occur, making the following number in the minuend four.

16 – 7 = 9

13 – 8 = 5

4 – 4 = 0

5 – 2 = 3

*Step 4:** Align the decimal point in the difference with the decimal points of the given.*

Write the decimal point in the difference between 0 and 5.

Hence, when we subtract 24.87 from fifty-five. 46, the difference is **30.59**.

*Checking: *

Remember that addition is the inverse of subtraction, so to can check our answer by adding the difference and the subtrahend to get the minuend as the total.

9 + 7 = 16

5 + 8 + 1 ( carry over ) = 14

1 ( carry over ) + 4 = 5

3 + 2 = 5

## Why can we insert more zeros?

In adding and subtracting decimals, inserting zeros as placeholders do not change or alter the value of the decimal. It means that the added zero may hold a place, but its place value is always zero.

Let us say, for instance, these numbers: 2.0, 11.00, and 135.000. The added zeros after the decimal points occupy the tenths, hundredths, or thousandths place, respectively, but the place value is zero.

Hence, it is safe to say that 2.0 is equal to 2, 11.00 is equal to 11, and 135.000 is equal to 135.

## More Examples

**Example 1:** Add 345.23 + 645.8 + 12

** Solution: **Properly align the decimal points of the given addends. Inserting zeros is necessary since these are unlike decimals. The image below shows the regrouping in columns.

3 + 0 + 0 = 3

2 + 8 + 0 = 10

5 + 5 + 2 + 1 ( carry over ) = 13

4 + 4 + 1 + 1 ( carry over ) = 10

3 + 6 + 1 ( carry over ) = 10

**Example 2:** The total of three decimals is 927.76. If the two decimals are 324.18 and 121.16, find the third decimal.

** Solution: **In this problem, we will use both addition and subtraction of decimals. Since we already know the two decimals, let us first get their sum.

8 + 6 = 14

1 + 1 + 1 ( carry over ) = 3

4 + 1 = 5

2 + 2 = 4

3 + 1 = 4

Now, let us subtract 445.34 from 927.76 to get the third number.

6 – 4 = 2

7 – 3 = 4

7 – 5 = 2

12 – 4 = 8

8 – 4 = 4

Let us check this by getting the sum of the three decimals, 324.18, 121.16, and 482. 42.

*Checking:*

8 + 6 + 2 = 16

1 + 1 + 4 + 1 ( carry over ) = 7

4 + 1 + 2 = 7

2 + 2 + 8 = 12

3 + 1 + 4 + 1 ( carry over ) = 9

Thus, the third number is **482.42.**

**Example 2:** A $3.75 pen is what Sophia wants to buy. If she already has $ 1.2, and her mother gave her $ 1.45, how much money does she need to buy the pen?

** Solution:** Let us start by calculating the total amount of money Sophie has, including her and her mother’s money.

0 + 5 = 5

2 + 4 = 6

1 + 1 = 2

Sophia has a total of $2.65 on hand. To find how much more she needs to buy the pen, let us subtract $2.65 from the pen cost, which is $3.75.

5 – 5 = 0

7 – 6 = 1

3 – 2 = 1

Therefore, Sophia needs **$1.10** more to buy the pen she wants.

**Example 3:** Following are the weights of the four students:

Maria – 32.6 kg. Joseph – 39. 75

Bernie – 29.5 kg. Thalia – 34. 25

- What is the total weight of Maria and Joseph?
- What is the total weight of Bernie and Thalia?
- What is the weight difference between Joseph and Maria?
- Find the difference between the two weights that are the heaviest and the lightest.

*Solution:*

- To find the total weight of Maria and Joseph, let us add 32.6 kg. and 39.75 kg.

0 + 5 = 5

6 + 7 = 13

2 + 9 + 1 ( carry over ) = 12

3 + 3 + 1 ( carry over ) = 7

The total of Maria and Joseph is **72.35 kg**.

- To find the total weight of Bernie and Thalia, let us add 29.5 kg. and 34.25 kg.

0 + 5 = 5

5 + 2 = 7

9 + 4 = 13

2 + 3 + 1 ( carry over ) = 6

The combined weight of Bernie and Thalia is **63.75 kg**.

- To find the difference between Joseph’s and Maria’s weight, let us subtract Maria’s weight (32.6 kg) from Joseph’s ( 39.75 kg. ).

5 – 0 = 5

7 – 6 = 1

9 – 2 = 7

3 – 3 = 0

Therefore, there is a **7.15 kg** weight difference between Joseph and Maria.

- Since Joseph has the heaviest weight and Bernie has the lightest weight, let us subtract Bernie’s weight ( 29.5 kg. ) from Joseph’s ( 39.75 kg. ) to get the difference.

Because of this, there is a **10.25 kg** difference in weight between the lightest and heaviest weights.

## Summary

- The steps in adding decimals are as follows:
*Step 1:*Insert placeholder zeros (0s) where needed.*Step 2:*Add the numbers as whole numbers.*Step 3:*Align the decimal point in the sum with the decimal points of the addends.*Step 4:*

- The following are the steps to subtract decimals:
*Step 1:*Insert placeholder zeros (0s) where needed.*Step 2:*Subtract the numbers as whole numbers.*Step 3:*Align the decimal point in the difference with the decimal points of the given.*Step 4:*

- Inserting zeros after the decimal place does not change or alter the value of the decimal number. It means that the added zero may hold a place, but its place value is always zero.
- Always line up the decimal points vertically when adding and subtracting decimals to align the numbers according to their place values properly.
- Perform carrying over or borrowing when needed in adding and subtracting decimals.
- When checking the answer in the subtraction of decimals, we can add obtained difference and the subtrahend, but the sum must equal the minuend.

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