## Key Concepts

- Addition and subtraction of decimals by lining up the decimals
- Adding decimals using properties of numbers and also Number Line
- Subtracting decimals using number line and also using partial difference method

### Introduction

In this chapter, we will learn about adding and subtracting decimals just like adding and subtracting whole numbers and using commutative and associative properties of numbers, partial addition, and subtraction, and also using number lines.

## Addition of Decimals Using strategies

Decimals can be added like the way we add whole numbers. There are different ways of adding decimals.

- Lining them using place values
- Using properties of numbers (Commutative property and Associative property)
- Using number line
- Using partial sums

### Let us learn one by one

#### Lining them using place value

In this method, we follow the following steps:

Write down the numbers one under the other, with the decimal points lined up

Add zeros to the right of the number so that the number to be added are of same digits

Then add using column addition and remember to put the decimal point in the answer

For example,

Add 3.456 to 2.4

**Step 1:** Line up the decimals

**Step 2:** Add zeros to the right side of the number if needed

**Step 3:** Add using column addition

### Using properties of numbers

The two properties of numbers are:

Commutative property of addition

Associative property of addition

Commutative property of addition means that you can switch the order of any of the numbers in an addition, the answer remains the same.

**For example:**

Sum of 4.2 + 3.5 = 7.7

By changing the order of the addends, 3.5 + 4.2 = 7.7

That is, 4.2 + 3.5 = 3.5 + 4.2 = 7.7

- Associative property of addition means that you can change the groupings of numbers being added and it does not change the result.

For example, (2.3 + 4.6) + 7.4 = 2.3 + (4.6 + 7.4)

Let us check

LHS = (2.3 + 4.6) + 7.4 = 14.3

RHS = 2.3 + (4.6 + 7.4) = 14.3

⸫ (2.3 + 4.6) + 7.4 = 2.3 + (4.6 + 7.4) = 14.3

#### Using number line

To add decimals using number line, labelling the number line with decimals is very important. We know how to label a number line using whole numbers.

The number line with decimals will look like this

Here, we started labelling the number line with zero and increase by 0.1. We can label the number line increased by 0.25 or 0.5 or 0.75 etc. We can also start the number line with the numbers given in the question.

**For example**:

Add 3.6 + 0.8

We can start labelling the number line starting with 3.6 and increase by 0.1

Then start adding 0.8 to 3.6, we get 4.4

⸫ 3.6 + 0.8 = 4.4

#### Using Partial sums

In this we use what we already know about adding decimals. But in partial addition, we

break the numbers up in the individual places and add.

**For example**:

Add 4.65 + 2.76 using partial addition

⸫ 4.65 + 2.76 = 7.41

### Subtraction of Decimals Using strategies

Decimals can be subtracted just like the subtraction of whole numbers. The different ways of subtracting decimals are:

- Lining them using place values
- Using number line
- Using partial differences

### Let us learn one by one

#### Lining them using place values

This is the same as addition. But instead of adding, we subtract the decimals.

To subtract decimals by lining them using place values, we follow the following steps:

**Step 1:** Line up the decimal points in a column. When needed add a zero to the left of the number to match the number of digits.

**Step 2:** Start on the right, and subtract each column in turn. Remember, we are subtracting digits in the same place value position.

**Step 3:** If the digit you are subtracting is bigger than the digit you are subtracting from, you have to borrow a group of ten from the column to the left.

**For example**:

Subtract 4.65 – 2. 49

### Using number line

To subtract decimal numbers using a number line. Start on the far-right side on the number line and label it backwards by tenths. This is nothing but counting back.

**For example**:

Subtract 4.8 – 0.9

Draw the number line labelling backward starting from 4.8

Then count backwards by tenths, 9 times

⸫ 4.8 – 0.9 = 3.9

#### Using Partial differences

Using partial differences helps you to subtract numbers that are difficult to subtract in one step in your head.

The steps to be followed while subtracting numbers using partial differences are as follows:

- When subtracting using partial differences, we write the numbers one below the other and start subtracting from left to right
- Then we start subtracting the whole number part by place values, then subtract the tenths digit and hundredths digit vice versa.
- If the number to be subtracted is greater than the number to be subtracted from, then swap the numbers in head, subtract the smaller one from the bigger one and put the negative sign in the answer.
- When writing these results in the answer column subtract the negative numbers from the number with positive sign.

For example:

- Subtract 43.85 – 21.63

Now, subtracting by place values

40 – 20 = 20

3 – 1 =

0.8 – 0.6 = 0.2

0.05 – 0.03 = 0.02

22.22

- Subtract 54.85 – 31.56

Now, subtracting by place values

50 – 30 = 20

4 – 1 = 3

0.8 – 0.5 = 0.3

0.05 – 0.06 = – 0.01 (⸪0.06 > 0.05)

Since there is a negative number, subtract that from the positive number

- 0.3 – 0.01 è 0.30 – 0.01 = 0.29

⸫ 54.85 – 31.56 = 20 + 3 + 0.29 = 23.29

## Exercise

- Add 6. 5+ 3.3 using number line
- Casey runs 9.5 miles, 13.2 miles then 11.5 miles the first week and 11.5 miles, 13.2 miles and 9.5 miles the next week. Which property is represented by Casey’s two weeks of running?
- Name the property illustrated below.

3.2+(a+5.6)=(3.2+a)+5.6 - Add 15.67 + 11.74 using partial sums
- Subtract 9.8 – 7.6 using number line
- Ms. Gracie is an electrician and has a length of wire that is 54.7m long . She has another length of wire that is 16.7 m long . How much longer is one wire than the other. Use any method to solve this problem
- Subtract 25.32 – 13.26 using partial differences.

### Concept Map

### What have we learned

- Adding Decimals using Number line, Lining up the decimal point, Partial sums and using properties of numbers.
- Subtracting decimals using Number line, lining up the decimals and partial differences

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