# Operations with Decimals

The numbers 7 and 8 are both whole numbers. What if you were told to represent a number equally between 7 and 8?

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Example 1, Aishah Amri - StudySmarter Originals

At first glance, we could definitely represent this number as a fraction. As we have seen from earlier articles, fractions express numbers that are not whole numbers. In this case, we would have the fraction $7\frac{1}{2}or\frac{15}{2}$. What if I told you that there is another way to write numbers that are not whole? That's right! These are called decimals.

In this article, we are going to explore decimals, its definition, operations on decimals, mixed operations and many examples.

## Decimals

Let us begin with the definition of a decimal.

A decimal is a number that contains a decimal point.

The decimal point is represented by a dot and separates a whole number from its fractional part. The value of the fractional part is always less than one. The number of digits formed by the fractional part determines the number of decimal places. Let us demonstrate the position of decimals on a number line to give us a clearer picture of this concept. This is displayed below.

Example 2, Aishah Amri - StudySmarter Originals

Another way to look at decimals is by representing them in a place value table. This will help us identify the number of decimal places for each decimal value given which is important to notice when performing operations involving decimals. Let us construct this place value table below.

 Thousands Hundreds Tens Ones Decimal Point (.) Tenths Hundredths Thousandths 1000 100 10 1 . 0.1 0.01 0.001

The digits before the decimals point represent the whole part of the number while the digits after the decimal point represent the decimal part of the number.

Given the decimal 12.45, the position for each number is defined below.

 Digit 1 2 . 4 5 Place Value Tens Ones Decimal place Tenths Hundredths The number associated with place value 10 2 . 0.4 0.05

In some textbooks, the fractional part is also referred to as the decimal part.

The decimal number 3.4512 has 3 as its whole number component and 0.4512 as its fractional part.

There are four decimal places here since there are four digits after the decimal point.

### Recap: Converting Fractions to Decimals

#### Method 1: Using A Calculator

The easiest way to convert a fraction to a decimal is by using a calculator. Simply divide the numerator by the denominator. Plugging these digits into your calculator will give you your answer.

#### Method 2: Long Division

Use long division to convert $\frac{7}{8}$into a decimal.

Therefore, $\frac{7}{8}=0.875$

### Recap: Converting Percentages to Decimals

Recall that a percentage takes the form $x%=\frac{x}{100}$. To convert a percentage to a decimal, all we have to do is remove the % sign and divide the remaining digit by 100. This is done by moving the decimal point two spaces to the left.

Convert 45% to a decimal.

Moving two spaces to the left, we obtain 45% = 0.45

Convert 3.67% to a decimal.

Moving two spaces to the left from the decimal point, we obtain 3.67% = 0.0367

### Order of Operations

When performing basic arithmetic operations on decimals such as addition, subtraction, multiplication and division, we must take into account the order of operations. This is particularly important when dealing with mixed operations involving decimals. The order of operations for decimals is abbreviated by the acronym PEMDAS, described in the table below.

 Order Letter Concept Explanation 1 P Parenthesis These are expressions inside a pair of parenthesis or brackets such as (x + y) and [x + y] or in the form of a grouping such as$\frac{x+y}{z}$ 2 E Exponent These are expressions of the form xy 3 M Multiplication These two operations have the same precedence, thus we simply execute the order of operation from left to right 4 D Division 5 A Addition These two operations have the same precedence, thus we simply execute the order of operation from left to right 6 S Subtraction

Always remember PEMDAS: Parenthesis, Exponent, Multiplication/Division, Addition/Subtraction.

### Adding and Subtracting Decimals

The addition and subtraction of decimals are similar to adding and subtracting integers. We always begin from the right-hand side and apply the column method. The column method helps maintain the place values and decimal places clearly to achieve an accurate summation or deduction between decimals.

1. Write down the numbers in column form such that their decimal points are lined up vertically;

2. Put in zeros as place-holders (if necessary) so that the numbers are of the same length;

3. Add or subtract using column addition or column subtraction;

4. Place the decimal point back into the sum or difference such that it lines up vertically with the numbers being added or subtracted.

Let us demonstrate this method with some worked examples below.

Do not forget to add the decimal point at the end of your answer!

Add 5.7 and 8.9

$5.7+8.914.6$

Subtract 2.3 from 4.8

$4.8-2.32.5$

#### Addition and Subtraction of Decimals with Whole Numbers

Remember to add in zeros as place-holders so that the numbers are of the same length! Below are some worked examples that show this.

Add 6 and 4.3

$6.\mathbf{0}+4.310.3$

Subtract 5 from 9.2

$9.2-5.\mathbf{0}4.2$

#### Addition and Subtraction of Two Unlike Decimals

As before, ensure that the decimals are of the same length by adding in zeros as place-holders. Here are two worked examples that exhibit this.

Add 5.43 and 6.678

$5.43\mathbf{0}+6.67812.108$

Subtract 2.3456 from 4.1

$4.1\mathbf{000}-2.34561.7544$

### Multiplying Decimals

To multiply decimals, we must consider the rules for multiplying decimals.

1. Remove the decimal point from the given numbers and carry out multiplication as usual;

2. Count the number of decimal places in each of the original numbers and find its sum;

3. The place value of the final answer is the sum of the decimal places found in Step 2;

4. Position the decimal point into the product calculated in Step 1.

Let us demonstrate this technique with the following worked examples.

Multiply 3.6 by 2.3

Ignoring the decimal places, we obtain 36 × 23 = 826.

Both decimals, 3.6 and 2.3 have one decimal place each.

The sum of these decimal places is two.

Thus, the product of 36 × 23 = 826 must have two decimal places.

Putting in the decimal point, we obtain 3.6 × 2.3 = 8.26.

#### Multiplication of Decimals with Whole Numbers

Multiply 5.7 by 8

Ignoring the decimal places, we obtain 57 × 8 = 456.

The decimal 5.7 has one decimal place while the whole number 8 has no decimal place.

The sum of these decimal places is one.

Thus, the product of 57 × 8 = 456 must have one decimal place.

Putting in the decimal point, we obtain 5.7 × 8 = 45.6.

#### Multiplication of Two Unlike Decimals

Multiply 2.165 by 9.1

Ignoring the decimal places, we obtain 2165 × 91 = 197015.

The decimals 2.165 and 9.1 have 3 decimal places and one decimal place respectively.

The sum of these decimal places is four.

Thus, the product of 2165 × 91 = 197015 must have four decimal places.

Putting in the decimal point, we obtain 2.165 × 9.1 = 19.7015.

#### Multiplication of Decimals by Powers of 10

When multiplying decimals by powers of 10, we simply move the decimal point to the right based on the number of zeros present in the power of 10. Below are two worked examples.

Multiply 3.87 by 100

There are two zeros in the number 100, so we must shift the decimal point of 3.87 to two places to the right.

3.87 × 100 = 387

Multiply 7.3956 by 1000

There are three zeros in the number 1000, so we must shift the decimal point of 7.3956 three places to the right.

7.3956 × 1000 = 7395.6

### Dividing Decimals

When dividing numbers in general, it is always easier to divide by a whole number.

The divisor is the number we divide by. For example, in the expression 32 ÷ 2.3, 32 is the dividend and 2.3 is the divisor.

Below are the essential steps for dividing decimals.

1. Convert the divisor into a whole number by multiplying it by a power of 10;

2. Multiply the dividend by the same power of 10 found in Step 1;

3. Divide these numbers using long division.

The following worked examples show the application of this approach.

The answer in Step 3 gives us the same yield as if e had divided the decimals.

Divide 4.62 by 0.12

Here, we have 4.62 ÷ 0.12 where the dividend is 4.62 and the divisor is 0.12.

To change the divisor into a whole number, we multiply it by 100. This yields 0.12 × 100 = 12.

Multiplying this same value to the dividend, we obtain 4.62 × 100 = 462.

Thus, we have 462 ÷ 12 which is the same as 4.62 ÷ 0.12.

Conducting long division, we obtain

Thus, 4.62 ÷ 0.12 = 38.5.

#### Division of Decimals with Whole Numbers

Divide 5.525 by 5

Here, we have 5.525 ÷ 5 where the dividend is 5.525 and the divisor is 5.

The divisor is already in the form of a whole number. However, the dividend is still in the form of a decimal.

To tackle this problem, we shall conduct long division while ignoring the decimal point of the dividend.

We shall now place the decimal point of the answer directly above the decimal point of the dividend as below.

Thus, 5.525 ÷ 5 = 1.105.

#### Division of Two Unlike Decimals

Divide 3.432 by 1.04

Here, we have 3.432 ÷ 1.04 where the dividend is 3.432 and the divisor is 1.04.

To change the divisor into a whole number, we multiply it by 100. This yields 1.04 × 100 = 104.

Multiplying this same value to the dividend, we obtain 3.432 × 100 = 343.2.

Thus, we have 343.2 ÷ 104 which is the same as 3.432 ÷ 1.04.

Although the divisor is already in the form of a whole number, the dividend is still in the form of a decimal.

As before, we shall conduct long division while ignoring the decimal point of the dividend.

We shall now place the decimal point of the answer directly above the decimal point of the dividend as below.

Thus, 3.432 ÷ 1.04 = 3.3.

#### Division of Decimals by Powers of 10

When dividing decimals by powers of 10, we simply move the decimal point to the left based on the number of zeros present in the power of 10. Below are two worked examples.

Divide 2.34 by 10

There is one zero in the number 10, so we must shift the decimal point of 2.34 one place to the left.

2.34 ÷ 10 = 0.234.

Divide 17635.6 by 1000

There are three zeros in the number 1000, so we must shift the decimal point of 17635.6 three places to the left.

17635.6 ÷ 1000 = 17.6356

### Mixed Operations with Decimals

In this section, we shall make use of the PEMDAS order of operations for decimals. Below are some worked examples.

Evaluate$0.123-4.5%$.

Solution

We begin by converting the percentage term

4.5% = 4.5 ÷ 100 = 0.045

We can now perform subtraction as usual

$0.123-0.0450.078$

Thus, the final answer is 0.078

Evaluate $\frac{3}{4}+0.57$.

Solution

We begin by converting the fraction term

$\frac{3}{4}=3÷4=0.75$

We can now perform addition as usual

$0.75+0.571.32$

Thus, the final answer is 1.32

Evaluate$\left(2.45+0.72\right)×3.12-1.99$.

Solution

We shall first calculate the expression inside the parenthesis

$2.45+0.723.17\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}3.17×3.12-1.99$

Next, we shall conduct multiplication

$3.17×3.12=9.8904\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}9.8904-1.99$

Finally, we shall perform subtraction

$9.8904-1.99007.9004$

Thus, the final answer is 7.9004.

Evaluate$4.6+\frac{1.3-0.7}{0.4}-2.3$.

Solution

We shall first calculate the expression in the grouping

$\frac{1.3-0.7}{0.4}=\left(1.3-0.7\right)÷0.4=0.6÷0.4=1.5\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}4.6+1.5-2.3$

For an expression in the form of a grouping, always remember to solve the numerator before the denominator

Next, we shall conduct addition

$4.6+1.56.1\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}6.1-2.3$

Finally, we shall perform subtraction

$6.1-2.33.8$

Thus, the final answer is 3.8.

Evaluate $0.{3}^{2}+\left(3.4-2.1\right)÷1.6$.

Solution

We shall first calculate the expression inside the parenthesis

$3.4-2.11.3\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}0.{3}^{2}+1.3÷1.6$

Next, we shall derive the exponent

$0.{3}^{2}=0.3×0.3=0.09\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}0.09+1.3÷1.6$

Following this, we shall conduct division

$1.3÷1.6=0.8125\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}0.09+0.8125$

Finally, we shall conduct addition

$0.0900+0.81250.9025$

Thus, the final answer is 0.9025.

## Operations with Decimals - Key takeaways

• The order of operations for decimals obeys PEMDAS: Parenthesis, Exponent, Multiplication/Division, Addition/Subtraction
• Steps for adding and subtracting decimals
1. Write down the numbers in column form. Make sure the decimal points are lined up

2. Put in zeros so that the numbers are of the same length

3. Add or subtract using column addition or column subtraction

• Steps for multiplying decimals
1. Remove the decimal points and multiply the numbers

2. Count the number of decimal places in each given number. Find its sum

3. The place value of the final answer is the sum of the decimal places derived in Step 2

• Steps for dividing decimals
1. Convert the divisor into a whole number by multiplying it by a power of 10

2. Multiply the dividend by the same power of 10 found in Step 1

3. Divide these numbers using long division

• When multiplying decimals by powers of 10, we simply shift the decimal point to the right according to the number of zeros in the power of 10.

• When dividing decimals by powers of 10, we simply shift the decimal point to the left according to the number of zeros in the power of 10.

#### Flashcards in Operations with Decimals 16

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##### Frequently Asked Questions about Operations with Decimals

How to do order of operations with fractions and decimals?

Convert the fraction into a decimal and execute the given operation using PEMDAS

What are the rules of decimals?

Operations on decimals obey the PEMDAS rule which is an acronym for Parenthesis, Exponents, Multiplication/Division, Addition/Subtracting

How do you do operations with decimals?

Operations with decimals include addition, subtraction, multiplication and division

Can order of operations have decimals?

Yes, order of operations can have decimals

How do you add, multiply and divide decimals?

• When adding decimals, the decimal of the numbers which are added from the other must be in-line
• To multiply decimals, multiply as though no decimal is present then count the number of digits after the decimal in each factor and write the exact number of digits behind the decimal of its product
• We apply long division when dividing decimals

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