Decimal Operations: Definition, Rules, Solved Examples

Decimal Operations: Decimal numbers are numbers that have a decimal point followed by digits that depict the fractional part. The number of digits in the decimal part determines the number of decimal places. In a decimal number, a decimal point is a dot. The digits after the decimal point denote a value less than one. 

For example, in a decimal number \(6.48,\) in which \(6\) is the whole number, while \(48\) is the decimal part. All basic mathematical operations like addition, subtraction, multiplication, and division can be performed on decimal numbers. This article will discuss the operation of decimal numbers, how the decimal operation works, decimal and fundamental operations on them, decimal BODMAS questions etc.

Operations on Decimal Numbers

The fundamental operations on decimals are addition, subtraction, multiplication, and division.

While adding and subtracting decimal numbers, we will first convert the decimal numbers to like decimals. Like decimals are those with the same number of decimal places. We will align the decimal points of the addends and, if necessary, add zeros to the end of one number to make the decimal places the same. Then continue to add (or subtract) as usual. The decimal point in the answer should be placed exactly where it is in the numbers being added (or subtracted).

While performing the multiplication, multiply the numbers as if they were all whole numbers. Then, count how many decimals places the two factors have and mark the total number of decimal places in the product. We can use the normal long division method when dividing decimals. Let us see each operation in detail.

Addition of Decimal Numbers

For the addition of decimal numbers 

1. We always begin from the right-hand side when adding decimals, just as we add two integer numbers.
2. Write the numbers in a way that the decimal points are exactly lined up.
3. Put zeroes at the end to convert all to like decimals.
4. Now add the decimal numbers together and determine the sum.

Let’s see an example of how decimals are added using the following steps.

Example: Add \(1.8\)  and \(2.6\)

Write the numbers in a way that the decimals are exactly lined up. The numbers will be written as follows: 

Put zeroes when the decimal numbers’ lengths differ. This step does not apply in this case as the length of the decimal numbers is the same.

Now add the decimal numbers together and determine the sum.

The sum obtained when adding the above two decimal numbers is \(4.4.\)

To avoid confusion, adding two decimal numbers can be viewed as a simple addition of two numbers with the decimal point added at the end in the same place of addends.

Addition of Decimals with Whole Numbers

Let see an example to add decimals with whole numbers.

Example: Add  \(1.86\) and \(2\).

Step-1: Write the numbers in a way that the decimals are exactly lined up. The numbers will be written as follows: 

Step-2: Put zeroes when the decimal numbers’ lengths differ. Because \(2\) is a whole number, two zeroes are added after the decimal point, resulting in the following addition:

Step-3: Now, add the decimal numbers together and determine the sum.

The answer to adding the above decimal numbers is \(3.86.\)

Addition of Two Unlike Decimal Numbers

Example: \(11.86\) and \(20.8.\)

Step-1: Write the numbers in a way that the decimals are exactly lined up. The numbers will be written as follows: 

Step-2: Put zeroes when the decimal numbers’ lengths differ. Because it is a decimal number, zero is added after the decimal point, resulting in the following addition:

Step-3: Now, add the decimal numbers together and determine the sum.

The answer to adding the above decimal numbers is \(32.66.\)

Subtraction of Decimal Numbers

For subtraction of decimal numbers 

1. We always begin on the right-hand side when subtracting decimals, just as we subtract two integers.
2. Write the numbers in a way that the decimal points are exactly lined up.
3. Put zeroes when the decimal numbers’ lengths differ.
4. Now subtract the decimal numbers together and determine the subtraction.

Let see an example of how decimals are subtracted using the following steps.

Example: Subtract \(1\) from \(2.89.\)

Step-1: Write the numbers in a way that the decimals are exactly lined up.

Step-2: Put zeroes when the decimal numbers’ lengths differ.

Step-3: Now, subtract the decimal numbers together and determine the subtraction.

Example: Subtract \(1.89\) from \(2\).

Step-1: Write the numbers in a way that the decimals are exactly lined up.

Step-2: Put zeroes when the decimal numbers’ lengths differ.

Step-3: Now, subtract the decimal numbers like we do for integers and determine the result.

Multiplication of Decimal Numbers

Multiplying Decimals with Whole Numbers

The primary difference between multiplying decimals with whole numbers and multiplying whole numbers is the decimal point placement.

Step-1: Ignore the decimal point at first and multiply the two numbers as usual.

Step-2: Count the number of decimal places in the decimal number after multiplication. The number of decimal places in the product received after multiplication will be the same.

Step-3: Place the decimal point in the obtained product in Step 2.

Multiplying Two Decimals Numbers

The primary difference between multiplying decimals with decimals numbers and multiplying decimal numbers is the decimal point placement.

Step-1: Ignore the decimal point at first and multiply the two numbers as usual.

Step-2: Count the number of decimal places in the decimal number after multiplication. The number of decimal places in the product received after multiplication will be the same.

Step-3: Place the decimal point in the obtained product in Step 2.

Multiplying Decimals by \(10, 100\) and \(1000\)

We simply shift the decimal point to the right as many places as the number of zeros in the power of \(10\) when multiplying any decimal by \(10, 100, 1000,\) or any other power of \(10.\)

1. If we multiply a decimal by \(10,\) we move the decimal point one place to the right because there is one zero after 1 in the number.
2. If we multiply a decimal by \(100,\) we move the decimal point two places to the right because there are two zeroes after 1 in the number.
3. Similarly, if we multiply a decimal by \(1000,\) we move the decimal point three places to the right because there are three zeroes after 1 in the number.

Examples:

\(1.23 \times 10=12.3\)
\(1.23 \times 100=123\)
\(1.23 \times 1000=1230\)

These are a few key points to remember when it comes to multiplying decimals.

1. The process for multiplying decimals is the same as for multiplying whole numbers or integers.
2. The decimal point should be put in the product so that the total number of decimal places equals the sum of decimal places in all multiplicands and multipliers.
3. Make sure you keep all the zeros in the product while placing the decimal point.
4. If the number of decimal places in the product exceeds the number of digits, zeros can be added to the left before inserting the decimal point.
5. The resultant product’s trailing zeros can be removed.

Division Using Decimals

The procedure of dividing decimals is identical to that of standard division, except that we must remember to position the decimal point correctly in the quotient. Because we are dealing with decimals, the steps below will assist us in grasping how to divide decimals.

1. The division of a decimal number by a whole number is done in the same way that it is done with whole numbers. We begin by dividing the two integers without considering the decimal point. The quotient’s decimal point is then placed in the same position as the dividend’s decimal point. 
2. To divide a decimal number by a decimal number, multiply the divisor by as many tens as needed until we reach a whole number. Then multiply the dividend by the same number of tens.

Dividing Decimals by Whole Numbers

The division of decimals by whole numbers is analogous to the division of whole numbers. A divisor is a whole number, and the dividend is a decimal. Thus the quotient will have the same number of decimals as the dividend.

Example: Divide \(7.68\) by \(3\).

Division of Decimals by 10, 100, 1000,…

When a number is divided by \(10\), we move the decimal point \(1\) place to the left.
For example,

\(86.5÷10=8.65\)

When a number is divided by \(100,\) we move the decimal point \(2\) places to the left.
For example,

\(86.5÷100=0.865\)

When a number is divided by \(1000,\) we move the decimal point \(3\) places to the left.
For example,

\(86.5÷1000=0.0865\)

When there is no digit in the one’s place, we write \(0\) as a placeholder. Thus, \(865\) is written as \(0.865.\)

Similarly, according to the number of zeros in the multiples of \(10,\) the decimal points will be shifted to the left while dividing a number by multiples of \(10.\)

Solved Examples: Operations on Decimal Numbers

Q.1. Divide \(876.345\) by \(100.\)
Ans: From the given \(876.345÷100.\)
We know,
When a number is divided by \(100,\) we move the decimal point \(2\) places to the left.
So, \(876.345÷100=8.76345\)
Hence, the obtained answer is \(8.76345.\)

Q.2. Find the product of \(2345.45\) and \(10.\)
Ans: From the given, \(2345.45 \times 10\)
We know,
If we multiply a decimal by \(10,\) we move one place to the right because there is one zero after \(1\) in the number.
So, \(2345.45 \times 10=23454.5\)
Hence, the obtained product is \(23454.5.\)

Q.3. Find the quotient of \(987.5 \div 1000.\)
Ans:  Given: \(987.5÷1000.\)
We know, 
When a number is divided by \(1000,\) we move the decimal point \(3\) places to the left.
So, \(987.5 \div 1000=0.9875\)
Hence, the obtained quotient is \(0.9875.\)

Q.4. Arun scored \(452.65\) marks out of \(600\) in the final examination. How many marks did he lose?
Ans: From the given, we can write it as 
Marks lost by Arun \(=600 – 452.65\)
\(=147.35\)
Hence, Arun lost \(147.35\) marks in his examination.

Q.5. Find the area of a square whose side is \(2.50\,{\rm{m}}.\)
Ans: Side of a square \(S = 2.50\,{\rm{m,}}\,{\rm{A = ?}}\)
We know,
Area of a square \(A = {\mathop{\rm side}\nolimits} \times {\mathop{\rm side}\nolimits} \)
\(\Rightarrow A=2.50 \times 2.50\)
\(\Rightarrow A=6.25 \mathrm{~m}^{2}\)
Hence, \(6.25 \mathrm{~m}^{2}\) is the area of a square.

Summary

A decimal number is defined as a decimal point separating the whole number and fractional parts. The fundamental operations on decimals are addition, subtraction, multiplication, and division. This article includes the basic operations of decimals, and steps to add, subtract, multiply and divide the decimal numbers with examples. While adding and subtracting, the decimal point is placed in line with the addends. 

When decimal numbers are multiplied, the total number of decimal places will be the sum of decimal places in multiplicand and multiplier. While dividing decimal numbers, is done the same way as dividing the whole numbers or integers using the long division method.

Frequently Asked Questions (FAQs)

Q.1. How do you do operations with decimals?
Ans: While adding and subtracting, the decimal point of the sum is placed in line with the addends. When decimal numbers are multiplied, the total number of decimal places will be the sum of decimal places in multiplicand and multiplier. While dividing decimal numbers, it is done the same way as dividing the whole numbers or integers using the long division method.

Q.2. Which operations do you line up the decimals for?
Ans: We used to line up the decimals for adding and subtracting the decimal numbers.

Q.3. What are the operations on decimals?
Ans: The operations on decimals are
1. Addition 
2. Subtraction 
3. Multiplication 
4. Division

Q.4. What are the rules for multiplying decimals?
Ans: The primary difference between multiplying decimals and multiplying whole numbers is the decimal point placement.
Step-1: Ignore the decimal point at first and multiply the two numbers as usual.
Step-2: Count the number of decimal places in the decimal number after multiplication. The number of decimal places in the product received after multiplication will be the same.
Step-3: Place the decimal point in the obtained product in Step 2.

Q.5. What are decimal places?
Ans: The number of digits after the decimal point in a decimal number is its number of decimal places.

We hope this detailed article on the operations of decimal numbers helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!