Addition & Subtraction of Decimal Numbers

Addition and subtraction are the two fundamental arithmetic operations in mathematics. The arithmetic operations such as multiplication and division are in fact derived from these two operations respectively.

As you can perform these operations on other types of numbers such as whole numbers and integers, similarly these operations can be performed with decimal numbers also.

Addition and Subtraction of Decimal Numbers

Adding and subtracting decimals is the same as the addition and subtraction of whole numbers keeping in mind that the decimal point needs to be in place. The length of the decimal numbers can be adjusted by adding or removing zeros from the decimal part. 

When you work with decimal numbers, you need to understand the difference between trailing zeros and leading zeros, as well as how they affect the value of the number.

Leading and Trailing Zeroes

Before moving further, let’s first understand these two basic concepts 

• Leading Zeroes
• Trailing Zeroes

You might know that you can attach zeros to the beginning of a whole number without changing its value. For example, these three numbers are all equal in value: $53$,  $053$, $0000053$

The reason for this becomes clear when you know about the place value of whole numbers. In the above example, we are attaching leading zeros to the value $53$.

As you can see $0000053 = 0 \times 1000000 + 0 \times 100000 + 0 \times 10000 + 0 \times 1000 + 0 \times 100 + 5 \times 10 + 3 \times 1$.

Zeros attached to the beginning of a number in this way are called leading zeros.

In decimals, this idea of zeros that don’t add value to a number can be extended to trailing zeros. A trailing zero is any zero that appears to the right of both the decimal point and every digit other than zero.

For example, $23.6$,  $23.60$,  $23.60000$.

All three of these numbers are the same. The reason becomes from the place value of decimals.

In this example, $23.60000$ means $23 + 6 \times \frac {1}{10} + 0 \times \frac {1}{100}+ 0 \times \frac {1}{1000} + 0 \times \frac {1}{10000} + 0 \times \frac {1}{10000}$. You can attach or remove as many trailing zeros as you want to without changing the value of a number.

When you understand trailing zeros, you can see that every whole number can be changed to a decimal easily. Just attach a decimal point and a 0 to the end of it. For example,$7 = 7.0$, $45 = 45.000$, $784 = 784.0$.

Note:

• Make sure that you don’t attach or remove any nonleading or non-trailing zeros because doing this changes the value of the decimal.
• For example, look at the number $0290.0060$. In this number, you can remove the leading and trailing zeros without changing the value, as $290.006$.
• The remaining zeros, however, need to stay where they are as placeholders between the decimal point and digits other than zero.

Addition of Decimal Numbers

The addition of decimals is done by starting from the right-hand side and then we move on to the left adding each column. But while adding one should make sure that the decimal points of all numbers are properly aligned.

Addition of Like Decimals

In the case of like decimals, the number of digits after the decimal point in each of the numbers is the same, and hence it is quite easy to align the decimal point in such numbers.

The first step here is to write the numbers in such a way that the decimal points of all the numbers come in one column (vertical).

Steps For Addition of Like Decimals

Step 1: Write the numbers one below the other such that they are aligned as per their place values and the decimal point is placed one below the other

Step 2: Now add the decimal numbers to get the sum

Examples

Ex 1: Add $17.4$ and $21.5$.

 $17.4 + 21.5 = 38.9$.

Ex 2: Add $13.7$ and $11.8$.

$13.7 + 11.8 = 25.5$.

Ex 3: Add $10.24$ and $24.32$.

$10.24 + 24.32 = 34.56$.

Ex 4: Add $43.37$ and $11.85$.

$43.37 + 11.85 = 55.22$.

Addition of Unlike Decimals

Unlike decimals are decimal numbers that have a different number of digits after a decimal point. For example $2.5$ and $8.62$, $9.632$ are all unlike decimal numbers.

The first step in addition of unlike decimal numbers is to make them like decimals by adding the required number of trailing zeroes.

Steps For Addition of Unlike Decimals

Step 1: Make all decimal numbers like decimals by adding the required number of trailing zeroes.

Step 2: Write the numbers one below the other such that they are aligned as per their place values and the decimal point is placed one below the other

Step 3: Now add the decimal numbers to get the sum

Examples

Ex 1: Add $32.4$ and $17.52$

$32.4$ and $17.52$ are unlike decimal numbers with the number of decimal places as $1$ and $2$ respectively. 

$32.4 = 32.40$, therefore, $32.4 + 17.52 = 32.40 + 17.52$

$32.4 + 17.52 = 49.92$

Ex 2: Add $53.5$ and $24.634$

$53.5$ and $24.634$ are unlike decimal numbers with the number of decimal places as $1$ and $3$ respectively. 

$53.5 = 53.500$, therefore, $53.5 + 24.634 = 53.500 + 24.634$

$53.5 + 24.634 = 78.134$.

Subtraction of Decimal Numbers

The subtraction of decimals is done by starting from the right-hand side and then we move on to the left subtracting each column. But while subtracting one should make sure that the decimal points of the numbers are properly aligned.

Subtraction of Like Decimals

As in the case of an addition, in the case of subtraction also with like decimals, the number of digits after the decimal point in each of the numbers is the same, and hence it is quite easy to align the decimal point in such numbers.

The first step here is to write the numbers in such a way that the decimal points of all the numbers come in one column (vertical).

Steps For Subtraction of Like Decimals

Step 1: Write the numbers one below the other such that they are aligned as per their place values and the decimal point is placed one below the other

Step 2: Now subtract the decimal numbers to get the difference

Examples

Ex 1: Subtract $12.2$ from $34.7$.

$34.7 – 12.2 = 22.5$

Subtraction of Unlike Decimals

As in the case of addition where the first step is to make unlike decimals to like decimals, here also the first step is to make them like decimals by adding the required number of trailing zeroes.

Steps For Subtraction of Unlike Decimals

Step 1: Make all decimal numbers like decimals by adding the required number of trailing zeroes.

Step 2: Write the numbers one below the other such that they are aligned as per their place values and the decimal point is placed one below the other

Step 3: Now subtract the decimal numbers to get the difference

Examples

Ex 1: Subtract $31.6$ from $52.841$.

$31.6$ from $52.841$ are unlike decimal numbers with the number of decimal places as $1$ and $3$ respectively. 

$31.6 = 31.600$, therefore, $52.841 – 31.6 = 52.841 – 31.600$.

Conclusion

Addition and subtraction of decimal numbers is same as that of whole numbers, with only one difference and it is to keep the decimal points aligned for the numbers. Another important point to remember is to make unlike decimals like decimals before adding or subtracting them.

Practice Problems

1. Add the following
   • $5.7$ and $2.1$
   • $6.32$ and $1.45$
   • $12.12$ and $43.34$
   • $25.75$ and $34.29$
   • $1.3$ and $3.62$
   • $14.72$ and $11.127$
2. Subtract the following
   • $3.5$ from $7.8$
   • $2.6$ from $9.75$
   • $4.52$ from $10.7$
   • $13.15$ from $27.876$
   • $11.62$ from $27.8$
   • $19.05$ from $34.806$

Recommended Reading

• Types of Decimal Numbers(With Examples)
• Decimal Fraction – Definition, Conversion & Operations (With Examples)
• Decimal Place Value Chart (Definition & Examples)

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