BCD Addition – How to add two BCD Numbers? | Solved Examples

Before delving into adding BCD addition, here is a quick understanding of a Binary Coded Decimal.

The decimal number system is made up of 10 digits (from 0 to 9). We know that we can form binary numbers for all these digits. The BCD system uses four binary digits to form the binary number (from 0001 to 1001).

Therefore, if a decimal number has one digit, then its equivalent binary form will have four binary digits. For 2-digit decimal numbers, the binary form will have 8 digits.

Here is a quick look at decimal numbers and their binary and BCD codes

BCD Conversion Table

Decimal NumberBinary FormBCD code
000000000
100010001
200100010
300110011
401000100
501010101
601100110
701110111
810001000
910011001
1010100001 0000
1110110001 0001
1211000001 0010
1311010001 0011
1411100001 0100
1511110001 0101

How to do BCD Addition?

Being a numerical code, there are certain rules for BCD addition as explained below:

1. Use the binary rules of addition to add the BCD numbers as shown below

Add 5(decimal) and 3 (decimal). Converting them into binary codes, we get

5 = 0101

3 = 0011

The BCD addition is done as follows:

0101 + 0011 = 1000

2. Check the result of the addition

If the result of the addition is less than 9, then it is a valid BCD number. However, if the result is greater than 9, then you need to convert it into a valid BCD code.

3. Make the result valid

If the result of the addition is greater than 9, then the result is an invalid BCD code. In such cases, you need to add 6 (0110) to the result to get a valid code. Here is an example:

Add 9 (decimal) and 6 (decimal). Converting them into binary codes, we get:

9 = 1001

6 = 0110

The BCD addition is:

1001 + 0110 = 1111 (decimal 15)

However, this is not a valid BCD code. Hence, we add 6 (0110) to the result as shown below:

1111 + 0110 = 1 0101 (also represented as 0001 0101 or decimal 15)

Why is 6 (0110) added to the result?

There are 6 invalid states of binary-coded decimal (from 10 to 15). In order to skip these states, we add 6 to the result of the invalid BCD addition.

Let us see some examples on BCD Addition

Solved Examples for BCD Addition

Example 1

(7)10 + (4)10

7 → 0111

4 → 0100

Therefore,

0111 + 0100 = 1011 → which is an invalid BCD number

Hence, we add 0110 to the result

1011 + 0110 = 1 0001 or 0001 0001 → (11)10

Example 2

Add (26)10 and (11)10

26 → 0010 0110

11 → 0001 0001

Therefore,

   0010 0110 + 0001 0001 = 0011 0111 → a valid BCD number (37)10

Since both 0011 (3)10 and 0111 (7)10 are valid BCD numbers, we don’t need to add 6 (0110) to the result.

Example 3

Add (599)10 and (984)10

599 → 0101 1001 1001

984 → 1001 1000 0100

Therefore,

0101 1001 1001 + 1001 1000 0100 = 1110 10001 1101

As you can see above, all three sums (1110, 10001, and 1101) are greater than 1010 (9)10. Hence, we add (6)10 to all three sums.

1110 10001 1101 + 0110 0110 0110 = 1 0101 1000 0111

The final result is:

0001 0101 1000 0111 → 1583

This can be verified by checking the decimal number addition:

599 + 984 = 1583.

Example 4

Add (1503)10 and (5623)10

The BCD code for 1503 → 0001 0101 0000 0011

The BCD code for 5623 →  0101 0110 0010 0011

Adding them, we get

0001 0101 0000 0011 + 0101 0110 0010 0011 = 0110 1011 0010 0110

As you can see from the results above, 1011 is an invalid BCD while other results are valid BCD codes. Hence, we add 0110 to the invalid code alone:

0110 1011 0010 0110 + 0000 0110 0000 0000 = 0111 0001 0010 0110

The final result is:

0111 0001 0010 0110 →  7126

This can be verified by checking the decimal addition:

1503 + 5623 = 7126

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