Understanding Number System Conversions

This question asks us to convert a decimal number, 76, into two different number systems: hexadecimal and BCD (Binary Coded Decimal). Understanding how to convert between different number systems is a fundamental concept in digital electronics and computer science.

Converting Decimal to Hexadecimal

The hexadecimal number system is a base-16 system, using digits 0-9 and letters A-F (where A represents 10, B is 11, C is 12, D is 13, E is 14, and F is 15). To convert a decimal number to hexadecimal, we repeatedly divide the decimal number by 16 and record the remainders. The hexadecimal number is formed by reading the remainders from bottom to top.

Step-by-Step Conversion of 76 Decimal to Hexadecimal

1. Divide the decimal number 76 by 16:

   \(76 \div 16\)

   The quotient is 4, and the remainder is 12.

   In hexadecimal, the remainder 12 is represented by the letter 'C'.

2. Divide the quotient (4) by 16:

   \(4 \div 16\)

   The quotient is 0, and the remainder is 4.

3. Read the remainders from bottom to top: The remainders are 4 and 12 (C).

   Reading from the last remainder up, we get 4 followed by C.

So, the decimal number 76 is 4C in hexadecimal.

Converting Decimal to BCD (Binary Coded Decimal)

The BCD (Binary Coded Decimal) system represents each decimal digit using its 4-bit binary equivalent. This is different from a straight binary conversion of the entire number. In BCD, each decimal digit (0-9) is represented by a unique 4-bit code.

BCD Codes for Decimal Digits (0-9)

Step-by-Step Conversion of 76 Decimal to BCD

To convert the decimal number 76 to BCD, we take each decimal digit separately and convert it to its 4-bit BCD code.

1. Consider the first digit, 7:

   The 4-bit BCD code for 7 is 0111.

2. Consider the second digit, 6:

   The 4-bit BCD code for 6 is 0110.

3. Combine the BCD codes for each digit:

   The BCD representation of 76 is the code for 7 followed by the code for 6.

   \(\text{BCD}(76) = \text{BCD}(7)\ \text{BCD}(6) = 0111\ 0110\)

So, the decimal number 76 is 0111 0110 in BCD.

Summary of Conversions

We found that:

• Decimal 76 is 4C in hexadecimal.
• Decimal 76 is 0111 0110 in BCD.

The question asks for the hexadecimal and BCD representation respectively. This means the answer should be the hexadecimal value first, followed by the BCD value.

The correct representation is 4C, 0111 0110.

Comparing with Options

Let's compare our results with the given options:

• Option 1: C4, 0111 0110 (Hexadecimal is incorrect)
• Option 2: 3E, 0010 0101 (Both are incorrect)
• Option 3: 4C, 0010 1100 (BCD is incorrect)
• Option 4: 4C, 0111 0110 (Matches our calculated hexadecimal and BCD values)

Therefore, the option that correctly represents the decimal number 76 in hexadecimal and BCD respectively is 4C, 0111 0110.

Revision Table: Number System Conversions

Additional Information: Number Systems Explained

Number systems are ways of representing numerical values using a set of digits or symbols. Different number systems are used in computing and electronics for various purposes.

• Decimal System (Base 10): This is the system we use in everyday life. It uses digits 0 through 9. Each position in a decimal number represents a power of 10. For example, \(76_{10} = 7 \times 10^1 + 6 \times 10^0\).
• Binary System (Base 2): This system uses only two digits, 0 and 1. It is the fundamental system used by computers. Each position represents a power of 2. For example, \(01001100_2\) is a binary number. Note that this is the straight binary of 76, not BCD. \(0 \times 2^7 + 1 \times 2^6 + 0 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 = 64 + 8 + 4 = 76_{10}\).
• Hexadecimal System (Base 16): This system uses digits 0-9 and letters A-F. It is often used as a shorthand for binary numbers because each hexadecimal digit corresponds to exactly four binary bits. This makes it easier to read and write large binary numbers.
• BCD System (Binary Coded Decimal): As explained earlier, BCD represents each decimal digit with a 4-bit binary code. While this is easy to convert from decimal, it is less efficient in terms of storage space compared to pure binary for larger numbers, as some 4-bit combinations (like 1010 to 1111) are not used. It's commonly used in digital displays and simple arithmetic circuits where direct correspondence with decimal digits is helpful.