Binary-Decimal-HexaDecimal Conversion Worksheet | Download Free ... - Free Printable

Let's solve each part of the problem step by step.

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Part 1: Convert Hexadecimal to Decimal and Binary

#### Hexadecimal to Decimal
To convert from hexadecimal (base-16) to decimal (base-10), multiply each digit by 16 raised to the power of its position (from right, starting at 0).

#### Hexadecimal to Binary
Each hexadecimal digit corresponds to a 4-bit binary number.

| Hexadecimal | Decimal | Binary |
|-------------|---------|--------|
| FF | | |
| A4 | | |
| 0F | | |
| BD | | |

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#### 1. FF
- Decimal:
$ F = 15 $, so:
$$
FF_{16} = 15 \times 16^1 + 15 \times 16^0 = 240 + 15 = 255
$$
- Binary:
$ F = 1111 $, so:
$$
FF_{16} = 1111\ 1111_2
$$

#### 2. A4
- Decimal:
$ A = 10 $, $ 4 = 4 $
$$
A4_{16} = 10 \times 16^1 + 4 \times 16^0 = 160 + 4 = 164
$$
- Binary:
$ A = 1010 $, $ 4 = 0100 $
$$
A4_{16} = 1010\ 0100_2
$$

#### 3. 0F
- Decimal:
$ 0 \times 16 + 15 = 15 $
- Binary:
$ 0 = 0000 $, $ F = 1111 $
$$
0F_{16} = 0000\ 1111_2 = 00001111_2
$$

#### 4. BD
- Decimal:
$ B = 11 $, $ D = 13 $
$$
BD_{16} = 11 \times 16 + 13 = 176 + 13 = 189
$$
- Binary:
$ B = 1011 $, $ D = 1101 $
$$
BD_{16} = 1011\ 1101_2
$$

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✔ Part 1 Table:

| Hexadecimal | Decimal | Binary |
|-------------|---------|------------|
| FF | 255 | 11111111 |
| A4 | 164 | 10100100 |
| 0F | 15 | 00001111 |
| BD | 189 | 10111101 |

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Part 2: Convert Decimal to Hexadecimal and Binary

#### Decimal → Hexadecimal
Divide by 16 repeatedly and note remainders (convert to hex digits).

#### Decimal → Binary
Repeated division by 2 or use powers of 2.

| Decimal | Hexadecimal | Binary |
|---------|-------------|------------|
| 162 | | |
| 238 | | |
| 16 | | |
| 11 | | |

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#### 1. 162
- Hexadecimal:
- $ 162 \div 16 = 10 $ remainder $ 2 $
- $ 10 = A $, so $ A2_{16} $
- Binary:
- $ 162 = 128 + 32 + 2 = 2^7 + 2^5 + 2^1 $
- So: $ 10100010_2 $

#### 2. 238
- Hexadecimal:
- $ 238 \div 16 = 14 $ rem $ 14 $
- $ 14 = E $, so $ EE_{16} $
- Binary:
- $ 238 = 128 + 64 + 32 + 16 + 4 + 2 + 2 $? Wait:
- $ 128 + 64 = 192 $, $ 238 - 192 = 46 $
- $ 32 + 16 = 48 > 46 $, so $ 32 $, $ 46 - 32 = 14 $
- $ 8 + 4 + 2 = 14 $
- So: $ 128 + 64 + 32 + 8 + 4 + 2 = 238 $
- Bits: $ 11101110_2 $

#### 3. 16
- Hexadecimal: $ 16 \div 16 = 1 $ rem $ 0 $ → $ 10_{16} $
- Binary: $ 10000_2 $

#### 4. 11
- Hexadecimal: $ 11 < 16 $ → $ B_{16} $
- Binary: $ 8 + 2 + 1 = 1011_2 $

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✔ Part 2 Table:

| Decimal | Hexadecimal | Binary |
|---------|-------------|------------|
| 162 | A2 | 10100010 |
| 238 | EE | 11101110 |
| 16 | 10 | 10000 |
| 11 | B | 1011 |

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Part 3: Convert Binary to Hexadecimal and Decimal

Group binary into 4-bit chunks (starting from the right), then convert each group to hex.

| Binary | Hexadecimal | Decimal |
|--------------|-------------|---------|
| 1000 0111 | | |
| 1111 0110 | | |
| 1011 1110 | | |
| 1010 1001 | | |

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#### 1. 1000 0111
- Hexadecimal:
- $ 1000_2 = 8 $, $ 0111_2 = 7 $ → $ 87_{16} $
- Decimal:
- $ 10000111_2 = 128 + 0 + 0 + 0 + 0 + 4 + 2 + 1 = 128 + 7 = 135 $

#### 2. 1111 0110
- Hexadecimal:
- $ 1111_2 = F $, $ 0110_2 = 6 $ → $ F6_{16} $
- Decimal:
- $ 11110110_2 = 128 + 64 + 32 + 16 + 0 + 4 + 2 + 0 = 254 $

#### 3. 1011 1110
- Hexadecimal:
- $ 1011_2 = B $, $ 1110_2 = E $ → $ BE_{16} $
- Decimal:
- $ 10111110_2 = 128 + 0 + 32 + 16 + 8 + 4 + 2 + 0 = 128 + 32 + 16 + 8 + 4 + 2 = 190 $

#### 4. 1010 1001
- Hexadecimal:
- $ 1010_2 = A $, $ 1001_2 = 9 $ → $ A9_{16} $
- Decimal:
- $ 10101001_2 = 128 + 0 + 32 + 0 + 8 + 0 + 0 + 1 = 128 + 32 + 8 + 1 = 169 $

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✔ Part 3 Table:

| Binary | Hexadecimal | Decimal |
|--------------|-------------|---------|
| 1000 0111 | 87 | 135 |
| 1111 0110 | F6 | 254 |
| 1011 1110 | BE | 190 |
| 1010 1001 | A9 | 169 |

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✔ Final Answer:

#### Part 1: Hex → Dec & Bin

| Hexadecimal | Decimal | Binary |
|-------------|---------|------------|
| FF | 255 | 11111111 |
| A4 | 164 | 10100100 |
| 0F | 15 | 00001111 |
| BD | 189 | 10111101 |

#### Part 2: Dec → Hex & Bin

| Decimal | Hexadecimal | Binary |
|---------|-------------|------------|
| 162 | A2 | 10100010 |
| 238 | EE | 11101110 |
| 16 | 10 | 10000 |
| 11 | B | 1011 |

#### Part 3: Bin → Hex & Dec

| Binary | Hexadecimal | Decimal |
|--------------|-------------|---------|
| 1000 0111 | 87 | 135 |
| 1111 0110 | F6 | 254 |
| 1011 1110 | BE | 190 |
| 1010 1001 | A9 | 169 |

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