Decimal to Binary Conversion Worksheet

Problem Overview:

The task involves converting decimal numbers to binary and binary numbers to decimal. Let's solve each part step by step.

---

Part 1: Converting Decimal to Binary

#### Step-by-Step Method for Decimal to Binary Conversion:
1. Divide the decimal number by 2.
2. Record the remainder (0 or 1).
3. Use the quotient for the next division.
4. Repeat until the quotient is 0.
5. The binary number is the sequence of remainders read from bottom to top.

#### Solutions:

1. 188₁₀ = ?
- 188 ÷ 2 = 94, remainder 0
- 94 ÷ 2 = 47, remainder 0
- 47 ÷ 2 = 23, remainder 1
- 23 ÷ 2 = 11, remainder 1
- 11 ÷ 2 = 5, remainder 1
- 5 ÷ 2 = 2, remainder 1
- 2 ÷ 2 = 1, remainder 0
- 1 ÷ 2 = 0, remainder 1
- Binary: 10111100₂

2. 180₁₀ = ?
- 180 ÷ 2 = 90, remainder 0
- 90 ÷ 2 = 45, remainder 0
- 45 ÷ 2 = 22, remainder 1
- 22 ÷ 2 = 11, remainder 0
- 11 ÷ 2 = 5, remainder 1
- 5 ÷ 2 = 2, remainder 1
- 2 ÷ 2 = 1, remainder 0
- 1 ÷ 2 = 0, remainder 1
- Binary: 10110100₂

3. 144₁₀ = ?
- 144 ÷ 2 = 72, remainder 0
- 72 ÷ 2 = 36, remainder 0
- 36 ÷ 2 = 18, remainder 0
- 18 ÷ 2 = 9, remainder 0
- 9 ÷ 2 = 4, remainder 1
- 4 ÷ 2 = 2, remainder 0
- 2 ÷ 2 = 1, remainder 0
- 1 ÷ 2 = 0, remainder 1
- Binary: 10010000₂

4. 245₁₀ = ?
- 245 ÷ 2 = 122, remainder 1
- 122 ÷ 2 = 61, remainder 0
- 61 ÷ 2 = 30, remainder 1
- 30 ÷ 2 = 15, remainder 0
- 15 ÷ 2 = 7, remainder 1
- 7 ÷ 2 = 3, remainder 1
- 3 ÷ 2 = 1, remainder 1
- 1 ÷ 2 = 0, remainder 1
- Binary: 11110101₂

5. 202₁₀ = ?
- 202 ÷ 2 = 101, remainder 0
- 101 ÷ 2 = 50, remainder 1
- 50 ÷ 2 = 25, remainder 0
- 25 ÷ 2 = 12, remainder 1
- 12 ÷ 2 = 6, remainder 0
- 6 ÷ 2 = 3, remainder 0
- 3 ÷ 2 = 1, remainder 1
- 1 ÷ 2 = 0, remainder 1
- Binary: 11001010₂

6. 231₁₀ = ?
- 231 ÷ 2 = 115, remainder 1
- 115 ÷ 2 = 57, remainder 1
- 57 ÷ 2 = 28, remainder 1
- 28 ÷ 2 = 14, remainder 0
- 14 ÷ 2 = 7, remainder 0
- 7 ÷ 2 = 3, remainder 1
- 3 ÷ 2 = 1, remainder 1
- 1 ÷ 2 = 0, remainder 1
- Binary: 11100111₂

7. 147₁₀ = ?
- 147 ÷ 2 = 73, remainder 1
- 73 ÷ 2 = 36, remainder 1
- 36 ÷ 2 = 18, remainder 0
- 18 ÷ 2 = 9, remainder 0
- 9 ÷ 2 = 4, remainder 1
- 4 ÷ 2 = 2, remainder 0
- 2 ÷ 2 = 1, remainder 0
- 1 ÷ 2 = 0, remainder 1
- Binary: 10010011₂

8. 138₁₀ = ?
- 138 ÷ 2 = 69, remainder 0
- 69 ÷ 2 = 34, remainder 1
- 34 ÷ 2 = 17, remainder 0
- 17 ÷ 2 = 8, remainder 1
- 8 ÷ 2 = 4, remainder 0
- 4 ÷ 2 = 2, remainder 0
- 2 ÷ 2 = 1, remainder 0
- 1 ÷ 2 = 0, remainder 1
- Binary: 10001010₂

---

Part 2: Converting Binary to Decimal

#### Step-by-Step Method for Binary to Decimal Conversion:
1. Write down the binary number.
2. Assign powers of 2 to each digit from right to left, starting with 2⁰.
3. Multiply each binary digit by its corresponding power of 2.
4. Sum all the results to get the decimal equivalent.

#### Solutions:

9. 1011100₂ = ?
- Binary: 1011100
- Powers of 2: 2⁶, 2⁵, 2⁴, 2³, 2², 2¹, 2⁰
- Calculation: \(1 \cdot 2^6 + 0 \cdot 2^5 + 1 \cdot 2^4 + 1 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 0 \cdot 2^0\)
- \(= 64 + 0 + 16 + 8 + 4 + 0 + 0 = 92\)
- Decimal: 92₁₀

10. 10100001₂ = ?
- Binary: 10100001
- Powers of 2: 2⁷, 2⁶, 2⁵, 2⁴, 2³, 2², 2¹, 2⁰
- Calculation: \(1 \cdot 2^7 + 0 \cdot 2^6 + 1 \cdot 2^5 + 0 \cdot 2^4 + 0 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0\)
- \(= 128 + 0 + 32 + 0 + 0 + 0 + 0 + 1 = 161\)
- Decimal: 161₁₀

11. 10001110₂ = ?
- Binary: 10001110
- Powers of 2: 2⁷, 2⁶, 2⁵, 2⁴, 2³, 2², 2¹, 2⁰
- Calculation: \(1 \cdot 2^7 + 0 \cdot 2^6 + 0 \cdot 2^5 + 0 \cdot 2^4 + 1 \cdot 2^3 + 1 \cdot 2^2 + 1 \cdot 2^1 + 0 \cdot 2^0\)
- \(= 128 + 0 + 0 + 0 + 8 + 4 + 2 + 0 = 142\)
- Decimal: 142₁₀

12. 10100000₂ = ?
- Binary: 10100000
- Powers of 2: 2⁷, 2⁶, 2⁵, 2⁴, 2³, 2², 2¹, 2⁰
- Calculation: \(1 \cdot 2^7 + 0 \cdot 2^6 + 1 \cdot 2^5 + 0 \cdot 2^4 + 0 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 0 \cdot 2^0\)
- \(= 128 + 0 + 32 + 0 + 0 + 0 + 0 + 0 = 160\)
- Decimal: 160₁₀

13. 11100010₂ = ?
- Binary: 11100010
- Powers of 2: 2⁷, 2⁶, 2⁵, 2⁴, 2³, 2², 2¹, 2⁰
- Calculation: \(1 \cdot 2^7 + 1 \cdot 2^6 + 1 \cdot 2^5 + 0 \cdot 2^4 + 0 \cdot 2^3 + 0 \cdot 2^2 + 1 \cdot 2^1 + 0 \cdot 2^0\)
- \(= 128 + 64 + 32 + 0 + 0 + 0 + 2 + 0 = 226\)
- Decimal: 226₁₀

14. 1010101₂ = ?
- Binary: 1010101
- Powers of 2: 2⁶, 2⁵, 2⁴, 2³, 2², 2¹, 2⁰
- Calculation: \(1 \cdot 2^6 + 0 \cdot 2^5 + 1 \cdot 2^4 + 0 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0\)
- \(= 64 + 0 + 16 + 0 + 4 + 0 + 1 = 85\)
- Decimal: 85₁₀

15. 11111101₂ = ?
- Binary: 11111101
- Powers of 2: 2⁷, 2⁶, 2⁵, 2⁴, 2³, 2², 2¹, 2⁰
- Calculation: \(1 \cdot 2^7 + 1 \cdot 2^6 + 1 \cdot 2^5 + 1 \cdot 2^4 + 1 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0\)
- \(= 128 + 64 + 32 + 16 + 8 + 4 + 0 + 1 = 249\)
- Decimal: 249₁₀

16. 11000111₂ = ?
- Binary: 11000111
- Powers of 2: 2⁷, 2⁶, 2⁵, 2⁴, 2³, 2², 2¹, 2⁰
- Calculation: \(1 \cdot 2^7 + 1 \cdot 2^6 + 0 \cdot 2^5 + 0 \cdot 2^4 + 0 \cdot 2^3 + 1 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0\)
- \(= 128 + 64 + 0 + 0 + 0 + 4 + 2 + 1 = 199\)
- Decimal: 199₁₀

---

Final Answers:

1. 188₁₀ = 10111100₂
2. 180₁₀ = 10110100₂
3. 144₁₀ = 10010000₂
4. 245₁₀ = 11110101₂
5. 202₁₀ = 11001010₂
6. 231₁₀ = 11100111₂
7. 147₁₀ = 10010011₂
8. 138₁₀ = 10001010₂

9. 1011100₂ = 92₁₀
10. 10100001₂ = 161₁₀
11. 10001110₂ = 142₁₀
12. 10100000₂ = 160₁₀
13. 11100010₂ = 226₁₀
14. 1010101₂ = 85₁₀
15. 11111101₂ = 249₁₀
16. 11000111₂ = 199₁₀

Boxed Final Answer:
\[
\boxed{
\begin{array}{ll}
1. & 10111100_2 \\
2. & 10110100_2 \\
3. & 10010000_2 \\
4. & 11110101_2 \\
5. & 11001010_2 \\
6. & 11100111_2 \\
7. & 10010011_2 \\
8. & 10001010_2 \\
9. & 92_{10} \\
10. & 161_{10} \\
11. & 142_{10} \\
12. & 160_{10} \\
13. & 226_{10} \\
14. & 85_{10} \\
15. & 249_{10} \\
16. & 199_{10} \\
\end{array}
}
\]