Binary, Denary, and Hexadecimal Conversion Practice

This lesson provides extensive practice in converting between binary, denary, and hexadecimal — a core skill tested in OCR J277 Section 2.6. Conversion questions appear regularly in Paper 1 and often carry 2-4 marks.

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Quick Reference: Conversion Routes

graph TD
  Bin["Binary"] <--> Hex["Hexadecimal"]
  Bin <--> Den1["Denary"]
  Hex <-->|"via binary or direct"| Den2["Denary"]
  Den1 <--> Den2

There are six possible conversions. The most efficient routes are:

From	To	Method
Binary	Denary	Add place values where bit = 1
Denary	Binary	Place value or successive division by 2
Binary	Hex	Group into nibbles (4-bit groups)
Hex	Binary	Expand each hex digit to 4 bits
Hex	Denary	Multiply digits by powers of 16
Denary	Hex	Divide by 16, or convert via binary

flowchart LR
    Binary((Binary - base 2)) -->|place values 128 64 32 ... 1| Denary((Denary - base 10))
    Denary -->|divide by 2 / place values| Binary
    Binary -->|group into 4-bit nibbles| Hex((Hex - base 16))
    Hex -->|expand each digit to 4 bits| Binary
    Hex -->|x16 then x1 place values| Denary
    Denary -->|divide by 16 OR via binary| Hex

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Worked Examples: Binary to Denary

Example 1: 01011010

128	64	32	16	8	4	2	1
0	1	0	1	1	0	1	0

64 + 16 + 8 + 2 = 90

Example 2: 11100001

128 + 64 + 32 + 1 = 225

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Worked Examples: Denary to Binary

Example 1: Convert 73 to binary

Using place values:

• 73 - 64 = 9 (1 in 64 column)
• 9 - 8 = 1 (1 in 8 column)
• 1 - 1 = 0 (1 in 1 column)

Result: 01001001

Check: 64 + 8 + 1 = 73. Correct!

Example 2: Convert 199 to binary

• 199 - 128 = 71
• 71 - 64 = 7
• 7 - 4 = 3
• 3 - 2 = 1
• 1 - 1 = 0

Result: 11000111

Check: 128 + 64 + 4 + 2 + 1 = 199. Correct!

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Worked Examples: Binary to Hexadecimal

Example 1: 10110100

Split: 1011 | 0100

• 1011 = B (8+2+1=11)
• 0100 = 4

Answer: B4

Example 2: 01111110

Split: 0111 | 1110

• 0111 = 7
• 1110 = E (8+4+2=14)

Answer: 7E

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Worked Examples: Hexadecimal to Binary

Example 1: Convert 9C to binary

• 9 = 1001
• C (12) = 1100

Answer: 10011100

Example 2: Convert D7 to binary

• D (13) = 1101
• 7 = 0111

Answer: 11010111

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Worked Examples: Hexadecimal to Denary

Example 1: Convert 5A to denary

• 5 x 16 = 80
• A (10) x 1 = 10

Answer: 90

Example 2: Convert E3 to denary

• E (14) x 16 = 224
• 3 x 1 = 3

Answer: 227

OCR Exam Tip: Always show your working in conversion questions. Even if you make an arithmetic error, you can gain method marks by demonstrating the correct process.

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Practice Set

Complete the following conversions:

Set 1: Binary to Denary

Binary	Denary
00101100	44
10010001	145
11110000	240
01010101	85

Set 2: Denary to Binary

Denary	Binary
100	01100100
250	11111010
37	00100101
192	11000000

Set 3: Binary to Hex

Binary	Hex
10101111	AF
01100011	63
11001010	CA
00010111	17

Set 4: Hex to Denary

Hex	Denary
2A	42
FF	255
80	128
C9	201

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Common Mistakes to Avoid

Mistake	Correction
Starting binary groups from the left	Always group from the right when converting to hex
Forgetting leading zeros	A nibble must always be 4 bits (e.g., 5 = 0101, not 101)
Confusing hex A-F with denary	A = 10, B = 11, C = 12, D = 13, E = 14, F = 15
Not checking the answer	Always convert back to verify your result
Using the wrong place values	Binary: powers of 2. Hex: powers of 16. Do not mix them up.

OCR Exam Tip: In the exam, if you have time, convert your answer back to the original number system to verify it. This takes seconds and can catch costly errors.

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Worked Example: Round-Trip Conversion of a Colour Code

Web developers regularly work with hex colour codes. This worked example converts the code #2F9 (a shade of green) through all three number systems to reinforce how conversions chain together.

Setup. The hex colour shorthand #2F9 is the three-digit form of the full code #22FF99. For this exercise, we focus on the middle byte FF so we can demonstrate a round-trip in 8 bits.

Step 1: Hex to binary.

• F (15) = 1111
• F (15) = 1111

Result: 11111111.

Step 2: Binary to denary.

Use the 8-bit place value table:

128	64	32	16	8	4	2	1
1	1	1	1	1	1	1	1

128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255 — the maximum value for 8 bits, which corresponds to 100% intensity of the green channel in an RGB colour.

Step 3: Denary back to hex.

Divide 255 by 16: 255 / 16 = 15 remainder 15. Both the quotient and remainder map to F, so 255 = FF. Round-trip confirmed.

Step 4: Apply to a different channel.

Suppose we want to convert the red channel value 8C to denary to check RGB sliders in a graphics tool.

• 8 x 16 = 128
• C = 12, so 12 x 1 = 12

Total: 128 + 12 = 140. In binary: 8 = 1000, C = 1100, so 8C = 10001100. Place value check: 128 + 8 + 4 = 140. All three representations agree.

Step 5: Compose a full RGB code.