3.3.2 Converting Between Bases — AQA 8525

Binary → Decimal

Use the place value table. Write out the binary number and multiply each digit by its place value, then add the results.

128

10110101₂ = 128 + 32 + 16 + 4 + 1 = 181₁₀

Decimal → Binary

Work from the largest place value down. Ask: "Is 128 ≤ my number?" If yes, put a 1 and subtract it.

Example — Convert 45₁₀ to binary

128 ≤ 45? No → 0

64 ≤ 45? No → 0

32 ≤ 45? Yes → 1, remainder = 45 − 32 = 13

16 ≤ 13? No → 0

8 ≤ 13? Yes → 1, remainder = 13 − 8 = 5

4 ≤ 5? Yes → 1, remainder = 5 − 4 = 1

2 ≤ 1? No → 0

1 ≤ 1? Yes → 1, remainder = 0

45₁₀ = 00101101₂

Binary → Hexadecimal

Split the binary number into groups of 4 (nibbles), working from the right. Convert each nibble to its hex digit.

Example — Convert 10110111₂ to hex

Split: 1011 | 0111

1011 = 8+2+1 = 11 = B

0111 = 4+2+1 = 7 = 7

10110111₂ = B7₁₆

Hexadecimal → Binary

Expand each hex digit into exactly 4 binary bits.

Example — Convert 3F₁₆ to binary

3 = 0011

F = 15 = 1111

3F₁₆ = 00111111₂

Hexadecimal → Decimal

Use place values for base 16: 16¹ = 16, 16⁰ = 1. For three digits: 16² = 256.

Example — Convert 2A₁₆ to decimal

(2 × 16) + (A × 1) = (2 × 16) + (10 × 1) = 32 + 10 = 42₁₀

Example — Convert 1F4₁₆ to decimal

(1 × 256) + (F × 16) + (4 × 1) = 256 + 240 + 4 = 500₁₀

Decimal → Hexadecimal

Repeatedly divide by 16. The remainders (read bottom to top) give the hex digits.

Example — Convert 200₁₀ to hex

200 ÷ 16 = 12 remainder 8 → digit: 8

12 ÷ 16 = 0 remainder 12 → digit: C

Read remainders bottom to top → C8₁₆

Check: (12 × 16) + 8 = 192 + 8 = 200 ✓

Quick Reference

Decimal	Binary (4-bit)	Hex
0	0000	0
5	0101	5
10	1010	A
12	1100	C
15	1111	F
16	0001 0000	10
255	1111 1111	FF

Exam tip: AQA exams regularly ask you to convert between these three bases. Always show your working — you can earn method marks even if the final answer has an arithmetic slip. The easiest path between hex and binary is direct (split/expand nibbles), not via decimal.

⚠️ Common Mistakes

Forgetting to pad nibbles to 4 bits — 6 in binary is 0110, not 110
Going via decimal unnecessarily when converting between binary and hex — split into nibbles directly
Mis-remembering hex values — A=10, B=11, C=12, D=13, E=14, F=15
Reading division remainders top-to-bottom instead of bottom-to-top

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Key points

Binary → decimal: multiply each bit by its place value and sum
Decimal → binary: subtract largest fitting power of 2 repeatedly
Binary ↔ hex: split into 4-bit nibbles — no decimal step needed
Hex → decimal: multiply each digit by 16ⁿ and sum

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Worksheet — 3.3.2 Converting Between Bases

8 questions · 18 marks

Q1Convert 10011010₂ to decimal. Show your working.[2]

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128+0+0+16+8+0+2+0 = 154 [working shown 1]; answer = 154₁₀ [1].

Q2Convert 73₁₀ to 8-bit binary. Show your working.[2]

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64+8+1=73; working shown [1]; 01001001 [1].

Q3Convert 11001110₂ to hexadecimal.[2]

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1100=12=C; 1110=14=E [1]; answer = CE₁₆ [1].

Q4Convert A9₁₆ to binary.[2]

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A=1010; 9=1001 [1]; answer = 10101001₂ [1].

Q5Convert 3C₁₆ to decimal. Show your working.[2]

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(3×16)+(12×1) [1]; 48+12=60₁₀ [1].

Q6Convert 156₁₀ to hexadecimal. Show your working.[3]

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156 ÷ 16 = 9 r 12 → digit C [1]; 9 ÷ 16 = 0 r 9 → digit 9 [1]; read bottom to top: 9C₁₆ [1].

Q7The HTML colour #7F00FF is written in hex. Convert each pair of hex digits to decimal: 7F, 00, FF.[3]

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7F = (7×16)+(15×1) = 112+15 = 127 [1]; 00 = 0 [1]; FF = (15×16)+(15×1) = 240+15 = 255 [1].

Q8A memory address is stored as 10110000 01011010 in binary. Write this as a hexadecimal address.[2]

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First byte: 1011=B, 0000=0 → B0 [1]; second byte: 0101=5, 1010=A → 5A; full address = B05A [1].

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Binary → Decimal

Decimal → Binary

Example — Convert 45₁₀ to binary

Binary → Hexadecimal

Example — Convert 10110111₂ to hex

Hexadecimal → Binary

Example — Convert 3F₁₆ to binary

Hexadecimal → Decimal

Example — Convert 2A₁₆ to decimal

Example — Convert 1F4₁₆ to decimal

Decimal → Hexadecimal

Example — Convert 200₁₀ to hex

Quick Reference

Key points

Worksheet — 3.3.2 Converting Between Bases

Mini Test — 3.3.2 Converting Bases