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Number Bases · Form 4

Number Bases: Practice Questions

Original SPM-style practice questions for Number Bases, each with full worked solutions, Paper 1 multiple-choice and Paper 2 structured.

Written by the spmmath.com.my editorial team.

Original practice questions for Number Bases, in the style of Mathematics Paper 1 (Objective) and Mathematics Paper 2 (Subjective). Each answer is worked so you can check your method, not just the result.

Multiple-choice (Paper 1 style)

Question 1

Convert 1101₂ to a number in base 10.

  1. A. 11
  2. B. 13
  3. C. 15
  4. D. 26

Question 2

Convert 45 (base 10) to a number in base 2.

  1. A. 101101₂
  2. B. 110011₂
  3. C. 101011₂
  4. D. 110101₂

Question 3

Convert 204₅ to a number in base 10.

  1. A. 29
  2. B. 54
  3. C. 104
  4. D. 154

Question 4

Find the value of 1011₂ + 110₂, giving the answer in base 2.

  1. A. 10001₂
  2. B. 1101₂
  3. C. 10011₂
  4. D. 11101₂

Question 5

Find the value of 432₅ − 143₅, giving the answer in base 5.

  1. A. 244₅
  2. B. 234₅
  3. C. 334₅
  4. D. 224₅

Question 6

Convert 11011₂ to a number in base 8.

  1. A. 27₈
  2. B. 33₈
  3. C. 35₈
  4. D. 63₈

Question 7

Find the value of 276₈ + 145₈, giving the answer in base 8.

  1. A. 443₈
  2. B. 421₈
  3. C. 533₈
  4. D. 413₈

Question 8

In the number 1301₅, what is the place value of the digit 3, expressed in base 10?

  1. A. 3
  2. B. 15
  3. C. 75
  4. D. 375

Question 9

Which of the following numbers has the greatest value?

  1. A. 111₂
  2. B. 12₈
  3. C. 1010₂
  4. D. 21₅

Structured (Paper 2 style)

Question 1 (4 marks)

(a) Convert 110110₂ to a number in base 8. (b) Convert 623₈ to a number in base 2.

  1. (a) First convert 110110₂ to base 10: 1×2⁵ + 1×2⁴ + 0×2³ + 1×2² + 1×2¹ + 0×2⁰ = 32 + 16 + 0 + 4 + 2 + 0 = 54.
  2. Then convert 54 to base 8 by division: 54 ÷ 8 = 6 remainder 6, and 6 ÷ 8 = 0 remainder 6, so reading remainders upward gives 66₈.
  3. (b) First convert 623₈ to base 10: 6×8² + 2×8¹ + 3×8⁰ = 384 + 16 + 3 = 403.
  4. Then convert 403 to base 2 by repeated division by 2, giving remainders that read as 110010011₂. Check: 256 + 128 + 16 + 2 + 1 = 403.

Question 2 (5 marks)

Evaluate, leaving each answer in the stated base: (a) 342₅ + 421₅, giving the answer in base 5. (b) 1000₅ − 231₅, giving the answer in base 5.

  1. (a) Convert to base 10: 342₅ = 3×25 + 4×5 + 2 = 97 and 421₅ = 4×25 + 2×5 + 1 = 111.
  2. Add: 97 + 111 = 208.
  3. Convert 208 back to base 5: 208 ÷ 5 = 41 r 3, 41 ÷ 5 = 8 r 1, 8 ÷ 5 = 1 r 3, 1 ÷ 5 = 0 r 1, giving 1313₅.
  4. (b) Convert to base 10: 1000₅ = 1×125 = 125 and 231₅ = 2×25 + 3×5 + 1 = 66.
  5. Subtract: 125 − 66 = 59.
  6. Convert 59 back to base 5: 59 ÷ 5 = 11 r 4, 11 ÷ 5 = 2 r 1, 2 ÷ 5 = 0 r 2, giving 214₅.

Question 3 (4 marks)

A number x satisfies 10110₂ + x = 100011₂. (a) Find the value of x in base 2.

(b) Express x in base 5.

  1. Rearrange to make x the subject: x = 100011₂ − 10110₂.
  2. Convert to base 10: 100011₂ = 32 + 2 + 1 = 35 and 10110₂ = 16 + 4 + 2 = 22.
  3. Subtract: 35 − 22 = 13.
  4. (a) Convert 13 to base 2: 13 = 8 + 4 + 1 = 1101₂.
  5. (b) Convert 13 to base 5: 13 ÷ 5 = 2 r 3, 2 ÷ 5 = 0 r 2, giving 23₅. Check: 2×5 + 3 = 13.

Question 4 (6 marks)

The number of chairs in a hall is 158 (in base 10). Express this number in (a) base 5, (b) base 8, (c) base 2.

  1. (a) Base 5 by repeated division: 158 ÷ 5 = 31 r 3, 31 ÷ 5 = 6 r 1, 6 ÷ 5 = 1 r 1, 1 ÷ 5 = 0 r 1, so reading remainders upward gives 1113₅. Check: 125 + 25 + 5 + 3 = 158.
  2. (b) Base 8 by repeated division: 158 ÷ 8 = 19 r 6, 19 ÷ 8 = 2 r 3, 2 ÷ 8 = 0 r 2, giving 236₈. Check: 2×64 + 3×8 + 6 = 158.
  3. (c) Base 2 by repeated division: 158 = 128 + 16 + 8 + 4 + 2, so the bits are 10011110₂. Check: 128 + 16 + 8 + 4 + 2 = 158.

Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)

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Frequently asked questions

What's the difference between the Paper 1 MCQ set and the Paper 2 structured set for number bases?

The Paper 1 set has quick multiple-choice questions on converting and calculating with number bases, good for building speed, while the Paper 2 set has longer structured questions requiring full working, such as multi-step conversions or operations shown clearly. Practise both formats, since SPM can test number bases either way.

Why shouldn't I check the worked solution right after reading the question?

Attempting the conversion or calculation yourself first, even if you make an error, shows you exactly which step of the number base method you're unsure of. Checking the solution immediately only shows you the right answer, not where your own understanding is weak, so try every step, then compare carefully against the solution.

What common pitfalls should I watch for in this number bases practice set?

Common pitfalls include mixing up place values when converting between bases, forgetting to carry or borrow correctly in a non-base-10 system, and confusing the base being converted from with the base being converted to. Slow down on each digit's place value and double-check your final answer is written in the base the question asked for.

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