These nine original questions cover the whole number bases section of SPM Mathematics. Work each one on paper first, then open the answer and compare your working.
Questions 1 to 3 are single conversions, 4 to 6 mix bases, and 7 to 9 add arithmetic and reasoning.
Practice questions
Question 1. Convert 47 to base 2.
Answer
47 ÷ 2 = 23 r 1, 23 ÷ 2 = 11 r 1, 11 ÷ 2 = 5 r 1, 5 ÷ 2 = 2 r 1, 2 ÷ 2 = 1 r 0, 1 ÷ 2 = 0 r 1.
Reading upwards: 101111₂. Check: 32 + 8 + 4 + 2 + 1 = 47.
Question 2. Convert 2034₅ to base ten. State the value of the digit 3.
Answer
2(125) + 0(25) + 3(5) + 4(1) = 250 + 15 + 4 = 269.
The digit 3 sits in the fives place, so its value is 3 × 5 = 15.
Question 3. Convert 156 to base 8.
Answer
156 ÷ 8 = 19 r 4, 19 ÷ 8 = 2 r 3, 2 ÷ 8 = 0 r 2.
Reading upwards: 234₈. Check: 2(64) + 3(8) + 4 = 128 + 24 + 4 = 156.
Question 4. Convert 11011₂ to base 8 by grouping in threes, then check by converting both to base ten.
Answer
Group from the right: 11 | 011, and pad the left group to 011. Each group gives 3, so the answer is 33₈.
Check: 11011₂ = 16 + 8 + 2 + 1 = 27, and 3(8) + 3 = 27.
Question 5. Convert 243₅ to base 2.
Answer
First to base ten: 2(25) + 4(5) + 3 = 50 + 20 + 3 = 73.
73 ÷ 2 = 36 r 1, 36 ÷ 2 = 18 r 0, 18 ÷ 2 = 9 r 0, 9 ÷ 2 = 4 r 1, 4 ÷ 2 = 2 r 0, 2 ÷ 2 = 1 r 0, 1 ÷ 2 = 0 r 1.
Reading upwards: 1001001₂. Check: 64 + 8 + 1 = 73.
Question 6. How many digits does 500 have when written in base 2? Explain without converting fully.
Answer
The powers of 2 are 256 and 512. Since 256 ≤ 500 < 512, the number has 9 digits.
An n-digit base 2 number lies between 2^(n−1) and 2^n − 1, and here n = 9 gives 256 to 511.
Question 7. Calculate 1011₂ + 110₂, giving the answer in base 2.
Answer
Line up the columns from the right. The units column gives 1 + 0 = 1.
Next, 1 + 1 = 10, so write 0 and carry 1, and 0 + 1 + 1 = 10 gives another 0 and carry 1. Then 1 + 0 + 1 = 10 gives 0 and carry 1, and the final carry gives 1.
Answer: 10001₂. Check: 11 + 6 = 17 = 16 + 1.
Question 8. Calculate 423₅ − 144₅, giving the answer in base 5.
Answer
Units: 3 is smaller than 4, so borrow 1 from the fives column, which adds 5 to give 8. Then 8 − 4 = 4.
Fives: 1 after the borrow, so borrow again from the 25s. 1 + 5 = 6, and 6 − 4 = 2.
25s: 4 − 1 = 3 after the borrow, and 3 − 1 = 2.
Answer: 224₅. Check: 113 − 49 = 64, and 2(25) + 2(5) + 4 = 64.
Question 9. A student converts 37 to base 5 and writes 72₅. Explain what is wrong and give the correct answer.
Answer
The digit 7 cannot appear in base 5, because only 0 to 4 are allowed.
The student stopped dividing when the quotient was 7. Continue: 37 ÷ 5 = 7 r 2, then 7 ÷ 5 = 1 r 2, then 1 ÷ 5 = 0 r 1.
The correct answer is 122₅. Check: 25 + 10 + 2 = 37.
If you got some wrong
Wrong answers to questions 1 to 3 point to converting base ten to another base. Errors in questions 4 and 5 suggest a review of converting between non-decimal bases.
Questions 6 and 9 are about auditing your own answer, which is the focus of checking place-value errors in base conversions. Record each slip in the mistake log tool so a pattern becomes visible.
If the same type of slip keeps returning, online one-to-one Mathematics tuition lets a teacher work on it with you directly.