Number Bases · Form 4
Number Bases: Paper 2 Answering Guide
How Number Bases appears in Mathematics Paper 2 (Subjective), and how to lay out your working so you earn every method mark.
How it is examined
This is usually a short, self-contained topic in Paper 1 and a structured question in Paper 2. Marks are lost less to difficulty than to carelessness, a digit five written in base five, or an arithmetic slip during conversion.
Neat, checked working scores well here.
Showing your working
Mathematics Paper 2 (Subjective) is worth 100 marks and gives marks for the steps, not only the answer. Write each line clearly: state the formula or rule, substitute the numbers, then simplify.
If the question carries units, carry them through to the final line.
The question
The number P = 143 (base five) and the number Q = 21 (base five). (a) Convert P to base ten.
(b) Find the value of P + Q, giving your answer in base five. (c) Express P in base two.
The working in the next section shows the layout that earns every method mark.
Mark-earning layout, line by line
- (a) State the rule: each digit is multiplied by its place value 25, 5, 1.
- (a) Substitute: P = 1x25 + 4x5 + 3x1.
- (a) Simplify: P = 25 + 20 + 3 = 48 (base ten).
- (b) State the rule: add in base five, carrying a group of five (not ten).
- (b) Units: 3 + 1 = 4. Fives: 4 + 2 = 6 = 11 (base five), so write 1 and carry 1. Twenty-fives: 1 + 1 = 2.
- (b) Answer: P + Q = 214 (base five). Check via base ten: 48 + 11 = 59, and 2x25 + 1x5 + 4 = 59.
- (c) State the rule: divide 48 repeatedly by 2 and record remainders. 48 / 2 = 24 r 0, 24 / 2 = 12 r 0, 12 / 2 = 6 r 0, 6 / 2 = 3 r 0, 3 / 2 = 1 r 1, 1 / 2 = 0 r 1.
- (c) Answer: reading the remainders upward, P = 110000 (base two). Check: 32 + 16 = 48.
Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)
Frequently asked questions
Do I need to show the base-ten step if the question only asks for base two?
If the starting number is already in base ten, no; divide straight into base two. But if you are asked to go from base five to base two, the base-ten stage is your safest bridge and should be shown, because it carries method marks and lets the marker follow your route.
Only skip it if you convert directly and confidently.
How should I lay out addition in a base to earn method marks?
Set it out vertically with digits aligned by column, exactly like ordinary addition, and write each carry small above the next column. Then add a one-line base-ten check beside it.
The aligned columns plus the visible carries are what a marker rewards; a bare final number with no column working risks getting only the answer mark, or none if it is wrong.
If part (b) uses my answer from part (a) and (a) was wrong, do I lose all the marks?
Usually not. Under follow-through marking, part (b) is judged on whether your method is correct given the figure you carried forward, so a single wrong value in (a) costs one mark, not the whole question.
This is another reason to show every step clearly: the marker can see your method is sound even when an earlier number is off.