Arithmetic and Algebraic Reasoning
What this domain tests
Arithmetic and Algebraic Reasoning carries the largest share of the Secondary 4 Cultural, Social and Technical option. The exam asks you to compute with real numbers, reason proportionally, simplify and factor algebraic expressions, solve equations and inequalities, and interpret relations and functions in context. Questions usually arrive as short real-life scenarios rather than bare calculations.
Core rules and concepts
- Real numbers: distinguish rational from irrational. A square root is irrational unless the radicand is a perfect square. Remember the priority of operations and the effect of a negative sign in front of a power: $(-3)^2=9$ but $-3^2=-9$.
- Proportional reasoning: a situation is proportional only if it passes through the origin. Set up a proportion $\frac{a}{b}=\frac{c}{d}$ and solve by cross-multiplication. Percentages are ratios out of 100; a discount followed by a tax is not the sum of the two rates.
- Algebraic expressions: apply exponent laws ($a^m\cdot a^n=a^{m+n}$, $\frac{a^m}{a^n}=a^{m-n}$, $(a^m)^n=a^{mn}$). Expand with the distributive property; factor by removing the common factor or by recognizing a difference of squares $a^2-b^2=(a-b)(a+b)$.
- Equations and inequalities: isolate the variable by inverse operations. When you multiply or divide both sides of an inequality by a negative number, reverse the inequality sign.
- Relations and functions: a relation is a function if each input has exactly one output. Read rate of change as slope and initial value as the y-intercept in a linear model.
Worked example
A phone plan costs a fixed $18 per month plus $0.12 per minute beyond 200 included minutes. A customer is billed $34.80. How many extra minutes were used?
Let $x$ be the extra minutes. The model is $18+0.12x=34.80$. Subtract 18: $0.12x=16.80$. Divide by 0.12: $x=140$. The customer used 140 extra minutes. Check: $18+0.12(140)=18+16.80=34.80$, which matches the bill.
Traps that cost marks
- Treating a non-proportional situation as proportional (for example, assuming a taxi fare is proportional when it has a base charge).
- Forgetting to reverse the inequality sign after dividing by a negative.
- Confusing the perimeter and area formulas, or using the wrong unit after a calculation.
- Rounding intermediate results too early and losing precision in the final answer.
- Giving a numerical answer without showing the model or verification when the question asks you to justify.