Arithmetic

Approximation, Error, and Tolerance

Every measurement contains inherent variation because no tool is perfectly precise. The tolerance interval is given by: measured value ± ½ × precision. Whenever you estimate, you introduce error. The exam tests whether you can measure this gap against the true figure: Percentage error = (|Measured Value - True Value| ÷ True Value) × 100. Always calculate the absolute difference first and ensure the denominator is the true (accurate) value, never the measured one. Do not round intermediate steps.

Compound Interest, Depreciation, and Rates

For compound interest, use the formula $F = P(1 + i)^n$ (in the booklet). To find the annual rate (i), rearrange to $i = (F/P)^{1/n} - 1$. To find time (n), use logs: $n = \log(F/P) \div \log(1 + i)$. Reducing-balance depreciation follows $F = P(1 - i)^n$. For AER (Annual Equivalent Rate), use $(1 + AER) = (1 + i)^t$, where $t$ is the number of periods per year. Always finish a comparison with a sentence: 'AER of Bank A > AER of Bank B, so Bank A gives the better return.' No sentence, no conclusion mark.

Present Value and Amortisation

The present value of an annuity (loans/mortgages) is $P = A[(1 + i)^n - 1] \div [i(1 + i)^n]$, where $A$ is the periodic repayment and $n$ is total periods. Method: (1) Convert to payment period (monthly rate $i$ as decimal, $n = \text{years} \times 12$). (2) State the formula before substituting. (3) Substitute and evaluate, rounding only at the end. An alternative accepted method is summing the geometric series of discounted repayments.

Costing, Wastage, and Break-even

The unit cost is the total manufacturing cost divided by the number of units made. Total cost = fixed costs + (variable cost per unit × units). If a percentage of production ($w\%$) is scrapped, you must make more than the order requires. Production needed = desired output ÷ $(1 - w/100)$. Do not multiply by $(1 + w/100)$ unless the question says material is sold with extra added. If the result is not a whole number, round up. Break-even = fixed costs ÷ (selling price per unit − variable cost per unit).

Value-Added Tax (VAT)

The exam will tell you the VAT rate to use; never try to recall rates. Formula: $P = \text{Final Price} \div (1 + r/100)$. VAT amount = Final Price − $P$. Common slip: calculating $r\%$ of the final price — this treats VAT as a percentage of the inclusive total, which is incorrect. Always state your currency sign.

Income Tax, USC, and PRSI

The standard rate cut-off point is the income taxed at the standard (20%) rate; excess is taxed at the higher (40%) rate. Gross tax is the sum of tax at both rates. Tax payable = gross tax − tax credit. Net pay = gross income − tax payable. A tax credit is subtracted from gross tax; an allowance increases the cut-off point. USC is charged in bands; split the income into bands, apply each rate to that portion, and sum. Never apply the top rate to total income.

Accuracy and Presentation

Keep full calculator accuracy during working; rounding early loses accuracy marks. Significant figures count from the first non-zero digit. For measurements, errors accumulate when multiplying: max area = max length × max width. Always include units (€, cm, degrees) and round exactly as instructed. A bare number often loses the final method mark.

Key terms

Percentage error

The absolute difference between measured and true values, divided by the true value.

AER

Annual Equivalent Rate; the rate compounded annually that equals the actual periodic compounding.

Standard rate cut-off point

The income threshold up to which earnings are taxed at the 20% standard rate.

Tax payable

Gross tax minus tax credits.

Amortisation

The process of paying off a debt (like a mortgage) over time in equal instalments.

Unit cost

Total manufacturing cost divided by the number of units produced.