CCSLC Mathematics cheat sheet

Percentage change = (new − old) ÷ old × 100; a rise from 40 to 50 is +25%, a fall from 50 to 40 is −20%.

Simple interest: I = P × r × t, where r is the annual rate as a decimal and t is in years; total amount = P + I.

Compound amount: A = P(1 + r/n)^(nt); for annual compounding A = P(1 + r)^t, so money grows faster than with simple interest.

Unit price = total cost ÷ quantity; comparing unit prices is the standard way to identify the better buy when pack sizes differ.

Currency conversion: amount in target currency = amount × rate; to reverse a conversion, divide by the rate rather than multiplying.

Scale on a map or plan: real length = drawn length × scale factor, e.g. 1 : 50 000 means 1 cm represents 0.5 km.

Mean = sum ÷ count; median = middle value of ordered data (average of the two middle values when n is even); mode = most frequent value.

Rounding rule: look at the first discarded digit - 5 or more rounds up, 4 or less rounds down; keep rounding until the end of a multi-step calculation.

Identify given information, the unknown, and the constraint before calculating; most errors come from answering a different question than the one asked.

Work backwards from the answer choices when a forward method is long: substitute candidates into the condition and eliminate failures.

Use a table or labelled diagram for multi-step problems; each row or label holds one quantity so no value is lost or double-counted.

Unitary method: find the value of one unit by dividing, then multiply by the required number of units (e.g. cost per item, then cost of n items).

Rate problems use the relationship distance = speed × time; rearrange to speed = distance ÷ time or time = distance ÷ speed.

Ratio sharing: total parts = sum of ratio terms; one share = total ÷ total parts × that term.

Check reasonableness by estimating first with rounded numbers; if the exact answer differs greatly from the estimate, an operation is wrong.

For 'at least' and 'at most' conditions, test the boundary value itself, since the boundary often decides whether the answer is included.

Correlation is not causation: two quantities rising together may both be driven by a third factor, so a trend alone does not prove a cause.

A percentage without its base is uninterpretable: 'increased by 50%' means different amounts depending on whether the base was 10 or 1 000.

Beware of misleading graph axes: a vertical axis that does not start at zero exaggerates small differences.

Sample bias: conclusions drawn from a self-selected or small group cannot be generalised to the whole population.

Average choice matters: mean is pulled by extreme values, median resists them, so a salary or house-price 'average' should state which is used.

Absolute versus relative risk: a rise from 1 in 10 000 to 2 in 10 000 is a 100% relative increase but only 0.01 percentage points absolute.

Check that units and time periods match before comparing figures; comparing a monthly cost with an annual one overstates one option.

A single example does not establish a rule; a claim needs a representative set of cases or a stated assumption before it is accepted.

Total cost of ownership = purchase price + running costs + maintenance − resale value; compare options over the same period.

Break-even point: fixed costs ÷ (price per unit − variable cost per unit) gives the number of units needed before profit begins.

A discount of a% followed by a further discount of b% is not (a + b)%; the combined multiplier is (1 − a/100)(1 − b/100).

Adding tax or VAT: final price = price × (1 + rate); removing tax already included: price ÷ (1 + rate).

Budget constraint: total spending must not exceed income; surplus = income − expenses, deficit = expenses − income.

Risk versus return: higher expected return usually carries higher uncertainty, so a decision should weigh both, not return alone.

When comparing loans, compare the total repayments and the annual effective rate, not only the monthly instalment.

State assumptions and a decision criterion (cheapest, fastest, safest) before comparing options, so the recommendation can be justified.