Mathematics: Arithmetic, Algebra, Ratios and Basic Statistics
Arithmetic and percentages
Follow order of operations (PEMDAS). Convert fractions, decimals and percentages freely. Percent change = (new - old) / old x 100. A 20 percent rise followed by a 20 percent fall is not zero change: multiply by 1.20 then 0.80 to get 0.96, a net 4 percent decrease. Dividing by a fraction is the same as multiplying by its reciprocal.
Algebra
Solve linear equations by undoing operations in reverse order and keeping both sides balanced. Handle exponent rules: x^a times x^b = x^(a+b), x^a / x^b = x^(a-b), x^(-a) = 1/x^a. Factor common terms before solving quadratics, and check answers by substitution.
Ratios and proportions
Set up a proportion with matching units in the same position, then cross-multiply. For scaling recipes, maps or dilutions, find the unit rate first (per one unit), then multiply.
Statistics
Mean is affected by extreme values; median is the middle value after sorting (average the two middle values when the count is even); mode is most frequent; range is max minus min. Standard deviation measures spread, not centre.
Traps to expect
Averaging two averages without weighting for group size; forgetting to sort data before finding the median; reversing percent increase and decrease; misreading what a ratio compares.
Worked example
A jacket costs 80 and rises 25 percent, then falls 20 percent. New price = 80 x 1.25 = 100, then 100 x 0.80 = 80. The final price equals the original, even though the percentages differ, because each percentage applies to a different base.