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The required sequence for evaluating expressions: Parentheses, Exponents, Multiplication and Division (left to right), then Addition and Subtraction (left to right) — remembered as PEMDAS.
Grouping symbols include brackets and fraction bars as well as parentheses — the numerator and denominator of a fraction are each evaluated as if wrapped in parentheses before dividing.
Adding and subtracting fractions requires a common denominator; multiplying multiplies straight across; dividing multiplies by the reciprocal of the second fraction.
Mixed numbers are usually easiest to convert to improper fractions before operating. To compare fractions quickly, cross-multiply or convert both to decimals.
Each digit's value depends on its position: places left of the decimal point are ones, tens, hundreds; places right of it are tenths, hundredths, thousandths.
Converting between fractions and decimals is a core TEAS move: divide numerator by denominator to get a decimal, and read a terminating decimal as a fraction over a power of ten (0.45 is 45/100, which reduces to 9/20).
A percent is a ratio out of 100; the core relationship is part = percent x whole, which can be solved for any one of the three quantities.
Common benchmark conversions worth memorizing: 1/8 = 12.5%, 1/4 = 25%, 1/3 is about 33.3%, 3/8 = 37.5%, and 2/3 is about 66.7%. They let you estimate answers before computing.
Percent change = (amount of change / original amount) x 100 — the denominator is always the ORIGINAL (starting) value, not the new one.
For successive changes, apply each multiplier in turn: a 10% increase followed by a 10% decrease is 1.10 x 0.90 = 0.99 of the original — a net 1% decrease, not zero.
A comparison of two quantities by division, written 3:4, 3/4, or '3 to 4'; ratios can compare part to part or part to whole.
Ratios scale like fractions — multiply or divide both terms by the same number. To split a total in a given ratio, add the ratio parts to find the number of shares, divide the total by that sum, then multiply back out.
An equation stating two ratios are equal (a/b = c/d), solved by cross-multiplication: ad = bc.
Proportions power most TEAS word problems: scaling recipes, map distances, medication dosing, and similar figures. Direct proportions rise together; in inverse proportion one quantity rises as the other falls (more workers, less time).