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A rate compares two quantities with different units (miles per hour, dollars per pound); a unit rate has a denominator of 1 and is found by dividing.
Distance = rate x time is the most-tested rate formula. Keep units visible while you work — if the answer should be in hours, the setup must cancel every other unit.
Solving for an unknown by performing the same inverse operation on both sides until the variable stands alone: undo addition with subtraction, multiplication with division.
When the variable appears on both sides, collect variable terms on one side and constants on the other. Distribute across parentheses and combine like terms before isolating the variable.
Statements comparing expressions with <, >, <=, or >=, solved like equations with one critical exception: multiplying or dividing both sides by a negative number reverses the inequality sign.
Inequality solutions are ranges, not single values. Translating words matters too: ‘at least 5’ means >= 5, ‘no more than 5’ means <= 5, and ‘between 3 and 5’ usually means 3 < x < 5.
A number's distance from zero on the number line, always non-negative: |5| = 5 and |-5| = 5.
Because absolute value measures distance, it appears in real-world error and tolerance problems, such as how far a measured dose may deviate from the ordered dose.
Rational numbers can be written as a fraction of two integers (including all terminating and repeating decimals); irrational numbers cannot — their decimals never terminate or repeat (pi, square root of 2).
The number families nest: whole numbers sit inside integers, integers inside rationals, and rationals plus irrationals together make the real numbers. To order mixed lists, convert everything to decimal form first.
Rounding replaces a number with a simpler one at a given place value — look one digit to the right: 5 or more rounds up, 4 or less rounds down — and estimation uses rounded numbers to approximate an answer.
Round only the final answer in multi-step problems — rounding mid-calculation compounds error. This matters clinically too, where dosage answers are typically rounded at the end, per the problem’s instructions.
Converting English phrases into math: 'sum' and 'more than' mean add, 'difference' and 'less than' mean subtract, 'of' and 'product' mean multiply, 'per' and 'quotient' mean divide, 'is' means equals.
Define the variable explicitly (let x = the number of nights) before writing the equation, and after solving, check that the answer makes sense in context — people, doses, and prices are rarely negative.