Word Problems

• Translate common English phrases into algebraic expressions (sum, difference, product, quotient, of, per, each)
• Recognize 'at least' as ≥ and 'at most' as ≤ and avoid the reversed-sign trap
• Convert 'X is Y% of Z' into 0.01·Y·Z = X without re-deriving the percent formula
• Set up age, rate, mixture, and ratio problems using a consistent variable labeling system
• Identify the unknown the question actually asks for before solving

Most 'hard' ACT Math problems are not mathematically hard. They are translation problems. The algebra is usually one or two steps once the equation is written. The difficulty is writing the equation. Memorize this dictionary: 'is/was/equals' → =, 'of' → × (multiplication), 'per/for each' → division or a rate, 'sum/total/combined' → +, 'difference/less than/fewer than' → − (watch order: 'x less than y' is y − x, not x − y), 'product/times' → ×, 'quotient' → ÷, 'more than' → +, 'twice' → 2·, 'half of' → (1/2)·. Inequality phrases trap students: 'at least' means ≥ (greater than or equal), 'at most' means ≤ (less than or equal), 'no more than' means ≤, 'no less than' means ≥, 'exceeds' means >. Percent phrases: 'X percent of Y' = (X/100)·Y. 'X is what percent of Y' = (X/Y)·100. A percent increase from A to B is (B − A)/A · 100, always divide by the original. For age problems, label the current age of one person as x and express everyone else in terms of x, then add or subtract years for past/future time. For rate problems, use distance = rate · time; for mixture problems, use amount of substance = concentration · volume. When two trips cover the same distance, time and speed are inversely related: going 3 times as fast takes 1/3 of the time, so write each time as distance ÷ speed before comparing. Two unknowns need two equations: for 'tickets sold and money taken in', write one equation for the count and one for the money. Before you start solving, underline the exact quantity the question asks for. 'How many more adult tickets than child tickets?' is not the same as 'How many adult tickets?'.

• Always define your variable in words first: 'Let x = the number of child tickets.' One line of writing saves you from solving for the wrong thing.
• When you see 'less than' or 'fewer than,' flip the subtraction order. 'five less than x' is x − 5, not 5 − x.
• For 'at least N,' write ≥ N. For 'at most N,' write ≤ N. Circle the inequality direction before you solve.
• If the problem has two unknowns, you need two equations. Re-read the problem until you find the second relationship.
• Plug your final answer back into the original English sentence, not the equation, to catch translation errors.