Pre-Algebra

• Convert between fractions, decimals, and percentages and calculate percent increase and decrease, including two changes in a row
• Set up and solve proportion equations (cross-multiplication)
• Translate ratio word problems into algebraic expressions
• Model a word problem: write the equation from the words, and handle rates and unit conversions by lining up units so they cancel
• Finish on one ACT-hard question and say what made it hard

A ratio compares two quantities (e.g., 3:5); a proportion states that two ratios are equal (3/5 = 6/10). To solve a proportion, cross-multiply: if a/b = c/d, then ad = bc. Percent change is calculated as (new − old) / old × 100. The ACT frequently wraps these concepts in word problems: 'If 40% of students are male and there are 240 males, how many students total?' translates to 0.40 × total = 240, so total = 600. When you set up a proportion, either orientation works as long as both ratios keep the same units in the same positions (miles over hours on both sides). Modeling means turning the words into an equation before doing any arithmetic: name the unknown, then translate phrase by phrase ('is' means equals, 'of' means times, 'per' means divided by). Rates follow distance = rate × time, and the same pattern covers work and cost (amount = rate × time). For unit conversions, multiply by a fraction equal to 1 and let the units cancel: 60 miles per hour × 5,280 feet per mile ÷ 3,600 seconds per hour = 88 feet per second. Being fluent with these translations is worth several easy points. When a percent or fraction problem never gives a total, pick a convenient one: 100 for percents, or a common multiple of the denominators for fractions (for thirds and quarters, start with 12), then work it through and convert back. Classify before you compute: on a wordy item, first name the one fact it hinges on, such as which amount a percent is taken of. The hardest questions, at the end of the section: two changes in a row and undoing a change. Changes multiply, so up 10% then down 10% is 1.10 × 0.90 = 0.99, a 1% decrease rather than no change; to undo a change, divide by the multiplier, so a price of $60 after a 20% increase started at 60 ÷ 1.20 = $50, not at 80% of $60 = $48.

• Percent increase of 25% means multiplying by 1.25; percent decrease of 25% means multiplying by 0.75.
• For ratio problems, assign a variable to the ratio unit: if the ratio is 3:5, let the parts be 3x and 5x.
• In unit conversions, write the units in every step. If the units do not cancel to what the question asks for, the setup is wrong.