ACT Math: Pre-Algebra and Elementary Algebra

TL;DR

Pre-Algebra and Elementary Algebra on the ACT cover fundamental math skills like operations with numbers, basic equations, and understanding variables. You'll need to be quick and accurate with arithmetic, ratios, percentages, and solving for unknowns. Mastering these basics sets you up for success in more advanced ACT Math topics.

1. The Mental Model

Think of Pre-Algebra and Elementary Algebra as your math foundation. If you understand these core concepts well, you'll be able to build on them for harder problems and spot patterns more easily. It's all about knowing the rules of numbers and how to manipulate simple expressions and equations.

2. The Core Material

These topics make up a significant portion of the ACT Math section (about 20-25% combined). You'll see questions on:

Arithmetic and Number Properties

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You need to be solid on order of operations (PEMDAS/BODMAS), understanding integers, fractions, decimals, and how to perform operations with them. Also, know about prime numbers, factors, multiples, absolute value, and basic exponents and roots.

Ratios, Proportions, and Percentages

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These show up a lot. A ratio compares two quantities (e.g., 3:4). A proportion states that two ratios are equal. Percentages are just ratios out of 100. You'll often convert between these forms.

Basic Equations and Inequalities

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This is where variables come in. You'll need to solve linear equations for one unknown, including those with fractions or decimals. You also need to understand and solve simple linear inequalities, remembering to flip the sign if you multiply or divide by a negative number.

Expressions and Polynomials

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You'll combine like terms, distribute, and evaluate algebraic expressions. Know how to add, subtract, and multiply basic polynomials (like (x+2)(x-3)).

Here's a quick look at the hierarchy of these skills:

graph TD
    A["Number Properties & Operations"] --> B["Fractions, Decimals, Percents"]
    B --> C["Ratios & Proportions"]
    C --> D["Expressions & Equations (Linear)"]
    D --> E["Inequalities"]
    D --> F["Basic Polynomial Operations"]

Word Problems

Many questions will be presented as word problems. The key is to translate the words into mathematical expressions or equations you can solve. Look for keywords like "is" (equals), "of" (multiply), "per" (divide), "more than" (add), "less than" (subtract).

3. Worked Example

Let's solve a multi-step problem:

Problem: A store is having a 20% off sale on all items. If a shirt originally costs $30, and you also have a coupon for an additional $5 off any purchase over $20, what's the final price of the shirt?

Solution:

1. Calculate the 20% discount:

   • Discount amount = 20% of $30 = 0.20 * $30 = $6
   • Price after discount = $30 - $6 = $24

2. Apply the coupon:

   • Since $24 is over $20, you can use the $5 coupon.
   • Final price = $24 - $5 = $19

The final price of the shirt is $19.

4. Key Takeaways

• Master the order of operations (PEMDAS/BODMAS) for all calculations.
• Be fluent in converting between fractions, decimals, and percentages.
• Practice solving linear equations with one variable quickly and accurately.
• Remember to flip the inequality sign when multiplying or dividing by a negative number.
• Carefully read word problems to translate them into the correct mathematical setup.
• Know your basic exponent rules and properties of numbers (e.g., even/odd, prime/composite).
• Don't forget absolute value represents a number's distance from zero, always positive.

Common Mistakes to Avoid:

• Forgetting to apply the order of operations, especially with parentheses or exponents.
• Mixing up percentage calculations (e.g., calculating the discount amount versus the final price).
• Making simple arithmetic errors under pressure; double-check calculations.
• Not flipping the inequality sign when appropriate, leading to an incorrect solution set.

5. Now Try It

Spend 15 minutes reviewing how to solve linear equations with fractions (e.g., (2/3)x + 5 = 11) and inequalities (e.g., -3x + 7 < 19). Practice at least two of each, focusing on showing all your steps. Success looks like consistently getting the correct answer, understanding why each step is taken, and being able to clearly explain your process.