ACT Math: Pre-Algebra Concepts

Pre-algebra accounts for roughly twenty to twenty-five percent of your ACT math score, making it the single largest content area on the test. Mastering these foundational concepts is non-negotiable for a high score because they underpin nearly every other math topic you'll encounter. A strong command of pre-algebra not only secures these points directly but also increases your speed and accuracy on more complex algebra and geometry problems that build upon these basics.

Foundational Operations: Fractions, Decimals, and Percents

The ACT expects fluency in moving between and operating with fractions, decimals, and percents. A fraction represents a part of a whole, written as $\frac{a}{b}$ (where $b\ne 0$). You must be able to add, subtract, multiply, and divide them seamlessly. Remember, division of fractions is performed by multiplying by the reciprocal. For example, $\frac{2}{3}÷\frac{4}{5}=\frac{2}{3}\times \frac{5}{4}=\frac{10}{12}=\frac{5}{6}$.

Decimals are base-10 representations of fractions. A key skill is converting between forms: to change a fraction to a decimal, divide the numerator by the denominator. To change a decimal to a percent, multiply by 100. For a percent problem, remember the basic relationship: $\text{Percent}=\frac{\text{Part}}{\text{Whole}}\times 100$. If a question states "30 is 40% of what number?", you set up the equation $30=0.40\times W$, then solve for $W=75$.

Ratios, Proportions, and Rates

A ratio is a comparative relationship between two quantities, often expressed as $a:b$ or $\frac{a}{b}$. A proportion is an equation stating that two ratios are equal: $\frac{a}{b}=\frac{c}{d}$. Solving proportion problems almost always involves cross-multiplication: $a\times d=b\times c$. These concepts are frequently tested in word problems involving scaling, similar figures, or unit rates.

For instance, if a map scale is 1 inch : 10 miles, and two towns are 3.5 inches apart on the map, the actual distance is found by setting up the proportion $\frac{1\text{ in}}{10\text{ mi}}=\frac{3.5\text{ in}}{x\text{ mi}}$. Cross-multiplying gives $1\times x=10\times 3.5$, so $x=35$ miles. Always ensure the units in your ratios correspond.

Absolute Value and the Number Line

The absolute value of a number, denoted $|x|$, is its distance from zero on the number line, always a non-negative value. Algebraically, $|x|=x$ if $x\ge 0$, and $|x|=-x$ if $x<0$. The ACT often tests this with equations like $|x-2|=5$. This means the expression inside the absolute value is 5 units from zero. Therefore, you must solve two equations: $x-2=5$ and $x-2=-5$, yielding solutions $x=7$ and $x=-3$. Remember to check both possibilities.

Working with Exponents and Scientific Notation

Exponent rules are essential. Know these core properties:

• Product Rule: $a^m\times a^n=a^{m+n}$
• Quotient Rule: $a^m÷a^n=a^{m-n}$
• Power Rule: $(a^m{)}^n=a^{m\times n}$
• Negative Exponent: $a^{-n}=\frac{1}{a^n}$
• Zero Exponent: $a^0=1$ (for $a\ne 0$)

A common test question involves simplifying an expression like $\frac{(2x^3{)}^4}{8x^{10}}$. Work step-by-step: $(2x^3{)}^4=2^4\times x^{12}=16x^{12}$. Then, $\frac{16x^{12}}{8x^{10}}=2x^2$.

Scientific notation expresses numbers as a product of a coefficient (between 1 and 10) and a power of 10: $a\times 10^n$, where $1\le a<10$. To multiply numbers in scientific notation, multiply the coefficients and add the exponents: $(3\times 10^4)\times (2\times 10^5)=6\times 10^9$. To divide, divide the coefficients and subtract the exponents. The ACT will ask you to perform these operations and then select the answer choice that is correctly written in scientific notation.

Basic Number Theory: Factors, Multiples, and Properties

This area deals with the properties of integers. A factor (or divisor) of an integer divides that number evenly (with no remainder). A multiple is the product of that number and any integer. Key concepts include:

• Greatest Common Factor (GCF): The largest number that divides evenly into two or more numbers.
• Least Common Multiple (LCM): The smallest number that is a multiple of two or more numbers.

For the numbers 24 and 36, the factors are:

• 24: 1, 2, 3, 4, 6, 8, 12, 24
• 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

The GCF is 12. The multiples are:

• 24: 24, 48, 72, 96...
• 36: 36, 72, 108...

The LCM is 72. Understanding these helps with problems about repeating cycles, simplifying fractions, or dividing items into groups.

Common Pitfalls

1. Misapplying Percent Change: The most common error is using the wrong "whole." Percent increase or decrease is always calculated based on the original amount. If a price increases from $50to$65, the increase is $15.Thepercentincreaseis$\frac{15}{50} = 0.30$or30$\frac{15}{65}$.
2. Forgetting the Two Solutions in Absolute Value Equations: Solving $|x|=k$ (where $k>0$) yields two solutions: $x=k$ and $x=-k$. Overlooking the negative solution is a frequent trap.
3. Mixing Up Exponent Rules: Confusing the product rule ($a^m\cdot a^n=a^{m+n}$) with the power rule ($(a^m{)}^n=a^{mn}$). Remember, you add exponents when multiplying like bases, and you multiply exponents when raising a power to a power.
4. Confusing Factors and Multiples: Students often reverse these. Remember that factors are smaller than or equal to the number (they divide into it), while multiples are larger than or equal to the number (the number divides into them).

Summary

• Pre-algebra is the highest-yield content area on the ACT Math test, forming the foundation for success.
• Fluency in converting and operating with fractions, decimals, and percents is essential; know the formula $\text{Percent}=\frac{\text{Part}}{\text{Whole}}\times 100$.
• Solve ratio and proportion problems reliably using cross-multiplication: if $\frac{a}{b}=\frac{c}{d}$, then $ad=bc$.
• Absolute value equations require solving for both the positive and negative cases of the expression inside the bars.
• Memorize and correctly apply the five core exponent rules, and practice operations with scientific notation.
• Distinguish between factors (numbers that divide another) and multiples (products of that number); know how to find GCF and LCM.