Pre-Algebra Basics

🔢 Number Properties

Part 1 of 7 — Factors, Multiples, Primes & Divisibility Rules

The ACT Math section gives you 45 questions in 50 minutes, each with 4 answer choices. Its official reporting categories are Preparing for Higher Math and Integrating Essential Skills — there is no separate "Pre-Algebra" score — but number properties, fractions, percents, and ratios run through both. Mastering them gives you quick, reliable points.

First 10 primes: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$

Remember: $1$ is neither prime nor composite, and $2$ is the only even prime.

Is $n$ prime? Test only the primes up to $\sqrt{n}$. If none of them divides $n$, then $n$ is prime (any factor larger than $\sqrt{n}$ would have to pair with one smaller than $\sqrt{n}$).

• $91$: $\sqrt{91} \approx 9.5$, so test $2, 3, 5, 7$. Since $91 = 7 \times 13$, it is not prime.
• $97$: $\sqrt{97} \approx 9.8$, so test $2, 3, 5, 7$. None divides $97$, so it is prime.

Divisibility Rules

Quick divisibility tests save time on the ACT:

Example 1: Is $891$ divisible by $9$?

$8 + 9 + 1 = 18 \quad\text{and}\quad 18 \div 9 = 2 \;✓$

Yes — $891 = 9 \times 99$.

Factors & Primes 🎯

LCM & GCF

Greatest Common Factor (GCF): the largest number that divides two or more numbers evenly.

Least Common Multiple (LCM): the smallest positive number that is a multiple of two or more numbers.

Example 2 — GCF: Find $\text{GCF}(24, 36)$.

$24 = 2^3 \times 3 \qquad 36 = 2^2 \times 3^2$

Take the lower power of each common prime: $2^2 \times 3 = 12$.

Example 3 — LCM: Find $\text{LCM}(8, 12)$.

$8 = 2^3 \qquad 12 = 2^2 \times 3$

Take the higher power of every prime: $2^3 \times 3 = 24$.

Shortcut: $\text{LCM}(a,b) = \frac{a \times b}{\text{GCF}(a,b)}$. So $\text{LCM}(8,12) = \frac{96}{4} = 24$ ✓

Every common multiple is a multiple of the LCM. The common multiples of $8$ and $12$ are exactly $24, 48, 72, 96, \ldots$ — the multiples of $24$. Example: two buses leave together at 9:00, one every $8$ minutes and one every $12$ minutes. They leave together again every $24$ minutes: at 9:24, 9:48, 10:12, …

Number Properties Practice 🧮

1. What is $\text{GCF}(18, 30)$?

2. What is $\text{LCM}(6, 10)$?

3. How many prime numbers are between 10 and 30?

Concept Check 🔍

ACT-Style Questions 📋

🍕 Fractions & Decimals

Part 2 of 7 — Operations, Converting & Ordering

Fraction and decimal arithmetic appears on virtually every ACT. You need to add, subtract, multiply, and divide with confidence.

Key fraction rules:

Always simplify your final answer.

Worked Examples

Example 1 — Addition: $\frac{2}{5} + \frac{1}{3}$

$\frac{2}{5} + \frac{1}{3} = \frac{6}{15} + \frac{5}{15} = \frac{11}{15}$

Example 2 — Subtraction with mixed numbers: $3\frac{1}{4} - 1\frac{2}{3}$

Convert to improper fractions:

$\frac{13}{4} - \frac{5}{3} = \frac{39}{12} - \frac{20}{12} = \frac{19}{12} = 1\frac{7}{12}$

Example 3 — Multiplication: $\frac{3}{8} \times \frac{4}{9}$

$\frac{3}{8} \times \frac{4}{9} = \frac{12}{72} = \frac{1}{6}$

Example 4 — Converting: $0.375 = \frac{375}{1000} = \frac{3}{8}$

Fraction Operations 🎯

Ordering & Comparing

To compare fractions, use one of these methods:

1. Common denominator: Convert all fractions to the same denominator and compare numerators.
2. Cross-multiply: To compare $\frac{a}{b}$ and $\frac{c}{d}$, check if $ad > bc$.
3. Convert to decimals.

Example 5: Order from least to greatest: $\frac{3}{5},\; \frac{2}{3},\; \frac{7}{10}$

Convert to decimals: $0.60,\; 0.\overline{6},\; 0.70$

$\frac{3}{5} < \frac{2}{3} < \frac{7}{10}$

Mixed Number Tip: To convert a mixed number $a\frac{b}{c}$ to an improper fraction: $\frac{ac + b}{c}$.

Fraction & Decimal Practice 🧮

1. Simplify $\frac{2}{3} + \frac{5}{6}$. Enter the answer as a fraction (e.g. 3/2).

2. What is $\frac{7}{8}$ as a decimal?

3. Convert the mixed number $2\frac{3}{5}$ to an improper fraction. Enter the numerator.

Concept Check 🔍

ACT-Style Questions 📋

💯 Percents

Part 3 of 7 — Percent of a Number, Increase/Decrease, Tax/Tip/Discount

"Percent" means "per hundred." So $45\%$ means $\frac{45}{100} = 0.45$.

Core conversions:

Percent of a number:

$\text{Part} = \text{Percent} \times \text{Whole}$

Example 1: What is $30\%$ of $250$?

$0.30 \times 250 = 75$

Percent Increase & Decrease

$\text{Percent Change} = \frac{|\text{New} - \text{Original}|}{\text{Original}} \times 100\%$

Example 2 — Increase: A price goes from $40 to $52. What is the percent increase?

$\frac{52 - 40}{40} \times 100\% = \frac{12}{40} \times 100\% = 30\%$

Example 3 — Decrease: A shirt originally costs $80 and is on sale for $60. What is the percent discount?

$\frac{80 - 60}{80} \times 100\% = \frac{20}{80} \times 100\% = 25\%$

Shortcut: For a $p\%$ increase, multiply by $(1 + \frac{p}{100})$. For a $p\%$ decrease, multiply by $(1 - \frac{p}{100})$.

Percent Calculations 🎯

Tax, Tip & Discount

These are the most common real-world percent problems on the ACT.

Example 4 — Sales Tax: A jacket costs $65 and the sales tax is $8\%$. What is the total?

$\text{Tax} = 0.08 \times 65 = 5.20 \implies \text{Total} = 65 + 5.20 = \$70.20$

Or use the multiplier: $65 \times 1.08 = 70.20$, i.e. $70.20.

Example 5 — Tip: A meal costs $42. You leave a $20\%$ tip. Total cost?

$42 \times 1.20 = \$50.40$

Example 6 — Discount: A $120 item is $35\%$ off. Sale price?

$120 \times (1 - 0.35) = 120 \times 0.65 = \$78$

Percent Practice 🧮

1. What is $40\%$ of $90$?

2. A $50 item is $20\%$ off. What is the sale price in dollars?

3. A value increased from $80$ to $100$. What is the percent increase?

Concept Check 🔍

ACT-Style Questions 📋

⚖️ Ratios & Proportions

Part 4 of 7 — Setting Up, Cross-Multiplying, Scaling & Unit Rates

A ratio compares two quantities: $a : b$ or $\frac{a}{b}$.

A proportion states that two ratios are equal: $\frac{a}{b} = \frac{c}{d}$.

Cross-multiplication is the go-to strategy:

$\frac{a}{b} = \frac{c}{d} \implies ad = bc$

Example 1: If $\frac{3}{5} = \frac{x}{20}$, find $x$.

$3 \times 20 = 5 \times x \implies 60 = 5x \implies x = 12$

Scaling & Part-to-Whole

When a ratio is $a : b$, the total parts are $a + b$.

Example 2: A class has boys and girls in the ratio $3 : 5$. If there are $40$ students, how many are boys?

Total parts $= 3 + 5 = 8$. Each part $= \frac{40}{8} = 5$.

$\text{Boys} = 3 \times 5 = 15$

Use a variable for the ratio. A ratio $a : b$ means the actual amounts are $ak$ and $bk$ for some multiplier $k$. Example: juice and water are mixed $4 : 7$, and there are $15$ more cups of water than juice. Write juice $= 4k$, water $= 7k$:

$7k - 4k = 15 \implies 3k = 15 \implies k = 5$

So juice $= 20$ cups and water $= 35$ cups. This handles differences, sums, and "after adding more" problems that a single proportion can't.

Example 3 — Scaling a recipe: A recipe uses flour and sugar in a $4 : 1$ ratio. If you use $12$ cups of flour, how much sugar?

$\frac{4}{1} = \frac{12}{x} \implies 4x = 12 \implies x = 3 \text{ cups}$

Example 4 — Map scale: On a map, $1$ inch represents $25$ miles. Two cities are $3.5$ inches apart. Actual distance?

$3.5 \times 25 = 87.5 \text{ miles}$

Ratios & Proportions 🎯

Unit Rates

A unit rate expresses a ratio per one unit of the second quantity.

Example 5: A car travels $210$ miles in $3.5$ hours. What is the speed in mph?

$\text{Rate} = \frac{210}{3.5} = 60 \text{ mph}$

Example 6 — Unit price: A 12-pack of soda costs $4.80. Price per can?

$\frac{4.80}{12} = \$0.40 \text{ per can}$

Example 7 — Better deal: Store A sells $5$ lb of apples for $6.25. Store B sells $3$ lb for $3.45. Which is cheaper per pound?

$A: \frac{6.25}{5} = \$1.25/\text{lb} \qquad B: \frac{3.45}{3} = \$1.15/\text{lb}$

Store B is cheaper.

Ratio & Rate Practice 🧮

1. Solve: $\frac{4}{9} = \frac{x}{27}$. What is $x$?

2. A printer prints $120$ pages in $8$ minutes. Pages per minute?

3. Markers and pens are in a $3 : 4$ ratio. If there are $28$ pens, how many markers?

Concept Check 🔍

ACT-Style Questions 📋

⚡ Exponents & Roots

Part 5 of 7 — Rules of Exponents, Square Roots & Order of Operations

Exponents appear frequently on the ACT. Know these rules cold:

Square Roots & Simplifying

$\sqrt{a} \text{ is the non-negative number whose square is } a.$

Perfect squares to memorise: $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144$

Simplifying radicals: Factor out the largest perfect square.

Example 1: $\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$

Example 2: $\sqrt{50} + \sqrt{18} = 5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2}$

Example 3: $\sqrt{12} \cdot \sqrt{3} = \sqrt{36} = 6$

Key property: $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ (for $a, b \geq 0$)

Exponents & Roots 🎯

Order of Operations (PEMDAS)

$\text{Parentheses} \to \text{Exponents} \to \text{Multiplication/Division} \to \text{Addition/Subtraction}$

Multiplication and division are done left to right (same level). Likewise for addition and subtraction.

Example 4: Evaluate $3 + 2 \times 4^2 - 10 \div 5$.

$= 3 + 2 \times 16 - 10 \div 5$ $= 3 + 32 - 2$ $= 33$

Example 5: Evaluate $\frac{(6 + 2)^2}{4} - 3^2$.

$= \frac{8^2}{4} - 9 = \frac{64}{4} - 9 = 16 - 9 = 7$

ACT Tip: Fraction bars act as grouping symbols — evaluate numerator and denominator separately.

Exponents & Roots Practice 🧮

1. Evaluate: $2^5$

2. Simplify $\sqrt{48}$. Enter in the form $a\sqrt{b}$ (e.g. 4sqrt3).

3. Evaluate: $5 \times 3 - 4^2 + 1$

Concept Check 🔍

ACT-Style Questions 📋

📊 Basic Statistics

Part 6 of 7 — Mean, Median, Mode, Range & Reading Charts

The ACT loves simple data-analysis questions. Know these four measures:

For an even number of data points, the median is the average of the two middle values.

Worked Examples

Example 1 — Mean: Test scores: $78, 85, 92, 88, 97$. Find the mean.

$\text{Mean} = \frac{78 + 85 + 92 + 88 + 97}{5} = \frac{440}{5} = 88$

Example 2 — Median (even count): Data: $4, 7, 9, 12$. Find the median.

$\text{Median} = \frac{7 + 9}{2} = 8$

Example 3 — Missing-value problem: The mean of five numbers is $20$. Four of them are $15, 18, 22, 25$. Find the fifth.

$\text{Sum} = 5 \times 20 = 100$ $\text{Known sum} = 15 + 18 + 22 + 25 = 80$ $\text{Fifth number} = 100 - 80 = 20$

ACT Tip: "Find the missing value given the mean" is a classic ACT question pattern.

Statistics Basics 🎯

Reading Charts & Tables

On the ACT, you may see bar graphs, pie charts, tables, or line graphs. The math is usually straightforward — the challenge is extracting the right numbers.

Strategy:

1. Read the title and axis labels first.
2. Identify what the question asks.
3. Pull the numbers and compute.

Example 4 — Table: A store sold the following units:

Average daily sales $= \frac{30 + 45 + 25 + 50 + 40}{5} = \frac{190}{5} = 38$ units.

Best day: Thursday ($50$ units).
Range: $50 - 25 = 25$ units.

Statistics Practice 🧮

1. Find the mean of $\{10, 14, 18, 22, 26\}$.

2. The mean of 4 numbers is $15$. Three of them are $12, 16, 20$. What is the fourth?

3. Find the range of $\{3, 8, 1, 15, 7\}$.

Concept Check 🔍

ACT-Style Questions 📋

📝 Review & Mixed Practice

Part 7 of 7 — Formula Cheat Sheet & Mixed ACT Pre-Algebra Problems

Pre-Algebra Formula Cheat Sheet

ACT Strategy Tips

1. Plug in answers — On multiple-choice, try each option if algebra feels slow.
2. Estimate — Eliminate clearly wrong answers before computing.
3. Watch units — Especially in rate and percent problems.
4. Read carefully — "Percent increase" is not the same as "new value."
5. Bank time — ACT Math gives you 50 minutes for 45 questions (about 67 seconds each, 4 answer choices per question). Pre-algebra items are usually quick, so finishing them fast leaves extra time for the harder problems.

Common traps:

• Confusing $-3^2 = -9$ with $(-3)^2 = 9$.
• Forgetting to simplify fractions.
• Mixing up LCM and GCF.
• Using the wrong base in percent-change problems.

Mixed Pre-Algebra 🎯

Worked Mixed Problems

Problem 1: Evaluate $\frac{3^2 + 4^2}{5}$.

$\frac{9 + 16}{5} = \frac{25}{5} = 5$

Problem 2: A bag has red and green marbles in a $2 : 5$ ratio. If there are $35$ total marbles, how many are red?

$\text{Total parts} = 7 \implies \text{each part} = 5 \implies \text{red} = 2 \times 5 = 10$

Problem 3: The median of $\{3, 7, x, 12, 18\}$ (already sorted) is $9$. What is $x$?

The median is the 3rd value: $x = 9$.

Problem 4: A sweater costs $64 after a $20\%$ discount. Original price?

$0.80P = 64 \implies P = 80$

Mixed Practice 🧮

1. Evaluate: $(-2)^3 + 5^2$

2. What is $\text{LCM}(9, 12)$?

3. A cyclist rides $45$ miles in $3$ hours. What is the speed in mph?

Final Concept Check 🔍

ACT-Style Final Questions 📋