🎯⭐ INTERACTIVE LESSON

Pre-Algebra and Elementary Algebra

Learn step-by-step with interactive practice!

Pre-Algebra and Elementary Algebra - Complete Interactive Lesson

Part 1: Number Properties

🔢 Number Properties

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

Pre-Algebra questions make up roughly 20–25 % of the ACT Math section. Mastering basic number properties gives you quick, reliable points.

Concept	Definition
Factor	A number that divides evenly into another
Multiple	The product of a number and any positive integer
Prime	A number greater than 1 with exactly two factors: 1 and itself
Composite	A number greater than 1 that is not prime

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.

Divisibility Rules

Quick divisibility tests save time on the ACT:

Divisor	Rule	Example
2	Last digit is even	$438$ → last digit $8$ ✓
3	Sum of digits divisible by 3	$627$ : $6+2+7 = 15$ ✓
4	Last two digits form a number divisible by 4	$316$ : $16 \div 4 = 4$ ✓
5	Last digit is 0 or 5	$745$ ✓
6	Divisible by both 2 and 3	$312$ : even and $3+1+2=6$ ✓
9	Sum of digits divisible by 9	$729$ : $7+2+9 = 18$ ✓

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 $\text{LCM}(8,12) = \frac{96}{4} = 24$ ✓

Number Properties Practice 🧮

What is $\text{GCF}(18, 30)$ ?
What is $\text{LCM}(6, 10)$ ?
How many prime numbers are between 10 and 30?

Concept Check 🔍

ACT-Style Questions 📋

Part 2: Fractions & Decimals

🍕 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:

Operation	Rule
Add / Subtract	Common denominator: $\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}$
Multiply	Straight across: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$
Divide	Flip and multiply: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

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:

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

Simplify $\frac{2}{3} + \frac{5}{6}$ . Enter the answer as a fraction (e.g. 3/2).
What is $\frac{7}{8}$ as a decimal?
Convert the mixed number $2\frac{3}{5}$ to an improper fraction. Enter the numerator.

Concept Check 🔍

ACT-Style Questions 📋

Part 3: Ratios & Proportions

💯 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:

From	To	Method
Percent → Decimal	$45\% = 0.45$	Divide by 100
Decimal → Percent	$0.08 = 8\%$	Multiply by 100
Fraction → Percent	$\frac{3}{4} = 75\%$	Divide then multiply by 100

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 🧮

What is $40\%$ of $90$ ?
A $50 item is $20\%$ off. What is the sale price in dollars?
A value increased from $80$ to $100$ . What is the percent increase?

Concept Check 🔍

ACT-Style Questions 📋

Part 4: Percentages

⚖️ 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 $= \frac{40}{8} = 5$ .

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

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 🧮

Solve: $\frac{4}{9} = \frac{x}{27}$ . What is $x$ ?
A printer prints $120$ pages in $8$ minutes. Pages per minute?
Markers and pens are in a $3 : 4$ ratio. If there are $28$ pens, how many markers?

Concept Check 🔍

ACT-Style Questions 📋

Part 5: Order of Operations

⚡ Exponents & Roots

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

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

Rule	Formula	Example
Product	$a^m \cdot a^n = a^{m+n}$	$2^3 \cdot 2^4 = 2^7 = 128$
Quotient	$\frac{a^m}{a^n} = a^{m-n}$	$\frac{5^6}{5^2} = 5^4 = 625$
Power of a Power	$(a^m)^n = a^{mn}$	$(3^2)^3 = 3^6 = 729$
Zero Exponent	$a^0 = 1$ (for $a \neq 0$ )	$7^0 = 1$
Negative Exponent	$a^{-n} = \frac{1}{a^n}$	$4^{-2} = \frac{1}{16}$

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 🧮

Evaluate: $2^5$
Simplify $\sqrt{48}$ . Enter in the form $a\sqrt{b}$ (e.g. 4sqrt3).
Evaluate: $5 \times 3 - 4^2 + 1$

Concept Check 🔍

ACT-Style Questions 📋

Part 6: Problem-Solving Workshop

📊 Basic Statistics

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

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

Measure	Definition	Example for $\{2, 3, 3, 7, 10\}$
Mean	Sum ÷ count	$\frac{2+3+3+7+10}{5} = 5$
Median	Middle value (sorted)	$3$
Mode	Most frequent value	$3$
Range	Max − Min	$10 - 2 = 8$

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:

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

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

Day	Mon	Tue	Wed	Thu	Fri
Units	30	45	25	50	40

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 🧮

Find the mean of $\{10, 14, 18, 22, 26\}$ .
The mean of 4 numbers is $15$ . Three of them are $12, 16, 20$ . What is the fourth?
Find the range of $\{3, 8, 1, 15, 7\}$ .

Concept Check 🔍

ACT-Style Questions 📋

Part 7: Review & Applications

📝 Review & Mixed Practice

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

Pre-Algebra Formula Cheat Sheet

Topic	Key Formulas
Factors / Multiples	$\text{LCM}(a,b) = \frac{a \times b}{\text{GCF}(a,b)}$
Fractions	Add: common denom; Divide: flip & multiply
Percents	Part $= \% \times$ Whole; Change $= \frac{
Proportions	$\frac{a}{b} = \frac{c}{d} \implies ad = bc$
Exponents	$a^m \cdot a^n = a^{m+n}$ ; $a^0 = 1$ ; $a^{-n} = \frac{1}{a^n}$
Roots	$\sqrt{ab} = \sqrt{a}\sqrt{b}$
Stats	Mean $= \frac{\text{sum}}{n}$ ; Median = middle; Mode = most common
PEMDAS	Parentheses → Exponents → Mult/Div → Add/Sub

ACT Strategy Tips

Plug in answers — On multiple-choice, try each option if algebra feels slow.
Estimate — Eliminate clearly wrong answers before computing.
Watch units — Especially in rate and percent problems.
Read carefully — "Percent increase" is not the same as "new value."
Don't overthink — Pre-algebra questions are meant to be quick; ~30 seconds each.

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 🧮

Evaluate: $(-2)^3 + 5^2$
What is $\text{LCM}(9, 12)$ ?
A cyclist rides $45$ miles in $3$ hours. What is the speed in mph?

Final Concept Check 🔍

ACT-Style Final Questions 📋

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