Number and Quantity (ACT Math)

What the ACT Tests

The ACT Math section includes Number and Quantity questions that test your understanding of:

• Real numbers and operations
• Rational and irrational numbers
• Number line concepts
• Absolute value
• Scientific notation
• Number properties and patterns
• Ratios and proportions
• Percentages

Real Number System

Types of Numbers

Natural Numbers (Counting Numbers): 1, 2, 3, 4, 5, ...

Whole Numbers: 0, 1, 2, 3, 4, 5, ... (natural numbers + zero)

Integers: ..., -3, -2, -1, 0, 1, 2, 3, ... (positive and negative whole numbers)

Rational Numbers: Numbers that can be expressed as $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$

• Examples: $\frac{1}{2}$, $-\frac{3}{4}$, $5$ (can write as $\frac{5}{1}$), $0.75$ (equals $\frac{3}{4}$)
• Includes terminating decimals: $0.5$, $0.125$
• Includes repeating decimals: $0.333...$ (equals $\frac{1}{3}$), $0.\overline{6}$

Irrational Numbers: Cannot be expressed as a fraction; non-terminating, non-repeating decimals

• Examples: $\pi$, $e$, $\sqrt{2}$, $\sqrt{3}$
• Note: $\sqrt{4} = 2$ is rational (it's a perfect square!)

Real Numbers: All rational and irrational numbers combined

Number Line

Key concepts:

Order: Numbers increase from left to right

• $-5 < -2 < 0 < 3 < 7$

Distance: The distance between two numbers $a$ and $b$ is $|a - b|$

• Distance from $-3$ to $5$: $|(-3) - 5| = |-8| = 8$

Midpoint: Between $a$ and $b$ is $\frac{a + b}{2}$

• Midpoint of $-4$ and $10$: $\frac{-4 + 10}{2} = \frac{6}{2} = 3$

Absolute Value

Definition: The distance from zero on the number line (always positive or zero)

$|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$

Examples:

• $|5| = 5$
• $|-7| = 7$
• $|0| = 0$
• $|-3.5| = 3.5$

Properties:

• $|x| \geq 0$ (always non-negative)
• $|x| = |-x|$ (same distance from zero)
• $|xy| = |x| \cdot |y|$
• $\left|\frac{x}{y}\right| = \frac{|x|}{|y|}$ (where $y \neq 0$)

Equations with absolute value:

Example: Solve $|x - 3| = 5$

Solution: Two cases

• Case 1: $x - 3 = 5$ → $x = 8$
• Case 2: $x - 3 = -5$ → $x = -2$

Answers: $x = 8$ or $x = -2$

Scientific Notation

Form: $a \times 10^n$ where $1 \leq |a| < 10$ and $n$ is an integer

Large numbers (positive exponent):

• $3,450,000 = 3.45 \times 10^6$
• $89,000 = 8.9 \times 10^4$

Small numbers (negative exponent):

• $0.00067 = 6.7 \times 10^{-4}$
• $0.0000002 = 2 \times 10^{-7}$

Operations in scientific notation:

Multiplication: Multiply coefficients, add exponents $(2 \times 10^5)(3 \times 10^7) = 6 \times 10^{12}$

Division: Divide coefficients, subtract exponents $\frac{8 \times 10^9}{4 \times 10^3} = 2 \times 10^{9-3} = 2 \times 10^6$

Properties of Numbers

Even and Odd

Even: Divisible by 2 (ends in 0, 2, 4, 6, 8)
Odd: Not divisible by 2 (ends in 1, 3, 5, 7, 9)

Rules:

• Even + Even = Even
• Odd + Odd = Even
• Even + Odd = Odd
• Even × Even = Even
• Odd × Odd = Odd
• Even × Odd = Even

Prime Numbers

Definition: A number greater than 1 with exactly two factors: 1 and itself

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

Note: 2 is the only even prime number!

Composite numbers: Have more than two factors

• Examples: 4, 6, 8, 9, 10, 12, 14, 15, ...

Divisibility Rules

Divisible by 2: Last digit is even
Divisible by 3: Sum of digits is divisible by 3
Divisible by 4: Last two digits form a number divisible by 4
Divisible by 5: Last digit is 0 or 5
Divisible by 6: Divisible by both 2 and 3
Divisible by 9: Sum of digits is divisible by 9
Divisible by 10: Last digit is 0

Example: Is 2,346 divisible by 3?
Sum of digits: $2 + 3 + 4 + 6 = 15$, and $15 ÷ 3 = 5$ → Yes! ✓

Ratios and Proportions

Ratios

A ratio compares two quantities

Forms:

• $3:2$ (ratio notation)
• $\frac{3}{2}$ (fraction form)
• "3 to 2" (words)

Example: If a recipe calls for 2 cups flour to 3 cups sugar, the ratio of flour to sugar is $2:3$ or $\frac{2}{3}$

Proportions

A proportion states that two ratios are equal

$\frac{a}{b} = \frac{c}{d}$

Cross multiply to solve: $a \cdot d = b \cdot c$

Example: If $\frac{x}{5} = \frac{12}{15}$, find $x$

Cross multiply: $15x = 5 \cdot 12 = 60$

Solve: $x = \frac{60}{15} = 4$

Direct Proportion

If $y$ varies directly with $x$: $y = kx$ for some constant $k$

Example: If $y = 12$ when $x = 3$, find $y$ when $x = 7$

Step 1: Find $k$: $12 = k(3)$ → $k = 4$

Step 2: Use $k$ to find new $y$: $y = 4(7) = 28$

Inverse Proportion

If $y$ varies inversely with $x$: $y = \frac{k}{x}$ for some constant $k$

Example: If $y = 6$ when $x = 4$, find $y$ when $x = 8$

Step 1: Find $k$: $6 = \frac{k}{4}$ → $k = 24$

Step 2: Use $k$ to find new $y$: $y = \frac{24}{8} = 3$

Percentages

Basic Percent Formula

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

Example: What is 35% of 80? $\text{Part} = 0.35 \times 80 = 28$

Percent Change

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

Example: A price increases from $50 to$65. What's the percent increase?

$\frac{65 - 50}{50} \times 100\% = \frac{15}{50} \times 100\% = 30\%$

Percent of Percent

Example: If 30% of a number is 60, what is 50% of that number?

Step 1: Find the number
$0.30x = 60$ → $x = 200$

Step 2: Find 50% of it
$0.50 \times 200 = 100$

ACT Question Strategies

Type 1: Classifying Numbers

Question: Which of the following is an irrational number?

Strategy:

• Perfect squares/cubes → rational
• $\pi$, $e$, non-perfect roots → irrational
• Fractions, integers, terminating/repeating decimals → rational

Type 2: Number Line Problems

Question: Point $P$ is halfway between $-7$ and $15$ on a number line. What is the coordinate of $P$?

Strategy: Use midpoint formula $\frac{-7 + 15}{2} = \frac{8}{2} = 4$

Type 3: Absolute Value

Question: What is $|-8| + |3| - |-5|$?

Strategy: Evaluate each absolute value first $8 + 3 - 5 = 6$

Type 4: Scientific Notation

Question: $(4 \times 10^8)(2 \times 10^{-3}) = ?$

Strategy:

• Multiply coefficients: $4 \times 2 = 8$
• Add exponents: $8 + (-3) = 5$
• Answer: $8 \times 10^5$

Type 5: Properties and Patterns

Question: If $n$ is an odd integer, which is always even?

Strategy: Test with examples

• $n + 1$: odd + 1 = even ✓
• $2n$: 2 × odd = even ✓
• $n^2$: odd × odd = odd ✗

Common ACT Mistakes

❌ Forgetting that $\sqrt{9} = 3$ is rational (perfect squares are rational)
❌ Not considering both solutions for $|x| = 5$ (must have $x = 5$ and $x = -5$)
❌ Miscounting decimal places in scientific notation
❌ Confusing direct and inverse variation
❌ Using wrong base for percent change (should be original value)
❌ Saying 0 is positive (it's neither positive nor negative)

Quick Tips for ACT

✓ Know your number types — rational vs irrational is common
✓ Absolute value creates TWO solutions — don't forget negative case
✓ Move decimal, adjust exponent for scientific notation
✓ Cross multiply for proportions — fast and reliable
✓ Percent change uses ORIGINAL as denominator
✓ Even × Odd = Even — useful for eliminating answers
✓ Prime numbers > 2 are odd — 2 is the exception

Practice Approach

1. Identify the concept being tested (absolute value, proportion, etc.)
2. Recall the rule or formula for that concept
3. Apply systematically — don't rush
4. Check reasonableness — does your answer make sense?
5. Eliminate obviously wrong answers first

Remember: Number and Quantity questions test fundamental concepts. Master these basics and you'll handle them quickly on test day!