ACT Math Flashcards

Proportion

A proportion sets two equal ratios to solve for missing dimensions, such as shadow heights or arc lengths.

Formula for Proportions

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

Recognizing Proportions

Look for problems involving scaling scenarios, shadow/height comparisons, or arc lengths.

Average

An average distributes a total sum equally across all items.

Average Formula

$$\text{Average} = \frac{\text{Sum of terms}}{\text{Total number of terms}}$$

Recognizing Average Problems

Recognize when asked for a final score based on a target average and partial sum.

Basic Probability

Probability measures successful outcomes against all possible outcomes.

Probability Formula

$$P(\text{Event}) = \frac{\text{Number of successful outcomes}}{\text{Total number of possible outcomes}}$$

Recognizing Probability Problems

Identify questions regarding item selection or outcomes of events.

Fundamental Counting Principle

Total combined outcomes for sequential choices are calculated by multiplying option counts.

Counting Principle Formula

$$N_{\text{total}} = n_1 \times n_2 \times n_3 \times \dots$$

Recognizing Counting Principle Problems

Questions about total combinations of choices.

Function

A mathematical expression representing a relationship between variables.

Circle Equation

Standard circle equation is $(x - h)^2 + (y - k)^2 = r^2$.

Solving for Circle Center

Change signs inside binomials to find center $(h, k)$.

Percent Change

Multi-step percent changes are applied sequentially and not added together.

Percent Change Formula

$$\text{Final} = \text{Initial} \times (1 \pm p_1) \times (1 \pm p_2)$$

Factoring Polynomials

Factor algebraic expressions using key pattern rules.

Difference of Squares

$a^2 - b^2 = (a + b)(a - b)$.

Exponent Rule: Product

$x^a \times x^b = x^{a+b}$.

Pythagorean Theorem

In a right triangle, $a^2 + b^2 = c^2$ where $c$ is the hypotenuse.

Midpoint Formula

$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.

Logarithm Definition

$\log_b(x) = n \iff b^n = x$.

Distance Formula

Calculate distance between two points: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Basic Trigonometry

$\sin(\theta) = \frac{\text{Opp}}{\text{Hyp}}, \cos(\theta) = \frac{\text{Adj}}{\text{Hyp}}, \tan(\theta) = \frac{\text{Opp}}{\text{Adj}}$.

Special Right Triangle Ratios

For $30^\circ-60^\circ-90^\circ$ triangles: $x : x\sqrt{3} : 2x$.

Perimeter of Circle

Circumference is given by the formula $C = 2\pi r$.

Angle Relationships with Transversals

Parallel lines create congruent angle pairs and supplementary angles.

Sign Flipping Rule in Inequalities

Flip the inequality sign when multiplying or dividing by negative numbers.