Numbers, Sequences, and Factors

Ever wonder why some math problems seem to follow patterns? The SAT tests your understanding of number relationships and sequences. Integers include all whole numbers and their negatives (...-3, -2, -1, 0, 1, 2, 3...), while real numbers include all integers plus fractions, decimals, and irrationals like √2 and π.

When solving complex expressions, remember PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction). For sequences, know that arithmetic sequences increase by adding the same number each time (like 3, 7, 11, 15...), while geometric sequences multiply by the same factor each time (like 3, 6, 12, 24...).

Understanding factors is crucial for simplifying fractions and solving equations. The prime factorization of a number breaks it into its prime number building blocks $like 200 = 2³ × 5²$. To find the Greatest Common Factor (GCF) of multiple numbers, multiply their shared prime factors. For the Least Common Multiple (LCM), look for the smallest number divisible by all numbers in question.

💡 When working with percentages, remember this simple formula: part = $percent/100$ × whole. This one formula can be rearranged to solve for any variable!

Averages, Counting, Statistics, and Probability

Averages show up everywhere on the SAT! The average (or mean) equals the sum of all values divided by the number of values. When working with speed problems, remember that average speed equals total distance divided by total time—not the average of individual speeds.

Statistical concepts are straightforward but important. The mode is simply the value that appears most often, while the median is the middle value when all numbers are arranged in order. If you have an even number of values, take the average of the two middle numbers.

Probability questions test your understanding of outcomes. The basic formula is probability = (desired outcomes)/(total outcomes). For independent events happening together, multiply their individual probabilities.

When working with powers and exponents, remember these key rules: when multiplying powers with the same base, add the exponents $x^a × x^b = x^(a+b)$. When dividing powers with the same base, subtract the exponents $x^a ÷ x^b = x^(a-b)$. Any number raised to the power of zero equals 1.

🔑 Understanding the Fundamental Counting Principle is crucial for probability questions: If one event can happen in N ways and another independent event in M ways, together they can happen in N × M ways.

Factoring and Solving Equations

Factoring is your secret weapon for solving quadratic equations! The FOIL method helps you expand expressions like $x+a$$x+b$ = x² + $a+b$x + ab. Going in reverse (reverse FOIL) lets you factor expressions like x² + 5x + 6 into $x+2$$x+3$ by finding numbers that multiply to give 6 and add to give 5.

To solve quadratic equations like x² + 4x + 3 = 0, factor the left side to get $x+3$$x+1$ = 0, then find the values that make each factor equal zero $x = -3 or x = -1$. For systems of linear equations, use substitution—express one variable in terms of the other, then plug that expression into the second equation.

Functions are simply rules that convert one number (input) to another number (output). Remember that for any given input value, a function can only have one output value. Directly proportional functions have the form y = kx, while inversely proportional functions look like y = k/x.

Absolute value expressions can be tricky. Remember that |x| < n means x is between -n and n, while |x| > n means x is either less than -n or greater than n.

🧠 When solving equations with absolute values, always consider both possibilities: the expression inside the absolute value bars could be positive OR negative!

Linear Functions and Parabolas

Mastering lines is essential for SAT success! The equation of a parabola takes the form y = ax² + bx + c. If a is positive, the parabola opens upward; if negative, it opens downward. The x-coordinate of the vertex is -b/2a, which is super helpful for finding maximum or minimum values.

For linear functions, know these key forms: point-slope form $y - y₁$ = m$x - x₁$ and slope-intercept form y = mx + b. The slope of a line is calculated as $y₂ - y₁$/$x₂ - x₁$ between any two points.

When working with lines in the coordinate plane, remember that parallel lines have equal slopes, while perpendicular lines have slopes that are negative reciprocals of each other $m₁ × m₂ = -1$. This comes up frequently on the SAT!

For intersecting lines, remember that opposite angles are equal, and angles along the same line add up to 180°. When a line crosses two parallel lines, it creates eight angles—four of one size and four of another.

👍 The distance formula is just the Pythagorean theorem in disguise! To find the distance between points (x₁, y₁) and (x₂, y₂), calculate √$(x₂ - x₁)² + (y₂ - y₁)²$.

Triangles

Right triangles are SAT favorites! Always remember the Pythagorean theorem: a² + b² = c², where c is the hypotenuse (longest side). The SAT loves to use special right triangles—particularly the 30°-60°-90° and 45°-45°-90° triangles—because their sides follow specific ratios.

In a 45°-45°-90° triangle, if the legs have length x, the hypotenuse has length x√2. In a 30°-60°-90° triangle, if the shortest leg has length x, the hypotenuse is 2x and the middle leg is x√3.

For any triangle, the area equals ½ × base × height. Remember that angles inside a triangle always add up to 180°, and an exterior angle equals the sum of the two remote interior angles.

Similar triangles have the same shape but different sizes. Their corresponding angles are equal, and their corresponding sides are proportional. This means you can set up ratios to find unknown sides. The classic 3-4-5 right triangle $and its multiples like 6-8-10$ appears frequently on the test.

⭐ The triangle inequality theorem is a powerful check: any side of a triangle must be less than the sum of the other two sides but greater than their difference. If this rule is broken, the triangle can't exist!

Circles, Polygons, and Solids

Circle problems appear regularly on the SAT. The area of a circle is πr², while the circumference is 2πr. For portions of a circle, remember that an arc length equals $n°/360°$ × 2πr, where n° is the central angle in degrees. Similarly, a sector's area equals $n°/360°$ × πr².

The equation of a circle with center (h, k) and radius r is $x - h$² + $y - k$² = r². This formula is useful when you need to find points on a circle or determine if points lie inside or outside a circle.

For rectangles and parallelograms, memorize the area formulas: rectangle area = length × width, and parallelogram area = base × height. In regular polygons (where all sides and angles are equal), the sum of interior angles equals $n - 2$ × 180°, where n is the number of sides.

Three-dimensional shapes also appear on the SAT. The volume of a rectangular solid is length × width × height, while its surface area is 2$lw + wh + lh$. For a cylinder, the volume is πr²h, and the surface area is 2πr² + 2πrh, or more simply 2πr$r + h$.

🔍 When working with 3D shapes, drawing a diagram can help you visualize the problem and identify which formula to use. Don't skip this important step!