A prime number has exactly two distinct factors: 1 and itself. A composite number has more than two factors. Note: 1 is neither prime nor composite. Divisibility rules help quickly test factors without long division.
Divisible by 3 → digit sum divisible by 3
Divisible by 9 → digit sum divisible by 9
Primes up to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
Is 51 prime? → digit sum = 5+1 = 6, divisible by 3. So 51 = 3 × 17 → NOT prime (composite)
How many prime numbers are there between 20 and 40 (exclusive)?
Fractions, decimals, and percents are three forms of the same value. Converting fluently between them is essential for SSAT problems involving comparisons, discounts, and ratios.
Decimal → Percent: × 100
Percent → Decimal: ÷ 100
1/4 = 0.25 = 25% | 1/3 ≈ 0.333 | 3/8 = 0.375
What is 3/8 as a percent? → 3 ÷ 8 = 0.375 → × 100 = 37.5%
A store reduces a price by 2/5. What is the percent discount?
Write as a whole number (e.g., 40).
A ratio compares two quantities. A proportion says two ratios are equal. Use cross-multiplication to solve for an unknown.
Part/Whole = %/100
Scale: map distance × scale factor = real distance
If 4 pencils cost $1.20, how much do 10 cost? → 4/1.20 = 10/x → x = 10 × 1.20 / 4 = $3.00
The ratio of boys to girls in a class is 3 : 5. If there are 24 boys, how many students are in the class in total?
To solve a linear equation, isolate the variable by performing inverse operations. Whatever you do to one side, do to the other to keep balance.
Distribute first: a(x + b) = ax + ab
Move variables to one side before solving
Solve: 3x − 7 = 14 → 3x = 21 → x = 7
Solve for x: 5(x − 3) = 2x + 6
Inequalities have a range of solutions, not just one. The key rule: when you multiply or divide both sides by a negative number, flip the inequality sign.
< or > → open circle on number line
≤ or ≥ → closed circle on number line
Solve: −2x > 8 → divide by −2 (flip!) → x < −4
What is the largest integer that satisfies: 3x − 5 < 10 ?
The most powerful word-problem tool: define a variable for the unknown, translate words into an equation, and solve. Rate × Time = Distance (D = R × T) is the foundation of motion problems.
Average speed = Total Distance / Total Time
For opposite directions: add speeds
A car travels 150 miles in 3 hours. Speed = 150/3 = 50 mph
Train A leaves at 60 mph. Train B leaves the same station 1 hour later at 90 mph in the same direction. After how many hours (from B's departure) does Train B catch Train A?
When two lines intersect or a transversal crosses parallel lines, specific angle relationships are formed. Recognizing these saves time on SSAT geometry questions.
Complementary angles: sum = 90°
Vertical angles: equal
Parallel lines cut by transversal:
→ Alternate interior angles = equal
→ Co-interior (same-side) = 180°
Two supplementary angles are in ratio 2:7. Smaller angle = 2/9 × 180 = 40°
Two parallel lines are cut by a transversal. One co-interior angle measures (3x + 20)° and the other measures (5x − 10)°. Find the value of x.
All triangles have an angle sum of 180°. The Pythagorean Theorem applies to right triangles. Memorizing special right-triangle ratios saves significant time on the SSAT.
Pythagorean: a² + b² = c²
Special triangles:
3-4-5 | 5-12-13 | 8-15-17
Exterior angle = sum of two non-adjacent interior angles
Triangle with base 8, height 5. Area = ½ × 8 × 5 = 20 sq. units
A right triangle has legs of length 9 and 12. What is the length of the hypotenuse?
All circle formulas are based on the radius (r). An arc is a fractional part of the circumference, and a sector is a fractional part of the area — both determined by the central angle.
Area = πr²
Arc length = (θ/360) × 2πr
Sector area = (θ/360) × πr²
π ≈ 3.14159
Circle with r = 5: Area = π(25) ≈ 78.54 sq. units
A circle has a diameter of 14 cm. What is its circumference? Leave your answer in terms of π.
Write like: 14π
The coordinate plane uses (x, y) pairs. Slope describes how steep a line is (rise over run). Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals.
Distance = √[(x₂−x₁)²+(y₂−y₁)²]
Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2)
y = mx + b (slope-intercept form)
Slope from (1, 2) to (4, 8): m = (8−2)/(4−1) = 6/3 = 2
Find the midpoint of the segment with endpoints (−4, 6) and (8, −2).
Write as (x, y) with no spaces, e.g., (2,3)
A function assigns exactly one output to each input. f(x) is read "f of x." To evaluate, substitute the given value for x. SSAT often uses composite notation like f(g(x)).
Composite: f(g(x)) → evaluate inside-out
Domain: all valid inputs (x-values)
Range: all possible outputs (y-values)
f(x) = 2x² − 3. Find f(4): = 2(16) − 3 = 32 − 3 = 29
If f(x) = 3x − 1 and g(x) = x² + 2, what is the value of f(g(2))?
In an arithmetic sequence, each term is found by adding a constant (common difference, d). In a geometric sequence, each term is found by multiplying a constant (common ratio, r).
Sum of first n terms: Sₙ = n(a₁+aₙ)/2
Geometric: aₙ = a₁ × rⁿ⁻¹
Common difference d = a₂ − a₁
Sequence: 5, 8, 11, 14, … → d = 3. 10th term = 5 + 9(3) = 32
An arithmetic sequence begins 4, 11, 18, 25, … What is the 15th term?
Measures of central tendency describe a data set. Mean is most affected by extreme values (outliers). Median is the middle value when sorted. Mode is the most frequent. Range shows spread.
Median = middle value (odd n) or average of two middle values (even n)
Mode = most frequent value
Range = max − min
Data: {3, 7, 7, 9, 12} → Mean = 38/5 = 7.6, Median = 7, Mode = 7
The test scores of 6 students are: 72, 85, 90, 68, 85, 94. What is the mean score? (Round to nearest whole number if needed.)
Probability = (favorable outcomes) ÷ (total outcomes). Values range from 0 (impossible) to 1 (certain). Independent events: outcomes don't affect each other.
P(A and B) = P(A) × P(B) [if independent]
P(A or B) = P(A) + P(B) − P(A and B)
P(not A) = 1 − P(A)
Roll a die: P(even) = 3/6 = 1/2
A bag contains 4 red, 6 blue, and 2 green marbles. If one marble is drawn at random, what is the probability it is NOT blue? Write as a simplified fraction.
Sets group elements together. The Inclusion-Exclusion Principle prevents double-counting when finding the total of two overlapping groups.
A ∩ B = elements in BOTH sets
A ∪ B = elements in EITHER set
Complement A' = everything NOT in A
|A|=12, |B|=9, |A∩B|=5 → |A∪B| = 12+9−5 = 16
In a class of 30 students, 18 play soccer and 14 play basketball. If 8 students play both, how many students play neither?
Exponent rules govern how to simplify expressions with powers. Negative exponents indicate reciprocals. Fractional exponents represent roots.
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
(aᵐ)ⁿ = aᵐⁿ
a⁰ = 1 (a ≠ 0)
a⁻ⁿ = 1/aⁿ | a^(1/2) = √a
Simplify 2³ × 2⁴ = 2⁷ = 128
Simplify: (3²)³ ÷ 3⁴
Give your answer as a single number.
The GCF (Greatest Common Factor) is the largest factor shared by two numbers. The LCM (Least Common Multiple) is the smallest multiple shared. A simple relationship connects them.
GCF: use prime factorization, take LOWEST powers
LCM: use prime factorization, take HIGHEST powers
LCM(a, b) = (a × b) / GCF(a, b)
GCF(12, 18): 12=2²×3, 18=2×3² → GCF = 2×3 = 6
What is the LCM of 12 and 20?
Absolute value |x| is the distance from zero — always non-negative. An absolute value equation |x| = k (k > 0) has TWO solutions: x = k or x = −k.
|x| < k → −k < x < k
|x| > k → x > k OR x < −k
|a − b| = distance between a and b on number line
|2x − 3| = 7 → 2x−3=7 → x=5, OR 2x−3=−7 → x=−2. Solutions: {5, −2}
Solve |3x + 6| = 15. What is the positive solution for x?
Counting problems require organized thinking. The Fundamental Counting Principle: if event A has m ways and event B has n ways, then A and B together have m × n ways. Combinations count selections where order doesn't matter.
Permutation (order matters): P(n,r) = n!/(n−r)!
Combination (order doesn't matter): C(n,r) = n!/[r!(n−r)!]
n! = n × (n−1) × … × 2 × 1
How many ways to choose 2 from 5? C(5,2) = 5!/(2!×3!) = 10
A restaurant offers 3 appetizers, 5 main courses, and 4 desserts. If a meal consists of one of each, how many different meals are possible?
Percent change measures relative growth or decline from an original value. SSAT often applies successive percent changes — be careful: a 20% increase followed by a 20% decrease does NOT return to the original value.
Increase by p%: multiply by (1 + p/100)
Decrease by p%: multiply by (1 − p/100)
Two successive changes: multiply factors together
$80 increased by 25%: 80 × 1.25 = $100
A jacket costs $120. It is first discounted by 20%, then an additional 10% is taken off the discounted price. What is the final price in dollars?