Digital SAT key facts

Math · Algebra

Linear equations in one variable

• Do the same operation to both sides of the equation.
• To clear fractions, multiply every term by the lowest common denominator.
• total = fixed amount + rate × number of units
• ax + b = cx + d has exactly one solution when a ≠ c.
• No solution: a = c but b ≠ d (for example 3x + 2 = 3x + 5).
• Infinitely many solutions: a = c and b = d (both sides are identical).
• If the question asks for an expression such as x + 3, find that, not just x.

Linear functions

• f(x) = mx + b, where m is the slope and b is the y-intercept.
• slope = change in output ÷ change in input: m = (y₂ − y₁)/(x₂ − x₁)
• b = f(0), the value when the input is 0.
• The x-intercept is where f(x) = 0.
• A positive slope means increasing; a negative slope means decreasing.
• In a table of a linear function, equal steps in x give equal steps in f(x).
• The unit of the slope is the output unit per input unit.

Linear equations in two variables

• y = mx + b: slope m, y-intercept (0, b).
• Ax + By = C has slope −A/B (rearrange to y = −(A/B)x + C/B).
• x-intercept: set y = 0. y-intercept: set x = 0.
• Line through (x₁, y₁) with slope m: y − y₁ = m(x − x₁).
• Parallel lines: same slope, different y-intercepts.
• Perpendicular lines: m₁ × m₂ = −1, so the second slope is −1/m₁.
• A point lies on a line only if its coordinates make the equation true.

Systems of two linear equations

• A solution must satisfy both equations; check it in each.
• For a₁x + b₁y = c₁ and a₂x + b₂y = c₂:
• One solution: a₁/a₂ ≠ b₁/b₂ (different slopes).
• No solution: a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (parallel lines).
• Infinitely many solutions: a₁/a₂ = b₁/b₂ = c₁/c₂ (the same line).
• If the question asks for x + y or x − y, try adding or subtracting the two equations first.
• In a word problem, write one equation for the count and one for the total.

Linear inequalities

• Adding or subtracting the same number keeps the sign the same.
• Multiplying or dividing by a negative number reverses the sign: −2x > 8 gives x < −4.
• at most, no more than, maximum: ≤
• at least, no less than, minimum: ≥
• Compound inequality a ≤ x ≤ b: do the same operation to all three parts.
• To test a point (x, y) in a system, substitute it into each inequality.
• < and > exclude the boundary value; ≤ and ≥ include it.

Math · Advanced Math

Equivalent expressions

• (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²
• Difference of squares: a² − b² = (a − b)(a + b)
• x^a × x^b = x^{a + b}, x^a ÷ x^b = x^{a − b}, (x^a)^b = x^ab
• x^−a = 1/x^a and x⁰ = 1 (x ≠ 0)
• x^1/n is the n-th root of x; x^m/n is the n-th root of x^m, so 8^2/3 = 4.
• Adding fractions: a/b + c/d = (ad + bc)/(bd)
• Quick check: put x = 2 into the original and into your answer; the values must match.

Nonlinear equations and systems

• Quadratic formula: x = (−b ± √(b² − 4ac))/(2a)
• b² − 4ac > 0: two real solutions; = 0: exactly one; < 0: no real solution.
• For ax² + bx + c = 0: sum of solutions = −b/a, product of solutions = c/a.
• If (x − p)(x − q) = 0, then x = p or x = q.
• |A| = k with k ≥ 0 means A = k or A = −k; if k < 0 there is no solution.
• A value that makes a denominator 0 can never be a solution.
• A square root sign means the non-negative root, so √(…) cannot equal a negative number.

Nonlinear functions

• Vertex form: y = a(x − h)² + k has vertex (h, k).
• For y = ax² + bx + c, the vertex is at x = −b/(2a); the y-intercept is c.
• Factored form y = a(x − p)(x − q) has zeros p and q; the vertex is at x = (p + q)/2.
• f(t) = a(b)^t: growth of r% per period means b = 1 + r/100; decay of r% means b = 1 − r/100.
• In a(b)^t/k, the quantity is multiplied by b once every k units of time.
• y = f(x − c) shifts the graph c units right; y = f(x) + d shifts it d units up.
• y = −f(x) reflects the graph across the x-axis; y = f(−x) reflects it across the y-axis.

Math · Problem-Solving and Data Analysis

Ratios, rates, proportions and units

• If a/b = c/d, then a × d = b × c.
• unit rate = total amount ÷ number of units
• distance = speed × time; average speed = total distance ÷ total time
• Ratio a to b: the parts are a/(a + b) and b/(a + b) of the whole.
• 1 hour = 60 minutes = 3,600 seconds
• Length scale factor k: areas are multiplied by k², volumes by k³.
• amount of substance in a mixture = concentration × total amount

Percentages

• part = (percent ÷ 100) × whole, so whole = part ÷ (percent ÷ 100)
• percent change = (new − original) ÷ original × 100
• Increase by r%: multiply by (1 + r/100). Decrease by r%: multiply by (1 − r/100).
• Original value = new value ÷ multiplier.
• Changes in a row: multiply the multipliers, e.g. +20% then −10% is 1.20 × 0.90 = 1.08, a rise of 8%.
• n periods at r% each: final = original × (1 + r/100)ⁿ
• A fall from 8% to 6% is 2 percentage points, but a 25% decrease.

One-variable data

• mean = sum of the values ÷ number of values, so sum = mean × number of values
• median = the middle value when the values are in order; with an even number of values, the mean of the two middle values
• range = greatest value − least value
• Standard deviation measures how far the values are from the mean: closer to the mean means a smaller standard deviation
• Add the same number c to every value: mean and median go up by c; range and standard deviation do not change
• Multiply every value by k (k > 0): mean, median, range and standard deviation are all multiplied by k
• A very large or very small outlier changes the mean and the range a lot and the median very little

Two-variable data

• Slope = the predicted change in y when x increases by 1 unit
• y-intercept = the predicted value of y when x = 0
• To predict, substitute the x-value into the equation of the line
• A point above the line has an actual y greater than the predicted y; a point below the line has an actual y less than predicted
• Linear model y = mx + b: equal differences in y for equal steps in x
• Exponential model y = a(b)^x: a is the value when x = 0, and y is multiplied by b for each increase of 1 in x
• b = 1 + r for growth at rate r, and b = 1 − r for decay (for example, b = 0.94 means a 6% decrease per step)

Probability

• probability = number of favorable outcomes ÷ total number of possible outcomes
• P(not A) = 1 − P(A)
• Independent events: P(A and B) = P(A) × P(B)
• P(A or B) = P(A) + P(B) − P(A and B)
• Conditional: P(A given B) = P(A and B) ÷ P(B); in a table, the cell that is both ÷ the total for B
• P(at least one) = 1 − P(none)
• Expected number = probability × number of trials or people

Samples, studies and margin of error

• Random sample: results can be generalized to the population the sample was taken from, and only to that population
• Random assignment to treatments (an experiment): a cause-and-effect conclusion is allowed
• Observational study or survey: shows an association only, not cause and effect
• Plausible range for the population value = estimate − margin of error to estimate + margin of error
• A larger random sample gives a smaller margin of error
• The margin of error is about the population mean or percent, not about each individual
• Estimated number in the population = sample proportion × population size

Math · Geometry and Trigonometry

Area and volume

• Rectangle: area = length × width; perimeter = 2(length + width)
• Triangle: area = ½ × base × height. Trapezoid: area = ½ × (sum of the parallel sides) × height
• Circle: area = πr²; circumference = 2πr
• Rectangular box: volume = lwh; surface area = 2(lw + lh + wh)
• Cylinder: volume = πr²h. Cone: volume = (1/3)πr²h
• Sphere: volume = (4/3)πr³; surface area = 4πr²
• Scale factor k: lengths × k, areas × k², volumes × k³

Lines, angles and triangles

• Angles on a straight line add to 180°; angles around a point add to 360°; vertical (opposite) angles are equal
• Parallel lines cut by a transversal: corresponding angles are equal, alternate interior angles are equal, same-side interior angles add to 180°
• The angles of a triangle add to 180°
• An exterior angle of a triangle = the sum of the two interior angles not next to it
• Isosceles triangle: the angles opposite the equal sides are equal
• Similar triangles: matching sides are in the same ratio k; areas are in the ratio k²
• Polygon with n sides: interior angles add to (n − 2) × 180°; exterior angles add to 360°

Right triangles and trigonometry

• Pythagorean theorem: a² + b² = c², where c is the hypotenuse
• Common whole-number triangles: 3-4-5, 5-12-13, 8-15-17, 7-24-25, and their multiples such as 6-8-10
• 45°-45°-90°: sides x, x, x√2 (hypotenuse = leg × √2)
• 30°-60°-90°: sides x, x√3, 2x (x is opposite 30°, 2x is the hypotenuse)
• sin = opposite ÷ hypotenuse; cos = adjacent ÷ hypotenuse; tan = opposite ÷ adjacent
• sin x° = cos(90° − x°): the sine of one acute angle equals the cosine of the other
• π radians = 180°: multiply degrees by π/180 to get radians, and radians by 180/π to get degrees

Circles

• Circumference = 2πr; area = πr²
• Arc length = (θ/360) × 2πr in degrees, or rθ with θ in radians
• Sector area = (θ/360) × πr² in degrees, or ½r²θ with θ in radians
• Circle with center (h, k) and radius r: (x − h)² + (y − k)² = r²
• Completing the square: x² + bx becomes (x + b/2)² − (b/2)²
• A tangent is perpendicular to the radius at the point where it touches the circle
• An inscribed angle is half the central angle on the same arc; an angle inscribed in a semicircle is 90°

Reading and Writing · Craft and Structure

Words in context

• Predict your own word before you look at the choices
• The clue is always in the text: an example, a definition, or a contrast
• but, yet, however, although, despite: the blank is the opposite of the idea nearby
• a colon, because, since, so, indeed: the blank agrees with the idea nearby
• Check the tone: positive, negative or neutral must match the text
• For an underlined word, a common word often has a less common meaning (to qualify a claim = to limit it)
• Put your choice back into the sentence and read the whole text again

Text structure and purpose

• Purpose = what the writer does with the whole text; function = the job of one sentence
• Answer choices begin with a verb (explain, describe, challenge, illustrate): check that the verb is right first
• Common structure: a common belief, then a correction (signals: in fact, however, yet)
• Common structure: a claim, then an example or evidence that supports it
• Common structure: a problem or question, then a solution or finding
• In poems and stories, the purpose is often to show a character's feeling or to describe a scene
• Reject a choice if any part of it is not in the text, even when the rest is correct

Cross-text connections

• Sum up each text in one short line before you read the options.
• Common relationships: full agreement, partial agreement with a caution, a different explanation, new evidence that supports or weakens the claim.
• Words such as "but," "however," "only if" and "yet" in Text 2 show where it parts from Text 1.
• The answer must fit both texts. An option that describes either text wrongly is out.
• Be careful with extreme options: "completely wrong," "never," "always," "proves."
• Answer for the person the question names, not for yourself.

Reading and Writing · Information and Ideas

Central ideas and details

• Main idea = the topic + what the author says about it.
• A main-idea answer must cover the whole text, not one sentence of it.
• Too narrow: true, but only a detail. Too broad: goes beyond the text. Both are wrong.
• After "but," "however" or "yet," expect the author's real point.
• Detail answer: stated in the text, usually in different words (a paraphrase).
• Reject any option that makes a claim the text never makes, even if it is true in real life.

Command of evidence: text

• The answer must match the whole claim, not half of it.
• Predict what the evidence should look like before you read the options.
• Support: the result is what the claim predicts.
• Weaken: the effect appears without the supposed cause, or the cause is present and the effect is missing.
• Being on the same topic is not enough. The option must be about the same link the claim makes.
• For quotations, match the meaning, not shared words.
• Check whether the question says support or weaken (undermine).

Command of evidence: data

• Read the claim before the table.
• For every number in an option, check the row, the column and the unit.
• Two tests: accurate (matches the table) and relevant (proves the claim).
• A claim with "increases," "decreases," "more than" or "less than" needs two or more values compared.
• A claim that a change caused an effect needs a comparison with a group that did not have the change.
• A percentage is not a count: 50% of 40 is 20, but 20% of 400 is 80.
• A rise from 40% to 46% is a rise of 6 percentage points.

Inferences

• Use every fact in the text. An answer that ignores one of them is usually wrong.
• The answer follows from the text alone, not from outside knowledge.
• Take one small step from the facts. Prefer the cautious option to the sweeping one.
• "If the hypothesis is correct": pick the result that the hypothesis predicts.
• A link between two things (correlation) does not prove cause: another factor may explain both.
• Read your choice back into the sentence. It must fit the words just before the blank.

Reading and Writing · Standard English Conventions

Boundaries

• Correct joins: "Sentence. Sentence." / "Sentence; sentence." / "Sentence, and sentence."
• Two sentences joined by a comma alone (a comma splice) is always wrong.
• A colon or a single dash follows a complete sentence and introduces a list, an example or an explanation.
• Semicolon + "however" or "therefore" + comma: "It rained; however, the game went on."
• Extra information: a comma on both sides or a dash on both sides. Do not mix the two.
• Needed information, and a title or job before a name, take no commas: "physicist Abdus Salam won."
• Apostrophes: the city's walls (one city), the cities' walls (several cities); its = belonging to it, it's = it is.

Form, structure and sense

• Ignore phrases between subject and verb: "The box of old letters is heavy."
• "Each," "every," "one of" and "the number of" take a singular verb. An -ing phrase used as a subject is singular: "Growing crops takes time."
• Two subjects joined by "and" take a plural verb. In a reversed sentence the subject comes after the verb: "Among the trees are two wells."
• Pronouns: its (one thing), their (more than one). it's = it is; they're = they are.
• Plural, no apostrophe: "the rivers flow." Possessive: "the river's banks" (one river), "the rivers' banks" (several).
• "Since" + a past date: has/have + past participle ("has grown since 2010"). An earlier event before another past event: had + past participle.
• After an opening describing phrase and its comma, name the person or thing it describes.

Reading and Writing · Expression of Ideas

Transitions

• Addition: in addition, moreover, furthermore, also.
• Contrast: however, in contrast, instead, on the other hand. Contrast after admitting a point: nevertheless, even so, still.
• Cause and result: therefore, consequently, as a result, thus, accordingly.
• Example or closer detail: for example, for instance, specifically.
• Saying it again or more strongly: that is, in other words, in fact, indeed. Summing up: in short.
• Time order: first, next, later, previously, finally.
• Similarity: similarly, likewise. "Granted" admits a point against the writer's own argument.

Rhetorical synthesis

• Read the goal first. It is the only test.
• Similarity: both things named, with "both," "like" or "similarly."
• Difference or contrast: both things named, with "while," "whereas," "but" or "unlike."
• Methodology: what the researchers did, not what they found.
• Result or finding: what was found, not how it was done.
• Introducing to a new audience: say who or what the subject is before giving any detail.
• A specific goal ("give the height," "give an example") needs that exact item from the notes.