The PSAT is your first look at the Digital SAT — and for a lot of 10th and 11th graders, it’s also the ticket to National Merit recognition. But here’s the good news: the PSAT tests a surprisingly short list of math formulas and grammar rules. Learn these 30, and you’ve covered the majority of what you’ll actually see on test day.
This quiz mixes three question styles: multiple choice for classic test practice, flip cards for fast formula review, and a few fill-in questions to make sure you can actually recall the answer under pressure. Your score updates live at the top of the page, and at the end you’ll get a Review Mistakes mode that only re-shows what you missed.
Algebra Essentials
5 questionsWhat is the slope formula for a line through two points (x₁, y₁) and (x₂, y₂)?
m = (y₂ − y₁) / (x₂ − x₁)
Example: For (2, 3) and (5, 9), slope = (9 − 3) / (5 − 2) = 6/3 = 2. Rise over run.
Which equation is in slope-intercept form with slope 3 and y-intercept −4?
Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. Slope 3 goes in the m slot, y-intercept −4 goes in the b slot: y = 3x − 4.
If 5x − 2 > 13, then which of the following must be true?
Solve like a normal equation: add 2 to both sides → 5x > 15. Divide by 5 → x > 3.
Solve for x: 3(x + 4) = 21. Enter just the number.
Distribute first: 3x + 12 = 21. Subtract 12: 3x = 9. Divide by 3: x = 3.
What is the distance formula between two points on a coordinate plane?
d = √((x₂ − x₁)² + (y₂ − y₁)²)
Example: distance from (1,2) to (4,6) = √(3² + 4²) = √25 = 5. It’s the Pythagorean theorem in disguise.
Geometry Must-Knows
5 questionsWhat is the area of a circle with radius r?
A = π r²
Example: for r = 5, area = π × 25 = 25π square units. On the PSAT, most answers stay in terms of π.
A right triangle has legs of length 6 and 8. What is the length of the hypotenuse?
Pythagorean theorem: a² + b² = c². So 6² + 8² = 36 + 64 = 100, and c = √100 = 10.
What are the side ratios of a 45-45-90 and a 30-60-90 triangle?
45-45-90: 1 : 1 : √2
30-60-90: 1 : √3 : 2
The 45-45-90 has two equal legs and a hypotenuse that’s √2 times a leg. The 30-60-90 has a short leg (opposite 30°), a long leg = short × √3, and a hypotenuse = short × 2.
The three interior angles of a triangle measure x°, 2x°, and 60°. What is the value of x?
Interior angles of any triangle add to 180°. So x + 2x + 60 = 180 → 3x = 120 → x = 40.
A circle has diameter 10. Its circumference is kπ. What is the value of k?
Circumference formula: C = π d (or 2π r). Here d = 10, so C = 10π, meaning k = 10.
Data & Advanced Math
5 questionsDefine mean, median, and mode.
Mean = sum / count
Median = middle value (sort first!)
Mode = most frequent value
For {3, 5, 5, 7, 10}: mean = 30/5 = 6, median = 5, mode = 5. If there are two middle numbers, average them.
A product’s price rose from $80 to $100. What is the percent increase?
Percent change formula: (new − old) / old × 100%. Here: (100 − 80) / 80 = 20/80 = 0.25 = 25%.
What is the quadratic formula to solve ax² + bx + c = 0?
x = (−b ± √(b² − 4ac)) / (2a)
The ± gives you both roots. If b² − 4ac (the discriminant) is negative, there are no real solutions.
Simplify: (x³)(x⁵)
When multiplying like bases, add the exponents: x³ · x⁵ = x⁸ (3 + 5 = 8).
A bag has 4 red marbles and 6 blue marbles. What is the probability of drawing a red marble? Enter as a decimal (e.g. 0.25).
Probability = favorable outcomes / total outcomes. There are 4 red and 10 total marbles: 4/10 = 0.40 (or 40%).
Sentence Structure
5 questionsChoose the correct verb: “The stack of books ___ heavy.”
The subject is stack (singular), not books. Prepositional phrases like “of books” don’t change the subject. So the verb must be singular: is.
Which version fixes the two independent clauses correctly?
A is a comma splice (two full sentences joined with just a comma). C is a run-on. D puts the comma in the wrong place. B uses a semicolon — the correct way to join two closely related independent clauses.
Which of the following is a complete sentence?
A complete sentence needs a subject AND a verb AND expresses a complete thought. C has all three (She = subject, won = verb, complete idea). A starts with “Because” (dependent clause). B has no main verb. D has no subject.
What makes a clause dependent (unable to stand alone)?
A subordinating conjunction at the start: because, although, when, if, since, while, unless, until, after, before
“Because she studied” = dependent (incomplete). “She studied” = independent (complete). Attach a dependent clause to an independent one with a comma: “Because she studied, she passed.”
A semicolon is correctly used to…
A semicolon does one main job on the PSAT: it joins two independent clauses (both must be able to stand alone as complete sentences). It can also separate items in a complicated list, but that’s rarer.
Punctuation Rules
5 questionsWhich sentence uses commas correctly?
An introductory phrase (like “After the exam”) that comes before the main clause gets a comma right after it. B is the only version that places the comma correctly at the boundary between the intro and the main clause.
Complete the sentence: “The dog wagged ___ tail.”
Its = possessive (like his, her). It’s = contraction of it is. Substitute “it is”: “The dog wagged it is tail” — nonsense. So no apostrophe.
Which sentence uses a colon correctly?
A colon must come after a complete sentence. “He brought three items” is a complete sentence, so the colon is fine. In A, “He brought” is not complete on its own (it needs the object).
What is the Oxford (serial) comma, and does the PSAT require it?
The Oxford comma is the comma before “and” or “or” in a list of three or more items.
“pencils, paper, and calculators” (with Oxford) vs. “pencils, paper and calculators” (without). The PSAT accepts either style, but requires the SAME style throughout a passage. Watch for inconsistency in a list.
Which sentence uses commas correctly around the nonessential clause?
A nonessential clause (extra info you could remove) must be surrounded by commas on both sides. “My brother is a teacher” still makes sense without “who lives in Texas,” so it needs bracketing commas.
Word Choice & Transitions
5 questionsIn formal SAT/PSAT style, complete: “Each student turned in ___ homework.”
Each is singular, so the pronoun that refers to it must also be singular. In formal PSAT/SAT style, that’s his or her. (In everyday writing, singular “they” is now widely accepted, but the PSAT still tests the traditional rule.)
Which completion avoids a dangling modifier? “Walking to school, ___”
The person doing the walking must come right after the comma. In A, C, and D, it sounds like the rain, the books, or the rain is walking. Only B has an actual person (I) walking.
Complete with parallel structure: “She likes running, swimming, and ___.”
Items in a list must all take the same grammatical form. The first two items use the -ing form (running, swimming), so the third must too: biking.
Name the four most common transition types the PSAT tests, with example words.
Contrast: however, although, but, yet
Cause/effect: because, since, therefore, thus
Example: for instance, such as, for example
Addition: furthermore, also, moreover, additionally
The PSAT’s #1 transition question: read the sentence before and after the blank. If they agree, use Addition. If they clash, use Contrast. If one causes the other, use Cause/effect.
Complete the question: “___ did you invite to the party?”
Rewrite the sentence and see whether the answer is he/she (use who) or him/her (use whom). “You did invite him” → use whom.
Nice work!
You’re building solid PSAT foundations. Review the ones you missed and try again in a couple of days — spaced practice locks it in.
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