Deliberate practice — one skill per drill, one reliable process per question type.
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How to Use This Review
Core approach + section-by-section strategy guide — click any section to expand
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Core Approach
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Practice with a plan
Focus on one skill per drill. Don't mix five question types in one sitting — isolate the skill, master the process, then raise the difficulty.
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Learn a process, then automate it
Build a reliable step-by-step routine for each question type. Drill it until it's fast enough to use automatically under test time pressure.
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Pace deliberately
Reading: ~1.5–2 min per passage. Math: 1–2 min per problem. Start with easy wins — flag hard ones and return if time allows.
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Eliminate aggressively
Rule out choices with extreme language, partial truth, or unsupported claims. You only need to find the best remaining answer — not a perfect one.
English
Standard English Conventions — Strategy
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Identify the error type first. Name the rule (subject-verb, pronoun clarity, punctuation, parallel structure, modifier placement) before reading the answer choices — this prevents the choices from distorting your thinking.
Strip interrupters to find the true subject. Cross out phrases beginning with of, along with, as well as, and in addition to. The verb must agree with what remains.
Pronoun ambiguity: if a pronoun could refer to two different people, it's wrong. The fix is almost always to replace the pronoun with the full noun.
Colon vs. comma splice: a colon is correct when Part 2 explains or defines Part 1. A comma alone joining two independent clauses is always a comma splice.
Parallel structure: for correlative pairs (not only/but also, either/or, both/and), identify the form of the first element and copy it exactly for the second.
Dangling modifier test: ask "who is doing the action in the opening phrase?" — that person or thing must be the grammatical subject immediately after the comma.
Re-read with your answer inserted. If the sentence flows naturally and you can name the rule that supports it, commit and move on.
Reading
Information & Ideas — Strategy
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Skim structure first. Read the first and last sentence of each paragraph to map the purpose and progression before tackling questions.
Identify the main idea per paragraph. Ask: "What is this paragraph trying to prove or explain?" Note topic sentences and transition words.
Map the author's stance. Distinguish objective fact from author opinion, inference, or tone shift. Ask: what is the author's claim, and what evidence supports it?
Evidence questions: locate the exact lines the passage uses to support a claim. The correct answer cites those same lines — not a paraphrase, not an addition from outside the text.
Inference questions: stay close to the text. The answer must be supportable by specific lines — never add facts or assumptions beyond what is written.
Function/organization questions: ask how a sentence contributes to the paragraph's argument or structure, not just what it says.
Vocabulary in context: use surrounding sentences to infer meaning. Pick the option that best fits the author's tone and the passage's logic, not the most common definition of the word.
Eliminate by the text: rule out choices with extreme language, partial truth, or information not directly supported by the passage.
Math
Geometry & Trigonometry — Strategy
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Always draw and label. Sketch the figure and label all given sides, angles, and relationships. A labeled diagram makes visual relationships that the text conceals.
SOHCAHTOA is your anchor. sin = Opp/Hyp, cos = Adj/Hyp, tan = Opp/Adj. Memorize 30-60-90 (1 : √3 : 2) and 45-45-90 (1 : 1 : √2) ratios as instant shortcuts.
Pythagorean theorem as a check: a² + b² = c² confirms side lengths. Use it to verify your trig answer is geometrically reasonable.
Circle essentials: circumference = 2πr, area = πr², arc length = (central angle / 360°) × 2πr, sector area = (central angle / 360°) × πr².
Parallel lines: corresponding angles are equal, alternate interior angles are equal, consecutive interior angles sum to 180°. These three facts unlock most multi-step angle problems.
Decompose complex shapes. Break irregular figures into rectangles, triangles, and semicircles. Calculate each part separately, then combine.
Sanity-check answers. If a side is longer than the hypotenuse, or an angle exceeds 180°, you've made an error — re-evaluate the model before recalculating.
Math
Problem Solving & Data Analysis — Strategy
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Read all labels before calculating. Extract units, axis scales, and column headers from graphs and tables first — misreading a label is the single most common trap.
Translate words into math. Convert percent changes to (new − old) / old. Convert proportions to fractions. Write the equation before solving.
Ratios and rates: set up a proportion with matching units on each side. Simplify before cross-multiplying to avoid arithmetic errors.
Weighted averages: when group sizes differ, you cannot simply average the averages. Multiply each average by its group size, sum the products, then divide by the total count.
Probability: P(event) = favorable outcomes / total outcomes. For "at least one" problems, use 1 − P(none).
Estimate to eliminate. Before calculating, make a rough estimate of the expected magnitude and use it to rule out clearly wrong answer choices.
Multi-step problems: break into parts, label each intermediate result, then combine. Writing down values at each step prevents compounding errors.
Always verify units in your final answer match what the question asks — especially when the table uses different units than the question stem.
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Section Strategy— Standard English Conventions
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Identify the error type first. Is it subject-verb agreement, pronoun clarity, punctuation, parallel structure, or modifier placement? Each has its own rule. Name the rule before looking at choices.
Isolate the true subject. Cross out prepositional phrases (of the data), appositive phrases, and interrupters (along with…, as well as…). The verb must agree with what remains.
Test by stripping the sentence. Remove the optional phrase, read only the core. If it still sounds wrong, your diagnosis is right.
Use process of elimination aggressively. One grammatical flaw disqualifies a choice entirely — you don't need to love the answer, just rule out the others.
Re-read with your chosen answer inserted. If the full sentence flows naturally and violates no rule you can name, you're done.
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Q1 of 5Subject-Verb Agreement
The analysis of water quality samples from sixteen urban waterways, along with detailed surveys conducted by three independent public health teams, that contaminant levels in lower-income neighborhoods consistently exceed federal safety standards.
Which choice correctly completes the sentence?
Step-by-Step Process
1
Find the true subject. Read left until you hit the first noun: The analysis. That is the subject — singular.
2
Cross out the interrupters. Mentally strike of water quality samples from sixteen urban waterways and along with detailed surveys conducted by three independent public health teams. These modify the subject but do not become it.
3
Test the stripped sentence:The analysis _____ that contaminant levels… — singular subject requires a singular verb.
4
Eliminate:suggest is plural ✗ · are suggesting is plural progressive ✗ · have been suggesting is plural perfect progressive ✗ → suggests ✓
The SAT routinely buries the true subject under long interrupters. Crossing them out is your most reliable defense — automate this step so it becomes reflex under time pressure.
Q2 of 5Pronoun Clarity
When Dr. Reyes reviewed her graduate student's dissertation, she told her that two chapters needed significant revision.
Which revision most clearly indicates that Dr. Reyes communicated the need for revision?
Step-by-Step Process
1
Diagnose the problem. Count the pronouns: she and her each appear once, but the sentence contains two female referents — Dr. Reyes and the graduate student. Every pronoun is ambiguous.
2
State the goal precisely. The question asks: which revision makes Dr. Reyes the unmistakable communicator? You need a choice that names Dr. Reyes as the acting subject.
3
Test each choice for remaining ambiguity. C and D still use she/her — still ambiguous ✗. B eliminates the communicating act entirely ✗. A names Dr. Reyes as subject and her graduate student as recipient, removing all ambiguity ✓.
When the SAT tests pronoun clarity, the fix is almost always to replace at least one pronoun with the full noun. Scan choices for the one that names the actor directly.
Q3 of 5Punctuation — Colon
Dr. Kwon's discovery had a single, stunning implication decades of conventional research had been built on a flawed assumption about cellular metabolism.
Which punctuation correctly fills the blank?
Step-by-Step Process
1
Label the two parts. Part 1: Dr. Kwon's discovery had a single, stunning implication. Part 2: decades of conventional research had been built on a flawed assumption. Ask: what is the logical relationship?
2
Part 2 reveals what that implication IS. This is an elaboration / definition relationship — exactly the case for a colon. Rule: a colon can connect two independent clauses when the second explains or defines the first ✓.
3
Eliminate the others: Comma alone (B) = comma splice — two independent clauses joined only by a comma ✗. ; yet (C) signals contrast — but Part 2 doesn't contradict Part 1 ✗. , however, (D) also signals contrast AND creates a comma splice ✗.
SAT colon questions hinge on one diagnostic: does Part 2 explain or define Part 1? If yes → colon. Watch for signal nouns like "implication," "reason," "truth," "result," and "conclusion" — they almost always precede a colon.
Q4 of 5Parallel Structure
The city's new transit plan aims not only to reduce traffic congestion in the downtown corridor but also air quality standards across the metropolitan region.
Which choice correctly completes the sentence?
Step-by-Step Process
1
Spot the correlative conjunction pair.not only … but also is the signal. This construction always requires the elements on each side to be in the same grammatical form — that's the rule, no exceptions.
2
Identify the first element's form. After not only you have: to reduce — an infinitive (to + base verb).
3
Copy the form. After but also you need another infinitive → to improve ✓. Verify aloud: "aims not only to reduce … but also to improve …" — smooth and parallel ✓.
4
Eliminate: A is a noun phrase ✗ · B is a gerund ✗ · D is a past-tense verb / adjective ✗.
For every correlative pair (not only/but also · either/or · both/and · neither/nor), find the form of the first element, then copy it exactly for the second. Automate this — it takes two seconds once the habit is built.
Q5 of 5Dangling Modifier
Having recently developed a cure for a rare autoimmune disorder, celebrated the breakthrough widely within the medical community.
Which choice correctly completes the sentence?
Step-by-Step Process
1
Identify the introductory participial phrase.Having recently developed a cure for a rare autoimmune disorder is a participial phrase. Ask: who performed this action?
2
Apply the modifier rule. The noun phrase immediately after the comma must be the one performing the action in the modifier. Whoever developed the cure must be the grammatical subject of the main clause.
3
Test each choice: A the researchers — researchers developed the cure ✓ and "the researchers celebrated…" is a complete clause ✓. B a breakthrough — a breakthrough didn't develop itself ✗. C the scientific community — too broad; the community didn't necessarily do the developing ✗. D it was the researchers who — "it" is an expletive, not the agent; modifier still dangles ✗.
Dangling modifiers are caught by one question: "Who/what is doing the action in the opening phrase?" That answer must be the subject of the main clause, placed right after the comma.
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Section Strategy— Information & Ideas
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Read for structure, not facts. On your first pass, identify the purpose of each paragraph (introduction / evidence / counterargument / conclusion). You can always return for specific details.
Main idea must span the whole passage. Eliminate choices that cover only one paragraph, or that are too extreme ("only," "always"). The correct answer captures the full arc of the author's argument.
For evidence questions, find data that directly proves the stated claim. Look for a specific number, study, or named finding — not a vague paraphrase or off-topic true statement.
For inference, stay anchored to the text. "Most directly supported" means the answer paraphrases exact lines. Any choice that adds information not written in the passage is wrong.
Function questions are structure questions. Ask what the paragraph does in the argument, then match your label to the choices. Don't let interesting content distract you from the structural role.
For vocab-in-context, cover the word and predict first. Substitute each choice back into the sentence. Pick the one that preserves the sentence's meaning and the author's tone.
Reading Passage — Urban Heat & Green Infrastructure
¶1Urban planners have long recognized that cities trap heat more efficiently than surrounding rural areas — a phenomenon known as the urban heat island (UHI) effect. The concentrated mass of asphalt, concrete, and steel absorbs solar radiation during the day and slowly releases it at night, keeping temperatures several degrees above those of adjacent suburbs. In low-income neighborhoods, where tree canopy coverage is often below fifteen percent, UHI conditions can be especially acute.
¶2In response, a growing number of municipalities have adopted green infrastructure programs: expanding parks, mandating reflective roofing materials, and planting street trees along commercial corridors. A 2022 study published in Environmental Health Perspectives found that neighborhoods with an average tree canopy of thirty percent or more experienced peak summer temperatures nearly 3°C lower than comparable areas with minimal vegetation. Researchers attribute this cooling primarily to evapotranspiration — the combined process by which plants release water vapor through their leaves, drawing heat from the surrounding air.
¶3Critics of green infrastructure initiatives argue that the costs of large-scale planting programs are prohibitive, particularly in cities already facing significant budget deficits. Proponents counter that the long-term savings from reduced emergency heat-related medical costs and lower municipal energy use outweigh the upfront investment.
Progress
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Q1 of 5Main Idea
Which choice best states the main idea of the passage?
Step-by-Step Process
1
Map the passage before reading choices. ¶1 = establishes the problem (UHI effect). ¶2 = presents evidence for green infrastructure's effectiveness. ¶3 = cost debate between critics and proponents. The main idea must reflect all three paragraphs.
2
Eliminate too-narrow choices. A covers only ¶1's detail about low-income areas ✗. C says "single most effective strategy" — the passage never ranks strategies and says "primarily," not exclusively ✗. D misrepresents ¶3 — budget deficits are mentioned as a concern, not as blocking "most cities" ✗.
3
Confirm B. It captures: green infrastructure reduces temperatures (¶2) + long-term financial argument (¶3 proponents) + ongoing cost debate (¶3 critics). Matches the full arc ✓.
Main idea choices that are too narrow (one paragraph only) or too extreme ("only," "single most," "prevents most") are almost always wrong. The correct answer honestly summarizes the whole passage — including any tension or debate the author presents.
Q2 of 5Evidence — Citing Detail
The passage claims that green infrastructure programs can measurably reduce urban temperatures.
Which detail from the passage most directly supports this claim?
Step-by-Step Process
1
Underline the claim's key words. "Measurably reduce urban temperatures." The word "measurably" means you need a specific, quantified result — a number or a study finding.
2
Test each choice: A describes low-income tree coverage — true but shows no temperature reduction ✗. C describes what programs cities adopted — no measurement given ✗. D is a cost criticism — irrelevant to temperature ✗. B cites the 2022 study showing a 3°C difference — that's a direct measured temperature reduction ✓.
3
Confirm B links tree canopy (green infrastructure) directly to a quantified temperature difference. It is the only choice that proves "measurably reduce" ✓.
For evidence questions, the correct answer almost always contains a specific measurement, statistic, or named study — not a general description of actions taken. When the claim uses a word like "measurably," treat it as a signal to look for a number in the passage.
Q3 of 5Function & Organization
"Critics of green infrastructure initiatives argue that the costs of large-scale planting programs are prohibitive, particularly in cities already facing significant budget deficits. Proponents counter that the long-term savings from reduced emergency heat-related medical costs and lower municipal energy use outweigh the upfront investment."
The third paragraph (¶3) primarily serves to...
Step-by-Step Process
1
Label the paragraph's structure before reading choices. Sentence 1: Critics raise a cost objection. Sentence 2: Proponents offer a financial rebuttal. The paragraph does two distinct things — you need a choice that reflects both.
2
Eliminate one-sided or inaccurate choices: A says "scientific evidence" — ¶3 contains no studies or data, only assertions ✗. C says "specific dollar amounts" — none are given ✗. D says the paragraph concludes against green infrastructure — it presents both sides without drawing a conclusion ✗.
3
Confirm B. "Introduce a cost-based objection … and present the financial case made by its supporters" — matches both sentences exactly ✓.
Function questions reward structural thinking. Before reading choices, label what each sentence in the paragraph does (objection / rebuttal / evidence / conclusion). Then pick the choice whose label matches your structural map — not the one with the most interesting content.
Q4 of 5Inference — Text-Based
¶2: "Researchers attribute this cooling primarily to evapotranspiration — the combined process by which plants release water vapor through their leaves, drawing heat from the surrounding air."
Based on the passage, which inference about evapotranspiration is most directly supported?
Step-by-Step Process
1
Re-read only the cited sentence. It says: evapotranspiration = plants releasing water vapor → draws heat from the air. That's the complete picture the passage gives you. Don't add anything from outside knowledge.
2
Eliminate choices that add new information the passage doesn't support: A says "nighttime hours" — no time-of-day information appears in the passage ✗. C says "equally effective in rural and urban environments" — no rural comparison is made ✗. D says evapotranspiration "replaced" roofing — the passage lists both as strategies and never ranks or compares them ✗.
3
Confirm B. It directly paraphrases the cited sentence — water vapor release draws heat from the air. No new claims, no outside knowledge ✓.
SAT inference means "what the text directly implies," not "what I can reasonably guess." If the answer adds any claim the passage doesn't explicitly make — even a claim that sounds logical — eliminate it.
Q5 of 5Vocabulary in Context
¶1: "In low-income neighborhoods, where tree canopy coverage is often below fifteen percent, UHI conditions can be especially acute."
As used in the first paragraph, "acute" most nearly means...
Step-by-Step Process
1
Cover the word and predict from context. "UHI conditions can be especially _______ " in areas with low tree coverage. The paragraph describes a problem that is worse in these neighborhoods — so the blank means intense, severe, or pronounced.
2
Substitute each choice and test: "especially severe" → heat conditions are especially intense here ✓. "especially perceptive" → perceptive = insightful; heat conditions can't be insightful ✗. "especially pointed" → pointed = sharp in shape; doesn't fit a heat phenomenon ✗. "especially brief" → brief = short in duration; the passage implies a persistent problem ✗.
3
Confirm A. "Severe" = extreme intensity — fits the grammatical slot and the contextual meaning perfectly ✓.
The vocab-in-context trap is that distractors are real definitions of the word. "Acute" genuinely means perceptive (an acute observer) and pointed (an acute angle) — but neither fits this sentence. Substitute first, choose second. Never pick a definition just because it's dictionary-correct.
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Section Strategy— Geometry & Trigonometry
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Always draw and label the figure. Sketch the diagram and mark every given side, angle, and relationship. Visual structure that's hidden in text becomes obvious once labeled.
SOHCAHTOA is your anchor. sin = Opp/Hyp · cos = Adj/Hyp · tan = Opp/Adj. Label the opposite, adjacent, and hypotenuse relative to the given angle before picking a ratio.
Know the special triangles cold. 30-60-90: sides 1 : √3 : 2. 45-45-90: sides 1 : 1 : √2. Recognize common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17.
Circle formulas from memory. Arc length = (θ/360°) × 2πr. Sector area = (θ/360°) × πr². Simplify θ/360° as a fraction first, then multiply.
Parallel lines — three rules. Corresponding angles = equal. Alternate interior = equal. Same-side interior (co-interior) = supplementary (sum to 180°). Draw a sketch and mark which type before computing.
Composite area — decompose first. Break the figure into rectangles, triangles, circles, or semicircles. Compute each area separately, then combine with addition or subtraction.
Sanity-check every numeric answer. No leg can exceed the hypotenuse; no angle in a triangle can exceed 180°. If your answer violates this, re-examine your model.
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Q1 of 5Right Triangle Trig
In right triangle $PQR$, angle $Q$ is a right angle, $\angle P = 38°$, and $PQ = 10$.
Which expression gives the length of $QR$?
Step-by-Step Process
1
Draw and label. Right angle at $Q$. Side $PQ = 10$ runs from $P$ to the right angle — it is adjacent to $\angle P$. Side $QR$ runs from the right angle to $R$ — it is opposite $\angle P$. $PR$ is the hypotenuse.
2
Pick the right ratio. You have the adjacent ($PQ = 10$) and want the opposite ($QR$). The ratio that links opposite and adjacent is tangent: $$\tan(\angle P) = \frac{\text{Opp}}{\text{Adj}} = \frac{QR}{PQ}$$
Eliminate the others: $10\cos(38°)$ = adjacent · cos = would give a different side ✗ · $10\sin(38°)$ links opposite to hypotenuse ✗ · $10/\sin(38°)$ = the hypotenuse $PR$ ✗.
Label every side as opposite, adjacent, or hypotenuse before choosing a ratio. The mistake is jumping to SOHCAHTOA without labeling first — that's when you pick sin instead of tan.
Q2 of 5Circle — Arc Length
A circle has a radius of $9$. A central angle of $80°$ intercepts an arc on the circle.
Simplify the fraction first. $\dfrac{80}{360} = \dfrac{2}{9}$, so: $\dfrac{2}{9} \times 18\pi = \dfrac{36\pi}{9} = 4\pi$ ✓.
4
Sanity check. The full circumference is $2\pi(9) = 18\pi$. The arc is $80/360 \approx 22\%$ of the circle. $22\% \times 18\pi \approx 4\pi$ ✓.
Always simplify $\theta/360°$ as a reduced fraction before multiplying — it keeps numbers small and prevents errors. Confirm your result is less than the full circumference; if it isn't, you inverted the fraction.
Q3 of 5Parallel Lines & Angles
Lines $l$ and $m$ are parallel. A transversal $t$ crosses both lines. At line $l$, the transversal forms a $72°$ angle. The question asks for the same-side interior angle (co-interior angle) at line $m$ — on the same side of $t$ as the $72°$ angle, between the two parallel lines.
What is the measure of the same-side interior angle at line $m$?
Step-by-Step Process
1
Recall the three parallel-line angle rules. (i) Corresponding angles = equal. (ii) Alternate interior angles = equal. (iii) Same-side interior (co-interior) angles are supplementary — they sum to $180°$.
2
Identify which rule applies. The question explicitly names "same-side interior" → supplementary rule. No need to look further.
3
Calculate: $180° - 72° = 108°$ ✓.
4
Eliminate the traps: $72°$ is correct for corresponding or alternate interior angles (not same-side) ✗. $162° = 180° - 18°$ (wrong arithmetic with wrong angle) ✗. $118° = 180° - 62°$ (wrong starting angle) ✗.
The moment you see "same-side interior" or "co-interior," write "= 180°" before doing anything else. Memorize which relationship gives equal angles (corresponding, alternate interior) vs. supplementary angles (same-side interior) — confusing them is the #1 mistake on these problems.
Q4 of 5Composite Area
A square has a side length of $10$. A circle is inscribed in the square, meaning the circle is tangent to all four sides of the square.
What is the area of the region inside the square but outside the circle?
Step-by-Step Process
1
Set up the decomposition. Shaded region = area of square $-$ area of circle. Label what you need: side of square = 10, radius of inscribed circle = ?
2
Find the radius. "Inscribed" means the circle touches all four sides. The diameter spans the full width of the square: $d = 10$, so $r = 5$.
Subtract: $100 - 25\pi$ ✓. Eliminate D: $25\pi \approx 78.5$, so $25\pi - 100 \approx -21.5$ — a negative area is impossible ✗.
Composite area always follows: large shape − small shape. Extract the radius from the geometry first — "inscribed" is a key word that means radius = half the side length. Never reverse the subtraction order.
Q5 of 5Pythagorean Theorem & Trig
In right triangle $ABC$, angle $C$ is a right angle, $AC = 5$, and $BC = 12$.
Label relative to $\angle A$. $AC = 5$ is between $\angle A$ and the right angle → adjacent. $BC = 12$ is across from $\angle A$ → opposite. $AB = 13$ is the hypotenuse.
Eliminate traps: $5/13 = \cos(A)$ (adjacent over hypotenuse) ✗ · $5/12 = \tan$ of the complementary angle ✗ · $12/5 > 1$ — impossible for any sine value ✗.
Two-step trig problems: (1) find the missing side with the Pythagorean theorem, then (2) label and apply SOHCAHTOA. Recognize the 5-12-13 triple on sight — it's one of the SAT's most common Pythagorean triples alongside 3-4-5 and 8-15-17.
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Section Strategy— Problem Solving & Data Analysis
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Read all labels before calculating. Extract units, axis scales, and column headers from graphs and tables first — misreading a label is the single most common PSDA trap.
Percent change formula: (new − old) / old × 100. Always divide by the original value, not the new one. Using the new value as denominator is the #1 percent-change error.
Set up proportions with matching units on each side. Write the same unit in numerator and denominator on both sides, then cross-multiply only after confirming the setup is correct.
Weighted averages require totals — never average the averages. Multiply each group's mean by its size, sum the products, then divide by the total count. Averaging group means only works when groups are equal in size.
For conditional probability, restrict the sample space. "Given that…" or "selected from those who…" changes your denominator to the size of that subgroup, not the whole population.
Estimate first to eliminate wrong choices. A quick mental estimate rules out clearly unreasonable answers before you do any exact arithmetic.
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Q1 of 5Mean of a Dataset
The exam scores for five students are: 72, 85, 91, 68, and 79.
What is the mean (average) score?
Step-by-Step Process
1
Write the mean formula. $$\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}$$
2
Sum the values in a chain. $72 + 85 = 157$, $157 + 91 = 248$, $248 + 68 = 316$, $316 + 79 = 395$.
3
Divide by the count: $395 \div 5 = 79$ ✓
4
Eliminate traps: 85 is the maximum score — never confuse max with mean ✗. 80 would require a total of 400, but $80 \times 5 = 400 \ne 395$ ✗.
Reverse-check: multiply your answer by the count and confirm it equals your sum. Here, $79 \times 5 = 395$ ✓. This one extra step catches arithmetic mistakes before they cost points.
Eliminate traps: 12% = raw dollar change with no division ✗. 23% ≈ $\frac{12}{52} \times 100$ — divided by the new price instead of old ✗. 130% = $\frac{52}{40} \times 100$ — the new price as a ratio of old, not the change ✗.
The denominator is always the ORIGINAL value. If your result exceeds 100%, you likely divided by the wrong number or forgot to subtract. Confirm: (52 − 40) / 40 = 0.30 = 30%.
Q3 of 5Ratio & Proportion
A recipe requires 3 cups of flour for every 2 cups of sugar. A baker decides to use 7.5 cups of flour.
How many cups of sugar are needed to maintain the same ratio?
Step-by-Step Process
1
Set up the proportion with matching units. Flour over sugar on both sides: $$\frac{\text{flour}}{\text{sugar}} = \frac{3}{2} = \frac{7.5}{x}$$
2
Cross-multiply: $3x = 2 \times 7.5 = 15$.
3
Solve for x: $x = 15 \div 3 = 5$ ✓
4
Verify: $\dfrac{3}{2} = 1.5$ and $\dfrac{7.5}{5} = 1.5$ ✓ — same ratio on both sides confirmed.
Write "flour / sugar = flour / sugar" before touching numbers. Cross-multiply only after confirming units align on each side. Then verify: plug your answer back in and check both sides simplify to the same value.
Q4 of 5Weighted Average
Group
Number of Students
Average Score
Class A
20
82
Class B
30
74
Combined
50
?
What is the combined mean score for all 50 students?
Step-by-Step Process
1
Never average the averages directly. Class B has 30 students vs. Class A's 20. The groups are unequal, so $(82 + 74)/2 = 78$ is wrong. The true mean must be pulled toward Class B's average (74) since that group is larger.
2
Recover each group's total score. $$\text{Class A total} = 20 \times 82 = 1{,}640$$ $$\text{Class B total} = 30 \times 74 = 2{,}220$$
Confirm direction makes sense. 77.2 lies between 74 and 82, closer to 74 — logical because Class B is larger. Choice C (78.0) is the unweighted-average trap ✗.
Weighted mean = Σ(group size × group mean) / total count. When groups differ in size, the larger group pulls the combined average toward its own mean. If your answer matches a simple average of the group means, you forgot to weight — multiply first, then divide.
Q5 of 5Conditional Probability
A survey of 100 students recorded whether each student studied and whether they passed the exam.
Passed
Did Not Pass
Total
Studied
48
12
60
Did Not Study
12
28
40
Total
60
40
100
A student is selected at random from those who passed. What is the probability that the selected student studied?
Step-by-Step Process
1
Identify the condition — this shrinks the sample space. "Selected from those who passed" → restrict to the Passed column: 60 students total. The denominator is 60, not 100.
2
Find the favorable outcomes within that group. Of the 60 who passed, how many studied? The Studied row, Passed column = 48.
Eliminate traps: 12/25 = 48/100 — used the total population as denominator (ignored the condition) ✗. 3/5 = 60/100 — P(passed overall), not conditional on studying ✗. 2/5 = 40/100 — P(did not study), wrong event entirely ✗.
Conditional probability habit: mentally cross out all rows/columns outside the given condition. You now have a smaller table. The denominator is the total of that restricted group (60 who passed), not the full population (100). This single habit eliminates the most common conditional probability trap on the SAT.