ACCUPLACER study guide

A topic names what a passage discusses. A main idea says what the passage establishes about that topic. Choose a statement that accounts for the whole passage, including any qualification or change in direction.

1. State the topic in a few words, then add what the writer wants you to understand about it.
2. Separate the central claim from examples, background and objections. Ask why each detail is present.
3. Check both ends of the passage. A conclusion may refine an opening claim rather than simply repeat it.
4. Test every important word in an answer against the passage. Reject a true detail if it is too narrow to summarize the whole text.

Common mistake: Selecting “Attendance doubled” because it is accurate. It is evidence within the explanation, not the passage’s complete point.

An inference connects clues that the writer has supplied. It can go beyond a directly stated sentence, but it cannot depend on an invented motive, an outside fact or an assumption the passage leaves open.

1. Find the specific action, contrast, description or result relevant to the question.
2. Put the clues together using the smallest extra assumption possible.
3. Distinguish a supported conclusion from something merely possible. Several explanations can fit one event, so use the surrounding details.
4. Keep the answer’s certainty and scope within the evidence. “May” and “some” do not justify “must” and “all.”

Common mistake: Concluding that Mei cannot speak any English or that her health has improved. Neither claim follows from the change in preparation.

Questions about rhetoric ask about a writer’s choices: why an example appears, how ideas are arranged, what an analogy clarifies, how a phrase establishes tone or whether evidence supports a claim. Consider what the material does and how well it supports the writer’s purpose, not just its subject.

1. Identify the claim or situation immediately before the detail in question.
2. Read the next sentence too. A detail can introduce an objection that the writer then limits or answers.
3. Name its job: illustrate, contrast, qualify, explain a cause, concede a point, introduce a problem or connect ideas.
4. Check whether the answer describes the writer’s actual purpose and tone. Mentioning a limitation does not necessarily reject the entire proposal.
5. When evaluating an argument, identify the exact claim and the evidence offered for it. Check whether that evidence supports the same scope, group and cause, or whether the reasoning needs an unstated assumption.

Common mistake: Confusing the content of a sentence with its role. “To describe young trees” ignores how the description supports the surrounding argument.

Two writers may discuss the same issue while differing in scope, priorities or proposed action. Summarize each position separately before looking for agreement or disagreement. A shared topic is not necessarily a shared conclusion.

1. Give each text a one-sentence summary that includes its main qualification.
2. Identify what each writer evaluates: a goal, a method, a cost, a risk or a specific setting.
3. For agreement questions, find evidence in both texts. A statement supported by only one text is insufficient.
4. For response questions, apply the second writer’s stated reasoning to the first writer’s claim; do not invent a personal attitude or an unrelated objection.

Common mistake: Treating Text 2’s qualification as complete opposition to Text 1. Both writers support digital access; they emphasize different limits and purposes.

A familiar word may have several meanings. A vocabulary-in-context answer must preserve the sentence’s meaning and tone here, even if another option is a correct dictionary definition elsewhere.

1. Read the sentence and the nearby explanation or contrast before considering the options.
2. Replace the word with your own plain-language meaning.
3. Substitute each option into the sentence. Check grammar, meaning and tone rather than matching one associated word.
4. Use the surrounding idea to distinguish close options. Inferences about the entire subject are less useful than the particular relationship in the sentence.

Common mistake: Choosing “calculated in numerical units” solely because “measured” often refers to quantities. The context describes the committee’s manner of responding.

A detail can be true and relevant to the general subject but still fail to support the writer’s specific point. Match the option to the purpose stated in the question.

1. Read the whole draft to identify its main point, audience, and level of certainty.
2. Restate the requested job: illustrate a claim, supply evidence, explain a cause, acknowledge a limit, or remove a distraction.
3. Check which option directly performs that job. Prefer specific evidence over a related fact that requires an unsupported assumption.
4. Reject options that repeat existing information, change the claim, or claim more than the evidence can show.
5. Read the chosen detail with the surrounding sentences to verify that references and tone still fit.

Common mistake: Choosing a fact merely because it mentions the same topic. Ask what the detail proves or illustrates, not just what it is about.

Sentence placement and transitions should reveal a logical relationship. Follow ideas and references, not repeated keywords alone.

1. Give each paragraph a brief role, such as problem, example, procedure, result, or limitation.
2. For an insertion, identify words that depend on earlier information: “this change,” “those records,” or “another explanation.”
3. Find a position where the needed information has just been established and the next sentence follows naturally.
4. For a transition, decide whether the relationship is cause, contrast, example, addition, or sequence before examining the options.
5. Read across both boundaries of an inserted sentence; a good link to the previous sentence must not disrupt the following one.

Common mistake: Choosing “however” merely because it sounds formal. Use it only when the next idea contrasts with the preceding one; use a cause or sequence transition when that is the actual relationship.

Choose wording that says exactly what the passage means, fits its tone, and avoids needless repetition. The shortest choice is useful only if it preserves the intended information.

1. Identify the sentence’s main claim and the facts that a revision must retain.
2. Read the neighboring sentences to check the intended relationship, tone, and reference of pronouns.
3. Remove repeated ideas and replace indirect wording with a precise word or phrase.
4. Reject choices that sound polished but change the claim, introduce unsupported details, or remove an essential qualification.

Common mistake: Choosing a shorter option that changes meaning—for example, replacing “may reduce delays” with “prevents delays.” Concision must not erase uncertainty or exaggerate a claim.

A complete sentence needs an independent clause. Connect complete clauses correctly, place modifiers beside what they describe, and keep related items in the same grammatical form.

1. Find the subject and finite verb in each clause. A dependent clause beginning with “because” or “although” needs an independent clause.
2. If two independent clauses meet, use a period, a semicolon, or a comma plus a coordinating conjunction such as “and” or “but.”
3. For an introductory modifier, identify who performs the action; that person or thing should be the subject immediately following it.
4. Check lists for parallel forms, such as “inspect, record, and report,” rather than “inspect, record, and reporting.”

Common mistake: Inserting a comma wherever the sentence sounds like it needs a pause. A comma alone cannot join two independent clauses, and a pause does not repair a missing main clause.

Find the grammatical job of the disputed word. Nearby nouns can distract you from the subject, and a pronoun’s form depends on how it functions in its clause.

1. For subject–verb agreement, identify the subject before choosing the verb. Temporarily set aside phrases beginning with “of,” “between,” or “from.”
2. Match a singular subject with a singular verb and a plural subject with a plural verb. “Each” is singular even in “each of the volunteers.”
3. With “either … or” or “neither … nor,” a mixed singular-and-plural subject normally takes a verb agreeing with the nearer subject.
4. For pronouns, identify the role: “they” performs an action, “them” receives an action or follows a preposition, and “their” introduces something possessed.
5. Check the surrounding time frame before changing verb tense. A grammatical form must also preserve the passage’s intended sequence and meaning.

Common mistake: Agreeing the verb with the closest noun instead of the subject: plural “neighborhoods” does not control the verb in the example.

Decide where complete clauses begin and end before choosing punctuation. Commas, semicolons, and apostrophes have different jobs; they are not interchangeable pause marks.

1. Check whether each side of a proposed boundary can stand as a complete sentence with a subject and a finite verb.
2. Join two independent clauses with a period, a semicolon, or a comma plus a coordinating conjunction such as “and” or “but.” A comma alone is a comma splice.
3. A transition such as “however” does not join independent clauses by itself: “The room was small; however, it met our needs” is correctly punctuated.
4. Keep a subject connected to its verb. Do not insert a lone comma between them merely because the subject contains a long descriptive phrase.
5. Use apostrophes for possession, not ordinary plurals. One baker’s address belongs to one baker; the bakers’ addresses belong to multiple bakers.
6. Read the entire proposed revision. A punctuation mark that fixes one boundary can still create an error elsewhere or change the meaning.

Common mistake: Choosing punctuation by how long you pause when speaking. Identify the clauses and the intended relationship instead.

Translate a situation into operations before calculating. Multiplication counts equal groups; division can give an exact share, a number of complete groups, or the number of containers needed. Those interpretations handle remainders differently.

1. Identify the total, the size of each group, and what the question asks you to find.
2. Use parentheses for a grouped calculation. Evaluate parentheses and powers, then multiplication and division from left to right, then addition and subtraction from left to right.
3. For complete groups, keep the whole-number quotient. For enough containers to hold everything, include one extra container whenever a remainder remains.
4. Estimate first and check the final quantity against the situation. A number of passengers cannot be a fraction of a person.

Common mistake: Rounding every division result down. Nine full bags are not enough if the instruction is to hold all 115 packs.

A denominator names the size of equal parts; the numerator counts them. Fractions can be added or subtracted only after the parts have the same size. Multiplication and division follow different rules.

1. Convert a mixed number to an improper fraction by multiplying the whole number by the denominator and adding the numerator.
2. For addition or subtraction, use a common denominator and change each numerator by the same factor used to change its denominator.
3. For multiplication, multiply numerators and denominators. Cancel common factors to make the arithmetic easier.
4. To divide by a nonzero fraction, multiply by its reciprocal. Reduce the answer and interpret any remainder or mixed number in context.

Common mistake: Subtracting denominators or applying a fraction to the wrong whole. One quarter of the original amount is different from one quarter of what remains.

Decimal digits represent tenths, hundredths, thousandths and smaller parts. Zeros can hold a place without changing a value: 4.7 and 4.70 are equal. Use the operation to decide how to align or scale the numbers.

1. For addition and subtraction, line up decimal points and fill empty places with zeros.
2. For multiplication, multiply as whole numbers, then restore the total number of decimal places from the factors.
3. For division, multiplying both dividend and divisor by the same power of ten makes the divisor a whole number without changing the quotient.
4. Round only when requested. Check the first digit after the required place; 5 or more rounds that place up for the positive quantities used here.

Common mistake: Aligning the final digits instead of decimal points when adding, or assuming that multiplying two numbers always makes the result larger.

Percent means per hundred. Every percent calculation has a base: the quantity representing 100%. Mark that base explicitly, especially when reversing a discount or applying a second change.

1. To find a part, multiply the whole by the percent written as a decimal.
2. To find a percent, divide the part by the whole and multiply by 100.
3. To recover a whole, divide the known part by the decimal proportion it represents.
4. For a percent increase or decrease, divide the change by the original value. For successive changes, multiply the successive remaining or increasing factors.

Common mistake: Adding 20% to the reduced price to undo a 20% discount. The original discount and that attempted increase use different bases.

Quantities written in different forms can represent the same value. Compare them by converting to a common form or by using exact fraction arithmetic. Keep the units and the whole being compared consistent.

1. Convert a percent to a decimal by dividing by 100. Convert a fraction by dividing its numerator by its denominator.
2. Compare decimals from left to right after the decimal points are aligned; add trailing zeros if helpful.
3. For two positive fractions, cross-products give an exact comparison: a/b is less than c/d when a × d is less than c × b.
4. For a best-value question, compare equal quantities or unit costs. A lower package price does not necessarily mean a lower price per unit.

Common mistake: Comparing only the numerators of fractions or assuming a decimal with more digits is larger. For example, 0.6875 is less than 0.69.

A rational number can be written as a fraction of integers with a nonzero denominator. Signed fractions and decimals follow the same number-line order and operation rules.

1. For comparison, rewrite values with a common denominator or as decimals. A negative number farther left is smaller, even if its absolute value is larger.
2. Add or subtract fractions only after making their denominators equal. Multiply numerators and denominators for products; multiply by the divisor’s reciprocal for division.
3. Use parentheses, exponents, multiplication and division, then addition and subtraction. The expressions −4² and (−4)² have different values.
4. A distance is an absolute difference; a midpoint is the average of the two coordinates. Check which the question requests.

Common mistake: Adding denominators or forgetting that dividing by a fraction can increase the magnitude of a number.

Ratios compare quantities; rates attach units to that comparison. Percent changes multiply a starting amount, and successive changes use the updated amount.

1. For a part-to-part ratio a:b, the first part is a/(a + b) of the total, not a/b of the total.
2. Write a unit rate with its units. Distance divided by time is speed; total cost divided by quantity is price per unit.
3. Convert an increase of p% to multiplication by 1 + p/100, or a decrease to multiplication by 1 − p/100. To recover the original amount, divide by the multiplier.
4. For inverse relationships such as worker count and completion time, keep total work constant. For mixtures, conserve the amount of the ingredient before and after mixing.
5. For overall averages of rates, use the appropriate totals: total distance/total fuel, for example, rather than averaging two miles-per-gallon values.

Common mistake: Subtracting the two percentages as though both were applied to the original amount.

Exponent rules describe repeated multiplication. Roots undo powers, and negative exponents represent reciprocals rather than negative values.

1. For the same nonzero base, multiply powers by adding exponents and divide powers by subtracting exponents.
2. For a power raised to another power, multiply the exponents. Apply an outside power to every factor inside a product.
3. Rewrite a negative exponent with a reciprocal: a⁻³ = 1/a³. A nonzero base to the zero power equals 1.
4. For a positive base a, a^(m/n) means the nth root of a raised to the power m. For example, 27^(2/3) = (∛27)² = 9.
5. Simplify square roots by taking out perfect-square factors. Combine radical terms only when their simplified radicals match.
6. In scientific notation, multiplication multiplies coefficients and adds exponents. Addition requires matching the powers of 10 first.

Common mistake: Multiplying exponents for x² × x⁵, or adding them for (x²)⁵. These are different operations.

An expression represents a quantity; it does not ask you to solve for an unknown unless an equation is also given. Terms can be combined only when their variable parts match.

1. Define what each variable measures before translating words into operations. Repeated per-item costs usually require multiplication.
2. Distribute a factor to every term inside parentheses, including a negative sign before parentheses.
3. Combine like terms, keeping variable terms separate from constants. A coefficient multiplies its variable; it is not the constant term.
4. To factor, identify the greatest common numerical factor and the lowest powers present in every term.
5. Substitute negative values using parentheses. Evaluate powers before multiplying and combining terms.

Common mistake: Applying a one-time fee or discount once per hour, or combining unlike terms such as 6h and 9 into 15h.

A linear equation balances two quantities. Apply the same operation to both sides; inequalities follow the same principle except that multiplying or dividing by a negative reverses their direction.

1. Distribute and combine like terms before isolating a variable. Clear fractions by multiplying every term by a common denominator.
2. Move variable terms to one side and constants to the other. If the variables cancel, decide whether the remaining statement is always true or impossible.
3. For a system, substitute one equation into the other or add multiples of the equations to eliminate a variable.
4. Reverse an inequality sign only when multiplying or dividing by a negative. For compound inequalities, apply the operation to all three parts.
5. Interpret the solution in context: whole-number purchases, included endpoints and strict budget limits may matter.

Common mistake: Rounding a maximum affordable whole-number quantity upward, or reversing an inequality while merely adding or subtracting.

A linear graph connects a constant rate of change with a starting value. Its axes and units tell you what slope and intercept mean in context.

1. Read each axis label and scale before using a plotted point. The horizontal coordinate comes first in (x, y).
2. Calculate slope as change in y divided by change in x, using the same point order in both differences.
3. In y = mx + b, m is the rate of change and b is the value when x = 0. A horizontal line has zero slope; a vertical line has undefined slope.
4. Parallel nonvertical lines have equal slopes. Perpendicular nonvertical, nonhorizontal lines have slopes whose product is −1.
5. An intersection satisfies both models. A scatterplot suggests a trend, but association by itself does not show causation.

Common mistake: Using y/x for slope when the line does not pass through the origin, or reversing only one of the coordinate differences.

A probability compares favorable outcomes with a clearly defined set of possible outcomes. Conditions and whether objects are replaced can change that set.

1. For equally likely outcomes, probability = favorable count/possible count. Use a table, list or tree to avoid missing outcomes.
2. For a complement, subtract from 1. For either of two events, add their probabilities and subtract the overlap counted twice.
3. A conditional probability restricts the denominator to the group named after “given that.” Count favorable outcomes inside that group only.
4. For independent events, multiply probabilities. Without replacement, update both the numerator and denominator for the next draw.
5. In set notation, an intersection means both conditions; a union means at least one; a complement means outside the set.

Common mistake: Using the entire population as the denominator after a condition has restricted selection to a smaller group.

A useful summary describes both the center and spread of data. Tables and statistical displays organize different information, so first identify what each mark or bar represents.

1. Find a mean by dividing the sum by the count. For a frequency table, multiply each value by its frequency before adding.
2. Order the data before finding a median. With an even count, average the two middle values.
3. Use range = maximum − minimum and interquartile range = third quartile − first quartile. A boxplot displays the quartiles and median, not each individual value.
4. Histogram bars count observations in intervals. The total frequency is the sum of the bar heights; exact values inside a bin are generally unknown.
5. An outlier can strongly change the mean without changing the median much. Scatterplots describe paired data and show direction, form and possible unusual points.

Common mistake: Averaging category values without accounting for different frequencies, or treating a boxplot’s box length as a count of observations.

Choose a geometry formula from the quantity requested: perimeter is length, area covers a surface, and volume fills space. Coordinate differences connect geometric figures to algebra.

1. Mark the given lengths and distinguish radius from diameter and perpendicular height from a slanted edge. Keep units consistent.
2. For composite areas, add nonoverlapping regions or subtract a missing region. A uniform border outside a rectangle increases each full dimension by twice the border width.
3. Use a² + b² = c² only for a right triangle, with c opposite the right angle. Coordinate distance uses horizontal and vertical differences as the legs.
4. Similar figures scale corresponding lengths by k and areas by k². Match corresponding sides before making a proportion.
5. A reflection across the y-axis changes (x, y) to (−x, y); across the x-axis it becomes (x, −y). A translation adds the movement to each coordinate.

Common mistake: Adding the two coordinate differences instead of finding the straight-line distance, or applying a length scale factor directly to area.

A linear model changes by a constant amount per unit. Its slope describes that rate; its intercept describes the value when the input is zero. A system asks where two conditions hold together.

1. Name the input and output, including their units. From two points, calculate slope as change in output divided by change in input.
2. Substitute one known point into y = mx + b to find the intercept. Check the other point.
3. For two competing models, set their outputs equal. For two independent conditions, use substitution or elimination.
4. Check whether the solution belongs to the context: a negative time or fractional number of whole objects may not be possible.

Common mistake: Using output divided by input as the slope ignores a possible fixed charge. Use the differences between two points instead.

Factoring rewrites a sum as a product. It helps simplify rational expressions, identify zeros, and solve polynomial equations without losing restrictions.

1. Remove the greatest common factor first, including a negative factor when it makes the remaining expression easier to read.
2. Look for difference of squares or a perfect-square trinomial. Otherwise, for ax² + bx + c, seek two terms whose coefficients multiply to ac and add to b.
3. Split the middle term and factor by grouping. Multiply your factors back together to check them.
4. Only use the zero-product rule when the equation has been written as a product equal to zero.

Common mistake: Canceling individual terms across addition is not factoring. For example, x cannot be canceled out of (x + 2)/(x + 3).

The same quadratic can reveal different features in different forms: ax² + bx + c shows the intercept, a(x − r)(x − s) shows the roots, and a(x − h)² + k shows the vertex.

1. Choose the feature the question asks for before doing algebra: zeros, maximum or minimum, intercept, or number of real roots.
2. Use factoring when factors are accessible. Otherwise use the quadratic formula, with discriminant b² − 4ac.
3. Find the vertex input with −b/(2a), or complete the square to obtain vertex form.
4. In a model, apply the allowed input interval. A mathematical root can be outside the physical situation.

Common mistake: The vertex input and the maximum output are different numbers. A question asking for the maximum value wants the y-value.

A function assigns one output to each allowed input. Composition feeds one function’s output into another, so both rules and their domain restrictions matter.

1. For an evaluation, replace every occurrence of the input variable, keeping negative or compound inputs in parentheses.
2. For f(g(x)), apply g first, then substitute the entire expression into f.
3. Exclude inputs that make denominators zero. For real square roots, require the radicand to be nonnegative; for logarithms, require a positive argument.
4. To find an inverse, exchange the roles of input and output and solve. Check that the original function is one-to-one on the stated domain.

Common mistake: f(g(x)) is not f(x)g(x), and the two possible composition orders usually give different functions.

Multiplying by a variable expression or squaring an equation can introduce invalid candidates. Recording restrictions and checking the original equation are part of solving it.

1. Before manipulating a rational equation, list every input excluded by its original denominators.
2. Clear denominators by multiplying every term by the least common denominator; do not drop terms.
3. For a radical equation, isolate the radical before squaring. Record any sign requirement imposed by the other side.
4. Substitute every candidate into the original equation and reject any that fail or make an expression undefined.

Common mistake: Squaring both sides preserves true solutions but can create extra ones. Solving the squared equation alone is not enough.

Degree and leading coefficient describe a polynomial’s end behavior. Factors identify zeros, while division and the remainder theorem connect a polynomial’s algebra to particular input values.

1. Combine like terms and put powers in descending order. Include a zero coefficient for a missing power during division.
2. The remainder when dividing P(x) by x − a is P(a). A zero remainder means x − a is a factor.
3. After removing a known factor, factor the quotient to locate additional zeros.
4. Use degree, leading sign, and each root’s multiplicity to check whether the graph’s broad behavior is plausible.

Common mistake: A factor x + 2 gives the zero −2, not 2. Set the entire factor equal to zero.

Exponential models multiply by a constant factor over equal intervals. A logarithm answers the inverse question: which exponent gives a particular positive value?

1. Distinguish an additive change from a multiplicative factor. A 6% decrease gives a factor of 0.94, not −0.06.
2. When possible, rewrite both sides of an exponential equation with the same positive base other than 1.
3. Use log_b(u) + log_b(v) = log_b(uv), but only when both u and v are positive. A logarithm of a sum cannot be split this way.
4. Solve the resulting equation, then check all original logarithm arguments and any context restrictions.

Common mistake: log(u + v) is not log(u) + log(v). Logarithm rules turn multiplication into addition, not addition into addition.

Advanced geometry links algebra to distance, circles, similar figures, and volume. Keep length, area, and volume units separate, and identify whether a scale factor must be squared or cubed.

1. Draw or label the figure, noting whether a stated measurement is a radius, diameter, vertical height, or slant height.
2. For similar figures with length scale factor k, use k² for areas and k³ for volumes.
3. Use the distance formula or Pythagorean theorem to connect horizontal and vertical changes.
4. Recognize a circle’s standard form (x − h)² + (y − k)² = r². Complete the square when necessary.

Common mistake: The right side of a circle equation is the radius squared. Similarly, a doubled length does not merely double area or volume.

Trigonometry connects angles to side ratios and periodic graphs. Right triangles give the basic ratios; the unit circle extends them to other angles, including negative values.

1. For a right triangle, identify opposite, adjacent, and hypotenuse relative to the named angle, then choose sine, cosine, or tangent.
2. Convert degrees and radians using 180° = π radians. Determine signs from the angle’s quadrant.
3. For A sin(Bx) + D or A cos(Bx) + D, amplitude is |A|, period is 2π/|B| for B ≠ 0, and midline is y = D.
4. For a non-right triangle, select the law of sines or cosines from the given sides and angles. Check that angles sum to 180°.

Common mistake: The number multiplying x changes the period inversely. It does not change the amplitude.