GRE General Test

1. Text Completion follows the logic

Learning goal: Infer the missing meaning before choosing a word.

Practice focus: editorial, not a measured error-frequency claim. This lesson uses original examples.

Verbal content includes Text Completion, Sentence Equivalence and passage reading [R4]. For completion, first identify the logical direction: contrast, support or cause. Predict a plain-language meaning before considering difficult vocabulary. A word you recognize is not correct unless it makes the entire sentence coherent.

In a multi-blank question, choices work together. The official scoring requires all blanks correct, with no partial credit [R4]. Test the completed sentence as a whole rather than treating each blank as an isolated definition quiz. These original single-blank exercises build the logic habit without pretending to reproduce the official interface.

Worked example

"Although the proposal looked complicated, its implementation was remarkably ___." Predict the blank.

1. Although sets up contrast with complicated.
2. Implementation should be relatively easy or simple.
3. "Straightforward" fits that prediction.
4. "Intricate" repeats the complexity instead of completing the intended contrast.

Mental model: Connectors constrain the missing meaning.

Common trap: Recognizing vocabulary is not the same as satisfying the sentence.

2. Sentence Equivalence needs two full fits

Learning goal: Find meaning-equivalent completions in context.

Practice focus: editorial, not a measured error-frequency claim. This lesson uses original examples.

Sentence Equivalence requires two choices that both fit the sentence and yield equivalent meanings; both must be correct, without partial credit [R4]. A synonym pair can still fail if neither word suits the context. Conversely, dictionary similarity is less important than the meaning of the completed sentence.

For our toy sentence "The instructions were brief and ____, leaving no ambiguity," concise and short repeat length but do not necessarily indicate clarity. Clear and unambiguous fit the stated result. Read the completed sentences aloud mentally and compare their implications rather than hunting a pair in isolation.

Worked example

Toy choices: clear, obscure, unambiguous, lengthy. "The guide was ____, so readers understood each step." Select the pair.

1. The result requires easy interpretation.
2. Clear fits; obscure contradicts understanding.
3. Unambiguous also fits and conveys the same clarity.
4. Lengthy alone says nothing about clarity. Choose clear and unambiguous.

Mental model: Two contextual fits must produce equivalent meanings.

Common trap: Do not select a synonym pair before reading the sentence.

3. Reading: distinguish support from inference

Learning goal: Map who claims what and how strongly.

Practice focus: editorial, not a measured error-frequency claim. This lesson uses original examples.

GRE Verbal includes Reading Comprehension and reasoning about text [R4]. Identify the author's claim, supporting observations and any qualification. A reported viewpoint may belong to another writer. An inference needs a route from the passage, while a detail question may need a precise restatement rather than a new conclusion.

When a passage says a pilot method helped some schools but the sample was small, retain both benefit and limit. The observation does not establish that all schools will benefit. These exercises are original mini-passages to isolate scope and attribution, not a replacement for longer reading practice.

Worked example

"Researchers observed improvement in three pilot schools. The author warns that larger studies are needed." What does the author support?

1. Three pilot schools improved according to the report.
2. The author asks for broader evidence.
3. The text does not guarantee the effect for all schools.
4. It supports promising pilot evidence with limited generalization.

Mental model: A good inference has a traceable support path.

Common trap: Outside familiarity is not passage evidence.

4. Quantitative Comparison needs boundary cases

Learning goal: Test all permitted kinds of numbers, not one favourite integer.

Practice focus: editorial, not a measured error-frequency claim. This lesson uses original examples.

Quantitative Comparison asks whether one quantity is always greater, the quantities are equal, or the relationship cannot be determined from the information. ETS advises trying varied values [R3]. With real x, compare x² and x: a fraction between zero and one behaves differently from a negative number or an integer above one.

A single test can refute an always claim, but cannot prove it unless justified for the whole domain. A domain restriction may make an algebraic argument possible. Keep the exact restrictions visible: positive still permits fractions, whereas positive integer does not.

Worked example

x is real. Compare A=x² and B=x.

1. x=1/4 gives A=1/16 < B=1/4.
2. x=-1 gives A=1 > B=-1.
3. x=1 gives equality.
4. Since permitted cases give different relations, the relationship cannot be determined.

Mental model: Use boundary and sign cases before choosing a fixed relation.

Common trap: Positive does not mean positive integer.

5. Rates add; times do not

Learning goal: Model simultaneous work in units per time.

Practice focus: editorial, not a measured error-frequency claim. This lesson uses original examples.

GRE Quantitative includes arithmetic, ratios and algebra [R3]. A worker completing a job in t hours has rate 1/t jobs per hour under a constant-rate assumption. Simultaneous rates add if the workers contribute independently and there is no leak or interference. The completion times themselves do not add or average.

The combined time should be shorter than either individual time for two positive filling rates. If a leak is supplied, subtract its rate rather than its time. These are explicitly simplified models, not descriptions of real pumps changing rate with pressure.

Worked example

A fills in six hours, B in three. Together, constant rates and no leaks. Time?

1. A rate = 1/6 tank/hour.
2. B rate = 1/3 tank/hour.
3. Combined rate = 1/2 tank/hour.
4. One tank / (1/2 tank/hour) = 2 hours.

Mental model: Add compatible work rates.

Common trap: Never average completion times for simultaneous work.

6. Probability: count the event and the space

Learning goal: Specify equal likelihood and replacement assumptions.

Practice focus: editorial, not a measured error-frequency claim. This lesson uses original examples.

Probability and data analysis are GRE quantitative topics [R3]. Count favourable outcomes over equally likely possible outcomes, or use a stated probabilistic model. Without replacement changes the second draw's denominator and composition. With replacement restores the original contents if mixing makes the draws independent under the model.

"At least one" can be easier to compute as one minus none. For a tiny sample space, enumerate outcomes to check the formula. These urns are toy models, not real-world frequencies. Hypothesis testing is outside this module's scope; this lesson teaches only the stated elementary probability model.

Worked example

A bag has three red and two blue tokens. Draw two without replacement. Probability both red?

1. First red probability = 3/5.
2. Given first red, remaining red probability = 2/4.
3. Multiply conditional probabilities: (3/5)(2/4)=3/10.
4. With replacement it would be (3/5)², a different model.

Mental model: Specify dependence before multiplying probabilities.

Common trap: A with-replacement shortcut fails without replacement.

7. A drawing is not a measurement

Learning goal: Use stated geometry and test scale-independent relations.

Practice focus: editorial, not a measured error-frequency claim. This lesson uses original examples.

GRE Quantitative includes geometry but not trigonometry or calculus [R3]. A sketch can help organize relations, yet apparent size is not a measurement unless the problem specifies it. A rectangle's area uses both side lengths; perimeter uses their sum. Choose the requested quantity before inserting values.

Scaling all lengths by k scales area by k², not k. This follows from multiplying two length dimensions. A quick units check distinguishes square units from linear units. The exercises here provide exact side lengths so no conclusion depends on whether the pictured shape looks long or square.

Worked example

A rectangle is 3 units by 8 units. Double both dimensions. New area?

1. Original area = 3×8 = 24 square units.
2. New sides = 6 and 16.
3. New area = 6×16 = 96 square units.
4. The factor is 2²=4, confirming 24×4=96.

Mental model: Use stated relations and the requested dimensions.

Common trap: Do not infer a right angle or equal length from appearance alone.

8. Multi-select means complete selection

Learning goal: Check every qualifying value, then stop.

Practice focus: editorial, not a measured error-frequency claim. This lesson uses original examples.

GRE question types include select-one-or-more and numeric entry [R3]. A multi-select question is not finished when you find one correct option. Evaluate each candidate against the condition. For numeric entry, carry exact fractions until the requested rounding step and state the unit; a correct number in the wrong unit is a wrong response.

ETS explicitly advises checking units and using calculators strategically [R3]. Simple arithmetic can be quicker by hand, but calculation aids do not decide which expression to use. These toy candidate lists make completeness visible without importing an official scoring model beyond the cited question-type description.

Worked example

Select every integer x in {-2,-1,0,1,2,3} satisfying x²<4.

1. -2 gives 4, not less than 4.
2. -1, 0 and 1 give 1, 0 and 1: all qualify.
3. 2 gives 4 and 3 gives 9: neither qualifies.
4. The complete set is {-1,0,1}.

Mental model: Completeness and units are part of correctness.

Common trap: Strict and non-strict inequalities have different boundary answers.

9. Data summaries do not tell the same story

Learning goal: Distinguish mean, median and weighted counts.

Practice focus: editorial, not a measured error-frequency claim. This lesson uses original examples.

Data analysis is part of GRE Quantitative [R3]. The mean divides a total by its count; the median is the middle of a sorted list, or the average of the two middle values for an even count. A large extreme value can move the mean substantially while leaving the median unchanged. Neither summary is automatically the right one for every question.

All numbers in this activity are toy values. Describe exactly what a chosen summary captures rather than treating it as a complete distribution. A mean of four observations need not equal any observed value, and a median does not disclose the spread. Inferential conclusions about populations require more than these descriptive calculations and are outside this lesson.

Worked example

Toy values 2,3,3,12. Find mean and median.

1. Sum = 20, count = 4; mean = 5.
2. Sorted middle values are 3 and 3.
3. Median = (3+3)/2 = 3.
4. The high value raises the mean above the median; this is not a claim about a real sample.

Mental model: Choose the summary the question requests.

Common trap: A descriptive mean is not a population guarantee.

10. Issue writing and current-section review

Learning goal: Build a supported position while using current navigation correctly.

Practice focus: editorial, not a measured error-frequency claim. This lesson uses original examples.

The shorter GRE includes one 30-minute Analyze an Issue essay [R1, R5, R8]. ETS rubrics evaluate task response, reasoning, support, organization and language [R7]; these descriptors are not evidence of how often candidates make an error. Follow the particular task directions, state a position and connect each example to a reason rather than adding unsupported statistics.

In V/Q sections you can skip, mark and revisit within that section; section-level adaptation affects the next section [R5]. This differs from current GMAT limited end review. The shorter GRE has no scheduled break and no unscored section despite older footnotes on the structure page. The time budget below is editorial, not an ETS-prescribed essay schedule.

Worked example

Toy issue: should cities prioritize parks over road expansion? Outline a qualified position.

1. Position: prioritize parks when basic transport access is already adequate.
2. Reason: public space supplies a different benefit from added road capacity.
3. Example: a labeled hypothetical district with adequate roads but no shared outdoor space.
4. Qualification: where essential transport is missing, that need may take priority. Explain why the condition changes the judgment.

Mental model: Support the position, address its limits, review before the section boundary.

Common trap: Do not mix shorter GRE rules with legacy sections or breaks.

11. Percent changes use different bases

Learning goal: Translate sequential percent changes into multipliers before comparing endpoints.

Percent questions become simpler when every percentage is attached to a named base. A ten percent increase in a price means the new price is 1.10 times the old price. A ten percent decrease means the new price is 0.90 times the old price. The two changes do not cancel because the second percentage is calculated on a different amount. Avoid adding percentage changes unless the question explicitly defines them on a common base.

For a price that rises from 80 to 100, the increase is 20 divided by 80, or 25 percent. Reversing the journey gives a decrease of 20 divided by 100, or 20 percent. The dollar difference is the same; the percentage is not. Write the denominator before doing the arithmetic. This is useful when comparing discount claims, growth rates and changes in populations in original quantitative problems.

An index can represent a starting value without inventing a dollar amount. Set the initial value to 100, apply each multiplier and interpret the endpoint. If a value rises by 20 percent and then falls by 20 percent, the index becomes 100 times 1.20 times 0.80, or 96. The net decrease is four percent. An unknown original amount does not prevent a proportional answer.

Practice set: calculate the endpoint after a 50 percent increase and a one-third decrease; then after two successive ten percent increases. The first returns to the start, because 1.5 times two-thirds equals one. The second produces a 21 percent increase, not 20 percent. Before a timed set, record whether your mistake was a wrong base, a wrong multiplier or a rounding error. Keep exact fractions until the final requested precision.

Source alignment: [R3]. All numerical data and practice questions here are original teaching examples, not official exam questions or observed results.

Worked example

An original price of 200 increases 25%, then decreases 20%. Find the final price.

1. Use the current price as each percentage base.
2. 200 × 1.25 = 250.
3. 250 × 0.80 = 200.
4. The reciprocal multipliers cancel here.

Mental model: Translate sequential percent changes into multipliers before comparing endpoints.

Common trap: Add successive percentages without checking their bases.

12. Linear equations can describe a whole family

Learning goal: Distinguish a unique solution from inconsistent equations or repeated information.

An equation expresses a condition, not necessarily a complete determination of every variable. With two unknown quantities, one equation usually leaves multiple possibilities. Substituting convenient values can reveal that freedom, but a general expression shows it more clearly. If x plus y equals ten, write y equals ten minus x. You can then ask whether the requested quantity stays fixed across the allowed values.

Two equations determine a unique solution when they provide independent information. If x plus y equals ten and x minus y equals two, adding them gives two x equals twelve. Thus x equals six and y equals four. Check both original equations, not only the simplified one. A sign error may satisfy the derived equation while failing the original pair.

If the second equation is twice the first, it gives no new information. x plus y equals ten and two x plus two y equals twenty describe the same line. By contrast, changing the second right-hand side to twenty-two creates a contradiction. The same left-hand side cannot equal twenty and twenty-two at once. Identify dependence before assuming every pair yields a unique intersection.

Practice set: compare three systems sharing x plus y equals eight. Pair it with x minus y equals two, with three x plus three y equals twenty-four, and with three x plus three y equals twenty-five. They have one solution, infinitely many solutions, and no solution respectively. State any domain restrictions: integer, positive or nonnegative constraints can reduce a family. On a quantitative comparison task, determine whether the requested expression is fixed even when the variables are not.

Source alignment: [R3]. All numerical data and practice questions here are original teaching examples, not official exam questions or observed results.

Worked example

x + y = 10 and 2x + 2y = 20. Is x uniquely determined?

1. Check whether the second equation adds information.
2. Divide the second equation by two.
3. Both equations become x + y = 10.
4. x is not uniquely determined without more restrictions.

Mental model: Distinguish a unique solution from inconsistent equations or repeated information.

Common trap: Count equations instead of checking independence.

13. Counting separates order from selection

Learning goal: Decide whether changing order creates a new outcome before counting.

Counting begins with a definition of an outcome. Selecting a president and a treasurer is different from selecting a two-person committee: roles make order matter in the first task. If four people are eligible for two distinct roles and no person can hold both, there are four choices for the first role and three for the second, giving twelve assignments. A committee counts each pair only once, giving six committees.

The multiplication principle applies when a complete outcome is built through successive choices. The number of available choices can change after each step. For a three-digit code made from five distinct symbols without repetition, the counts are five, four and three. If repetition is allowed, each position has five choices. These are different sample spaces, not alternative methods for the same task.

For an unordered selection of two different people, divide ordered assignments by two because each committee was counted in two orders. For three selected people, divide by six because the same triple has six orders. Explain the duplication instead of mechanically dividing by a memorized number. A restriction, such as two named people not serving together, is often easier to handle by subtracting prohibited outcomes from the unrestricted total.

Practice set: from five people, form a two-person committee. There are ten possible committees. If one particular pair is prohibited, nine remain. Now choose a president and treasurer from the same five people: twenty assignments are possible. If the named pair cannot serve together in either order, two assignments are prohibited and eighteen remain. The difference reveals why a committee formula cannot be used unchanged for distinct roles. Write the outcome description above your arithmetic.

Source alignment: [R3]. All numerical data and practice questions here are original teaching examples, not official exam questions or observed results.

Worked example

Five candidates, two distinct roles, no person holds both. Count assignments.

1. Name the roles so order matters.
2. Choose the first role in five ways.
3. Choose the second in four ways.
4. Multiply to get twenty assignments.

Mental model: Decide whether changing order creates a new outcome before counting.

Common trap: Use the ordered count for an unordered committee.

14. Weighted means need counts

Learning goal: Recover totals before combining averages from unequal-sized groups.

An average is a total divided by a count. When combining groups, the group averages alone are insufficient unless their sizes are known or equal. A group of two students averaging seventy and a group of eight averaging ninety do not combine to eighty. The larger group contributes more observations. Reconstruct each total, add the totals, and divide by the combined number of observations.

In this example the totals are two times seventy, or 140, and eight times ninety, or 720. Their sum is 860, spread across ten students, producing a mean of eighty-six. This weighted mean must lie between the two group means. That bound is a quick check, but it does not replace the calculation: many incorrect answers also lie between seventy and ninety.

Weighted means also appear in mixtures, portfolios and test-score problems. Define the weight in the units actually used: liters, dollars, students or observations. A statement that two solutions have different concentrations does not specify the resulting concentration without their quantities. If quantities are equal, the simple average works because the equal weighting is a fact, not a universal rule.

Practice set: combine three observations averaging ten with two observations averaging twenty-five. The total is thirty plus fifty, so the mean is sixteen. Then ask how large a second group averaging twenty must be to combine with four observations averaging ten into a mean of fifteen. Solve (40+20n)/(4+n)=15 to obtain n=4. A missing value may be solved algebraically rather than guessed. Record whether the requested result is a total, a mean or a missing count.

Source alignment: [R3]. All numerical data and practice questions here are original teaching examples, not official exam questions or observed results.

Worked example

Two observations average 70 and eight average 90. Find the combined mean.

1. Recover each group total.
2. 2×70=140 and 8×90=720.
3. Add totals to get 860 and counts to get 10.
4. 860÷10=86.

Mental model: Recover totals before combining averages from unequal-sized groups.

Common trap: Average averages without checking equal weights.

15. Inference stops where the passage stops

Learning goal: Keep a supported conclusion narrower than a tempting generalization.

Reading questions ask you to use the passage, not to supply missing research from your own experience. A supported inference may not repeat a sentence word for word, but it must follow from the information provided. Distinguish what the author reports, what another person believes, and what the author concludes. An opinion attributed to a critic is not automatically the position of the passage author.

Quantifiers and modal words often determine scope. Some is weaker than all; may is weaker than must; a pattern in one sample is weaker than a rule about every setting. When an answer strengthens the passage, identify the additional evidence that would be needed. If the passage says three tested materials resisted moisture, it does not establish that every material of that type will do so.

Consider an original passage: a pilot library extended evening hours at two branches. Attendance increased at one branch and stayed stable at the other. The report did not measure changes in nearby branches. The data support a statement about differing outcomes at the two tested branches. They do not show that the policy increased citywide attendance, nor that evening hours never affect attendance. Both extremes overreach.

Practice set: label four claims as supported or unsupported. Outcomes differed between the tested branches is supported. Every branch would gain visitors is unsupported. The stable branch proves the policy is useless is unsupported. Changes elsewhere were not measured is supported. For a timed set, paraphrase the author conclusion in one sentence and underline the evidence offered for it. When eliminating an option, explain the scope mismatch rather than merely calling it too extreme.

Source alignment: [R4]. All numerical data and practice questions here are original teaching examples, not official exam questions or observed results.

Worked example

A study of two branches reports attendance up at one and stable at the other. What follows?

1. Restrict the conclusion to the measured branches.
2. Different outcomes were observed.
3. No citywide total was measured.
4. A universal effect is not established.

Mental model: Keep a supported conclusion narrower than a tempting generalization.

Common trap: Replace limited evidence with a universal claim.

16. Mixed practice turns errors into a next step

Learning goal: Review a small mixed set by error type rather than by the answer key alone.

A practice set is useful only if its review changes the next attempt. Separate a content gap from a misread condition, a calculation error and an inefficient method. They need different responses. A content gap needs a short explanation and a fresh example. A misread condition needs a reading checkpoint. An arithmetic error needs an independent calculation. A correct but slow method needs a simpler representation.

Try this original three-part set before reading the solutions. First, a price of 120 increases by twenty-five percent and then decreases by twenty percent. Second, a group of two observations has mean six and another group of three has mean sixteen. Third, a statement says some reviewed reports contained errors. Does it establish that every reviewed report contained errors? Work independently and write a one-line explanation for each.

The first answer is 120 because 120 times 1.25 times 0.80 returns to the start. The second answer is twelve because the combined total is twelve plus forty-eight, divided by five. The third answer is no because some does not entail all. These questions mix arithmetic and verbal scope deliberately; the mixed set is an editorial exercise, not a miniature official test or a score predictor.

After checking, redo only the missed skill with changed values or wording. For percentages use an increase of ten percent followed by a ten percent decrease. For means use three observations averaging four and one averaging twelve. For scope change some to all and ask what follows about at least one report when the reviewed set is nonempty. The new answers are a one percent decrease, six, and yes respectively. Record your actual reasoning, not simply that you now remember the answer. Use the official ETS practice interface separately for timing and navigation.

Source alignment: [R3]. All numerical data and practice questions here are original teaching examples, not official exam questions or observed results.

Worked example

Combine two observations averaging 6 and three averaging 16.

1. Translate averages to totals.
2. 2×6=12 and 3×16=48.
3. Total 60 over five observations.
4. The mean is 12.

Mental model: Review a small mixed set by error type rather than by the answer key alone.

Common trap: Read the key without diagnosing the method.