GMAT

1. Build the equation before computing

Learning goal: Translate a word relationship into a solvable model.

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

Quantitative Reasoning uses arithmetic and algebra Problem Solving, without a calculator [G2]. Start with a named unknown and translate each relationship before doing arithmetic. A ratio relates quantities multiplicatively; a difference relates them additively. Writing x and x+12 for two prices preserves a fixed difference, whereas x and 12x would invent a ratio.

Check your solution against the original story, not just the final algebra line. Units and positivity can reject an algebraically valid but contextually impossible value. The activities are original skill drills; they are not a complete QR syllabus or evidence of a measured candidate error rate.

Worked example

Two items cost $84 total; the second costs $12 more than the first. Find the first price.

1. Let first price be x dollars; second is x+12.
2. x + (x+12) = 84.
3. 2x = 72, so x = 36.
4. Check: second is 48; 36+48=84 and 48-36=12.

Mental model: Translate first; calculate second; check the story.

Common trap: Do not replace a difference with a ratio.

2. Percent change has a base

Learning goal: Keep consecutive multipliers and percentage points distinct.

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

A percentage change is computed relative to a specified starting amount. A 20% increase followed by a 20% decrease is not zero net change: the decrease uses the larger new base. Use multipliers 1.20 and 0.80; their product is 0.96. Percentage points instead subtract two percentage values directly.

Data Insights advice tells candidates to check units and use only supplied data [G6]. In a toy table, a conversion rate moving from 20% to 30% has a ten-point increase and a 50% relative increase. Both are correct descriptions of different quantities; the question determines which you must report.

Worked example

A toy price of $100 rises 20% then falls 20%. Final price?

1. After increase: 100 × 1.20 = 120.
2. The decrease is 20% of 120, or 24.
3. 120 - 24 = 96.
4. Net change is (96-100)/100 = -4%, not zero.

Mental model: A percent must name its base.

Common trap: Equal percentage changes need not cancel.

3. Weight the rate by its denominator

Learning goal: Combine counts before calculating an overall average.

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

A rate is a numerator divided by a denominator. An overall purchase rate sums purchases and visits before dividing. An unweighted average of channel rates gives each channel equal influence even when one has twice the traffic. DI includes tables and graphics, so filters, populations and units must stay explicit [G2, G6].

The combined rate must lie between the group rates when the denominators are positive. It lies closer to the larger group's rate. This range check catches an impossible arithmetic output but does not substitute for calculating the correct weighted value. All channel numbers here are toy data, not real marketing performance.

Worked example

Toy channels: A has 200 visits and 60 purchases; B has 400 visits and 80 purchases. Overall conversion?

1. A rate = 60/200 = 30%; B = 80/400 = 20%.
2. Total purchases = 60+80 = 140.
3. Total visits = 200+400 = 600.
4. Overall = 140/600 = 7/30 = 23 1/3%; use 23.3% only if rounding is requested.

Mental model: Combine the denominator before combining rates.

Common trap: A filtered rate cannot use the unfiltered denominator.

4. Sufficiency means one answer

Learning goal: Test each statement alone before combining them.

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

Data Sufficiency asks whether the data determine the answer, not whether you can find one convenient solution. A statement is insufficient if two admissible cases give different answers. GMAC explicitly advises distinguishing a single value from a range and avoiding unsupported assumptions [G6]. Test each statement independently; do not carry statement 1 into statement 2.

For a value question, uniqueness is the target. For a yes/no question, a consistently yes or consistently no answer can be sufficient. A data set that gives both answers is insufficient. These activities use descriptive answer choices rather than claiming an exact official response-button order.

Worked example

x,y are positive reals. What is x? (1) x+y=10. (2) x/y=3/2.

1. (1) admits (x,y)=(6,4) and (5,5), so x is not fixed.
2. (2) admits (3,2) and (6,4), so scale is not fixed.
3. Together x=1.5y and 2.5y=10, so y=4.
4. x=6. Together sufficient; neither alone.

Mental model: Prove uniqueness or exhibit two different answers.

Common trap: Never combine statements before testing them alone.

5. Domains change sufficiency

Learning goal: Use the exact restrictions, especially for squares.

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

The same equation can be sufficient or insufficient depending on the stem's domain. If x is real, x²=9 allows -3 and 3; if x is positive, it permits only 3. You may use restrictions in the stem for both statements, but cannot add a positivity assumption simply because an example uses positive quantities.

A yes/no target can collapse several numerical possibilities into one answer. With x²=9, the question "Is x nonzero?" is determined even though "What is x?" is not over the reals. DI assesses sufficiency rather than mechanical equation solving [G2, G6]. Always rewrite the target before evaluating statements.

Worked example

x is real. Is x>0? (1) x²=9. (2) x<0. Evaluate each alone.

1. (1) allows x=3 (yes) and -3 (no), so insufficient.
2. (2) requires negative x, so the answer is always no.
3. A fixed no is a determined answer.
4. Statement 2 alone is sufficient; statement 1 alone is not.

Mental model: The target and domain decide sufficiency.

Common trap: Do not add a domain restriction the stem omits.

6. Filters change the table you answer from

Learning goal: Apply conditions before forming a numerator and denominator.

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

DI Table Analysis and Multi-Source Reasoning require integrating information without changing populations mid-calculation [G2]. If a question asks about mobile visitors in one month, remove desktop visitors and other months from both numerator and denominator. A plausible percentage computed on the wrong cohort is still wrong.

Write a denominator sentence before computing: "Among the 80 mobile visitors in May..." This makes a hidden mismatch visible. If one source reports purchases and another reports visits, confirm matching period and group definitions. These small tables are toy exercises; they do not reproduce the full official DI interface.

Worked example

Toy May table: mobile 80 visits/20 purchases; desktop 120/12. Find mobile conversion.

1. The requested population is mobile visitors in May.
2. Use 20 purchases and 80 visits.
3. 20/80 = 0.25 = 25%.
4. All-device rate would be 32/200 = 16%, a different question.

Mental model: A rate needs a matched cohort and period.

Common trap: Do not turn missing source data into zero.

7. Critical Reasoning: name the bridge

Learning goal: Identify the missing connection rather than repeat the evidence.

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

Verbal Reasoning contains Critical Reasoning and Reading Comprehension, not legacy Sentence Correction [G2]. A recommendation based on a benefit needs a connection between the proposed action and that benefit. Repeating the evidence can sound reassuring without repairing the gap. A weakening answer may instead show a cost, selection effect or ineffective route.

Consider replacing a billing system because it could reduce errors. The argument needs that the replacement actually addresses errors in this setting; it does not automatically need every employee to prefer the interface. Evaluate the option relative to the conclusion and the stem, not its general business appeal.

Worked example

"Use software X because it will reduce duplicate entries." Which assumption is needed?

1. The claim is adoption of X based on duplicate reduction.
2. If X cannot handle the entries used here, the promised benefit fails.
3. A required bridge is that X can address these duplicates in this workflow.
4. Universal employee enthusiasm is stronger than the reasoning requires.

Mental model: Find the bridge the conclusion needs.

Common trap: Topic relevance alone does not fix an argument.

8. Evaluate by asking a two-way question

Learning goal: Choose information whose answers change the argument's force.

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

An evaluation question is useful when one possible answer strengthens the reasoning and another weakens it. An interesting question about the topic is not enough. For a plan to increase profit through lower prices, ask whether increased volume offsets the lower contribution per unit. Either answer affects the claimed result.

This two-way test keeps you from selecting a question whose answers all lead to the same interpretation. It also prevents importing a desired answer. GMAT CR evaluates reasoning on supplied information [G2]; all business figures below are toy assumptions, not forecasts or financial advice.

Worked example

Toy plan: reduce price from $10 to $9; unit variable cost remains $6; expected volume rises from 100 to 120. Does contribution rise?

1. Old unit contribution = 10-6 = 4; total = 400.
2. New unit contribution = 9-6 = 3; total = 360.
3. Volume rises but contribution falls by 40.
4. Before endorsing profit growth, evaluate volume and costs; fixed costs are unchanged here.

Mental model: An evaluation question must discriminate between relevant outcomes.

Common trap: Revenue growth does not guarantee profit growth.

9. RC: inference with a scope ceiling

Learning goal: Keep the author's certainty and range intact.

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

GMAT RC tests passage understanding and supported inference [G2]. "A method can reduce delays in small offices" does not establish that it always works in every institution. Preserve qualifiers, sample scope and distinctions between the author's view and cited opposing views. An inference should be traceable to a sentence or combination of statements.

A familiar management principle can be true outside the passage and still be the wrong answer. Use a counterexample to test an overstrong inference: a large office in which the method fails can coexist with the given small-office claim. This module isolates inference, not complete official-length passages or adaptive item difficulty.

Worked example

Mini-passage: "Pilot offices cut delays after adding a queue tool. The author cautions that complex offices may require different designs." What is supported?

1. The pilot gives evidence of benefit in those offices.
2. The caution limits transfer to complex offices.
3. A universal solution claim conflicts with the qualification.
4. The supported reading is conditional benefit with a stated scope limit.

Mental model: Inference cannot be stronger than its support.

Common trap: Do not replace passage evidence with general business knowledge.

10. Finish the section before planning edits

Learning goal: Respect current navigation and time constraints.

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

The current GMAT has three 45-minute sections. Quantitative Reasoning has 21 questions, Verbal 23 and Data Insights 20 [G1]. Section order is selectable, and one optional ten-minute break occurs after section one or two. This is a current-format lesson, not a legacy GMAT guide.

Bookmark as you work. Review opens only after all section questions are answered, with time remaining; you may review as many as time permits but edit up to three answers per section [G1]. Unanswered questions carry a penalty [G4]. A pacing average is an editorial planning aid, not a mandate to give every item the same time or a simulation of the scoring algorithm.

Worked example

If you reserve three minutes of a 45-minute QR section for end review, how much time remains for 21 initial answers?

1. Initial-answer budget = 45-3 = 42 minutes.
2. Average planning time = 42/21 = 2 minutes per item.
3. This is a budget, not equal-time instruction.
4. Review still requires finishing all answers; no remaining time means no review opportunity.

Mental model: Complete first; review with remaining time; edit within the limit.

Common trap: A pace average is not an adaptive scoring rule.

11. Overlapping sets need an intersection

Learning goal: Avoid counting members of two sets twice.

An overlapping-set question describes membership, not necessarily a sequence of choices. Draw two circles or make a four-cell table: A only, B only, both, and neither. Everyone belongs in exactly one cell of that partition. Adding the total in A to the total in B counts people in the intersection twice. Subtract the intersection once to recover the union.

In an original group of sixty people, thirty study French, twenty-five study Spanish, and ten study both. Forty-five study at least one because thirty plus twenty-five minus ten equals forty-five. Fifteen study neither. The French-only cell contains twenty; Spanish-only contains fifteen. Check that twenty plus fifteen plus ten plus fifteen returns sixty. This check catches a missing cell.

If the intersection is not supplied, the question may ask for a range or whether available statements determine it. The intersection cannot exceed the smaller set, and the union cannot exceed the total group. These bounds can constrain the result without fixing it. Treat a reported percentage as a count only after identifying the population to which it applies.

Practice set: forty people, twenty in A, eighteen in B, and five in both. Union thirty-three and neither seven. Now change both to eight: union thirty and neither ten. The same marginal totals permit different neither counts, so a question about neither requires overlap information. The practice model fits general quantitative and data-sufficiency reasoning, not a claimed official item. In review, label the error as duplication or a missing total rather than a vague counting mistake.

Source alignment: [G2]. All numerical data and practice questions here are original teaching examples, not official exam questions or observed results.

Worked example

60 people; 30 in French, 25 in Spanish, 10 in both. Find neither.

1. Add the two set totals.
2. Subtract the intersection once: 45 in the union.
3. Subtract the union from 60.
4. 15 study neither.

Mental model: Avoid counting members of two sets twice.

Common trap: Count overlapping members twice.

12. Remainders encode a family of integers

Learning goal: Translate a remainder into an equation with an integer quotient.

When a nonnegative integer n leaves remainder r on division by a positive integer d, write n equals d times q plus r, where q is an integer and zero is no greater than r but r is less than d. The remainder is a constraint, not the number itself. If n leaves remainder two on division by five, possible values include two, seven, twelve and seventeen.

A second restriction may select one value or leave several. If n is between ten and twenty, exclusive, and has remainder two when divided by five, n may be twelve or seventeen. Asking whether n is even now has two different outcomes. The range and remainder do not alone answer that yes/no question. Test representative values while preserving every condition.

Adding a second remainder can resolve the case. If n also leaves remainder two when divided by three, twelve fails because its remainder is zero; seventeen passes. You do not need a sophisticated modular theorem for this small range. List the candidates from the first restriction and filter them using the second. For a large range, use the same structure algebraically rather than attempting an endless list.

Practice set: n is greater than ten and less than thirty and leaves remainder one upon division by four. Candidates are thirteen, seventeen, twenty-one, twenty-five and twenty-nine. If n is a multiple of three, only twenty-one remains. Now remove the range: infinitely many values can satisfy both congruences. A unique result in a bounded example does not justify uniqueness when a condition is removed. In data sufficiency, the domain and boundaries are part of the evidence.

Source alignment: [G2]. All numerical data and practice questions here are original teaching examples, not official exam questions or observed results.

Worked example

10<n<20; n leaves remainder 2 upon division by 5 and by 3. Find n.

1. List values matching the range and first remainder.
2. 12 and 17 are candidates.
3. 12 fails the division-by-three condition.
4. 17 satisfies both conditions.

Mental model: Translate a remainder into an equation with an integer quotient.

Common trap: Treat a remainder as a unique value.

13. A graph needs its scale and denominator

Learning goal: Read what each plotted value represents before comparing changes.

A graph is a representation of quantities, not a substitute for their definitions. Inspect the axis label, unit, scale and denominator before deciding what grew fastest or what is largest. A line showing revenue per customer is different from one showing total revenue. Increasing the number of customers can raise total revenue while average revenue falls.

Use an original two-period table. Period one has ten customers and revenue of 500. Period two has twenty customers and revenue of 800. Total revenue rises by 300, or sixty percent of the earlier total. Revenue per customer falls from fifty to forty, a twenty percent decline. Neither statement contradicts the other; they describe different quantities.

Truncated scales may exaggerate visual differences. A bar drawn from ninety to one hundred is not a tenfold change in the measured quantity. Use the labels to calculate ratios. If a chart is indexed to a base of one hundred, it shows change relative to the base, not an absolute dollar amount unless that base amount is supplied. Data Insights officially includes graphic interpretation and table analysis; these models train the reading step without copying official material.

Practice set: twelve orders produce revenue 360 in one period, and fifteen orders produce 420 in the next. Mean revenue per order moves from thirty to twenty-eight. Total revenue rises while the mean falls. Now compare a count of active accounts with a percentage of active accounts: a falling percentage can coincide with a rising count if the total account population grows. Before choosing an option, write the quantity in a sentence, including the denominator. If an axis is missing, state the missing information rather than infer it from the shape.

Source alignment: [G2]. All numerical data and practice questions here are original teaching examples, not official exam questions or observed results.

Worked example

10 customers produce 500 revenue; 20 produce 800. Find period-two revenue per customer.

1. Identify revenue per customer as total/count.
2. Use period-two values:800 and 20.
3. 800/20=40.
4. Do not use the period-one count.

Mental model: Read what each plotted value represents before comparing changes.

Common trap: Read a visual slope without checking the plotted quantity.

14. Causal evaluation uses a discriminating result

Learning goal: Choose information whose opposite answers would affect the argument differently.

To evaluate a causal argument, locate the proposed cause, the measured outcome and any alternative explanation. A useful question is not merely related to the topic. Its possible answers should change how strongly the evidence supports the conclusion. Ask what a yes would do and what a no would do before judging an evaluate-the-argument option.

An original company changes its training and then sees fewer errors. Management attributes the decline to the new training. During the same period, the company may also have replaced complex tasks with simple ones. Asking whether task complexity changed tests an alternative explanation. If it changed sharply, the training claim is less secure. If it did not, that particular alternative is weaker, though other causes may remain.

An irrelevant question can sound interesting: did staff prefer the room color? Unless the argument links color to errors, either answer leaves the central causal explanation largely untouched. Another trap asks whether managers believe training matters. Belief is not the outcome evidence needed to separate causes. Match the proposed question to a concrete link in the reasoning chain.

Practice set: a town adds cycle lanes and reports fewer crashes. Evaluate whether the number of cyclists changed, whether reporting rules changed and whether the measured crashes included all roads. Each addresses exposure, measurement or scope. A question about the mayor favorite sport is not useful without a stated causal link. For a timed review, write both branches for the best question. Do not treat a favorable answer as proof of causation; it may only remove one competing explanation. The original cases practice the Critical Reasoning scope described by GMAC.

Source alignment: [G2]. All numerical data and practice questions here are original teaching examples, not official exam questions or observed results.

Worked example

Training changed and errors fell. Test whether tasks also became easier.

1. State the causal conclusion.
2. Name easier tasks as an alternative cause.
3. A large complexity change weakens the attribution.
4. No complexity change removes only this alternative.

Mental model: Choose information whose opposite answers would affect the argument differently.

Common trap: Confuse removal of one rival with proof of the cause.

15. Sufficiency can answer no

Learning goal: Judge whether the yes/no answer is fixed, not whether it is favorable.

A yes/no data-sufficiency problem is answered when every permitted case yields the same answer, even if that answer is no. Sufficient does not mean a statement confirms the proposition. It means it resolves the requested question. Restate the question exactly before using each statement and preserve domain restrictions such as integer, positive or nonzero.

Consider an original question: is integer n even? Statement one says n is an odd integer. That statement is sufficient because the answer is always no. Statement two says n is greater than ten. It is insufficient because eleven and twelve satisfy it but give different answers. A numerical value of n is unnecessary when parity alone answers the question.

Counterexamples are a direct test of insufficiency. Find two cases satisfying the same statement with different answers to the target question. They need not be large or complicated. If the question asks whether x squared exceeds one and x is positive, use one-half and two to show that positivity alone does not resolve it. If x is greater than two, the answer is fixed as yes.

Practice set: question whether n is positive, with integer n. A statement n equals minus three is sufficient with answer no. A statement n squared equals nine is insufficient because n can be three or minus three. A statement n squared equals zero is sufficient with answer no because n must be zero. Check the original question rather than whether you can calculate a magnitude. These examples use the sufficiency idea in current Data Insights; they are editorial exercises, not official response-choice replicas.

Source alignment: [G2]. All numerical data and practice questions here are original teaching examples, not official exam questions or observed results.

Worked example

Is integer n even? Statement: n is odd. Is it sufficient?

1. Ask whether the yes/no answer is fixed.
2. Every allowed n is odd.
3. The answer to even is always no.
4. The statement is sufficient.

Mental model: Judge whether the yes/no answer is fixed, not whether it is favorable.

Common trap: Require a yes answer before calling evidence sufficient.

16. Integrated practice audits the model first

Learning goal: Review a mixed quantitative and argument set by the reasoning step that failed.

This original mixed set combines overlapping sets, proportional interpretation and evaluation of a causal claim. It does not reproduce the official timing, adaptivity or scoring. Use it to identify whether your error came from a missing denominator, a duplicated count or a weak inference. A correct answer reached by an unsupported shortcut still needs review.

Question one: forty people include twenty in A, eighteen in B and five in both. Find neither. Question two: ten customers produce 500 revenue and twenty customers produce 800. Find the change in mean revenue per customer. Question three: training changed while task complexity fell; can the fall in errors be uniquely attributed to training from these observations alone? Try all three before reading the next paragraph.

The first answer is seven because the union is thirty-three. The second is a fall from fifty to forty, a twenty percent decrease relative to fifty. The third answer is no because the simultaneous complexity change remains an alternative explanation. The quantitative answers depend on named denominators; the argument answer depends on the evidence boundary. None of these results supplies a GMAT score estimate.

For follow-up, change the overlap to eight, producing ten people in neither set. Change period-two revenue to 1000, making mean revenue fifty and leaving it unchanged despite the larger total. Finally hold task complexity constant while keeping the same training result: one alternative is removed but causation is still not proven. Write a short rule you will apply next time, then solve a new example without viewing the model. Use the official GMAC practice interface separately for current section navigation and review mechanics. A small set teaches a method; it does not establish full exam readiness.

Source alignment: [G2]. All numerical data and practice questions here are original teaching examples, not official exam questions or observed results.

Worked example

40 people,20 in A,18 in B,5 in both. Find neither.

1. Compute union:20+18-5=33.
2. Subtract from total 40.
3. Neither is 7.
4. Check four disjoint cells sum to 40.

Mental model: Review a mixed quantitative and argument set by the reasoning step that failed.

Common trap: Treat answer-key review as proof of a working method.