GMAT Focus Edition · GRE & GMAT Analytical Reasoning
GRE Quantitative Reasoning
Arithmetic, algebra, geometry and data analysis at up to a second algebra level.
Seven concepts spanning what GRE Quantitative Reasoning actually asks: the three response formats, number properties, percent change, averages, rates, algebra with exponents and quadratics, and reading a shared table or graph. Geometry is named in the syllabus blurb and is not a concept here — the legacy notes never taught it, and inventing a chapter would be fabrication.
- GMAT Focus Edition
- Medium level
- 7 concepts
- 5 practice questions
1Three response formats on one measure
The Quant measure mixes three formats on a shared arithmetic–algebra–data syllabus: Quantitative Comparison (which of two quantities is larger, or whether that cannot be determined), multiple-choice with one or more correct answers, and numeric entry typed into a box. The on-screen calculator is available for messy arithmetic; simple estimates are still faster by hand.
Format is not content. The same percent-change skill appears as a comparison of two quantities, as a multi-select stem, and as a blank that wants a bare number. Recognising the format tells you how many answers to commit and whether "cannot be determined" is even an option — it does not change the arithmetic.
Figure. Format is not content: the same percent-change skill can appear as a comparison, a multi-select stem, or a typed number. Equal bars mark equal syllabus weight, not equal frequency.
How to read the stem
- Name the formatComparison of two labelled quantities, a choice list that may allow more than one answer, or a blank for a typed number.
- Commit the right number of answersMulti-select means each option is true or false on its own; numeric entry wants one value, not a letter.
- Save the calculator for messUse it when the arithmetic is ugly; do not reach for it on a two-digit percent or a clean fraction.
| Format | What you produce | Trap to watch |
|---|---|---|
| Quantitative Comparison | A, B, C (equal), or D (cannot tell) | D is impossible when both sides are fixed numbers |
| Multiple-choice (one or more) | Every option that must be true | Test each option independently — more than one can be correct |
| Numeric entry | A typed number (fraction or decimal as asked) | No answer choices to back-solve from |
A stem asks which of three expressions must be odd when n is even, and says to select all that apply. The format is
- Multiple-choice with one or more answers
- Quantitative Comparison
- Numeric entry
"Select all that apply" is multi-select: each option is judged true or false on its own. Quantitative Comparison always pits Quantity A against Quantity B with four fixed choices, and numeric entry wants a typed number with no options to tick.
2Parity, signs, factors and remainders
Factors, multiples, remainders, and the even/odd and positive/negative rules underlie a large share of GRE arithmetic. Parity is closed under multiplication and addition in fixed ways: even + odd is odd, even × anything is even, and an even square is even. Sign rules are the twin: a negative times a negative is positive, and an odd power of a negative stays negative.
When a stem says "must be", one counter-example kills an option. When it says "could be", one working example keeps it. Plugging in 0, 1 and −1 is often enough to settle both — those three values are the ones the tricks list names first.
Figure. Sign and parity are closed under multiplication and addition in fixed ways — plug 0, 1 and −1 when a stem says must be or could be.
How to settle parity
- Translate the given"n is even" means n = 2k for an integer k; write that before expanding expressions.
- Apply one rule at a timeEven + odd = odd; even × integer = even; even × even = even. Do not jump to the whole expression at once.
- Kill options with a counter-exampleFor "must be odd", a single even value of the expression drops that option.
| Operation | Result | Why it matters |
|---|---|---|
| even + odd | odd | Adding 1, 3, 5… flips parity |
| even + even | even | Parity preserved |
| even × anything | even | A factor of 2 survives |
| odd × odd | odd | No factor of 2 appears |
| even² | even | Still carries the factor 2 |
Which must be odd
If n is an even integer, which of the following must be odd: n+3, 2n, n^2?
- n even, so n + 3 = even + oddodd
- 2n = 2 × (even integer)even
- n² = even × eveneven
- Must be odd among the threeonly n + 3
Pro tip. For parity questions, apply the even/odd rules directly instead of plugging in several numbers. Plug-in is the right tool when the claim is algebraic and the parity rules do not decide it alone.
If n is even, which expression must be odd?
- n + 3
- 2n
- n²
Even + odd is odd, so n + 3 is odd. 2n is a multiple of 2, and n² is even × even, so both stay even. The trap is treating "must be" as "could be" and keeping 2n because some even numbers look odd when misread.
3Percent change, always over the old value
Percent change is (\text{new} - \text{old}) / \text{old} \times 100\%. The numerator is the difference; the denominator is always the value you are changing from — never the new value, and never whichever number happens to be larger.
That choice of base makes the relationship asymmetric. A rise from 80 to 100 is a 25% increase, but the fall from 100 back to 80 is a 20% decrease — the same gap of 20 measured against two different bases. Any stem that walks a quantity up and then asks for the reverse percentage is testing this and nothing else.
Figure. Both bars grow from zero; the dashed rule marks the old value that every percent change divides by. The gap from 80 to 100 is 20 in absolute units, but that gap is 25% of the shorter bar and only 20% of the taller one — the figure shows why the two percentages cannot match.
How it works
- Name the baseUnderline the value after "more than", "less than" or "compared with" — that is the denominator.
- Take new minus oldKeep the sign; a negative result is a decrease, not a mistake.
- Divide by the base, then × 100Divide by the base you named, not by the number that makes the arithmetic prettier.
Rise, then the reverse fall
A quantity rises from 80 to 100. What is the percent increase? What percent decrease returns it from 100 to 80?
- Increase: (100 − 80) / 800.25
- 0.25 × 100%25% increase
- Decrease: (80 − 100) / 100−0.20
- −0.20 × 100%20% decrease
Pro tip. The same gap of 20 is 25% of 80 and only 20% of 100. If a stem asks for both directions, write both denominators before computing either percentage — swapping them is the whole trap.
A value rises from 80 to 100. The percent increase is
- 25%
- 20%
- 15%
(100 − 80)/80 × 100% = 25%. Answering 20% divides by the new value instead of the old one — the reverse-direction percentage. 15% is 12/80 and invents a different gap.
4The mean is the sum divided by the count
The average (arithmetic mean) is \text{sum} / \text{count}. Every GRE average question is either computing that quotient, recovering the sum as mean × count, or adjusting a mean when one value joins or leaves the set.
Recovering the sum is the move that unlocks the rest: if five numbers average 15, their total is 75, and any new constraint is a constraint on that total. Working with the mean alone, without converting back to a sum, is how people lose a mark on an otherwise one-step stem.
Figure. Five bars at the five values, with a dashed rule at the mean of 15. The mean is not one of the data points that has to appear — here it happens to match the middle bar — and the eye check is that the area above the rule balances the area below it.
How it works
- Write sum = mean × countConvert every stated average into a total before adding or removing a value.
- Adjust the totalAdd a new member's value to the sum, or subtract a removed one.
- Divide by the new countOnly after the total is right do you form the new mean.
Five numbers, one mean
The numbers 12, 18, 15, 21 and 9 have what mean? Confirm by recovering the sum from that mean.
- Sum = 12 + 18 + 15 + 21 + 975
- Count5
- Mean = 75 / 515
- Check: 15 × 575
Pro tip. When a stem gives the mean and asks what must be true of the numbers, multiply back to the sum first. Constraints on "the average" are almost always constraints on that total.
Five numbers average 15. Their sum is
- 75
- 15
- 3
Sum = mean × count = 15 × 5 = 75. Leaving the answer at 15 confuses the mean with the total; dividing instead of multiplying gives 3.
5Rates add; times do not
Distance–rate–time is d = r \cdot t, and the same algebra runs work problems: a machine's rate is bottles per minute, and two machines working together add their rates. The job size divided by the combined rate is the time together.
Times never add. A 4-minute machine and a 6-minute machine do not take 10 minutes or 5 minutes together — they take 2.4, which is less than either alone. That inequality is the fastest sanity check on a combined-rate answer.
Figure. Three rate bars in bottles per minute: A at 15, B at 10, and the combined bar at 25 — exactly the sum. The figure is about rates stacking; the times 4 min and 6 min do not appear as bar heights, because adding those times is the mistake the concept exists to kill.
How it works
- Convert every time into a rateBottles per minute, or job-fraction per day, before combining anything.
- Add the ratesHelpers add; only then do you have one combined rate.
- Divide the job by the combined rateTime = job size ÷ combined rate — the last step, never an intermediate one.
Two machines, sixty bottles
Machine A fills 60 bottles in 4 minutes and Machine B fills 60 bottles in 6 minutes. Working together, how long to fill 60 bottles?
- Rate of A = 60 / 415 bottles/min
- Rate of B = 60 / 610 bottles/min
- Combined rate = 15 + 1025 bottles/min
- Time = 60 / 252.4 minutes
Pro tip. Add rates, not times, when things work together; then divide the job size by the combined rate. Any answer at or above 4 minutes here is wrong before you finish the arithmetic, because a second machine cannot slow the first one down.
Machine A fills 60 bottles in 4 minutes and B fills 60 in 6 minutes. Together they fill 60 bottles in
- 2.4 minutes
- 5 minutes
- 10 minutes
Rates are 15 and 10 bottles/min, so together 25 bottles/min and 60/25 = 2.4 minutes. 5 averages the times and 10 adds them; both fail the check that two machines together must beat the faster machine alone (4 minutes).
6Exponents, linear equations and the quadratic formula
Algebra on the GRE means solving linear and quadratic equations, applying exponent rules, and translating word problems into equations. The exponent identities that earn marks are a^m \cdot a^n = a^{m+n} and (a^m)^n = a^{mn} — multiply exponents when a power is raised to a power, add them when the bases match and you multiply.
A quadratic ax^2 + bx + c = 0 is solved by factoring when the integers cooperate, or by x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} when they do not. Inequalities use the same algebra with one extra rule: multiplying or dividing by a negative reverses the inequality sign.
Figure. Matching bases multiply by adding exponents: the stacked 2+3 bar equals the single a⁵ bar. Raising a power to a power multiplies the exponents instead.
How to finish a quadratic
- Try factoring firstTwo integers that multiply to c and add to b turn the quadratic into a product of linear factors.
- Else use the quadratic formulaCompute the discriminant b^2 - 4ac, then split into the two roots with \pm.
- Check in the originalSubstitute each root back; a sign slip in −b is the usual miss.
| Rule | Example | Result |
|---|---|---|
| a^m \cdot a^n = a^{m+n} | 2^3 \cdot 2^4 | 2^7 = 128 |
| (a^m)^n = a^{mn} | (3^2)^3 | 3^6 = 729 |
| a^0 = 1 (a ≠ 0) | 7^0 | 1 |
| a^{-n} = 1/a^n | 2^{-3} | 1/8 |
Solve by the quadratic formula
Solve x^2 - 5x + 6 = 0 using the quadratic formula, then confirm by factoring.
- a = 1, b = −5, c = 6; discriminant (−5)² − 4(1)(6)25 − 24 = 1
- x = (5 ± √1) / 2(5 ± 1) / 2
- Rootsx = 3 or x = 2
- Factor check: (x − 2)(x − 3)x² − 5x + 6
Pro tip. When the discriminant is a perfect square, factoring and the formula agree. Use the formula on the exam when the integers do not jump out in under ten seconds — do not spend a minute hunting factors you can compute instead.
The solutions of x^2 - 5x + 6 = 0 are
- 2 and 3
- −2 and −3
- 1 and 6
Discriminant 1 gives x = (5 ± 1)/2, so 3 and 2. The negative pair would solve x² + 5x + 6 = 0, and 1 and 6 multiply to 6 but add to 7, not 5.
Notes
- Question Types: The Quant measure mixes Quantitative Comparison, multiple-choice (one or more answers), and numeric-entry questions with an on-screen calculator.
- Arithmetic and Number Properties: Master factors, multiples, remainders, and properties of even/odd and positive/negative numbers, which underlie many problems.
- Algebra Foundations: Solve linear and quadratic equations, work with exponents and inequalities, and translate word problems into equations.
- Data Interpretation: A shared graph or table feeds several questions requiring reading values, computing percentages, and comparing quantities.
Formulas
- Percent change: \dfrac{\text{new}-\text{old}}{\text{old}}\times100\%
- Average (mean): \dfrac{\text{sum}}{\text{count}}
- Distance-rate-time: d = r\cdot t
- Quadratic formula: x = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a}
- Exponent rules: a^m\cdot a^n = a^{m+n}, (a^m)^n = a^{mn}
Exam traps & shortcuts
- Plug in smart numbers for variable-expression problems, especially 0, 1, and -1, to test which answer holds.
- Use the on-screen calculator only for messy arithmetic; simple estimates are faster mentally.
- For 'must be true' multiple-answer questions, test each option independently since more than one can be correct.
Reference tables
| Name | Formula | Watch |
|---|---|---|
| Percent change | (\text{new}-\text{old})/\text{old}\times 100\% | Denominator is always the old value |
| Mean | \text{sum}/\text{count} | Recover the sum as mean × count |
| Distance–rate–time | d = r\cdot t | Same algebra as work rates |
| Combined work rate | Add rates; time = job ÷ combined rate | Never add the individual times |
| Quadratic formula | x = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a} | Factor when the discriminant is a tidy square |
| Exponent product | a^m\cdot a^n = a^{m+n} | Bases must match |
| Exponent tower | (a^m)^n = a^{mn} | Multiply the exponents |
Recap
Read only this the night before.
- Formats
- QC (A/B/C/D), multi-select, numeric entry. Format decides how you answer; it does not change the arithmetic. D is impossible when both QC sides are fixed numbers.
- Parity
- Even + odd = odd; even × anything = even. For "must be", one counter-example kills an option. Try 0, 1 and −1 when the claim is algebraic.
- Percent
- (new − old)/old × 100. Rise 80→100 is 25%; fall 100→80 is 20%. Same gap, different base.
- Mean
- sum/count. Convert every average back to a total before adding or removing a value.
- Rates
- Add rates, invert at the end. 15 + 10 = 25 bottles/min → 60/25 = 2.4 min. Any combined time above the faster alone is wrong.
- Algebra
- aᵐ·aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ. Quadratic: factor or (−b ± √(b²−4ac))/(2a). Multiplying an inequality by a negative flips the sign.
- DI
- Re-read labels every stem. Percent change divides by the earlier period; percent of whole divides by the total of all parts.
Practise GRE Quantitative Reasoning
Reading is free and needs no account. Practice, mocks and progress live in the app.
- 5 exam-style questions on this topic, with explanations
- A 5-question practice set that ends the chapter
- Timed mocks scored with the real marking scheme
- Readiness tracked per topic, kept on your device