GRE General Test Quantitative Reasoning Syllabus

Quantitative Reasoning syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Quantitative Reasoning in GRE General Test, not a summary of it.

1. 1

   Arithmetic

   6 topics

   • Integers, Factors, and Multiples
   • Fractions, Decimals, and Percent
   • Ratios and Proportions
   • Exponents and Roots
   • Number Properties and Divisibility
   • Estimation and Number Line
2. 2

   Algebra

   6 topics

   • Algebraic Expressions and Operations
   • Linear Equations and Inequalities
   • Quadratic Equations
   • Functions and Coordinate Geometry
   • Word Problems and Translation
   • Sequences and Patterns
3. 3

   Geometry

   6 topics

   • Lines and Angles
   • Triangles
   • Quadrilaterals and Polygons
   • Circles
   • Three-Dimensional Figures and Volume
   • Coordinate Geometry
4. 4

   Data Analysis

   5 topics

   • Descriptive Statistics (Mean, Median, Mode, Range)
   • Standard Deviation and Distributions
   • Counting Methods and Permutations
   • Probability
   • Data Interpretation Sets
5. 5

   Quantitative Comparison

   4 topics

   • Comparing Two Quantities
   • Using the Four Answer Choices
   • Plugging In and Testing Cases
   • Avoiding Calculation Traps
6. 6

   Problem Solving and Numeric Entry

   4 topics

   • Multiple-Choice (Single Answer)
   • Multiple-Choice (Multiple Answers)
   • Numeric Entry Questions
   • On-Screen Calculator Use
7. 7

   Quant Test Strategy and Pacing

   4 topics

   • Section Adaptive Format and Timing
   • Backsolving and Plugging In Numbers
   • Estimation and Approximation
   • Common Errors and Time Management

Quantitative Reasoning flashcards for GRE General Test

25 of 50 cards from the Quantitative Reasoning deck — real questions with worked answers.

1. What is a prime number, and what is the smallest prime?

   A prime is an integer greater than $1$ whose only positive divisors are $1$ and itself. The smallest prime is $2$, which is also the only even prime. Note that $1$ is NOT prime.

2. For two positive integers $a$ and $b$, what is the relationship between their greatest common divisor (GCD) and least common multiple (LCM)?

   $$\gcd(a,b) \times \operatorname{lcm}(a,b) = a \times b$$ Example: for $12$ and $18$, $\gcd = 6$, $\operatorname{lcm} = 36$, and $6 \times 36 = 216 = 12 \times 18$.

3. How do you count the number of positive divisors of an integer from its prime factorization?

   If $n = p^{a} \cdot q^{b} \cdot r^{c}$, the number of positive divisors is $(a+1)(b+1)(c+1)$. Example: $72 = 2^{3} \cdot 3^{2}$ has $(3+1)(2+1) = 12$ divisors.

4. State the even/odd rules for addition and multiplication of integers.

   Addition: even $\pm$ even $=$ even; odd $\pm$ odd $=$ even; even $\pm$ odd $=$ odd. Multiplication: even $\times$ anything $=$ even; odd $\times$ odd $=$ odd.

5. What are the divisibility rules for $3$ and for $9$?

   An integer is divisible by $3$ if the sum of its digits is divisible by $3$, and divisible by $9$ if the sum of its digits is divisible by $9$. Example: $486$ has digit sum $18$, so it is divisible by both $3$ and $9$.

6. Write the quotient–remainder form for dividing an integer $n$ by a positive divisor $d$, including the constraint on the remainder.

   $$n = dq + r, \quad 0 \leq r < d$$ where $q$ is the quotient and $r$ the remainder. Example: $17 = 5 \cdot 3 + 2$, so $17$ divided by $5$ leaves remainder $2$.

7. What is the formula for percent change (increase or decrease)?

   $$\text{Percent change} = \frac{\text{new} - \text{original}}{\text{original}} \times 100\%$$ Always divide by the ORIGINAL (starting) value.

8. How do you handle successive percent changes, e.g., a $20\%$ increase followed by a $20\%$ decrease?

   Multiply the change factors: $1.20 \times 0.80 = 0.96$, a net $4\%$ decrease. Successive percent changes are never simply added; a rise and fall of the same percent always yields a net decrease.

9. What is the rule for dividing one fraction by another?

   Multiply by the reciprocal of the divisor: $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$$

10. If two quantities are in the ratio $a : b$, what fraction of the total does each represent?

    The total has $a + b$ parts, so the quantities are $\frac{a}{a+b}$ and $\frac{b}{a+b}$ of the total. Example: ratio $3:5$ means the first is $\frac{3}{8}$ of the whole.

11. What is the cross-multiplication property of a proportion?

    $$\frac{a}{b} = \frac{c}{d} \iff ad = bc \quad (b, d \neq 0)$$ Use it to solve for an unknown term in any proportion.

12. Write the equations for direct proportion and inverse proportion between $y$ and $x$.

    Direct proportion: $y = kx$ (the ratio $\frac{y}{x}$ is constant; both grow together). Inverse proportion: $y = \frac{k}{x}$, equivalently $xy = k$ (one grows as the other shrinks).

13. State the three basic exponent laws for products, quotients, and powers with the same base $a$.

    $$a^{m} \cdot a^{n} = a^{m+n}, \qquad \frac{a^{m}}{a^{n}} = a^{m-n}, \qquad (a^{m})^{n} = a^{mn}$$

14. What do a zero exponent, a negative exponent, and a fractional exponent mean?

    $a^{0} = 1$ (for $a \neq 0$); $\; a^{-n} = \frac{1}{a^{n}}$; $\; a^{m/n} = \sqrt[n]{a^{m}} = (\sqrt[n]{a})^{m}$. Example: $8^{2/3} = (\sqrt[3]{8})^{2} = 4$.

15. State the multiplication and division rules for square roots, and the common trap involving sums.

    $$\sqrt{ab} = \sqrt{a}\,\sqrt{b}, \qquad \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \quad (a, b \geq 0)$$ But in general $\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}$. Example: $\sqrt{9+16} = 5$, not $3 + 4 = 7$.

16. For a number $x$ with $0 < x < 1$, how do $x^{2}$, $x$, and $\sqrt{x}$ compare?

    $$x^{2} < x < \sqrt{x}$$ Squaring a number between $0$ and $1$ makes it smaller; taking its square root makes it larger. Example: for $x = \frac{1}{4}$: $\frac{1}{16} < \frac{1}{4} < \frac{1}{2}$.

17. Define absolute value, and give its interpretation for the distance between two points on the number line.

    $|x| = x$ if $x \geq 0$ and $|x| = -x$ if $x < 0$; it is the distance from $x$ to $0$. The distance between $a$ and $b$ on the number line is $|a - b|$. Example: distance between $-3$ and $5$ is $|-3 - 5| = 8$.

18. What approximate values of $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, and $\pi$ should be memorized for estimation?

    $$\sqrt{2} \approx 1.41, \quad \sqrt{3} \approx 1.73, \quad \sqrt{5} \approx 2.24, \quad \pi \approx 3.14$$ These let you place radicals and circle values quickly on a number line or in quantity comparisons.

19. State the three key algebraic identities: square of a sum, square of a difference, and difference of squares.

    $$(a+b)^{2} = a^{2} + 2ab + b^{2}$$ $$(a-b)^{2} = a^{2} - 2ab + b^{2}$$ $$a^{2} - b^{2} = (a+b)(a-b)$$

20. How do you factor a quadratic of the form $x^{2} + bx + c$?

    Find two numbers whose sum is $b$ and whose product is $c$; then $x^{2} + bx + c = (x + m)(x + n)$. Example: $x^{2} + 7x + 12 = (x+3)(x+4)$ since $3 + 4 = 7$ and $3 \cdot 4 = 12$.

21. What happens to an inequality when both sides are multiplied or divided by a negative number?

    The inequality sign reverses. Example: from $-2x < 6$, dividing by $-2$ gives $x > -3$. (Adding or subtracting any number, or multiplying by a positive number, does not flip the sign.)

22. How do you rewrite the absolute value inequalities $|x| < a$ and $|x| > a$ (for $a > 0$) without absolute value?

    $|x| < a \iff -a < x < a$ (a single interval). $|x| > a \iff x > a \text{ or } x < -a$ (two rays). More generally, $|x - c| < a$ means $x$ is within distance $a$ of $c$.

23. State the quadratic formula for solving $ax^{2} + bx + c = 0$.

    $$x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$$

24. How does the discriminant of $ax^{2} + bx + c = 0$ determine the number of real solutions?

    The discriminant is $D = b^{2} - 4ac$. If $D > 0$: two distinct real roots; if $D = 0$: exactly one real root (a repeated root); if $D < 0$: no real roots.

25. For the quadratic $ax^{2} + bx + c = 0$, what are the sum and product of the roots?

    $$\text{Sum of roots} = -\frac{b}{a}, \qquad \text{Product of roots} = \frac{c}{a}$$ Example: $x^{2} - 5x + 6 = 0$ has roots summing to $5$ with product $6$ (roots $2$ and $3$).

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