GRE General 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, not a summary of it.

1. 1

   Arithmetic

   2 topics

   • Properties of Integers
     • Even and odd properties
   • Fractions and Ratios
     • Simplifying fractions
     • Ratio concepts
2. 2

   Algebra

   2 topics

   • Equations and Inequalities
     • Solving linear equations
     • Solving inequalities
   • Functions and Graphs
     • Interpreting function graphs
     • Graph transformations
3. 3

   Geometry

   2 topics

   • Lines and Angles
     • Parallel lines
     • Angle properties
   • Circles and Polygons
     • Circle properties
     • Polygon properties
4. 4

   Data Analysis

   2 topics

   • Statistics
     • Mean, median, mode
     • Standard deviation
   • Probability
     • Basic probability rules
     • Independent and dependent events

Quantitative Reasoning flashcards for GRE General

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

1. What is the definition of an integer, and which numbers are included?

   An integer is any whole number without a fractional or decimal part, including positive whole numbers, negative whole numbers, and zero: $\{\dots, -2, -1, 0, 1, 2, \dots\}$.

2. State the divisibility rule for 3 and for 9.

   A number is divisible by 3 if the sum of its digits is divisible by 3; it is divisible by 9 if the sum of its digits is divisible by 9.

3. What are the rules for the parity (even/odd) of sums and products of integers?

   $\text{even} \pm \text{even} = \text{even}$, $\text{odd} \pm \text{odd} = \text{even}$, $\text{even} \pm \text{odd} = \text{odd}$. For products, a product is even if at least one factor is even, and odd only if all factors are odd.

4. What is a prime number, and what is the only even prime?

   A prime number is an integer greater than 1 whose only positive divisors are 1 and itself. The only even prime number is $2$.

5. For integers, how are the quotient and remainder defined in the division algorithm?

   For integer $a$ and positive integer $d$, there exist unique integers $q$ (quotient) and $r$ (remainder) with $a = qd + r$ and $0 \leq r < d$.

6. How do you find the greatest common divisor (GCD) and least common multiple (LCM) using prime factorization?

   The GCD is the product of each shared prime raised to its lowest power; the LCM is the product of each prime raised to its highest power. Also $\text{GCD}(a,b)\times\text{LCM}(a,b)=|a\cdot b|$.

7. What is the rule for the sign of a product or quotient of two nonzero numbers?

   Like signs give a positive result ($+\cdot+=+$, $-\cdot-=+$); unlike signs give a negative result ($+\cdot-=-$). The same rule applies to division.

8. State the exponent rules for multiplying, dividing, and raising a power to a power.

   $a^{m}\cdot a^{n}=a^{m+n}$, $\dfrac{a^{m}}{a^{n}}=a^{m-n}$, and $(a^{m})^{n}=a^{mn}$.

9. What do a zero exponent and a negative exponent mean?

   $a^{0}=1$ for $a\neq 0$, and $a^{-n}=\dfrac{1}{a^{n}}$ for $a\neq 0$.

10. How do you add or subtract fractions with different denominators?

    Rewrite each fraction with a common denominator, then add or subtract the numerators: $\dfrac{a}{b}\pm\dfrac{c}{d}=\dfrac{ad\pm bc}{bd}$.

11. How do you multiply and divide fractions?

    Multiply straight across: $\dfrac{a}{b}\cdot\dfrac{c}{d}=\dfrac{ac}{bd}$. To divide, multiply by the reciprocal: $\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\cdot\dfrac{d}{c}=\dfrac{ad}{bc}$.

12. What is a ratio, and how does it relate to a proportion?

    A ratio compares two quantities, written $a:b$ or $\dfrac{a}{b}$. A proportion is a statement that two ratios are equal, $\dfrac{a}{b}=\dfrac{c}{d}$, which can be solved by cross-multiplication: $ad=bc$.

13. How do you convert a fraction to a percent and find a percent of a number?

    Convert a fraction to a percent by multiplying by $100\%$. To find $p\%$ of a number $N$, compute $\dfrac{p}{100}\cdot N$.

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

    $$\text{percent change}=\frac{\text{new value}-\text{old value}}{\text{old value}}\times 100\%.$$ A positive result is an increase, a negative result is a decrease.

15. How do you compare two fractions such as $\dfrac{a}{b}$ and $\dfrac{c}{d}$ (with positive denominators)?

    Cross-multiply: compare $ad$ with $bc$. If $ad>bc$ then $\dfrac{a}{b}>\dfrac{c}{d}$; if $ad<bc$ then $\dfrac{a}{b}<\dfrac{c}{d}$.

16. What is the general form of a linear equation in one variable, and how is it solved?

    A linear equation has the form $ax+b=0$ with $a\neq 0$. Its solution is $x=-\dfrac{b}{a}$, found by isolating $x$ using inverse operations.

17. What is the quadratic formula, and when do you use it?

    For $ax^{2}+bx+c=0$ with $a\neq 0$, the solutions are $$x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}.$$ Use it to solve any quadratic that does not factor easily.

18. What does the discriminant of a quadratic tell you about its roots?

    For $ax^{2}+bx+c=0$, the discriminant is $\Delta=b^{2}-4ac$. If $\Delta>0$ there are two distinct real roots; if $\Delta=0$ one repeated real root; if $\Delta<0$ no real roots.

19. What are the key algebraic identities for $(a+b)^{2}$, $(a-b)^{2}$, and $a^{2}-b^{2}$?

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

20. What special rule applies when multiplying or dividing both sides of an inequality by a negative number?

    You must reverse the direction of the inequality sign. For example, if $-2x<6$ then dividing by $-2$ gives $x>-3$.

21. What does $|x|<a$ (with $a>0$) mean as a compound inequality?

    $|x|<a$ is equivalent to $-a<x<a$. Correspondingly, $|x|>a$ means $x<-a$ or $x>a$.

22. How do you solve a system of two linear equations by substitution and by elimination?

    Substitution: solve one equation for a variable and substitute into the other. Elimination: add or subtract multiples of the equations to cancel one variable. Both yield the intersection point $(x,y)$.

23. What is the definition of a function, and what are its domain and range?

    A function assigns exactly one output to each input. The domain is the set of allowed input values ($x$); the range is the set of resulting output values ($y=f(x)$).

24. What is the slope-intercept form of a line, and what does each term represent?

    $y=mx+b$, where $m$ is the slope (rate of change) and $b$ is the $y$-intercept (the value of $y$ where the line crosses the $y$-axis).

25. What is the formula for the slope of a line through points $(x_1,y_1)$ and $(x_2,y_2)$?

    $$m=\frac{y_2-y_1}{x_2-x_1},\quad x_2\neq x_1.$$

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