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GRE General Quantitative Reasoning Syllabus
The Quantitative Reasoning syllabus for GRE General in full: 4 chapters, 8 topics and 15 sub-topics, every one listed below, plus 50 Quantitative Reasoning flashcards with answers.
Quantitative Reasoning syllabus: all 4 chapters
Expand any chapter to see its topics and sub-topics.
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Arithmetic
2 topics- Properties of Integers
- Even and odd properties
- Fractions and Ratios
- Simplifying fractions
- Ratio concepts
- Properties of Integers
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Algebra
2 topics- Equations and Inequalities
- Solving linear equations
- Solving inequalities
- Functions and Graphs
- Interpreting function graphs
- Graph transformations
- Equations and Inequalities
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Geometry
2 topics- Lines and Angles
- Parallel lines
- Angle properties
- Circles and Polygons
- Circle properties
- Polygon properties
- Lines and Angles
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Data Analysis
2 topics- Statistics
- Mean, median, mode
- Standard deviation
- Probability
- Basic probability rules
- Independent and dependent events
- Statistics
GRE General Quantitative Reasoning flashcards — 50 questions with answers
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: $\{\dots, -2, -1, 0, 1, 2, \dots\}$.
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.
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.
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 $2$.
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 $0 \leq r < d$.
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 $\text{GCD}(a,b)\times\text{LCM}(a,b)=|a\cdot b|$.
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.
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 $(a^{m})^{n}=a^{mn}$.
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 $a\neq 0$.
How do you add or subtract fractions with different denominators?
Rewrite each fraction with a common denominator, then add or subtract the numerators: $\dfrac{a}{b}\pm\dfrac{c}{d}=\dfrac{ad\pm bc}{bd}$.
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: $\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\cdot\dfrac{d}{c}=\dfrac{ad}{bc}$.
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: $ad=bc$.
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 $\dfrac{p}{100}\cdot N$.
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.
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 $\dfrac{a}{b}<\dfrac{c}{d}$.
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.
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.
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.
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 $a^{2}-b^{2}=(a+b)(a-b)$.
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 $x>-3$.
Show the other 30 Quantitative Reasoning flashcards
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 $x>a$.
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 $(x,y)$.
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 ($y=f(x)$).
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).
What is the formula for the slope of a line through points $(x_1,y_1)$ and $(x_2,y_2)$?
$$m=\frac{y_2-y_1}{x_2-x_1},\quad x_2\neq x_1.$$
What is the distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$?
$$d=\sqrt{(x_2-x_1)^{2}+(y_2-y_1)^{2}}.$$
What is the midpoint formula for the segment joining $(x_1,y_1)$ and $(x_2,y_2)$?
$$\left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right).$$
How are the slopes of parallel lines and of perpendicular lines related?
Parallel lines have equal slopes ($m_1=m_2$). Perpendicular lines have slopes that are negative reciprocals: $m_1\cdot m_2=-1$.
What is the general shape and vertex form of a quadratic function's graph?
The graph of $y=ax^{2}+bx+c$ is a parabola. In vertex form $y=a(x-h)^{2}+k$, the vertex is $(h,k)$; it opens upward if $a>0$ and downward if $a<0$.
How do you compute a composite function value $f(g(x))$?
First evaluate the inner function $g(x)$, then substitute that result into $f$. For example, if $f(x)=x^{2}$ and $g(x)=x+1$, then $f(g(3))=f(4)=16$.
What are complementary and supplementary angles?
Complementary angles sum to $90^{\circ}$; supplementary angles sum to $180^{\circ}$.
What is the relationship between vertical angles formed by two intersecting lines?
Vertical (opposite) angles are congruent—they have equal measure.
When a transversal crosses two parallel lines, what is true of corresponding and alternate interior angles?
Corresponding angles are equal, alternate interior angles are equal, and co-interior (same-side interior) angles are supplementary, summing to $180^{\circ}$.
What is the sum of the interior angles of any triangle, and of a quadrilateral?
The interior angles of a triangle sum to $180^{\circ}$; those of a quadrilateral sum to $360^{\circ}$.
What is the sum of the interior angles of a polygon with $n$ sides?
$$\text{sum}=(n-2)\times 180^{\circ}.$$
State the Pythagorean theorem and name a common Pythagorean triple.
For a right triangle with legs $a,b$ and hypotenuse $c$: $a^{2}+b^{2}=c^{2}$. A common triple is $3\text{-}4\text{-}5$.
What is the circumference of a circle with radius $r$?
$$C=2\pi r=\pi d,$$ where $d=2r$ is the diameter.
What is the area of a circle with radius $r$?
$$A=\pi r^{2}.$$
How do you find the arc length and sector area for a central angle of $\theta$ degrees?
Arc length $=\dfrac{\theta}{360^{\circ}}\cdot 2\pi r$ and sector area $=\dfrac{\theta}{360^{\circ}}\cdot \pi r^{2}$.
What is the inscribed angle theorem for a circle?
An inscribed angle is half the central angle that subtends the same arc. Consequently, an angle inscribed in a semicircle is a right angle ($90^{\circ}$).
What is the area of a triangle, and the area of a parallelogram?
Triangle: $A=\dfrac{1}{2}bh$. Parallelogram: $A=bh$, where $b$ is the base and $h$ is the perpendicular height.
What is the area of a trapezoid with parallel sides $b_1$ and $b_2$ and height $h$?
$$A=\frac{1}{2}(b_1+b_2)h.$$
How do you compute the arithmetic mean (average) of a data set?
$$\text{mean}=\frac{\text{sum of all values}}{\text{number of values}}=\frac{\sum x_i}{n}.$$
How do you find the median of a data set?
Order the values; the median is the middle value if $n$ is odd, or the average of the two middle values if $n$ is even.
What are the mode and the range of a data set?
The mode is the value that appears most frequently. The range is the difference between the largest and smallest values: $\text{range}=\max-\min$.
How is standard deviation interpreted, and what does a larger value indicate?
Standard deviation measures how spread out data values are around the mean. A larger standard deviation indicates greater dispersion; a smaller one indicates values cluster near the mean.
What are the quartiles and the interquartile range (IQR)?
Quartiles $Q_1$, $Q_2$ (median), and $Q_3$ divide ordered data into four equal parts. The interquartile range is $\text{IQR}=Q_3-Q_1$, the spread of the middle $50\%$ of the data.
What is the basic formula for the probability of an event $E$ with equally likely outcomes?
$$P(E)=\frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}},\quad 0\leq P(E)\leq 1.$$
What is the addition rule for the probability of $A$ or $B$, and how does it simplify for mutually exclusive events?
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$. If $A$ and $B$ are mutually exclusive, $P(A\cap B)=0$, so $P(A\cup B)=P(A)+P(B)$.
What is the multiplication rule for independent events, and the formula for the complement of an event?
For independent events, $P(A\cap B)=P(A)\cdot P(B)$. The complement rule gives $P(\text{not }A)=1-P(A)$.
GRE General Quantitative Reasoning FAQ
What is in the GRE General Quantitative Reasoning syllabus?
GRE General Quantitative Reasoning is split into 4 chapters: Arithmetic, Algebra, Geometry and Data Analysis. Together they contain 8 topics and 15 sub-topics.
How many chapters are in GRE General Quantitative Reasoning?
4 chapters, containing 8 topics and 15 sub-topics. The largest chapter is Arithmetic, with 2 topics.
Are there flashcards for GRE General Quantitative Reasoning?
Yes — a 50-card Quantitative Reasoning deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.