GRE General Test Quantitative Reasoning Syllabus

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.)

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 $c$.

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

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

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.

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 $3$).

In the composition $f(g(x))$, which function is applied first? Evaluate $f(g(2))$ for $f(x) = x^{2}$ and $g(x) = x + 1$.

The inner function $g$ is applied first, then $f$. Here $g(2) = 3$, so $f(g(2)) = f(3) = 9$. Order matters: $g(f(2)) = g(4) = 5 \neq 9$.

What is the formula for the slope of the line through points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$?

$$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$$ (rise over run). A horizontal line has slope $0$; a vertical line has undefined slope.

In the slope-intercept form $y = mx + b$, what do $m$ and $b$ represent, and how do you find the intercepts of a line?

$m$ is the slope and $b$ is the $y$-intercept (the value of $y$ when $x = 0$). To find the $x$-intercept, set $y = 0$ and solve; to find the $y$-intercept, set $x = 0$.

What is the relationship between the slopes of parallel lines and of perpendicular lines?

Parallel lines have equal slopes: $m_{1} = m_{2}$. Perpendicular lines have slopes that are negative reciprocals: $m_{1} \cdot m_{2} = -1$. Example: a line of slope $\frac{2}{3}$ is perpendicular to one of slope $-\frac{3}{2}$.

State the distance formula and the midpoint formula for points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$.

Distance: $$d = \sqrt{(x_{2}-x_{1})^{2} + (y_{2}-y_{1})^{2}}$$ Midpoint: $$\left( \frac{x_{1}+x_{2}}{2},\; \frac{y_{1}+y_{2}}{2} \right)$$

What is the equation of a circle in the coordinate plane with center $(h, k)$ and radius $r$?

$$(x - h)^{2} + (y - k)^{2} = r^{2}$$ Centered at the origin this becomes $x^{2} + y^{2} = r^{2}$.

State the fundamental distance–rate–time relationship and its rearrangements.

$$d = rt, \qquad r = \frac{d}{t}, \qquad t = \frac{d}{r}$$ Distance equals rate (speed) times time; keep units consistent.

If one worker finishes a job in $a$ hours and another in $b$ hours, how long do they take working together?

Add rates: $\frac{1}{a} + \frac{1}{b} = \frac{1}{t}$, which gives $$t = \frac{ab}{a+b}$$ Example: $3$ hours and $6$ hours alone gives $\frac{3 \cdot 6}{3+6} = 2$ hours together.

How is average speed over a whole trip computed, and why is it usually NOT the average of the two speeds?

$$\text{Average speed} = \frac{\text{total distance}}{\text{total time}}$$ Example: driving equal distances at $30$ and $60$ mph gives average $\frac{2d}{\frac{d}{30} + \frac{d}{60}} = 40$ mph, not $45$, because more time is spent at the slower speed.

How do the phrases "of", "is", "less than", and "$x$ percent of $y$" translate into algebra in word problems?

"of" $\to$ multiply; "is" $\to$ $=$; "$a$ less than $b$" $\to$ $b - a$ (order reverses); "$x$ percent of $y$" $\to$ $\frac{x}{100} \cdot y$. Example: "$5$ less than twice $x$ is $11$" becomes $2x - 5 = 11$.

Give the formulas for the $n$th term of an arithmetic sequence and of a geometric sequence.

Arithmetic (common difference $d$): $$a_{n} = a_{1} + (n-1)d$$ Geometric (common ratio $r$): $$a_{n} = a_{1} \cdot r^{\,n-1}$$

What is the sum of the first $n$ terms of an arithmetic sequence, and the sum of the first $n$ positive integers?

$$S_{n} = \frac{n}{2}(a_{1} + a_{n}) = n \times (\text{average of first and last terms})$$ In particular, $1 + 2 + \cdots + n = \frac{n(n+1)}{2}$. Example: $1 + 2 + \cdots + 100 = 5050$.

What are the key angle facts for intersecting lines and for parallel lines cut by a transversal?

Vertical (opposite) angles are equal; angles on a straight line sum to $180^{\circ}$. With parallel lines and a transversal: corresponding angles and alternate interior angles are equal, and co-interior (same-side) angles sum to $180^{\circ}$.

What is the sum of the interior angles of a triangle, and what does the exterior angle theorem say?

The interior angles of any triangle sum to $180^{\circ}$. An exterior angle equals the sum of the two remote (non-adjacent) interior angles. Also, the larger side is always opposite the larger angle.

State the triangle inequality for the possible length of a triangle's side.

Each side must be less than the sum and greater than the (positive) difference of the other two: $$|a - b| < c < a + b$$ Example: sides $5$ and $8$ force the third side to satisfy $3 < c < 13$.

State the Pythagorean theorem and list the most common Pythagorean triples to memorize.

In a right triangle with legs $a, b$ and hypotenuse $c$: $$a^{2} + b^{2} = c^{2}$$ Common triples: $3\text{-}4\text{-}5$, $5\text{-}12\text{-}13$, $8\text{-}15\text{-}17$, and their multiples (e.g., $6\text{-}8\text{-}10$).

What are the side ratios of the two special right triangles: $45^{\circ}$–$45^{\circ}$–$90^{\circ}$ and $30^{\circ}$–$60^{\circ}$–$90^{\circ}$?

$45$–$45$–$90$: sides in ratio $1 : 1 : \sqrt{2}$ (hypotenuse $= \text{leg} \times \sqrt{2}$). $30$–$60$–$90$: sides in ratio $1 : \sqrt{3} : 2$, opposite the $30^{\circ}$, $60^{\circ}$, and $90^{\circ}$ angles respectively.

What is the area formula for a general triangle, and for an equilateral triangle with side $s$?

General triangle: $$A = \frac{1}{2} b h$$ (base times perpendicular height). Equilateral triangle: $$A = \frac{s^{2}\sqrt{3}}{4}$$

What is the sum of the interior angles of an $n$-sided polygon, and the measure of each interior angle of a regular $n$-gon?

Interior angle sum: $(n-2) \times 180^{\circ}$. Each angle of a regular $n$-gon: $\frac{(n-2) \times 180^{\circ}}{n}$. Example: a regular hexagon has angle sum $720^{\circ}$ and each angle $120^{\circ}$.

Give the area formulas for a parallelogram and for a trapezoid.

Parallelogram: $$A = bh$$ (base $\times$ perpendicular height). Trapezoid: $$A = \frac{1}{2}(b_{1} + b_{2})h$$ (average of the parallel bases times the height).

State the formulas for the circumference and the area of a circle of radius $r$.

$$C = 2\pi r = \pi d, \qquad A = \pi r^{2}$$ where $d = 2r$ is the diameter.

For a central angle of $\theta$ degrees in a circle of radius $r$, what are the arc length and sector area?

$$\text{Arc length} = \frac{\theta}{360} \cdot 2\pi r, \qquad \text{Sector area} = \frac{\theta}{360} \cdot \pi r^{2}$$ The fraction $\frac{\theta}{360}$ is the portion of the full circle.

How does an inscribed angle relate to the central angle subtending the same arc, and what is special about a triangle inscribed in a semicircle?

An inscribed angle is half the central angle subtending the same arc: $\text{inscribed} = \frac{1}{2} \times \text{central}$. A triangle inscribed in a semicircle (one side is a diameter) is always a right triangle, with the right angle opposite the diameter.

For a rectangular solid with dimensions $l$, $w$, $h$, give the volume, surface area, and length of the interior (space) diagonal.

$$V = lwh, \qquad SA = 2(lw + lh + wh), \qquad \text{diagonal} = \sqrt{l^{2} + w^{2} + h^{2}}$$ For a cube of edge $s$: $V = s^{3}$, $SA = 6s^{2}$, diagonal $= s\sqrt{3}$.

Give the volume and total surface area of a right circular cylinder with radius $r$ and height $h$.

$$V = \pi r^{2} h, \qquad SA = 2\pi r^{2} + 2\pi r h$$ (two circular ends plus the lateral "unrolled rectangle" of area $2\pi r h$).