GRE General Test Quantitative Reasoning Flashcards

24 sample cards from the Quantitative Reasoning deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. 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.

2. 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 $x = 0$.

3. 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 $-\frac{3}{2}$.

4. 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: $$\left( \frac{x_{1}+x_{2}}{2},\; \frac{y_{1}+y_{2}}{2} \right)$$

5. 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 $x^{2} + y^{2} = r^{2}$.

6. 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.

7. 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.

8. 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.

9. 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 $2x - 5 = 11$.

10. 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 $r$): $$a_{n} = a_{1} \cdot r^{\,n-1}$$

11. 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: $1 + 2 + \cdots + 100 = 5050$.

12. 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 $180^{\circ}$.

13. 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.

14. 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 $3 < c < 13$.

15. 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., $6\text{-}8\text{-}10$).

16. 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.

17. 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: $$A = \frac{s^{2}\sqrt{3}}{4}$$

18. 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 $120^{\circ}$.

19. 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).

20. 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.

21. 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.

22. 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.

23. 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 $= s\sqrt{3}$.

24. 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 $2\pi r h$).

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Quantitative Reasoning flashcards FAQ