What is the Midpoint Formula for the segment connecting (x1, y1) and (x2, y2)?

Midpoint M = ((x1 + x2) / 2, (y1 + y2) / 2). It is the average of the x-coordinates and the average of the y-coordinates.

What are the rules for multiplying, dividing, and raising powers with exponents?

• (x^a) * (x^b) = x^(a + b) • (x^a) / (x^b) = x^(a - b) • (x^a)^b = x^(a * b) • (x * y)^a = x^a * y^a • (x / y)^a = x^a / y^a.

What are the side length ratios for the two high-yield special right triangles on the GRE?

• 45°-45°-90° (Isosceles right): 1 : 1 : sqrt(2) (hypotenuse = side * sqrt(2)). • 30°-60°-90°: 1 : sqrt(3) : 2 (hypotenuse = 2 * short side; long leg opposite 60° = short side * sqrt(3)).

What are the Area and Perimeter / Circumference formulas for Rectangles, Triangles, and Circles?

• Rectangle: Area = l * w; Perimeter = 2(l + w) • Triangle: Area = 0.5 * base * height • Circle: Area = pi * r^2; Circumference = 2 * pi * r = pi * d.

How do you solve an absolute value equation |ax + b| = c (for c >= 0)?

Split into two linear equations: ax + b = c OR ax + b = -c. Solve each equation independently. If c < 0, there is NO solution (absolute values cannot be negative).

What is the standard equation of a Circle with center (h, k) and radius r?

(x - h)^2 + (y - k)^2 = r^2. If centered at the origin (0, 0), the equation simplifies to x^2 + y^2 = r^2.

What are the Volume and Total Surface Area formulas for a Right Circular Cylinder?

• Volume: V = pi * r^2 * h • Total Surface Area: SA = 2 * pi * r^2 (two circular bases) + 2 * pi * r * h (lateral surface).

In GRE coordinate geometry, what does a line equation of the form x = c represent?

A vertical line with undefined slope passing through (c, y). It is perpendicular to the x-axis and parallel to the y-axis.

How do you quickly compare two fractions a/b and c/d without common denominators?

Cross-multiplication method: Multiply numerator of one by denominator of the other. Compare a * d with b * c (for positive denominators). If a*d > b*c, then a/b > c/d. Example: 5/8 vs 7/11 => 5*11 = 55 vs 8*7 = 56; since 56 > 55, 7/11 > 5/8.

How do you solve ratio and proportion problems with part-to-part vs part-to-whole relationships?

If ratio of boys to girls is 3 : 5, total parts = 3 + 5 = 8. Fraction of boys is 3/8, fraction of girls is 5/8. To find actual quantities from a total N, multiply N by the respective fractions.

What is the formula for the interior and exterior angle sums of an n-sided convex polygon?

• Sum of interior angles = (n - 2) * 180°. In a regular n-gon, each interior angle is [(n - 2) * 180°] / n. • Sum of exterior angles (taking one exterior angle at each vertex) = 360° for any convex polygon.

What are the rules for zero and negative exponents in GRE math?

• For x != 0: x^0 = 1 (0^0 is undefined in GRE). • x^(-n) = 1 / (x^n). • (a / b)^(-n) = (b / a)^n.

What is the formula for the length of an arc and area of a sector with central angle theta (in degrees)?

• Arc Length = (theta / 360°) * 2 * pi * r • Sector Area = (theta / 360°) * pi * r^2.

What is the value of sqrt(x^2) in GRE algebra?

sqrt(x^2) = |x|, NOT necessarily x. If x = -4, sqrt((-4)^2) = sqrt(16) = +4 = |-4|. The radical symbol always denotes the non-negative principal root.

What is the Area formula for a Trapezoid?

Area = [(b1 + b2) / 2] * h, where b1 and b2 are the parallel bases and h is the perpendicular height.

What is the formula for Percentage Change on the GRE?

Percent Change = [(New Value - Original Value) / (Original Value)] * 100%. The denominator must ALWAYS be the baseline/original value.

What is the relationship between the average and median of an Arithmetic Sequence?

In any set of evenly spaced numbers (an arithmetic progression): Mean = Median = (First term + Last term) / 2. Also: Sum = (Number of terms) * Mean.

What is the Slope Formula between two coordinate points (x1, y1) and (x2, y2)?

Slope m = (y2 - y1) / (x2 - x1) = Rise / Run. A horizontal line has m = 0; a vertical line has undefined slope.

What is the Distance Formula between two points (x1, y1) and (x2, y2) in the Cartesian plane?

d = sqrt[(x2 - x1)^2 + (y2 - y1)^2], which is a direct application of the Pythagorean theorem.

What algebraic identities define the difference of squares and perfect square binomials?

• a^2 - b^2 = (a - b)(a + b) • (a + b)^2 = a^2 + 2ab + b^2 • (a - b)^2 = a^2 - 2ab + b^2.

How do you simplify radical expressions like sqrt(48)?

Factor out the largest perfect square factor: sqrt(48) = sqrt(16 * 3) = sqrt(16) * sqrt(3) = 4 * sqrt(3).

What are the slope relationships for parallel and perpendicular lines in the coordinate plane?

• Distinct parallel non-vertical lines have equal slopes (m1 = m2) and different y-intercepts (distinct vertical lines x = c1 and x = c2 are also parallel with undefined slope). • Perpendicular non-vertical/non-horizontal lines have negative reciprocal slopes (m1 * m2 = -1); a horizontal line (m = 0) is perpendicular to a vertical line (undefined slope).

How do you rationalize a fraction with a radical denominator: 5 / (sqrt(3) - 1)?

Multiply numerator and denominator by the conjugate (sqrt(3) + 1): [5 * (sqrt(3) + 1)] / [(sqrt(3) - 1)(sqrt(3) + 1)] = [5(sqrt(3) + 1)] / (3 - 1) = [5(sqrt(3) + 1)] / 2.

What is the Triangle Inequality Theorem?

The length of any side of a triangle must be strictly greater than the positive difference and strictly less than the sum of the other two sides: |a - b| < c < a + b.

What are the common Pythagorean Triples tested on the GRE?

3-4-5, 5-12-13, 8-15-17, 7-24-25, and their integer multiples (e.g., 6-8-10, 9-12-15, 10-24-26).

What are the roots of x^2 = 36 versus the value of sqrt(36)?

• x^2 = 36 has TWO solutions: x = +6 or x = -6. • sqrt(36) has ONE unique value: +6. Radical signs never yield negative numbers on the GRE.

What is the Space Diagonal formula for a rectangular prism with dimensions l, w, h?

Diagonal d = sqrt(l^2 + w^2 + h^2). For a cube with edge length s, d = s * sqrt(3).

What is the GRE divisibility rule for 11?

Subtract and add the digits in an alternating pattern from left to right (e.g., for d1d2d3d4: d1 - d2 + d3 - d4). If the resulting difference is divisible by 11 (including 0), the original number is divisible by 11.

What is the difference between (-x)^n and -x^n for even integers n?

• (-3)^4 = (-3)*(-3)*(-3)*(-3) = +81 (negative sign is inside the parentheses and raised to an even power). • -3^4 = -(3^4) = -81 (order of operations: exponentiation happens before negation).

What are the signs of x and y in the four quadrants of the coordinate plane?

• Quadrant I: (+, +) • Quadrant II: (-, +) • Quadrant III: (-, -) • Quadrant IV: (+, -).

What are the Slope-Intercept and Point-Slope forms of a linear equation?

• Slope-Intercept: y = m*x + b (m is slope, b is y-intercept at (0, b)). • Point-Slope: y - y1 = m*(x - x1).

What is the mathematical translation of 'x is 300% of y' versus 'x is 300% MORE than y'?

• 'x is 300% of y': x = 3.00 * y. • 'x is 300% more than y': x = y + 3.00 * y = 4.00 * y (an increase of 300%).

GRE Quantitative Reasoning: Arithmetic, Algebra, Coordinate & Plane Geometry

32 GRE Quantitative Reasoning review cards on fractions, exponents, radicals, percentages, ratios, arithmetic sequences, linear and absolute-value equations, coordinate geometry, triangles, circles, polygons, cylinders, and rectangular solids.

Level: basic
Languages: en → en
Cards: 32
Markdown: GRE Quantitative Reasoning: Arithmetic, Algebra, Coordinate & Plane Geometry (.md)

Flashcards (32)

#	Front (Term / Question)	Back (Definition / Answer)
1	What is the Midpoint Formula for the segment connecting (x1, y1) and (x2, y2)?	Midpoint M = ((x1 + x2) / 2, (y1 + y2) / 2). It is the average of the x-coordinates and the average of the y-coordinates.
2	What are the rules for multiplying, dividing, and raising powers with exponents?	• (x^a) * (x^b) = x^(a + b) • (x^a) / (x^b) = x^(a - b) • (x^a)^b = x^(a * b) • (x * y)^a = x^a * y^a • (x / y)^a = x^a / y^a.
3	What are the side length ratios for the two high-yield special right triangles on the GRE?	• 45°-45°-90° (Isosceles right): 1 : 1 : sqrt(2) (hypotenuse = side * sqrt(2)). • 30°-60°-90°: 1 : sqrt(3) : 2 (hypotenuse = 2 * short side; long leg opposite 60° = short side * sqrt(3)).
4	What are the Area and Perimeter / Circumference formulas for Rectangles, Triangles, and Circles?	• Rectangle: Area = l * w; Perimeter = 2(l + w) • Triangle: Area = 0.5 * base * height • Circle: Area = pi * r^2; Circumference = 2 * pi * r = pi * d.
5	How do you solve an absolute value equation \|ax + b\| = c (for c >= 0)?	Split into two linear equations: ax + b = c OR ax + b = -c. Solve each equation independently. If c < 0, there is NO solution (absolute values cannot be negative).
6	What is the standard equation of a Circle with center (h, k) and radius r?	(x - h)^2 + (y - k)^2 = r^2. If centered at the origin (0, 0), the equation simplifies to x^2 + y^2 = r^2.
7	What are the Volume and Total Surface Area formulas for a Right Circular Cylinder?	• Volume: V = pi * r^2 * h • Total Surface Area: SA = 2 * pi * r^2 (two circular bases) + 2 * pi * r * h (lateral surface).
8	In GRE coordinate geometry, what does a line equation of the form x = c represent?	A vertical line with undefined slope passing through (c, y). It is perpendicular to the x-axis and parallel to the y-axis.
9	How do you quickly compare two fractions a/b and c/d without common denominators?	Cross-multiplication method: Multiply numerator of one by denominator of the other. Compare a * d with b * c (for positive denominators). If ad > bc, then a/b > c/d. Example: 5/8 vs 7/11 => 511 = 55 vs 87 = 56; since 56 > 55, 7/11 > 5/8.
10	How do you solve ratio and proportion problems with part-to-part vs part-to-whole relationships?	If ratio of boys to girls is 3 : 5, total parts = 3 + 5 = 8. Fraction of boys is 3/8, fraction of girls is 5/8. To find actual quantities from a total N, multiply N by the respective fractions.
11	What is the formula for the interior and exterior angle sums of an n-sided convex polygon?	• Sum of interior angles = (n - 2) * 180°. In a regular n-gon, each interior angle is [(n - 2) * 180°] / n. • Sum of exterior angles (taking one exterior angle at each vertex) = 360° for any convex polygon.
12	What are the rules for zero and negative exponents in GRE math?	• For x != 0: x^0 = 1 (0^0 is undefined in GRE). • x^(-n) = 1 / (x^n). • (a / b)^(-n) = (b / a)^n.
13	What is the formula for the length of an arc and area of a sector with central angle theta (in degrees)?	• Arc Length = (theta / 360°) * 2 * pi * r • Sector Area = (theta / 360°) * pi * r^2.
14	What is the value of sqrt(x^2) in GRE algebra?	sqrt(x^2) = \|x\|, NOT necessarily x. If x = -4, sqrt((-4)^2) = sqrt(16) = +4 = \|-4\|. The radical symbol always denotes the non-negative principal root.
15	What is the Area formula for a Trapezoid?	Area = [(b1 + b2) / 2] * h, where b1 and b2 are the parallel bases and h is the perpendicular height.
16	What is the formula for Percentage Change on the GRE?	Percent Change = [(New Value - Original Value) / (Original Value)] * 100%. The denominator must ALWAYS be the baseline/original value.
17	What is the relationship between the average and median of an Arithmetic Sequence?	In any set of evenly spaced numbers (an arithmetic progression): Mean = Median = (First term + Last term) / 2. Also: Sum = (Number of terms) * Mean.
18	What is the Slope Formula between two coordinate points (x1, y1) and (x2, y2)?	Slope m = (y2 - y1) / (x2 - x1) = Rise / Run. A horizontal line has m = 0; a vertical line has undefined slope.
19	What is the Distance Formula between two points (x1, y1) and (x2, y2) in the Cartesian plane?	d = sqrt[(x2 - x1)^2 + (y2 - y1)^2], which is a direct application of the Pythagorean theorem.
20	What algebraic identities define the difference of squares and perfect square binomials?	• a^2 - b^2 = (a - b)(a + b) • (a + b)^2 = a^2 + 2ab + b^2 • (a - b)^2 = a^2 - 2ab + b^2.
21	How do you simplify radical expressions like sqrt(48)?	Factor out the largest perfect square factor: sqrt(48) = sqrt(16 * 3) = sqrt(16) * sqrt(3) = 4 * sqrt(3).
22	What are the slope relationships for parallel and perpendicular lines in the coordinate plane?	• Distinct parallel non-vertical lines have equal slopes (m1 = m2) and different y-intercepts (distinct vertical lines x = c1 and x = c2 are also parallel with undefined slope). • Perpendicular non-vertical/non-horizontal lines have negative reciprocal slopes (m1 * m2 = -1); a horizontal line (m = 0) is perpendicular to a vertical line (undefined slope).
23	How do you rationalize a fraction with a radical denominator: 5 / (sqrt(3) - 1)?	Multiply numerator and denominator by the conjugate (sqrt(3) + 1): [5 * (sqrt(3) + 1)] / [(sqrt(3) - 1)(sqrt(3) + 1)] = [5(sqrt(3) + 1)] / (3 - 1) = [5(sqrt(3) + 1)] / 2.
24	What is the Triangle Inequality Theorem?	The length of any side of a triangle must be strictly greater than the positive difference and strictly less than the sum of the other two sides: \|a - b\| < c < a + b.
25	What are the common Pythagorean Triples tested on the GRE?	3-4-5, 5-12-13, 8-15-17, 7-24-25, and their integer multiples (e.g., 6-8-10, 9-12-15, 10-24-26).
26	What are the roots of x^2 = 36 versus the value of sqrt(36)?	• x^2 = 36 has TWO solutions: x = +6 or x = -6. • sqrt(36) has ONE unique value: +6. Radical signs never yield negative numbers on the GRE.
27	What is the Space Diagonal formula for a rectangular prism with dimensions l, w, h?	Diagonal d = sqrt(l^2 + w^2 + h^2). For a cube with edge length s, d = s * sqrt(3).
28	What is the GRE divisibility rule for 11?	Subtract and add the digits in an alternating pattern from left to right (e.g., for d1d2d3d4: d1 - d2 + d3 - d4). If the resulting difference is divisible by 11 (including 0), the original number is divisible by 11.
29	What is the difference between (-x)^n and -x^n for even integers n?	• (-3)^4 = (-3)(-3)(-3)*(-3) = +81 (negative sign is inside the parentheses and raised to an even power). • -3^4 = -(3^4) = -81 (order of operations: exponentiation happens before negation).
30	What are the signs of x and y in the four quadrants of the coordinate plane?	• Quadrant I: (+, +) • Quadrant II: (-, +) • Quadrant III: (-, -) • Quadrant IV: (+, -).
31	What are the Slope-Intercept and Point-Slope forms of a linear equation?	• Slope-Intercept: y = mx + b (m is slope, b is y-intercept at (0, b)). • Point-Slope: y - y1 = m(x - x1).
32	What is the mathematical translation of 'x is 300% of y' versus 'x is 300% MORE than y'?	• 'x is 300% of y': x = 3.00 * y. • 'x is 300% more than y': x = y + 3.00 * y = 4.00 * y (an increase of 300%).