GRE - Comprehensive Quantitative Section (420 Questions )

Identify why vertically opposite angles in intersecting lines are equal, showing that X equals Z and that quantity A equals quantity B, guiding the choice in a GRE comparison question.

Apply inequality properties: multiply or divide by a positive number, or add or subtract the same value, preserves order; use 100 to clear decimals, showing 1.76 > 0.176.

Use the rule that even powers of minus one equal plus one and odd powers equal minus one to compare quantities and determine which is larger.

Solve a GRE comparison inequality by simplifying expressions with a positive B, recognizing common factors, and showing C > 1 to conclude that quantity A is greater, selecting option eight.

The remainders are two for both divisions, so the quantities are equal; hence B is the correct answer.

Compute the share of humanities faculty that are non-adjunct by combining 17% of 200 and 14% of 250, yielding about 51%.

Solve a data interpretation problem on percentages of Bronx Zoo animals, determine the total animal count, and compute the minimum births to make birds at least 20 percent.

Analyze a data interpretation problem on zoo animal composition by calculating the share of birds raised in the wild after removing insects and fish.

practice data interpretation with a zoo animal category breakdown (birds, reptiles, amphibians, mammals, insects, fish, others) and compute captive-raised versus wild percentage components.

Compute the data interpretation ratio of engineering students to engineering faculty from the given counts. Reveal that 275 students and 34 faculty yield an approximate 8:1 ratio.

Determine the height range in the chart for the fourth U.S. presidents, with a maximum of 193 cm and a minimum of 163 cm, yielding a 30 cm range.

Compute the percent of U.S. presidents at least 185 cm tall by using the 10 of 43 ratio and converting to a percentage, illustrating data interpretation strategies.

Determine the median in a 43-item data set using the (n+1)/2 rule for odd counts and the average of the middle terms for even counts; example yields 22nd term, 182.

Solve a 1998 data interpretation question about towels, identifying not imported from China. Multiply 22 million by 12 to get 264 million, confirming option B.

Develop data interpretation skills by analyzing monthly imports from China for 2000, estimating the nine-month average, and projecting a 12-month total of about 57.6 million doses.

Practice data interpretation by computing the average annual decline in student population from 46 million to 42 million over eight years, yielding 500,000 per year.

Learn to interpret data on rescue squad calls by category, determine the not-listed category by subtracting the sum of listed categories from the total 1000, yielding 90.

Analyze data interpretation by calculating percentage relationships; determine that 32 percent of 50000 equals 16000, illustrating how percentages compare income contributions.

analyze a data interpretation scenario comparing mortgage expense (24%) and utilities (8%) within a total of 50,000 to compute percent relationships and determine the utilities share.

In this data interpretation lesson, multiply 24% of annual sales by 29% of plant sales to estimate herbs account for about 7 percent of total plants sold in April 2008.

Solve a data interpretation problem about four tomato varieties—red giants, mortgage lifters, beefsteaks, sun gods—using 1000 plants, 29% and 40% calculations to find possible beefsteak counts.

Analyze which statements about calories burned during various activities are possible for different weights, comparing hourly and total calories to determine which choices could be true.

Use the data interpretation chart to identify the combination of activities with the fewest total calories per hour, noting that typing and crocket burn fewer calories, as per choice e.

Calculate the percentage of days in San Luis in July with a maximum temperature of 94 or more, using 44 total days and 15 qualifying days, yielding about 34 percent.

Learn to differentiate actual revenue numbers from percentage changes and apply per customer revenue analysis across stores to determine which statement must be true in data interpretation questions.

Analyze data interpretation in the gre comprehensive quantitative section by comparing 1976 private versus public health expenditures, showing a near 90 to 60 billion ratio, simplified to 3:2.

Analyze a gre data interpretation question on 2016 downloads, calculating six to eleven million inclusive out of 40 million to identify b and f.

Interpret a data interpretation question on private four-year enrollment, calculate 1972 value as 22% of five million, and apply a 50% increase to estimate about 1.65 million in 1995.

Compute the percentage of African countries with a GDP between 10 billion and 20 billion that also have a population between 10 million and 20 million, yielding about 23 percent.

The lecture explains how to find the median from a frequency distribution of televisions per household, using a 100-household example and even-number handling.

Compute the average number of televisions per household by summing the products 0×16, 1×8, 2×24, 3×20, 4×20, then divide by 100; the five or more category prevents a definite mean.

Explore data interpretation strategies for a GRE quantitative question on households with televisions, using ratio setup, cross-multiplication, and percentage comparisons to determine the number of households with four televisions.

Analyze a 1984 data interpretation chart to determine which of seven categories exceed nine percent of a family's gross annual income, with the solution indicating five categories.

Analyze a data interpretation question by identifying two categories totaling 49 percent in 1983, then compute 39 percent of a $45,000 income in 1984 to find the total amount.

practice data interpretation by converting percentages to actual amounts, using yearly incomes of 50,000 and 45,000 to find increases from 1,500 to 4,500, a 3,000 gain.

Apply the percentage decrease formula (decrease amount over original value times 100) to determine Family X's savings decline from 1983 to 1984, yielding about 56.8 percent.

Explore a GRE data interpretation problem where 1983 incomes are 25,000 each, Mr. X rises to 27,500 in 1984, leaving Mrs. X with 17,500 and a 30 percent decrease.

In 1973, total health expenditures were about 9 percent of gross national product, as shown by the data interpretation of expenditures near 40 billion.

Calculate the percentage of 25 million out of 150 million in the 2001 workforce pie chart to identify categories above 16.6%, revealing agriculture, clerical, and manufacturing.

This data interpretation question derives the 2001 to 2015 ratio by 18 percent of 150 million and 24 percent of 175 million, yielding 9:14.

Explore data interpretation of projected workforce changes from 2001 to 2015 across multiple categories using percentages and totals. The analysis shows all categories increase in workers over the period.

Analyze data interpretation by comparing per-student expenditure to student population across years to identify the year with the highest per-student expenditure.

Compute per-student expenditures by dividing 80 billion by 42 million, yielding about $1,900, with 2000 as the closest option.

Analyze year-over-year population changes of at least one million in a data interpretation question, compare gradients and identify when changes exceed one million, selecting option B as correct.

Analyze a data interpretation question to identify the third least category on a graph of rescue squad calls, noting overdose as a reference point.

Compute the percent increase in total school expenditures from 1973 to 1980, using 45 million to 80 million, with an increase of 35 million and about 77.8%.

Analyze the 1980 profit-division data to compute the average of 10% and 20%, yielding 4.5 million from a 30 million total.

Assess data interpretation of profits and division contributions: interpret 17% of 20 million and 11% of 30 million into dollars, yielding about 3.4 million and 3.3 million.

Analyze a data interpretation question by comparing division contributions, noting division P at 30% and division Q at 3%, then compute the percent relationship to conclude 1000%.

Practice data interpretation by comparing six divisions' profit contributions across 1979 and 1980, using percentage rankings to determine how many divisions dropped in rank.

Determine which divisions added more dollars to profits in 1980 by comparing percent contributions against rising total profits, revealing four divisions increased their dollar contributions.

Analyze a data interpretation item to determine the percentage of African countries with population under 20 million by evaluating the intersection of two sets, yielding 62%.

Solve a data interpretation question by using a pie chart showing training 6%, first aid 10%, paramedic 8% of expenses, with total expenses $50,000, yielding $12,000.

Explore data interpretation by calculating annual earnings from a fixed monthly increase in salary using an arithmetic progression; a 12-month sequence from 400 to 950 sums to 8,100.

Compute the percent by which exports exceed imports in 2008 by dividing the 7 billion difference by the 5 billion import and multiplying by 100, yielding 140 percent.

Practice data interpretation by correcting an unemployment rate from 8.5% to 6% and calculating the ten-year average, yielding a 0.25 percentage point difference.

Data interpretation focuses on estimating the number of intellectual property lawyers under 30 who specialize in copyright law, with 26 percent in industrial property and 74 percent in copyright.

Compute the mean from a relative frequency distribution by summing x times its relative frequency, with a worked example yielding 2.03.

Compute the mean of the random variable x from its relative frequency distribution by summing x times its relative frequency; yields 2.03 for the given data.

Convert the relative frequencies to counts for 100 trials, then apply the even-N median rule. The 50th and 51st observations fall in 1, so the median is 1.

Estimate the gross national product in 1968 by interpreting health expenditures as a percent of gross national product and use cross-multiplication to approximate nine hundred and fifty billion dollars.

Interpret a data interpretation question about 1969 private health expenditure, identifying an approximate value of four to five billion dollars.

Identify the first year when public health expenditures reach at least 30 billion dollars; the data show 1960–69 below the threshold and 1970 meets it.

Explore data interpretation using a box and whisker plot to determine the range by identifying the minimum value of 105 and the maximum of 146, yielding a range of 41.

Learn to read a box and whisker plot to identify the lower quartile at 114 and upper quartile at 126, and compute the quartile range of 12.

Identify the data set's statistics by locating the minimum 105, maximum 146, median 118, the third quartile range 12, and the total range 41.

Learn to interpret a box plot by using quartiles and percentiles to estimate counts, noting that 5% of 800 equals about 40 measurements between the 75th and 80th percentiles.

Identify the category with the greatest percent increase from 2003 to 2004: miscellaneous expenses rose from 3% to about 9%, the largest increase among seven categories.

Compare the slopes of the lines on the total expenditures graph to identify the greatest increase, which occurs from 1997 to 1998.

Compute the percent of total expenditures represented by private school expenditures in 2001 by forming and simplifying the fraction to 18.75%, rounded to 19%.

Learn to read a chart of countries by GDP brackets and population bands, and sum category counts to answer data interpretation questions.

Explore a data interpretation approach to finding a combined mean: merge two groups of 20 and 30 values with means 85 and 75 to get a total mean of 79.

Learn how to determine the median in data interpretation, ordering numbers and handling even or odd sets. Merging groups reveals the overall median cannot be determined without exact values.

Compute the mean of nine airline passenger counts by summing the numbers and dividing by nine. Confirm the sum is 270, yielding a mean of 30.

Sort the nine heights in ascending order and identify the median as the fifth value; with nine numbers, the median is twenty eight.

Sort passenger counts to reveal the median and quartiles, then compute the interquartile range, with lower quartile 21.5 and upper quartile 38.5, yielding 17.

Analyze how the percent of total health expenditure differs across categories between 1979 and 1950; three categories show greater 1979 values, so option B is correct.

Analyze the data interpretation graph of health expenditures to infer dollar amounts and percentages for 1950 and 1979, focusing on construction, physician, and dental services.

Identify the category with the least share of total health expenditures in 1979, as the caption states that research is the least.

Analyze 1979 health expenditure data to determine how many categories fall below 21 billion dollars or below 10 percent of total expenditures, using approximate percentages from the graph.

Calculate the ratio of hospital care expenditures in 1979 versus 1950, convert percentages to totals, and approximate to identify the correct multiple-choice answer.

Determine the approximate difference between the energy costs range and the tax range by max minus min; energy costs range is about 12, tax range about 7, difference about 5.

Analyze a data-interpretation task comparing male to female lawyer counts across age brackets using a 2015 graph, where the over-70 group shows a near-equal ratio and is the lowest.

Analyze a data interpretation scenario by applying 11% to 400,000 lawyers aged 51–60. Estimate the tax-law subset as about 44,000, and identify closest answer from the options.

Explore data interpretation in the GRE quantitative section by analyzing stress factors, difficulty factors, and average time per question to identify the highest stress scenario.

Solve for k from the given equation by substituting x with 2k, obtaining k^2 = 8, and taking square roots to get k = ±2√2, with only positive option shown.

Demonstrate exponent rules for roots, showing how to combine powers and simplify expressions like k^(1/6) = m^2, concluding with k = 12.

Demonstrates that a 25% price increase followed by another 25% increases the original price by 56.25 percent, illustrated with an example using price 16.

Apply modular arithmetic to a GRE quantitative problem: find A with A ≡ 2 mod 5 and A ≡ 5 mod 6, A < 40, then compute A mod 7.

Solve a GRE comprehensive quantitative section question by converting division to multiplication, flipping a fraction, and simplifying 10/45 to 2/9.

Use the difference of squares rule, a^2 - b^2 = (a-b)(a+b), to 49 and 35, yielding 14 and 84, cancel 14, and obtain 84.

Using pairwise averages, B plus C equals ten and C plus D equals twenty. Subtract to find B minus D equals minus ten, showing how arithmetic mean solves variable relationships.

Explore prime factorization of numbers like 12 and 54, apply base exponent rules to combine terms, use square roots, and arrive at a final result of 944.

Learn to determine the median of 24 consecutive odd integers when their average arithmetic mean is 48 by averaging the two central terms.

Reveal the prime factorization of 462 as 2, 3, 7, and 11, and show that 22 is a valid divisor, illustrating divisibility and factor-based problem solving.

Apply exponent rules to combine powers of ten and coefficients, showing the result equals two point four times ten to the power fifty one.

Analyze why even powers preserve positivity of negative inputs, and compare F(-1)^2 and F(2) by substitution, showing twelve is greater than three-fourths, concluding the correct choice.

Apply exponent rules to relate x and y by taking reciprocal sixteenth powers on both sides, use (a^b)^c = a^{bc} to simplify, cancel the exponents, and conclude y = x^4.

Pick values for P and Q within the given ranges to solve a GRE quantitative multiple choice question, showing how the closest pair leads to option C.

This lecture explains how to obtain an equivalent linear inequality by isolating x, moving terms, and dividing by a negative number, which flips the inequality sign, yielding x > 10.

Evaluate one minus x squared for multiple candidate values of x, simplify step by step, and show that the expression equals 81 across all cases.

Solve k^3 minus five equals three to get k^3 equals eight; take the cube root to conclude k equals two.

Apply the laws of exponents for like bases to combine powers via multiplication and division, simplify the problem, and reach the solution 1.

Solve a GRE quantitative mean problem by using the given average of X, Y, and 20 to find X+Y, then compute the average of 2X+3, 2Y-4, and 8, yielding 11.

Solve a GRE quantitative problem on cross-multiplication with fractions, noting denominators cannot be zero and X cannot be minus two or seven, leading to X equals two.

The lecture demonstrates simplifying a rational expression by factoring and cross-multiplication to show it equals (x+6)/(x+2) for defined values of x.

Compute the percent decrease from 2105 to 1705, a 400 drop relative to the original 2105, illustrating how to estimate a roughly 19 percent decrease.

Apply the distance from the origin in the xy-plane using Pythagoras: x^2 + y^2 = 40, identifying coordinates like (6,2) that satisfy the equation.

Solve a GRE problem by handling three x equals two y equals five, substitute x = 5/3 and y = 5/2, and compute 24x and y squared to get 250.

Illustrates how squaring both sides and comparing k squared to 839 shows that 28 squared < k squared < 29 squared, so k lies between 28 and 29.

Convert percentage to fraction and apply reciprocal multiplication to divide fractions, showing that 0.25% equals 1/400.

Compute the percent of non-white marbles by using the total X and white marbles Y: percent non-white equals (X − Y) divided by X, times 100.

Compute the probability of drawing the numbers one, two, three, four without replacement in strict order, yielding 1/24 as the final result in the GRE comprehensive quantitative section.

Solve a GRE quantitative problem by using Y = X/5 to form the 2-to-3 ratio for Y, substitute to express X or Y, and compute the ratio, yielding 10/3.

Apply a two-step recurrence to compute the fifth term: a_n = a_{n-1}^2 − 2 a_{n-2} with a1 = 2 and a2 = 3, yielding a5 = 351.

Learn how standard deviation measures how numbers spread around the mean. Compare sets with different gaps; the largest spread yields the greatest standard deviation, indicating higher dispersion.

Relate 75% of X to 125% of Y, simplify to 3X = 5Y, derive X = 5 and Y = 3, hence Y is 60% of X.

Maximize the expression by assigning A, B, C from 2, 3, 5 so largest number is in the numerator and the smallest in the denominator, e.g., A=5, B=3, C=2.

Analyze key GRE quantitative concepts, including absolute value, modulus, and sign rules for sums, products, and division.

Solve a remainder problem by dividing by six with remainder three, and express K as 6x+3. Apply even-odd parity rules to determine which expression cannot be an even integer.

Solve a GRE quantitative item using substitution and elimination on a linear system, deriving Y equals four times X from simplified equations.

Master evaluating expressions with negative exponents and fractions by applying exponent rules, converting powers to reciprocals, finding common denominators, and expanding fractions for simplification.

Solve the linear relation 2x - 3y = 6 by multiplying by -1 and then by 2, using distribution to show that 6y - 4x = -12.

Convert 0.000125 to a fraction by expressing it as 125 times 10 to the power minus six, which equals 1/8000.

Compute the missing score in a ten-student distribution using the mean of 62, with scores 40, two at 55, three at 70, and four at X.

Apply the difference of squares rule, a^2 - b^2 = (a-b)(a+b). Substitute a = 2√3 and b = √5 to get 12 - 5 = 7.

Analyze a recurrence sequence with t1 = -2 and t2 = -1, deriving t3 = 1 and t4 = 5, and note sign rules: opposite signs minus, same signs plus.

Practice solving a percent-proportion problem where P percent of 160 equals Q percent of 40, using percent definitions and cross-multiplication to relate P and Q.

Apply the difference of squares and algebraic simplifications to evaluate a GRE quantitative question involving x and y, illustrating how to simplify complex expressions to a final value.

Analyze ratio problems by identifying integer counts of girls and boys summing to 24, testing splits like 3:5 and 1:1 to determine feasible ratios.

Show that a two-digit number with digits A and B, whose digits are reversed to form K, must be divisible by eleven.

Factor the difference of squares and set X minus Y equal to 4 to derive X plus Y equal to 3; solve the two-equation system to obtain X equals 3.5.

Learn how to minimize the product of four distinct integers from -6 to 10 by analyzing sign patterns and the zero possibility.

Analyze how to find the remainder modulo 10 for x^2+4x+9 when x+2 is divisible by 10, using x=8 as an example; determine the final remainder is 5.

Explore how adding a fifth number changes the average of five numbers, set up the sum relationships, and solve the resulting algebraic equations using cross-multiplication and expansion.

Determine the area of an isosceles right triangle with hypotenuse eight by applying Pythagoras to find equal legs and using the area formula.

Compute the ratio of the small shaded region to the large shaded region using square areas: second minus first (18-12) and fourth minus third (50-32), yielding 6:18 or 1:3.

Compare the area of triangle DC to the area of rectangle ABCDE by using base times height over two, and cancel terms to reveal a ratio of 1/2.

Solve linear angle equations from the GRE quantitative section by equating 2x+40 to 5x-35 and using the supplementary relation 2x+40 + 2y+24 = 180. Derive x=25 and y=33.

Analyze a coordinate-plane line with positive slope that avoids the first quadrant, using acute and obtuse angles to test if line g is perpendicular to hedge or intersects line j.

Identify the line equation using slope-intercept and point-slope forms, verify candidate points against y = mx + c, and determine which point does not lie on the line.

Use the difference of squares to relate the shaded area to the rectangle’s dimensions, showing the area equals (Y-X)(Y+X). Determine the rectangle’s width as Y-X via the identity Y^2-X^2=(Y-X)(Y+X).

Locate A and B on the x-axis in the x-y coordinate system, determine base AB and height from C’s ordinate, then compute area as base times height over two.

Learn to compute the area of a triangle from coordinates by identifying base and height using the base times height over 2 formula, with a worked example.

This lecture derives line K's slope from a triangle with height 12 and base 5, yielding x-intercept 5, y-intercept 12, and a negative slope -12/5 due to an obtuse angle.

Derive the possible equation of a line given two parallel lines and a point, determine slope m as 2/5, and obtain y = (2/5) x + 2.

Calculate the slope of line K from two given points and apply the two-point formula to find the third point's x-coordinate, yielding x = 6.

Solves a line passing through the origin by finding k, using distance ratios and slope relations, yielding k = -24.5.

Find the x-intercept of the line 3y = 2x + 6 by setting y to zero, solving for x, which yields x = -3; hence the intercept A equals -3.

Analyze lines P and Q with positive slopes and positive y-intercepts, determine their intersection, and identify that the intersection can lie in the first, second, or third quadrant.

Apply Pythagoras and triangle similarity to relate X and Y and identify the shaded region. Compute its area as root three over two X squared minus Y squared.

Apply Pythagoras to a right triangle, form X^2 - X - 6 = 0, and obtain X = 3 using 3-4-5 and 5-12-13 triangles; the answer is B.

Derive the equation of a line from its intercepts: x-intercept minus two and y-intercept three, yielding x/(-2) + y/3 = 1, which simplifies to -3x + 2y = 6.

Derive how a volume ratio of 1:8 between cubes translates to a side ratio of 1:2 using cube roots, then compute the surface area ratio as 1:4.

Compute the shaded region as the square area minus the circle area, using Pythagoras to link sides; square area is 64, circle area is 16 pi.

Solve a circle-based geometry problem using radii to form an equilateral triangle, identify 30-60-90 relationships, and compute area with sqrt(3)/4 a^2; for side six, area is 9 sqrt(3).

Use a 30-60-90 triangle inside a circle to find the diameter as eight, then compute the area as pi r^2 with radius four, yielding 16 pi.

Compute coordinates inside a square using 30-60-90 triangles, relating side lengths to the hypotenuse and opposite angles to determine the x and y.

Apply the exterior angles sum of 360 degrees and the equal interior angles of a regular hexagon to determine angles M and Q, using isosceles triangle reasoning.

Apply the Pythagorean theorem to a right triangle and bound x by comparing squares, showing that x lies between seven and eight.

In this geometry problem, solve the perimeter of the ABCDE pentagon using symmetry and a 30-60-90 triangle, finding each side length as 12 to obtain a perimeter of 60.

In the Cordner system airball problem, determine Limpy's equation by checking intercepts: x-intercept 6 and y-intercept 2.5 confirm option a.

solve a geometry-based gre question by angle chasing with parallel lines, using the interior angles sum of a triangle, to find x as 75 degrees.

Determine the side length of the smaller shaded square from a big square area of 25 and the nine times area relationship, yielding a smaller side of 5/3.

Compute the shaded region's area in a large square subdivided into smaller squares, using triangles and rectangles, for a GRE quantitative question, arriving at 24 square centimeters.

Apply supplementary angle relationships to an 85-degree CFT to determine angle a as 50 degrees. Solve for x using a + x = 180 degrees, yielding x = 130 degrees.

The lecture explains ranking points P, Q, and R by their distance to the origin using d = sqrt(x^2+y^2), comparing x^2+y^2 values to order from smallest to largest.

Compute angle measures in a regular pentagon by applying exterior-angle sums and isosceles properties, concluding X = 36° and A = 54°.

Apply the circle circumference formula, using diameter sqrt(5) x, to compute the circumference as pi times sqrt(5) x.

Using a circle area of 9 pi and 30-60-90 triangle ratios, the lecture derives an equilateral side length of 4 and applies the area formula (sqrt(3)/4)s^2 to obtain 4 sqrt(3).

Solve a geometry problem about outer square area four x squared and inner square area two x squared, using a 45-45-90 triangle.

Solve for W in terms of X and Y by applying triangle angle sums and algebra, deriving expressions from interior angles totaling 180 degrees.

Determine the rectangle's sides from a perimeter of 20 and a diagonal of 9 using a+b=10 and a^2+b^2=81. Use (a+b)^2 = a^2+2ab+b^2 to find ab=19/2, giving the area 19/2.

This lecture uses a circle with diameter 10 and a 6-length segment to apply Thales' theorem and Pythagoras, showing the angle inside exceeds 90 degrees and BC is under 8.

Solve a GRE quantitative geometry problem using parallel lines, isosceles properties, and interior angle sums to determine x in a quadrilateral, yielding x = 80.

Determine the probability that a random point in a four by three rectangle lies above the line y equals x, yielding 3/8.

Apply Pythagoras to a composite figure, using 3-4-5, 5-12-13, and 7-24-25 right triangles to find base, height, and area, then sum triangle and rectangle areas.

Explore a sphere inside a cylinder and identify the complete set of contact points. Reveal that the cylinder touches the sphere at two end points and along a central circle.

The lecture demonstrates how a 50% side increase affects the area of an equilateral triangle, using the area formula (√3/4) a^2 and 30-60-90 triangle reasoning.

Analyze the diagram to determine the unknown angle using degree measures, equal angles, and the 180-degree angle sum, yielding x equals 45.

Compute the area of triangle abc using coordinate geometry; determine base and height from the coordinates and apply base times height over two.

Count triangles of any orientation formed from a grid of equally spaced points. Sum per row to reach a total of 120 triangles.

an isosceles triangle problem uses angle relationships and the triangle sum to express y in terms of x, solving for the angle measures.

Calculate the percent decrease in volume when all linear dimensions shrink by 60 percent in a cylinder with diameter equal to height; volume decreases by 93.6 percent.

Count right triangles formed by three dots in an 11 by 11 grid by analyzing rectangle shapes and diagonals, yielding 360 triangles per orientation and 720 total.

Explore how a retailer's 80 percent wholesale-to-retail markup converts 100 into 180, and how a 30 percent price drop equals a 26 percent wholesale increase to achieve the same effect.

Solve a multi-team ratio problem by converting 3:2 and 4:5 into a common base, use a total of 300 fans, and deduce that Mets fans equal 80.

Solve a three-kind high school distribution across districts using totals and equal-size constraints to determine the number of private independent schools in district a.

Convert 90 kilometers per hour to meters per second, apply the distance equals velocity times time formula, and determine that 600 meters takes about 24 seconds.

Determine the original price after two sequential discounts of 25% and 50% with a final price of $60, yielding an original price of $160.

In the GRE comprehensive quantitative section, multiple choice question 106 shows dividing 3.6e-8 amps by 1000 to yield 3.6e-11 amps in a delicate circuit.

This GRE quantitative problem asks for the least wins to reach 60 percent to qualify for year end tournament, given 14 of 18 won and 30 remaining; conclude five more.

Compute the overall MBA rate by weighting 30% male and 50% female MBA with a 60/40 gender split. The result is that 62% of employees do not have an MBA.

Solve a system of linear equations to find the chocolate bar price from gumball and lollipop costs, using elimination, yielding the chocolate price of 0.71 dollars.

Explore a GRE quantitative problem on a console price increase, use cross-multiplication and ratios to find the original price from a 50 percent rise and a 240 dollar purchase.

Solve a multiple choice quantitative problem where three friends' contributions, including Techland's four dollars, match fractional shares of the total to yield a sixty-dollar gift price.

Determine the maximum a child can receive when distributing 100 candies to 10 children, each at least one and all different, by giving nine children 1–9 and 55 to last.

Compute the combined pumping rates of two pumps (3 and 2 hours to empty a pool) and show they empty five sixths of pool per hour, finishing in 75 minutes.

Solve a linear problem to find Nina's money. Equate six widgets at price x to eight widgets at x minus 1.25, giving x = 5 and money = 30 dollars.

Solve a four-person age problem using algebra, with Ben as three times Ron, Jack and Ken younger by seven, and a total of 161 to reveal Ron’s age of 22.

Set up 85 + 5(x - 1) = 365 to model the first kilometer cost plus the remaining distance, solving yields x = 57 kilometers.

A produces 350 widgets per hour and B produces 250, for a combined 600 widgets per hour. To reach 1000 widgets, they need 1 hour 40 minutes.

Form and solve a linear population equation from Appleton's comparison, 3X+400 = 3(X−900), to determine X = 1550.

Solve a GRE quantitative problem by setting the dog's weight as three times the average weight of the five kids, yielding the fraction 3/8 of the total six animals.

Use proportional reasoning to relate marble volume to water rise, then cross-multiply to determine that 22 marbles raise the water by 2 3/4 inches.

Solve a GRE ratio problem by cross multiplying to estimate how many liters of white paint match 350 liters of black paint, concluding the amount is less than 175 liters.

Use a universal diagram to determine how many of 50 students study both French and Spanish, given 31 French, 17 Spanish, and 10 neither, resulting in eight.

Determine weekly allowance by tracking fractions spent: three fifths at the arcade, three fifths of the remaining at the toy store, and 0.8 at the candy store; solve for total.

Compute the ratio of boys to girls from their weights: 60 for boys, 48 for girls, and an overall average of 50. Derive B:G = 1:5.

Solve a GRE quantitative problem about a barrel's capacity: from one-fifth full, add liquid to form a linear equation and express volume V in terms of K.

practice calculating compound interest with semiannual compounding, converting a 3.96% annual rate to about 4%, to estimate future value after two years for a $10,000 investment.

Compute Cape Town's share of the total population by modeling Cape Town as one fourth of the other cities, resulting in a 20 percent share.

Solve a GRE quantitative problem by modeling a ratio and percent growth: four times as many apples as oranges, 15% and 10% growth, totaling 420 trees, to find x.

Determine the television advertising expenditure as 50x when 60 percent of total revenue is allocated to advertising and five sixth of that advertising is spent on television.

Applies the distance equals speed times time to a GRE quantitative section problem about a two-part bike trip totaling 120 miles, with speeds of 25 mph and 50 mph.

Compute the three vice presidents' average salary by using 15 executives at 80k, 82 data-entry staff at 25k, and the overall 44.5k across 100 employees, yielding 400k annually.

Compare compounding frequencies for a $5 million principal at 7% over two years; daily compounding yields the largest total, while annual yields the least.

Convert six minutes to hours and convert kilometers to meters, then multiply velocity by time to determine the bridge length in meters, and the result is 800 meters.

Explore how a price per transaction and the number of transactions change by a percent, and compute the resulting percent increase in revenue from the prior year.

Resolve a weighted average speed problem: five hours at 120 km/h, remainder at 180 km/h, overall average 180 km/h, giving a total trip duration of 30 hours.

Explore a probability problem using a standard deck of 52 cards: determine the chance that the first heart appears on the third draw or later, with replacement after non-hearts.

Determine the return speed required to obtain a 6 km/h overall average for a round trip from A to B, given a 5 km/h outbound leg.

Tackle a two-leg 100-mile trip with p miles at 30 mph and the rest at 50 mph, and derive the overall average speed by calculating total time.

Calculate the amount of water to add to a four-quart alcohol and four-quart water mixture to achieve a 3:5 alcohol-to-water ratio; add 8/3 quarts of water.

Solve a time-distance problem: determine the usual speed x for a 280 km trip when a 30-minute late departure and a 7/6 times faster speed yield regular arrival.

Calculate the percentage of all students who walked to school by combining two classes, using 40% of 35 and 80% of 45, to obtain 62.5%.

Apply the distance equals speed times time formula to a GRE comprehensive quantitative section two-leg journey with speeds 115 and 135 km/h. Compute A to B time in minutes.

Examine how a retailer's 8% markup becomes a 180% sale price, and how a 30% markdown yields a final price of 126% of wholesale, a 26% increase.

Calculate the total Sara must pay for shoes, earrings, and a dance ticket, including an eight percent tax on shoes and earrings, arriving at 93.04 dollars.

Compare Peter's 12 percent annual interest paid at year end to Martha's monthly-compounded $100,000, and estimate the approximate difference after one year.

Solve a GRE quantitative ratio problem with pigs, cows, and chickens in 7:8:10, total 300, yielding 120 chickens.

Solve a GRE quantitative problem to set the collection price P that nets a target profit Z, given total cost W, J crates, and Q gift boxes per crate.

Count positive integers below 100 that leave remainder 2 when divided by 30 by listing numbers of the form 30k+2 and counting until 100.

This question shows that a 25% increase from 2005 to 2006 makes the account balance rise from 5,000 to 6,250 dollars, based on earlier 2004–2005 growth.

Calculate the wholesale price per cup from a bulk order of 692 dollars for 80 cups, compare it to the single cup price of 12.50, and determine the difference.

Rearrange a division with remainder problem to isolate E, showing that E equals Q minus W as the subject of the equation.

this gre quantitative quiz models seven years ago age relationships, deriving bob’s present age b = 4k + 7 in terms of k, with kate currently 11.

Analyze chords on a circle of radius 2 to show that AB exceeds 2 when the central angle is greater than 120 degrees, yielding a probability of 2/3.

Determine the radius of a circle tangent to each other and to the sides of an equilateral triangle of side two, using 30-60-90 relationships; the answer is E.

Fifteen equally spaced points on a circle yield 455 total triangles when choosing any three points. Five equilateral triangles exist, so 455 minus 5 equals 450 non-equilateral triangles.

Analyze a concert revenue problem with orchestra tickets at $50 and balcony tickets at $30, where balcony revenue share is B; set up equations and solve for B.

Solve a ratio-based problem converting eight minutes for 30 potatoes into an hourly rate using cross-multiplication, arrive at 225 potatoes, and memorize square numbers to speed calculations.

Solve a GRE quantitative word problem by using algebra to compare album sales, determine counts divisible by five, and identify options exceeding seven thousand five hundred.

Examine a GRE problem aligning shift cycles: Edward's 10-day cycle (7 work, 3 off) and Clara's 7-day cycle (5 work, 2 off) using linear equations to find common off days.

Practice evaluating expressions with brackets and basic arithmetic to determine which expressions equal twenty one.

determine which three negative numbers yield a product less than minus one by comparing their modulus values and signs. select the correct triple from the options.

Covers exponent rules for the GRE comprehensive quantitative section, including power of a power and base conversion, to evaluate expressions like eight to the twenty-four and pick the correct answer.

Analyze and compare means, medians, standard deviations, and ranges for two data sets A and B, with examples using consecutive numbers, to determine which statements are true.

Practice converting twelve point twelve times ten to the minus three into decimal form. Shift the decimal appropriately and compare options to identify all values equal to the expression.

Analyze which statements about the signs of A, B, and C must be true given even powers and positive products, concluding that C must be correct regardless of B's sign.

Walks through evaluating and simplifying equations with radicals, tests answer choices by substitution, squares terms, and cancels radicals to identify the correct options.

Solve a GRE quantitative problem by converting divisions to multiplications using reciprocals, simplifying fractions and negatives, and selecting the option pair that yields a sum between 1 and 2.

Analyze a GRE quantitative data set of 149 scores, exploring mean, median, distribution shapes, and counterexamples; assess statements about normality and mean–median equality.

The lecture explains how multiplying by powers of ten shifts the decimal point, using positive and negative exponents to find equivalent representations of a small number.

Explore solving proportional can-count problems using variable x, ratios, and integer totals; identify seven-multiple totals and select valid options in a GRE quantitative context.

Analyze a normal distribution with Jamal at mean plus two standard deviations and Charlie at the fifth percentile, compare their distances to the mean, and identify statements for 500 students.

Show that if A, B, C are multiples of three, then A+B+C and A-B+C are divisible by three, since A=3X, B=3Y, C=3Z with X, Y, Z integers.

Factor X^2−Y^2 as (X−Y)(X+Y) to derive X=0 or X=±Y, and use modulus properties to show X^2/Y^2=1, concluding that statements B and C are true.

Examine how A and B signs affect the inequality 3A > 4B, testing positive and negative cases to determine which relationships could be true.

Analyze a GRE quantitative question involving non-zero integers X, Y, Z and inequalities. Learn how multiplying or dividing by negatives flips the inequality.

Analyze inequalities to determine whether there are both positive and negative solutions. Use root analysis at -1, 0, and 1 and sign analysis to identify negative intervals.

Compare the sale price of two notebooks at 99 cents with the regular 59-cent price, showing a 19-cent saving per pair and concluding that 11 notebooks exceed 95-cent savings.

Solve a GRE quantitative problem on ratios and percentages in a dressing recipe, using cross-multiplication to compare vinegar, oil, and water volumes and evaluate statements.

Determine which statements give the necessary information to state that Y is a factor of X, using divisibility, X = Yk, and prime-factor considerations.

Learn to use slope, distance, and midpoint to locate points on a line, apply Pythagoras, and verify coordinates for a GRE quantitative problem.

Analyze normally distributed variables X and Y, noting that Y has greater spread and different mean positioning. Evaluate two-standard-deviation probabilities for X and Y.

Explore bulk configurations of four six-by-three-by-two inch boxes and apply the surface-area formula 2(lw+lh+wh) to compare possible total surface areas.

The lecture analyzes a data sufficiency-style problem about penguin heights, using chinstrap 13.2 cm and gentoo 15.4 cm ranges to determine the overall height range.

Analyze how modulus and squaring interact for complex numbers to determine true statements about Z within -1 to 1, including comparisons like Z^2 versus Z.

Translate percent statements into algebraic expressions, using 'percent of' as multiplication and multiplying by 100 to form fractions. Identify options, such as B and E, for the GRE quantitative question.

We compare two machines on 24-hour and 48-hour cycles (20/4 and 40/8) and trace rest overlaps from Monday noon to Saturday noon.

Explore a GRE quantitative section problem on inequalities and exponent rules. Analyze X between minus one and zero, powers, and comparisons to determine which statements hold.

Convert mixed numbers to improper fractions, multiply and simplify to find equivalent fractions, and identify correct choices (A, B, E) in a GRE quantitative problem.

Learn to solve a rational equation by factoring quadratics, canceling common factors, and applying domain restrictions to exclude x = -12 and x = 2.

Compare binomial expansion terms to determine the greatest value between options B and C, showing that the final result favors A as the greatest value.

Practice solving GRE quantitative multiple-choice questions on digits and discounting zeros, evaluating which options are true, with examples illustrating how zeros affect answers.

Identify which values of a could be the square of an integer, noting that four, nine, sixteen, and twenty-five are not squares, and apply square-root and difference-of-squares ideas.

Explore how multiplying by a negative number reverses an inequality, and show that A is the correct equivalent for all nonzero A, B, C.

Explore P(A or B) given P(A)=1/2 and P(B)=1/3, with B a subset of A. Determine that P(A or B) ranges from 1/2 to 5/6, identifying possible values within this interval.

Identify when two equations determine X and Y, using examples like X+Y=4 and X-Y=4, and explain why inequalities may not suffice.

Explains how to identify operations that preserve equivalent fractions, using cross multiplication, reciprocals, and multiplication or division by a factor to confirm two fractions are equal.

Calculate triangle area using base times height over two, with base five and height four yielding area ten; vertex C on the line y=4 preserves height and area.

Apply probability principles to determine the chance a sea turtle lays eggs at John's house, given different street layouts and counts of beachside and landside houses.

Solve a three-set distribution problem for L, M, and Eliz by using totals 9300, 7100, and 5200 to determine overlap and verify which statements are true.

Test x values such as −5, −4, 0, 3, and 5 to determine which points lie on y = (3x^2 + 2)/(x − 1) and satisfy the equation.

This lecture uses the average age formula for a city block, with four buildings at 2 years and none beyond 80, to determine feasible building counts.

Solve a GRE quantitative problem on integers by using the average of three numbers, A+B+C=33, and parity reasoning to identify viable values for A, B, and C.

Explain a GRE quantitative problem on divisibility involving 41k and y, show how k must be a factor and deduce that only option B must be divisible by y.

Master prime factorization and the divisor-counting formula to determine the number of positive divisors, and recognize when that count is odd.

apply percent changes across years by using a 10 percent increase followed by a 5 percent decrease to compute price bounds and identify feasible 2008 car prices.

Learn how to count lunch orders using combinations by selecting 2, 3, or 4 lunches from 10 options, calculating 10 choose r (45, 120, 210).

Identify the unit digit of 57^N for positive integer N by recognizing a four-step cycle of unit digits 7, 9, 3, 1; the result repeats every four powers.

Explore how to manipulate inequalities, apply negative multiplication rules, and determine when expressions like x+y are positive, with practice on comparing x and y.

Resolve a GRE quantitative puzzle by converting a ratio into a product, applying reciprocal, cancellation, and inequality reasoning to bound 3x and identify equal choices B and C.

Explore rectangle area under variable length and width bounds in gre quantitative section, compute minimum and maximum area, and determine percentage changes from the nominal six square inches.

Compute the four-student sum as 340 from the 85 average, then bound the fifth score X with 84 < (340+X)/5 < 86, giving 80 < X < 90.

Analyze 200 length measurements with a 17 cm range and a 49.5 cm value to determine cases where 49.5 is max or min, yielding values like 33 or 34 cm.

Solve a GRE quantitative problem using one fourth and one third of gross income to bound x with inequalities, deducing mortgage and expense ranges and identifying valid answer choices.

Analyze data using the mean and standard deviation to identify values between two and three standard deviations from the mean, solving interval calculations for GRE quantitative questions.

Compare sphere, hemisphere, cylinder, cube, and rectangular prism volumes to meet a 100 cm^3 aluminium requirement, using diameter 7.5 cm, radius 3.75 cm, and pi.

Apply triangle inequality to sides 1, x, and x^2, determine valid x values, identify isosceles and equilateral cases, and conclude that options b and c form valid triangles.

The lecture shows solving for K by factoring the equation with N as an integer, using factors of 16, and identifying candidate values such as 17, -17, 8, and -8.

Identify the city as positive integers that are multiples of the least common multiple of 72 and 216, and find its common factors (2, 3, 6) via prime factorization.

Apply the triangle inequality to determine possible perimeters for a triangle with sides 12, 18, and x. Determine 6 < x < 30, so the perimeter lies between 36 and 60.

Explore how the average of consecutive integers is determined by endpoints A and B, using A+B equals 2 times the mean, to identify valid start and end pairs.

Solve GRE with four distinct integers x, y, z, w, where x = yz + w and w < y, yielding yz = 4 and x = 6 or 7.

Determine the number of polygon sides from the interior angle sum using exterior angles (sum 360) and the 180(n−2) formula, showing sums under 540 limit to triangles or quadrilaterals.

Analyze factor pairs of 14 to find possible arithmetic means of two integers whose product is 14, yielding 7.5, 4.5, -7.5, and -4.5.

Analyze the region in the xy plane bounded by x=0, y=0, and 4x+3y=60 with x>0 and y>0, using point tests to identify inside points, such as B and C.

Analyze counts for French and geography with a universal set, computing only French, only geography, both, and neither, concluding geography but not French (option C).

Apply triangle inequality to bound the third side between x+3 and 5x+5 for x>0, then verify options x+2, 6x+1, 5x+5, 2x+17; A, C, and E emerge as possible.

solve a gre quantitative problem on tank capacity using initial half fill and 30% usage (3/10). add eight gallons and derive the remaining expression to identify the correct answer.

Use the recurrence a_{n+1} = 2a_n - 3 to deduce earlier terms from a4 = 19, revealing a1 = 5 and values like 5, 7, 11, 19, 35.

Practice solving a GRE style quantitative word problem: with orange price X and apple price 2X, identify apple orange combinations whose total equals 20X.

Explore permutations and the importance of order as Jeff chooses 1, 2, or 3 of 4 movies, yielding 4, 12, and 24 arrangements respectively.

Determine the possible sixth test score for six biology tests using the arithmetic mean of 89 and a total of 534 with five scores between 90 and 100.

Identify which lines intersect x = 3 between the points (3,1) and (3,2). Evaluate options to determine lines with y between 1 and 2 at x = 3.

Translate the constraints into bounds: x lies between 150 and 200, and y lies between 20% and 50% of x. This yields feasible values between 30 and 100 inclusive.

Analyze clerks' salary data from 1990 to 2000 to compute percent increases using original values and actual increases, revealing a range from about 27% to 175%.

Explore coordinate geometry concepts, including points A, B, C in quadrants, triangle area calculation, and how the maximum area of triangle ABC is determined with a point on the x-axis.