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Quantitative Reasoning Concepts and Tips
Note: Due to the limitations of current web technology, the appearance of mathematical equations, symbols, graphs and diagrams may be slightly distorted. It may help to view this material in Arial 18 pt. font.
The Quantitative Reasoning section of the General Test is designed to measure basic mathematical skills and understanding of elementary mathematical concepts as well as the ability to reason quantitatively and to solve problems in a quantitative setting.
In general, the mathematics required does not extend beyond that usually covered in high school. It is expected that examinees are familiar with conventional symbolism, such as x < y (x is less than y), (x is not equal to y), m || n (line m is parallel to line n), m n (line m is perpendicular to line n), and the symbol for a right angle in a figure:
Also, standard mathematical conventions are used in the test questions unless otherwise indicated. For example, numbers are in base 10, the positive direction of a number line is to the right and distances are nonnegative. Whenever nonstandard notation or conventions are used in a question, they are explicitly introduced in the question.
Many of the questions are posed as word problems in a real-life setting, with quantitative information given in the text of a question or in a table or graph of data. Other questions are posed in a pure-math setting that may include a geometric figure, a graph, or a coordinate system. The following conventions about numbers and figures are used in the quantitative section.
All numbers used are real numbers.
Numbers are to be used as exact numbers, even though in some contexts they are likely to have been rounded. For example, if a question states that "30 percent of the company's profit was from health products," then 30 percent is to be used as an exact percent. It is not to be used as a rounded number obtained from, say, 29 percent or 30.1 percent.
An integer that is given as the number of objects in a real-life or pure-math setting is to be taken as the total number of these objects. For example, if a question states that "a bag contains 50 marbles, and 23 of the marbles are red," then 50 is to be taken as the total number of marbles in the bag and 23 is to be taken as the total number of red marbles in the bag. Therefore, the other 27 marbles are not red.
Questions may involve units of measurement such as English units or metric units. If an answer to a question requires converting one unit of measurement to another, then the relationship between the units is provided, unless the relationship is a common one such as minutes to hours or centimeters to meters.
Geometric figures that accompany questions provide information useful in answering the questions. However, unless a note states that a geometric figure is drawn to scale, you should solve these problems not by estimating sizes by sight or by measurement, but by reasoning about geometry.
Geometric figures consist of points, lines (or line segments), curves (such as circles), angles, regions, etc., and labels that identify these objects or their sizes. (Note that geometric figures may appear somewhat jagged on a computer screen.)
Geometric figures are assumed to lie in a plane unless otherwise indicated.
Points are indicated by a dot, a label or the intersection of two or more lines or curves.
Points on a line or curve are assumed to be in the order shown; points that are on opposite sides of a line or curve are assumed to be oriented as shown.
Lines shown as straight are assumed to be straight (though they may look jagged on a computer screen). When curves are shown, they are assumed to be not straight.
Angle measures are assumed to be positive and less than or equal to 360°.
To illustrate some of these conventions, consider the following geometric figures:
From the figures, it can be determined that:
From the figures, it cannot be determined whether:
When a square, circle, polygon or other closed geometric figure is described in words with no picture, the figure is assumed to enclose a convex region. It is also assumed that such a closed geometric figure is not just a single point. For example, a quadrilateral cannot be any of the following:
When graphs of real-life data accompany questions, they are drawn as accurately as possible so you can read or estimate data values from the graphs (whether or not there is a note that the graphs are drawn to scale).
Standard conventions apply to graphs of data unless otherwise indicated. For example, a circle graph represents 100 percent of the data indicated in the graph's title and the areas of the individual sectors are proportional to the percents they represent. Scales, gridlines, dots, bars, shadings, solid and dashed lines, legends, etc., are used on graphs to indicate the data. Sometimes, scales that do not begin at zero are used, as well as broken scales.
Coordinate systems such as number lines and xy-planes are generally drawn to scale.
The questions in the quantitative sections include four broad content areas: arithmetic, algebra, geometry and data analysis.
Questions that test arithmetic include those involving the following topics: arithmetic operations (addition, subtraction, multiplication, division and powers) on real numbers, operations on radical expressions, the number line, estimation, percent, absolute value and properties of integers (for example, divisibility, factoring, prime numbers and odd and even integers).
Some facts about arithmetic that may be helpful
For any two numbers on the number line, the number on the left is less than the number on the right; for example, -4 is to the left of -3, which is to the left of 0.
The sum and product of signed numbers will be positive or negative depending on the operation and the signs of the numbers; for example, the product of a negative number and a positive number is negative.
Division by zero is undefined; that is, is not a real number for any x.
If n is a positive integer, then xn denotes the product of n factors of x; for example, 34 means (3)(3)(3)(3) = 81. If , then x0 = 1.
Squaring a number between 0 and 1 (or raising it to a higher power) results in a smaller number; for example,
and (0.5)3 = 0.125.
An odd integer power of a negative number is negative, and an even integer power is positive; for example,
(-2)3 = -8 and (-2)2 = 4.
The radical sign means "the nonnegative square root of"; for example, and . The negative square root of 4 is denoted by . If x < 0, then is not a real number, for example, is not a real number.
The absolute value of x, denoted by |x|, is equal to x if and equal to -x if x < 0; for example, |8| = 8 and |-8| = -(-8) = 8.
If n is a positive integer, then n! denotes the product of all positive integers less than or equal to n; for example, 4! = (4)(3)(2)(1) = 24.
The sum and product of even and odd integers will be even or odd depending on the operation and the kinds of integers; for example, the sum of an odd integer and an even integer is odd.
If an integer P is a divisor (also called a factor) of another integer N, then N is the product of P and another integer, and N is said to be a multiple of P; for example, 3 is a divisor (or a factor) of 6, and 6 is a multiple of 3.
A prime number is a positive integer that has only two distinct positive divisors: 1 and itself. For example, 2, 3, 5, 7 and 11 are prime numbers, but 9 is not a prime number because it has three positive divisors: 1, 3 and 9.
Questions that test algebra include those involving the following topics: rules of exponents, factoring and simplifying algebraic expressions, concepts of relations and functions, equations and inequalities and coordinate geometry (including slope, intercepts and graphs of equations and inequalities). The skills required include the ability to solve linear and quadratic equations and inequalities and simultaneous equations; the ability to read a word problem and set up the necessary equations or inequalities to solve it; and the ability to apply basic algebraic skills to solve problems.
Some facts about algebra that may be helpful
If ab = 0, then a = 0 or b = 0; for example, if (x - 1)(x + 2) = 0, it follows that either x - 1 = 0 or x + 2 = 0; therefore, x = 1 or x = -2.
Adding a number to or subtracting a number from both sides of an equation preserves the equality. Similarly, multiplying or dividing both sides of an equation by a nonzero number preserves the equality. Similar rules apply to inequalities, except that multiplying or dividing both sides of an inequality by a negative number reverses the inequality. For example, multiplying the inequality 3x - 4 > 5 by 4 yields the inequality 12x - 16 > 20; however, multiplying that same inequality by -4 yields -12x + 16 < -20.
The following rules for exponents may be useful. If r, s, x and y are positive numbers, then:
The rectangular coordinate plane, or xy-plane, is shown below:
The x-axis and y-axis intersect at the origin O, and they partition the plane into four quadrants, as shown. Each point in the plane has coordinates (x, y) that give its location with respect to the axes; for example, the point P(2, -8) is located 2 units to the right of the y-axis and 8 units below the x-axis. The units on the x-axis are the same length as the units on the y-axis, unless otherwise noted.
Equations involving the variables x and y can be graphed in the xy-plane. For example, the graph of the linear equation y = x - 2 is a line with a slope of and a y-intercept of -2, as shown below:
Questions that test geometry include those involving the following topics: properties associated with parallel lines, circles, triangles (including isosceles, equilateral, and 30° - 60° - 90° triangles), rectangles, other polygons, area, perimeter, volume, the Pythagorean Theorem and angle measure in degrees. The ability to construct proofs is not measured.
Some facts about geometry that may be helpful
If two lines intersect, then the opposite angles (called vertical angles) are equal; for example, in the figure below, x = y.
If two parallel lines are intersected by a third line, certain angles that are formed are equal. As shown in the figure below, if , then x = y = z.
The sum of the degree measures of the angles of a triangle is 180.
The square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the two legs (Pythagorean Theorem).
The sides of a 45° - 45° - 90° triangle are in the ratio and the sides of a 30° - 60° - 90° triangle are in the ratio
Drawing in lines that are not shown in a figure can sometimes be helpful in solving a geometry problem; for example, by drawing the dashed lines in the pentagon below, the total number of degrees in the angles of the pentagon can be found by adding the number of degrees in each of the three triangles: 180 + 180 + 180 = 540.
The number of degrees of arc in a circle is 360.
If O is the center of the circle in the figure below, then the length of arc ABC is times the circumference of the circle.
The volume of a rectangular solid or a right circular cylinder is the product of the area of the base and the height; for example, the volume of a cylinder with a base of radius 2 and a height of 5 is
Questions that test data analysis include those involving the following topics: basic descriptive statistics (such as mean, median, mode, range, standard deviation and percentiles), interpretation of data given in graphs and tables (such as bar and circle graphs and frequency distributions) and elementary probability. The questions assess the ability to synthesize information, to select appropriate data for answering a question and to determine whether or not the data provided are sufficient to answer a given question. The emphasis in these questions is on the understanding of basic principles (for example, basic properties of a normal distribution) and reasoning within the context of given information.
Some facts about descriptive statistics and probability that may be helpful
In a distribution of n measurements, the (arithmetic) mean is the sum of the measurements divided by n. The median is the middle measurement after the measurements are ordered by size if n is odd or it is the mean of the two middle measurements if n is even. The mode is the most frequently occurring measurement (there may be more than one mode). The range is the difference between the greatest measurement and the least measurement. Thus, for the measurements 70, 72, 72, 76, 78 and 82, the mean is 450 ÷ 6 = 75, the median is (72 + 76) ÷ 2 = 74, the mode is 72 and the range is 12.
The probability that an event will occur is a value between 0 and 1, inclusive. If p is the probability that a particular event will occur, then and the probability that the event will not occur is 1 - p. For example, if the probability is 0.85 that it will rain tomorrow, then the probability that it will not rain tomorrow is 1 - 0.85 = 0.15.