GED Math Concept Review

Overview

This chapter covers the number skills that sit underneath every other math question: comparing and ordering fractions and decimals, finding factors and multiples, working with exponents and roots, using absolute value, and doing multi-step arithmetic with positive and negative numbers. On the GED test a calculator is provided for most questions, but not all of them, so these skills need to work by hand as well.

To compare numbers written in different forms, change them all to one form first, usually decimals. Factors are numbers that divide evenly into a number, and multiples are what you get when you multiply it by a whole number (1, 2, 3, ...). The greatest common factor helps with cutting things into equal pieces, while the least common multiple tells you when repeating events line up again.

Exponents count repeated multiplication. When the same base is multiplied, the exponents are added, and a power raised to a power multiplies the exponents. For a number that is zero or positive, a square root undoes squaring; a cube root undoes cubing for any number. Absolute value is a distance from zero, so it is never negative, and the distance between two numbers on a number line is the absolute value of their difference.

For long calculations, follow the order of operations: parentheses, exponents, multiplication and division from left to right, and then addition and subtraction from left to right. A fraction with zero on the bottom has no value, so it is called undefined.

Overview

This chapter covers rates, ratios, proportions, scale drawings, unit conversions, and percent problems such as tax, tips, discounts, markups, commissions, percent change, and simple interest. The GED assessment targets name real-world percent problems such as tips, commissions, and percent increase and decrease.

A unit rate tells how much of one thing for one unit of another, such as dollars per ounce or miles per hour. To compare prices, divide each total price by the number of units and pick the smallest. A proportion sets two equal ratios side by side, and cross-multiplying solves for the missing value. Scale drawings work the same way: multiply each drawing length by the scale to get the real length.

To find a percent of a number, change the percent to a decimal and multiply. Percent change is the amount of change divided by the original amount. When two percents are applied one after the other, such as a discount and then tax, the second percent is taken of the new amount, not the original.

Simple interest uses the formula I = Prt from the formula sheet, where P is the amount borrowed or saved, r is the yearly rate as a decimal, and t is the time in years.

Overview

This chapter covers flat shapes: the perimeter (distance around) and area (space inside) of rectangles, squares, triangles, parallelograms, trapezoids, circles, and shapes made by combining them. It also covers the Pythagorean theorem for right triangles. The GED formula sheet lists these formulas, so the main job is choosing the right one and putting in the right numbers.

Perimeter is measured in plain units such as feet, while area is measured in square units such as square feet. A triangle's area is half of base times height. For a circle, the radius is the distance from the center to the edge and the diameter is twice the radius. The circumference is the distance around the circle, and the test uses 3.14 for pi.

For a composite shape, split it into simple shapes, find each area, and add them, or subtract a piece that has been cut out. When an area or perimeter is given, you can work backward to find a missing side or radius.

In a right triangle, the two shorter sides (legs) and the longest side (hypotenuse) are linked by a² + b² = c². This is useful for ladders, ramps, diagonals of rectangles, and TV screen sizes.

Overview

This chapter covers solid shapes: rectangular boxes, right prisms, cylinders, pyramids, cones, spheres, and solids built from them. You will find volume (how much a shape holds) and surface area (how much material covers the outside), and you will work backward to find a missing height, radius, or side. The GED formula sheet lists all of these formulas.

Volume is measured in cubic units, such as cubic feet. For any prism or cylinder, the volume is the area of the base times the height. A pyramid or cone with the same base and height holds exactly one-third of that amount. A sphere uses its own formula, four-thirds times pi times the radius cubed.

Surface area adds up the areas of all the outside faces. For a closed cylinder, that means the curved side plus the two circular ends. For a composite solid, such as a cylinder with a cone on top, find the volume of each part and add them.

Overview

This chapter covers reading data from tables, bar graphs, circle graphs, dot plots, box plots, and scatter plots, and describing data with the mean, median, mode, and range. It also covers weighted averages, counting the number of possible outcomes, and finding probabilities.

The mean is the total divided by the number of values. The median is the middle value after the data are put in order; with an even number of values it is the mean of the two middle values. The mode is the value that appears most often, and the range is the largest value minus the smallest. A box plot shows the middle half of the data inside the box, and the length of the box is the interquartile range.

A weighted average multiplies each value by its weight before adding. When a needed average is given, multiply it by the number of values to get the total needed, then subtract the values you already have to find the missing one.

To count choices made in steps, multiply the number of options at each step. When order matters, as with different jobs, the count is a permutation; when order does not matter, as with a committee, divide out the repeated orders to get a combination. Probability is the number of favorable outcomes divided by the total number of equally likely outcomes.

Overview

This chapter covers algebraic expressions: writing them from word descriptions, evaluating them by substituting numbers, combining like terms, and adding, subtracting, multiplying, and factoring polynomials. It also covers simple rational expressions, which are fractions with variables in them.

Like terms have the same variable raised to the same power, and only like terms can be combined. The distributive property multiplies the number outside the parentheses by every term inside, and the sign goes along. When one polynomial is subtracted from another, the sign of every term in the second polynomial changes.

To multiply two binomials, multiply each term in the first by each term in the second, often remembered as FOIL (first, outer, inner, last), and then combine like terms. Factoring reverses multiplication: take out the greatest common factor first, then look for two numbers that multiply to the constant term and add to the middle coefficient.

A rational expression is simplified by factoring the top and bottom and canceling a common factor. Only whole factors cancel, never single terms in a sum. To add rational expressions, rewrite them with a common denominator first.

Overview

This chapter covers solving one-variable linear equations, including ones with fractions, decimals, parentheses, and variables on both sides. It also covers writing equations from real situations, solving a formula for one letter, and solving systems of two equations in two unknowns.

An equation is a balance, so whatever is done to one side must be done to the other. Clear parentheses with the distributive property, collect the variable terms on one side and the numbers on the other, and then divide by the number in front of the variable. Always check by putting the answer back into the original equation.

In word problems, a fixed amount is added once and a rate is multiplied by the unknown. When two quantities are unknown, such as adult and child tickets, write one equation for the count and one for the money. A system can be solved by substitution, where one equation is put into the other, or by elimination, where the equations are added so one variable cancels.

Overview

This chapter covers linear inequalities in one variable: solving them, showing their solutions on a number line, and writing them from real situations with words such as "at least" and "at most."

An inequality is solved with the same steps as an equation, with one extra rule: when both sides are multiplied or divided by a negative number, the inequality sign flips. On a number line, a closed circle means the endpoint is included (≤ or ≥) and an open circle means it is not (< or >). Shading to the right shows greater values and shading to the left shows smaller values.

In word problems, "at most" and "no more than" mean ≤, while "at least" and "no less than" mean ≥. When the answer must be a whole number, such as a number of guests, round in the direction that keeps the inequality true.

Overview

This chapter covers quadratic equations, which have an x² term. You will solve them by taking square roots, by factoring, and by using the quadratic formula, and you will write quadratic equations for situations involving area, consecutive numbers, and objects thrown in the air.

Before factoring, move every term to one side so the equation equals zero. If a product equals zero, at least one factor must be zero, so each factor is set equal to zero and solved. An equation such as x² = 49 has two answers, 7 and −7, because both square to 49.

When an equation does not factor easily, use the quadratic formula from the formula sheet. Identify a, b, and c from the standard form ax² + bx + c = 0 and substitute carefully, especially when b or c is negative. In real situations, throw away answers that make no sense, such as a negative length.

Overview

This chapter covers the coordinate plane, slope, equations of lines, parallel and perpendicular lines, function notation, and reading key features of graphs and tables. Many of the algebra targets in the GED assessment guide involve graphs and functions.

A point (x, y) is plotted by moving x units left or right and then y units up or down from the origin. The four quadrants are numbered counterclockwise starting from the upper right. Slope measures steepness: rise over run, or the change in y divided by the change in x. A line going up from left to right has a positive slope.

In slope-intercept form, y = mx + b, m is the slope and b is the y-intercept, where the line crosses the y-axis. In real situations, the slope is a rate such as dollars per hour, and the y-intercept is a starting amount such as a flat fee. The point-slope form, y − y₁ = m(x − x₁), writes the equation of a line from one point and a slope. Parallel lines have equal slopes, and perpendicular slopes are negative reciprocals.

A function gives exactly one output for each input. Function notation f(5) means put 5 in for x. Key features of a graph include its intercepts, where it rises or falls, and its highest or lowest point. To compare two functions shown in different ways, find each one's rate of change and starting value.