GRE Math Algebra

Application Word Problem Examples

1. Ellen has received the following scores on 3 exams: 82, 74, and 90. What score will Ellen need to receive on the next exam so that the average (arithmetic mean) score of the 4 exams will be 85? Solution: let x represent the score on Ellen's next exam. This initial step of assigning a variable to the quantity that is sought is an important beginning to solving the problem. Then in terms of x, the average of the 4 exams is 82+74+90+x / 4= 85 solving for the x, you get x= 94 which is the score Ellen will need to attain on the next exam. 2. A mixture of 12 ounces of vinegar and oil is 40 percent vinegar, where all of the measurements are by weight. How many ounces of oil must be added to the mixture to produce a new mixture that is only 25 percent vinegar? Solution: Let x represent the number of ounces of oil to be added. Then the total number of the ounces of the new mixture will be 12+x, and the total number of ounces of vinegar in the new mixture will by (0.40)(12). Since the new mixture must be 25 percent vinegar, (0.40)(12) / 12+x= 0.25 Solve for x and you get 7.2 ounces of oil that must be added to produce a new mixture that is 25 percent vinegar. 3. In a driving competition, Jeff and Dennis drove the same course at average speeds of 51 miles per hour and 54 miles per hour, respectively. If it took Jeff 40 minutes to drive the course, how long did it take Dennis? Solution: Let x be the time, in minutes, that it took Dennis to drive the course. The distance d, in miles is equal to the product of the rate r, in miles per hour, and the t, in hours; that is, d=rt Note that since the rates are given in miles per hour, it is necessary to express the times in hours; for example, 40 minutes equals 40/60 of an hour. Thus, the distance traveled by Jeff is the product of his speed and his time, (51)(40/60) miles, and the distance traveled by Dennis is similarly represented by (54)(x/60) miles. Since the distances are equal, (51)(40/60)=(54)(x/60) Solve for x. x=37.8 minutes that it took for Dennis to drive the course. 4. Working alone at its constant rate, machine A takes 3 hours to produce a batch of identical computer parts. Working alone at its constant rate, machine B takes 2 hours to produce an identical batch of parts. How long will it take the two machines, working simultaneously at their respective constant rates, to produce an identical batch of parts? Solution: Since machine A takes 3 hours to produce a batch, machine A can produce 1/3 of the batch in 1 hour. Similarly, machine B can produce 1/2 of the batch in 1 hour. If we let x represent the number of hours it takes both machines, working simultaneously, to produce the batch, then the two machines will produce 1/x of the job in 1 hour. When the two machines work together, adding their individual production rates, 1/3 and 1/2, gives their combined production rate 1/x. Therefore, (1/3)+(1/2)= 1/x Solve for x. x= 6/5 hours, or 1 hour and 12 minutes, for both machines to produce a batch of parts. 5. At a fruit stand, apples can be purchase for $0.15 each and pears for $0.20 each. At these rates, a bag of apples and pears was purchased for $3.80. If the bag contained 21 pieces of fruit, how many of the pieces were pears? Solution: If a represents the number of apples purchased and p represents the number of pears purchased, the infomation can be translated into the following system of equations 0.15a+0.20p=3.80 (total cost) a+p=21 (total number of fruit) From the second equation, a=21-p. Substituting 21-p into the first equation for a gives us p=13. Of the 21 pieces of fruit, 13 were pears. 6. To produce a particular radio model, it costs a manufacturer $30 per radio, and it is assumed that if 500 radios are produced, all of them will be sold. What must be the selling price per radio to ensure that the profit (revenue from the sales minus the total production cost) on the 500 radios is greater than $8,200? Solution: If y represents the selling price per radio, then the profit is 500(y-30). Therefore, we set 500(y-30)>8,200 Solve for y and you get y> 46.4 The selling price must be greater that $46.40 to ensure that the profit is greater than $8,200

Inequality

A mathematical startement that uses one of the following inequality signs. See picture. Inequalities can involve variables are are similar to equations, except that the two sides are related by one of the the inequality signs instead of the equality sign used in equations. For, examply, the inequality 4x-1≤7 is a linear inequality in one variable, which states that "4x-1 is less than or equal to 7." To solve an inequality means to find the set of all values of th variable that make the inequality true. This set of values is also know as the solution set of an inequality. Two inequalities that have the same solution set are call equivalent inequalities.

Identity

A statement of equality between two algebraic expressions that is true for all possible values of the variables involved

Equation

A statement of equality between two algebraic expressions that is true for only certain values of the variables involved. The values are called the solutions of the equation. 3x + 5= -2 x - 3y= 10

Function

An algebraic expression in one variable can be used to define a function of that variable. Functions are usually denoted by letters such as f, g, and h. For example, the algebraic expression 3x+5 can be used to define a function f by f(x)=3x+5 where f(x) is called the value of f at x and is obtained by substituting the value of x in the expression above. For example, if x=1 is substituted in the expression about the result is f(1)=8. Another way of thinking of a function f is that it is a machine that takes an input, which is the value of the variable x, and produces the corresponding output, f(x). For any function, each input x gives exactly one output f(x). However, more than one value of x can give the same output f(x). For example, if g is the function defined by g(x)=x²-2x+3, then g(0)=3 and g(2)=3.

Linear Equation

An equation involving one or more variables in which each term in the equation is either a constant term or a variable multiplied by a coefficient. None of the variables are multiplied together or raised to a power greater than 1. i.e., 2x+1= 7x and 10x-9y-z= 3 are linear equations, but x+y²= 0 and xz= 3 are not

Quadratic Equation

An equation that can be written in the form ax²+bx+c= 0 where a, b, and c are real numbers and a does not equal 0. When such an equation has solutions, they can be found using the quadratic formula. *See Picture* where the notation ± is shorthand for indicating two solutions- one that uses the plus sighn and the other that uses the minus sign.

Algebraic Expression

An expression that has one or more variables, and can be written as a single term or as a sum of terms. Examples: 2x y-1/4 8/ n + p

Simple Interest

Based only on the intial deposit, which serves as the amount on which interest is computed, called the principal, for the entire time period. If the amount P is invested at a simple annual interest rate of r percent, then the value V of the investment at the end of t years is given by the formula V=P(1+ rt/100) where P and V are in dollars. Example: If $10,000 is invested at a simple annual interest rate of 6 percent, what is the value of the investment after half a year? Solution: According to the formula for simple interest, the value of the investment after 1/2 year is $10,000(1+0.06(1/2))= $10,000(1.03)= $10,300

Rules of Exponents

Here are the basic rules of exponents, where the bases x and y are nonzero real numbers and the exponents a and b are integers. 1. x^-a= 1/ x^a Ex: 4^-3= 1/4^3= 1/64, x^-10= 1/x^10, and 1/2^a= 2^a 2. (x^a)(x^b)= x^a+b Ex: (3^2)(3^4)= 3^2+4= 3^6= 729 and (y^3)(y^-1)= y^2 3. x^a/ x^b= x^a-b= 1/ x^b-a Ex: 5^7/ 5^4= 5^7-4= 5^3= 125 and t^3/ t^8= t^-5= 1/t^5 4. x^º= 1 Ex: 7º= 1 and (-3)º= 1. Note that 0º is not defined. 5. (xª)(yª)= (xy)^ª Ex: (2^3)(3^3)= 6^3= 216 and (10z)^3= 10^3 z^3= 1,000z^3 6. (x/y)ª= xª/ yª Ex: (3/4)^2= 3^2/ 4^2= 9/16 and (r/4t)^3= r^3/ 64t^3 7. (x^a)^b= x^ab Ex: (2^5)^2= 2^10= 1,024 and (3y^6)^2= (3^2)(y^6)^2= 9y^12

Exponents

In the algebraic expression xª, x is called the base and a is called the exponent. The following property is useful: If xª= x^b, then a= b. This is true for all positive numbers x, except x= 1, and for all integers a and b. For example, if 2^y= 64, then since 64 is 2^6, you have 2^y= 2 ^6, and you can conclude that y= 6.

Elimination

In the elimination method, the object is to make the coefficients of one variable the same in both equations so that one variable can be eliminated either by adding the equations together or subtracting one from the other. Let's use the system of equation: 4x+3y= 13 x+2y= 2 Multiply both sides of the second equation by 4 so that you have two equations with the same coefficient of x. So then you get... 4x+3y= 13 4x+8y= 8 If you subtract the second equation from the first, the result is -5y= 5. Thus, y= -1, and substituting -1 to y in either of the original equations yields x= 4.

Solving a Quadratic Equation using the Quadratic Formula Example

In the quadratic equation 2x²-x-6=0, we have a= 2, b= -1, and c=-6. Therefore, the quadratic formula yields (refer to picture for the quadratic formula) x= [-(-1) ±√ (-1)²-4(2)(-6) / 2(2) = 1±√49 / 4 = 1±7/ 4 Hence the two soultions are x= 1+7/4= 2 and x= 1-7/4= -3/2 Quadratic equations have at most two real solutions, as in the example above. However, some quadratic equations have only one real solution. For example, the quadratic equation x²+4x+4=0 has only one solution, which is x= -2. In this case, the expression under the square root symbol in the quadratic formula is equal to 0, and so adding or subtracting 0 yields the same result. Other quadratic equations have not real solutions; for example, x²+x+5=0. In this case, the expression under the square root symbol is negative, so the entire expression is not a real number.

Substitution

In the substitution method, one equation is manipulated to express one variable in terms of the other. Then the expression is substituted in the other equation. Ex: to solve the system of equations 4x+3y= 13 x+2y= 2 you can express x in the second equation in terms of y as x= 2-2y. Then substitute 2-2y for x in the first equation to find the value of y. 4(2-2y)+3y= 13 8-8y+3y= 13 8-8-5y= 13-8 -5y= 5 y= -1 Then -1 can be substituted for y in either equation to find the value of x. We use the second equation: x+2y=2 x+2(-1)=2 x-2= 2 x= 4 So the ordered pair is (4, -1)

Compound Interest

Interest is added to the principal at regular time intervals, such as annually, quarterly, and monthly. Each time interest is added to the principal, the interest is said to be compounded. After each compounding, interest is earned on the new principal, which is the sum fo the preceding principal and the interest just added. If the amount P is invested at an annual interest rate of r percent, compounded annually, then the value V of the investment at the end of t years is given by the formula V=P(1+ r/100)^t Example: If an amount P is to be invested at an annual interest rate of 3.5 percent, compounded annually, what should be the value of P so that the value of the investment is $1,00 at the end of 3 years? Solution: According to the formula for 3.5 percent annual interest, compounded annually, the value of the investment after 3 years is P(1+ 0.035)^3= $1,000 To find the value of P, we divide both sides of the equation by (1+0.035)^3 P= $1,000/ (1+0.035)^3 approximately equals $901.94 Thus, an amount of approximately $901.94 should be invested. If the amount P is invested at an annual interest rate of r percent, compunded n times per year, then the value V of the investment at the end of t years is given by the formula V=P(1+ r/100n)^nt Example: A college student expects to earn at least $1,00 in interest on an initial investment of $20,000. If the money is invested for one year at interest compounded quarterly, what is the least annual interest rate that would achieve the goal? Solution: According to the formula for r percent annual interest, compounded quarterly, the value of the investment after 1 year is $20,000(1+ r/400)^4 By setting this value greater than or equal to $21,000 and solving for r, we get $20,000(1+r/400)^4≥ $21,000 (1+ r/400)^4≥1.05 we can use the fact that taking the positive fourth root of each side of an inequality preserves the direction of the inequality. This is also true for the positive square root or any other positive root. 1+ r/400≥ fourth root of 1.05 r≥400 (fourth root of 1.05 -1) For the fourth root, you can solve it by doing square root twice. So the least annual interest rate is approximately 4.91 percent.

Examples of Domain of Function

Let f be the function defined by f(x)= 2x/ x-6. In this case, f is not defined at x=6 because 12/0 is not defined. Hence, the domain of f consists of all real numbers except for 6. Let g be the function defined by g(x)=x^3+ √(x+2) -10. In this case, g(x) is not a real number x<-2. Hence, the domain of g consists of all real numbers x such that x≥-2. Let h be the function defined by h(x)= |x|, the absolute value of x, which is the distance between x and 0 on the number line. The domain of h is the set of all real numbers. Also, h(x)=h (-x) for all real numbers x, which reflects the property that on the number line the distance between x and 0 is the same as the distance between -x and 0.

Interest

Some applications involve computing interest earned on an investment during a specified time period. The interest can be computed as simple interest or compound interest.

Solving a Quadratic Equation by Factoring

Some quadratic equations can be solved more quickly by factoring. For example, the quadratic equation 2x²-x-6=0 can be factored as (2x+3)(x-2)=0. When a product is equal to 0, at least one of the factors must be equal to 0, which leads to two cases: either 2x+3=0 or x-2=0. Therefore, 2x+3=0 OR x-2=0 2x=-3 x=2 x=-3/2 and the solutions are -3/2 and 2. Here is another example of a quadratic equation that can be easily facored. 5x^2+3x-2=0 (5x-2)(x+1)=0 Therefore, 5x-2=0 OR x+1=0 5x=2 x=-1 x=2/5

Finding the Distance between Two Point in the xy-Plane

The distance between two point in the xy-plane can be found by using the Pythagorean theorem. For example, the distance between the two points Q(-2, -3) and R(4,1.5) is the length of the line segment QR. To find the distance, construct a right triangle and then not that the two shorter sides of the triangle have lengths QS= 4-(-2)= 6 and RS=1.5-(-3)=4.5. Since the line segment QR is the hyptoenuse of the triangle, you can applying the Pythagorean theorem: QR= square root of 6^2+4.5^2= square root of 56.25= 7.5

Domain of a Function

The domain of a function is the set of all permissible inputs, that is, all permissible values of the variable x. For the functions f and g defined above, the domain is the set of all real numbers. Sometimes, the domain of the function is given explicitly and is restricted to a specific set of values of x. For example, we can define the function h by h(x)=x^2-4 for -2≤x≤2. Without an explicit restriction, the domain is assumed to be the set of all values of x for which f(x) is a real number.

Solving a Linear Inequality≤

The inequality -3x+5≤17 can be solved as follows -3x+5≤17 -3x+5-5≤17-5 = -3x≤12 -3x/-3≥12/-3 *dividing -3 reverses the sign* x≥-4 Therefore, the solution set of -3x+5≤17 consists of all real numbers greater than or equal to -4. Another example: (4x+9)/11 < 5 11* (4x+9) < 5*11 = 4x+9 < 55 4x+9-9 < 55-9 4x< 46 x < 46/4 x=11.5 There fore, the sulution set of (4x+9)/11 < 5 consists of all real numbers less than 11.5.

Rules of Solving Linear Inequalities

The procedure used to solve a linear inequality is similar to that used to solve a linear equation, which is to simplify the inequality by isolating the variable on one side of the inequality, using the folowing two rules. When the same constant is added to or subtracted from both sides of an inequality, the direction of the inequality is preserved and the new inequality is equivalent to the original When both sides of the inequality are multiplied or divided by the same nonzero constant, the direction of the inequality is preserved if the constant is positive but the direction is reversed if the constant is negative. In either case, the new inequality is equivalent to the original.

Rules for Solving Equivalent Equations

When the constant is added to or subracted from both sides of an equation, the equality is preserved and the new equation is equivalent to the original equation. When both sides of an equation is multiplied or divided by the same nonzero constant, the equality is preserved and the new equation is equivalent to the original equation.

Example Problems of Slope-Intercept Lines

1. In the xy-plane, the slope of the line passing through the points Q(-2,-3) and R(4,1.5) is 1.5-(-3) / 4- (-2)= 4.5 / 6= 0.75 Line QR appears to intersect the y-axis close to the point (0,-1.5), so the y-intercept of th eline must be close to -1.5. To ge the exact value of the y-intercept, substitute the coordinates of any point on the line, say Q (-2, -3), in the the equation y=0.75x+b, and solve it for b y=0.75x+b -3=0.75 (-2)+b b=-3+(0.75)(-2) b=-1.5 There fore, the equation of line QR is y=0.75x-1.5 You can find the x-intercept by setting y=0 in an equation of the line and solving it for x 0=0.75x-1.5 1.5= 0.75x x=1.5/0.75= 2 2. Consider the system of linear equations in two variables 4x+3y=13 x+2y=2 Solving each equation for y in terms of x yields y=-4/3x+13/3 y=-1/2x+1 If you graphed the two equations, the solution of the system of equations is the point at which the two graphs intersect, which is (4, -1) 3. Consider the following system of linear inequalities x-3y≥-6 2x+y≥-1 Solving each inequality for y in terms of x yields y ≤1/3x+2 y ≥-2x-1 Each point (x,y) that satisfies the first inequality y≤1/3x+2 is either on the line y=1/3x+2 or below the line because the y-coordinate is either equal to or less than 1/3x+2. Therefore the graph of y≤1/3x+2 consists of the line y=1/3x+2 and the entire region below it. Similarly, the graph of y≥-2x-1 consists of the line y=-2x-1 and the entire region above it. Thus, the solution set of the system of inequalitites consists of all the points that lie in the shaded region. (See picture of an example of how such a graph would look like) See more of the section of section 2.8 in the pdf beginning on page 37

Variable

A letter that represents a quantity whose value is unknown. The letters x and y are often used as variables, though any letter can be used.

Linear Equations in Two Variables

A linear equation in two variables, x and y, can be written in the form ax+by= c 3x+2y= 8 where a, b, and c are real numbers and a and b are not both zero. A solution of such an equation is an ordered par of numbers (x, y) that makes the equation true when the values of x and y are substituted into the equation. For examply, both (2, 1) and (-2/3, 5) are solutions of the equation 3x+2y= 8, but (1,2) is not a solution. A linear equation in two variables has infinitely many solutions. If another linear equation in the same variables is given, it is usually possible to find a unique solution of both equations.

The Slope-Intercept Form

Equations in two variables can be represented as graphs in the coordinate plane. In the xy-plane, the graph of an equation in the variables x and y is the set of all points whose ordered pairs (x,y) satisfy the equation. The graph of a linear equation of the form y=mx+b is a straight line in the xy-plane, were m is called the slope of the line and b is called the y-intercept. The x-intercepts of a graph are the x-coordinates of the points at which the graph intersects the x-axis. Similarly, the y-intercepts of the graph are the y-coordinates of the points at which the graph intersects the y-axis. Sometimes the term x-intercept and y-intercept refer to the actual intersection points. The slope of a line passing through two points Q(y2- y1) / (x2- x1) This ratio is often called "rise over run," where rise is the change in y when moving from Q to R and run is the change of x when moving from Q to R. A horizontal line has a slop of 0, since the rise is 0 for any points on the line. So the equation of every horizontal line has the form y=b, where b is the y-intercept. The slope of a vertical line is not defined, since the run is 0. The equation of every vertical line has the form x=a , where a is the x-intercept. Two lines are parallel if their slopes are equal. Two lines are perpendicular if their slops are negative reciprocals of each other. For example, the line with the equation y=2x+5 is perpendicular to the line with equation y=-1/2x +9.

Parts of an Algebraic Expression

Terms Coefficient Variable Constant

Graphing Functions

The coordinate plane can be used for graphing functions. To graph a function in the xy-plane, you represent each input x and its corresponding output f(x) as a point (x,y), where y= f(x). In other words, you use the x-axis for the input and the y-axis for the output. See more of the section 2.9 in the pdf beginning on page 39

Linear Equations in One Variable

To solve a linear equation in one variable, simplify each side of the equation by combining like terms. Then use the rules for producing simpler equivalent equations. 11x-4-8x= 2(x+4)-2x 3x-4= 2x+8-2x 3x-4= 8 3x-4+4= 8+4 3x=12 3x/3= 12/3 x= 4 You can always check your solution by substituting it into the original equation. Note that it is possible for a linear equation to have no solutions. Ex: 2x+3= 2(7+x) has not solution, since it is equivalent to the equation 3= 14, which is false. Also it is possible that what looks to be a linear equation turns out to be an identity when you try to solve it. Ex: 3x-6= -3(2-x) is true for all values of x, so it is an identity

Application Examples and Notes

Translating verbal descriptions into algebraic expressions is an essential initial step in solving word problems. Some examples are given below. If the square of the number x is multiplied by 3, and then 10 is added to that product, the result can be represented by 3x²+10. If John's present salary s is increased by 14 percent, then his new salary is 1.14s. If y gallons of syrup are to be distributed among 5 people so that one particular person gets 1 gallon and the rest of the syrup is divided equally among the remaining 4, then each of the 4 people will get y-1 / 4 gallons of syrup.

Equivalent Equations

Two equations that have the same solutions i.e., x+1=2 and 2x+2= 4 are equivalent equations; both are true when x= 1 and are false otherwise.

System of Equations

Two equations with the same variables are called a system of equations, and the equations in the system are called simultaneous equations. To solve a system of two equations means to find an ordered pair of numbers that satisfies both equations in the system. There are two basic methods for solving systems of linear equations, by substitution or by elimination.

Rectangular Coordinate System, xy-Coordinate System, or xy-Plane

Two real number lines that are perpendicular to each other and that intersect at their respective zero points, or often called the xy- coordinate systen or xy-plane. The horizontal number line is called the x-axis and the vertical number line is called the y-axis. The point where the two axes intersect is called the origin, denoted by 0. The positive half of the x-axis is to the right of the origin, and the positive of the y-axis is above the origin. The two axes divide the plan into four regions called quadrants I, II, III, and IV.