Title

GRE Math Practice Test I
Answer Key with Explanantions

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 This page contains the answers to Practice Test I with detailed explanations.

 1

 Column A Column B

 Answer: B   Hey, they look a lot alike, but they are not exactly the same.  Can you see the difference between the two?? Hopefully, you noticed that in Column B, there was a ( ) around the - and the 3. In Column A, there is no (  ). This means the - is NOT part of the base, so it will not get expanded like it did in Column B.  It is interpreted as finding the negative or opposite of 3 to the fourth power. Hence, Column B's value is greater than Column A's value. Need more help on this topic?  Exponents

 2.    x and y are integers:

 Column A x Column B y

 Answer: D   The reason that the answer is D on this problem is that opposites have the same absolute value.  Since x and y are integers, x could be a positive or negative value or zero.  Since y is set equal to the absolute value of x, then y’s value will always be nonnegative.   Based on the information given, x could be a negative value, and in that case, y would always be greater.  But if x is a nonnegative value, then x and y will be the same value.   For example, x could be -5, then y = 5.  But x could be 5, where y = 5.  So, the relationship cannot be determined from the information given. Need more help on this topic?  Absolute Value

 3

 Column A .25% of .25 Column B

 Answer: A *Move decimal on .25% two places to the left     *Move decimal 5 places to the LEFT

 Since .000625 > .0000625, the answer is Column A’s quantity is greater. Need more help on these topics?  Decimals, Percents, and Scientific Notation

 4.  If 25 students in one class had an average of 93% and 20 students from another class had an average of 98%, approximately what is the average in percent of all 45 students?

 Answer: D In general, when finding the average of tests, you add the tests together and then divide by the number of tests.  On this problem we have two separate averages to start with. So we need to find the total number of points we have using that information and then divide by the number of students, which in this case is 45. If 25 students averaged 93, then the total number of points would be (25)(93) = 2325.  If 20 students averaged 98, then the total number of points would be (20)(98) = 1960. So the total number of points is 2325 + 1960 = 4285. To find the average, take the number of points and divide by the number of students: The overall average percent is 95.2%. Need more help on this topic?  Averages (Mean)

 5.  A water treatment plant is built with two cylindrical tanks to contain water for a town.  Each tank has a radius of 10 feet and a depth of 20 feet.  If there are about 7.5 gallons in a cubic foot of water, approximately how many gallons of water can be treated at the plant at any one time?

 Answer: C First, let’s find the volume of each cylindrical tank.  The volume of a cylindrical tank is  Filling in 10 for radius and 20 for height we get: Next we want to consider the fact that we have two tanks. Multiplying the volume by two we get: Next we want to consider that there are about 7.5 gallons in a cubic foot of water.  Multiplying the volume of the two cylinders by 7.5 we get: Next notice that none of the answers has the pi symbol in them. That means we will have to put in 3.14 for pi and see what we get: 94,200 gallons of water can be treated at the water plant at one time. Need more help on this topic?  Formulas for Three-Dimensional Figures

 6

 Column A Probability of randomly selecting  a given name out of a hat  containing 12 different names. Column B .08

 Answer: A In Column A, each name has an equally likely chance of being picked. The probability of randomly selecting a given name out if a hat containing 12 different names can be found by taking the number of names being selected, which in this case is 1, and put that over the total number of names, which is 12: Since .08333... is larger than .08, then the probability of randomly selecting a given name out if a hat containing 12 different names is a larger value than .08. Need more help on this topic?  Probability

 7.  If     what is the value of ?

 Answer: A The first step here is to find the value of x.  If we know that, then we can plug that in to the second expression and find out what 3x + 5 is. Solving the equation for x we get:

 *Mult. BOTH sides by the LCD of -2   *Inverse of sub. 10 is add 10 to BOTH sides

 Now be careful, this is not our final answer.  We still need to plug in -6 for x in the given expression:

 *Plug in -6 for x

 So when , the expression 3x + 5 is equal to -13. Need more help on these topics?  Solving Linear Equations and Operations with Algebraic Expressions

 8.   x is a positive integer:

 Column A 5 Column B y

 Answer: D If we did a cross multiplication we would get: It is given that x is a positive integer.  That could mean x is 1, 2, 3, ....  If x is 1, then y would equal 5.  However, if x is an integer greater than or equal to 2 then y would be greater than 5.  Since we don’t know x’s exact value, then we cannot determine the relationship between 5 and y.  Need more help on this topic?  Solving Linear Equations

 9

 Column A Column B

 Answer: C Rationalizing the denominator in Column A we get:

 *Mult. num. and den. by square root of 5

 The two quantities are equal to each other. Need more help on this topic?  Square Root

 10.  What is the area of the rectangle above?

 Answer: E The area of a rectangle is length times width.  So, we need the length and the width to get our answer.  It looks like we have the length which is 10, but we still need the width.  Note how we have the diagonal of the rectangle which is 12.  The diagonal, length and width of the rectangle make up an right triangle.  Anytime you are missing one side of a right triangle, you can use the Pythagorean theorem to find that long lost side.  Using the Pythagorean Theorem we get:

 *Pythagorean Theorem *Plugging in 10 for one leg and 12 for the hypotenuse *Subtract 100 form BOTH sides   *Take the square root of BOTH sides *Square root of 4 is 2

 2 square root of 11 is the width of the rectangle.  Putting the length and the width values of the rectangle into the area formula we get:

 *Plugging in 10 for length and 2 sq.root of 11 for width

 The area of the rectangle is 20 square root of 11. Need more help on this topic?  Formulas for Two-Dimensional Figures

 11

 Column A Column B

 *Factor the Perfect Square Trinomial *The sq. root of an expression squared is the abs. value of that expression *Since x squared plus 7 is ALWAYS positive it is equal to its abs. value

 The two values are the same. Need more help on these topics?  Factoring and Square Root

 12.  What is the area of a circle that is inscribed in a square whose area is 81 square inches?

 Answer: D Ultimately, we need to find the area of the circle.  That means we need the area of a circle formula which is  Since we were not given the radius of the circle, it looks like we will have to go out and find it. Since the area of a square is found by taking the side squared, this means the side of our square is going to be 9 inches, since its area is 81 square inches. Let's look at a visual of this: It looks like the side of the square is also the diameter of the circle. Since the diameter is 9, the radius, which is half of the diameter, is going to be 9/2 or 4.5. Putting the radius into the area formula we get:

 *Plugging in 4.5 for the radius

 The area of the circle is 20.25 pi square inches. Need more help on these topics?  Formulas for Two-Dimensional Figures and Inscribed Figures

 13.  Being a conscientious driver, Suzy stayed at or below the speed limit while traveling down the interstate.  Overall, she went an average rate of 65 mph and it took her 10 hours to complete her journey.  If she traveled for 6 of her 10 hours at 70 mph, what constant speed did she go for the remaining 4 hours to obtain the overall 65 mph average?

 Answer: C This problem has several parts to it. We will need to use the formula distance = (rate)(time) to help us out with this. First let see how many miles Suzy traveled overall. Plugging in the rate of 65 mph and the time of 10 hours we get: Suzy traveled a total of 650 miles. Next we start breaking it down.  It says that she traveled for 6 hours at 70 mph.  From this we can figure out how many of the 650 miles she traveled at this speed: She traveled 420 miles at the 70 mph.  How many miles does that leave us for the unknown speed? For the last part, we are again going to use the distance/rate formula, but now we are looking for the rate.  Plugging in 230 for the distance and 4 for the time we get: Suzy traveled 57.5 mph for 4 hours of her trip. Need more help on this topic?  Word Problems

 14

 *Change the division to mult. of the recip.     *Find the products

 The answer is 1/4. Need more help on these topics?  Fractions and Order of Operations

 15.  Find the area of the graph above enclosed by figure ABCD.

 Answer: B From the picture, we can see that figure ABCD is a quadrilateral but doesn’t fit into a form like rectangle, parallelogram, etc...  We will have to take the area of triangle ABE and subtract from it the area of triangle CDE, this will leave us with figure ABCD.  Lets look at triangle ABE first.  The area of a triangle is ½ (base)(height).  So we need to know the base and the height of this triangle.  Since it is a right triangle, the base and height are going to be the lengths of the two legs, which are sides AB and AE.  Side AB can be found by taking the absolute value of the difference of the y values of their ordered pairs, which would be 12 - 0 = 12. Side AE can be found by taking the absolute value of the difference of the x values of their ordered pairs, which would be 6 - 0 = 6. Plugging 6 in for the base and 12 in for the height into the formula for the area of a triangle we get: The area of triangle ABE is 36. Now lets look at triangle CDE.  The area of a triangle is ½ (base)(height).  So we need to know the base and the height of this triangle. The base of this triangle is side DE.  The height can be found by making a perpendicular line from C to the base. Side DE can be found by taking the absolute value of the difference of the x values of their ordered pairs, which would be 6 - 3 = 3. The height can be found by taking the absolute value of the difference of the y value of C and 0 (the y value on the base), which would be 4 - 0 = 4. Plugging 3 in for the base and 4 in for the height into the formula for the area of a triangle we get: The area of triangle CDE is 6. Last, we need to take the difference of the areas of the two triangles: The area of the figure ABCD is 30. Need more help on these topics?  Formulas for Two-Dimensional Figures and Coordinate Geometry

 16

 Column A Column B -10

 Answer: D Note how 6x - 2y is two times 3x - y.  Taking 2 times the given inequality we get:

 *Taking 2 times BOTH sides

 It looks like 6x - 2y is less than -8.   6x - 2y could be -8.5 or it can be -12, in other words 6x - 2y can be either less than or greater than -10.  Since this is our only given information, the relationship cannot be determined from the given information. Need more help on this topic?  Linear Inequality

 17.  If account codes for a certain company are assigned as follows: two letters and then three one digit numbers, how many different account codes can be made?  Note that letters and digits can be repeated.

 Answer: B We can use the fundamental counting principle to answer this question.  Basically we need to take the product of the number of ways each event can occur. There are 5 stages or events:   letter 1,  letter 2, digit 1, digit 2, and digit 3. In general, there are 26 letters in the alphabet.  Neither letter has any restriction, so there are 26 possibilities for each letter.  In general there are 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.  There are no restriction on any of the digits, so each one of those has 10 possibilities.  Putting that all together we get:

 Letter 1 Letter 2 Digit 1 Digit 2 Digit 3 Total 26 x 26 x 10 x 10 x 10 = 676000

 There are 676,000 different account codes possible. Need more help on this topic?  Counting Principle

 18.  In the figure above, XY is a line segment.  What is the value of ?  Note that a and b are measured in degrees.

 Answer: D Since 4 angles of measure a make up XY and  XY is a line segment that means, the 4 a angles would have to make up 180 degrees. Similarly, the 3 b angles would be 180 degrees. This would give us enough information to find out what a and b are equal to. Lets start with a:

 *The 4 a angles = 180 degrees   *Divide BOTH sides by 4

 Next lets look at b:

 *The 3 b angles = 180 degrees   *Divide BOTH sides by 3

 Putting those values in for a and b we get:

 *Plug in 45 for a and 60 for b

 Our answer to this problem is -7.   Need more help on these topics?  Basic Geometry and Operations with Algebraic Expressions

 19

 *Use the FOIL method to mult. two binomials *Make sure you add exponents when you mult. like bases

 Need more help on this topic?  Multiplying Polynomials

 20

 Column A The ratio of juniors to seniors at City College in the  Fall 2000 semester. Column B The ratio of seniors to juniors at  City College in the  Fall 2000 semester.

 Answer: A You can do this problem two different ways.  You can either take the ratio of the percents for each classification or you can find the number of students that go with each one and take that ratio.  Either way you would end up with the same answer. I’m going to use the percents that are given to set up each ratio. For the  ratio of juniors to seniors at City College in the Fall 2000 semester, we need to make sure that the percent attached to juniors goes on top and the percent for seniors goes on bottom: For the  ratio of seniors to juniors at City College in the Fall 2000 semester, we need to make sure that the percent attached to seniors goes on top and the percent for juniors goes on bottom: It looks like the ratio of juniors to seniors is greater than the ratio of seniors to juniors. Need more help on this topic?  Percent and Problem Solving

 21.  If enrollment was 17% higher for freshmen at City College in the Fall 2000 semester than the Fall 1999 semester, approximately what was the enrollment of freshman in the Fall 1999 semester?

 Answer: A We need to be careful here, the temptation is to just take 40% - 17% = 23% of the enrollment of 10,500.   However, keep in mind that we are talking about 17% higher in terms of enrollment, not compared to the other classifications and enrollment may or may not have been the same overall for 1999 and 2000. The first thing we need to find is the actual enrollment for freshmen in 2000.  According to the chart, freshmen made up 40% of the 10,500 enrolled.  Taking 40% of 10,500 we get: 4200 is 17% higher than the enrollment of freshmen in 1999.  Letting x represent the 1999 freshmen enrollment and putting this into an equation we get:

 *17% higher means we ADD 17% (or .17) of the amount   *Divide BOTH sides by 1.17

 Rounding this up we get 3590 for our answer. Need more help on this topic?  Percent and Problem Solving

 22.  If the areas of sectors in the circle graphs are drawn in proportion to the percentages shown, what is the measure, in degrees, of the central angle sector representing the percentage of juniors enrolled in the Fall 2000 semester?

 Answer: D On this problem, the key is to know that a circle measures 360 degrees. So if we know the percentage of the circle that a sector represents, then we can take that percentage of 360 degrees and find the measure of just that sector.  Since juniors were 20% of the enrollment, we need to take 20% of 360: The measurement of the central angle sector representing the percentage of juniors enrolled in the Fall 2000 semester is 72 degrees. Need more help on this topic?  Percent and Problem Solving

 23.   3, 3, 4, 4, 5, 5

 Column A Mode of the numbers listed Column B Standard deviation of the numbers listed

 Answer: A The mode of a set of data is the value(s) that occurs most often.  In this case, all three values occur twice.  So the mode is 3, 4, and 5. For the standard deviation, we need to first find the mean:

 *(sum of values)/(number of values)

 The mean of these values is 4. Next we need to find the difference between the mean and each separate value of the data set,  AND  square each difference found AND add up all of the squared values found:

 x x - 4 3 -1 1 3 -1 1 4 0 0 4 0 0 5 1 1 5 1 1 SUM: 4

 Next we need to divide the sum found by the number of data values in the set AND find the nonnegative square root of the quotient found. On the test you will not have a calculator, so on a problem like this you need to estimate.  You can see that you are taking the square root of a number that is between 0 and 1, so your answer will be between 0 and 1.    This means that the mode, whether it is 3, 4, or 5, is always greater than the standard deviation. Need more help on this topic?  Central Tendencies

 24.    What is the mean of the advertising costs from 1999 to 2002?

 Answer: A Since we are looking for the mean of advertising costs, we need to look at the 2nd bar graph.  Keep in mind that the advertising costs are in thousands of dollars.  In 1999, the advertising cost is \$50000, in 2000 it is \$30000, in 2001 it is \$55000, and in 2002 it is \$60000.  To find the mean we need to sum up all the values and then divide by the number of values.

 *(sum of values)/(number of values)

 The mean of the advertising costs from 1999 to 2002 is \$48,750. Need more help on these topics?  Central Tendencies and Reading Graphs

 25.   The percent decrease in profits from 1999 to 2000 was approximately what percent?

 Answer: C Since we are looking at the decrease in profits, we need to look at the 1st bar graph.  1999 had a profit of \$250,000,000 and 2000 had a profit of \$150,000,000. We are needing to find the percent decrease in profits from 1999 to 2000.  First lets see what the decrease in dollar amount was: There was a \$100,000,000 decrease in profits from 1999 to 2000.  From this we need to find out what percent 100,000,000 is of 250,000,000, since that is the amount we are decreasing from:

 *x represents the percent we are looking for

 \$100,000,000 is 40% of \$250,000,000.  Need more help on these topics?  Percents and Reading Graphs

 26.  Looking at the figure above, if triangle ABC is an equilateral triangle and line BC is parallel to line DE, what is the measure of angle 5?

 Answer: C Lets see what the information that was given to us means.  If ABC is an equilateral triangle, that means all three sides are equal and all three angles have the same measurement.  That means, angles 1, 2, and 3 are each 60 degrees.  The reason for that is the degrees of all three angles of any triangle sum up to be 180 degrees.  If all three angels are the same, then they would each have to be 60 degrees, because 60 + 60 + 60 = 180. Now we have a value to work with. We have to work our way to angle 5.  Next, we can use the fact that lines BC and DE are parallel to each other and ray AE is a transversal that intersects these two lines.  This means angles 3 and 4 are equal to each other because they are corresponding angles.  So this means angle 4 is also 60 degrees.  We are getting closer.  Note how angles 4 and 5 together make a straight angle.  This means their sum is 180 degrees.  Well if angle 4 is 60 degrees, this means angle 5 must equal 180 - 60 = 120 degrees. Angle 5 is 120 degrees. Need more help on this topic?  Basic Geometry

 27

 Column A Slope of the line on graph above. Column B The y value of the y-intercept of the line on the graph above.

 Answer: B The slope can be found by using rise/run.  I prefer to go left to right.  I’m going to start with the point that is on the x-axis, which is (-4, 0) and then step up to the point that is on the y-axis (0, 2).  If we do that, we rise up 2 and go over to the right 4.  So the slope is rise/run which is  2/4 = 1/2. The y-intercept is the point where the graph crosses the y-axis.  In this case that is (0, 2).  The y-value of that point is 2. This means the y value of the y-intercept of the graph is greater than the slope of the line. Need more help on this topic?  Coordinate Geometry

 28.  What is the solution(s) to ?

 *Multiply BOTH sides by the LCD 10       *Factor out a GCF of 9 *Factor the difference of squares *Set 1st factor = 0         *Set 2nd factor = 0

 This problem has two solutions, x = 2 and x = -2. Need more help on this topic?  Solving Quadratic Equations

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