PRAXIS MATH final study
At a yard sale. Tensile sold drinking glasses for $2.00 each and plates for $3.50 each. Nicholas spent a total of $18.00 on drinking glasses and plates at Tenille's yard sale. If Nicholas bought at least one glass and one plate, how many drinking glasses did he buy? A. 1 B. 2 C. 3 D. 4
(B) The question requires an understanding of equations and the ability to translate a word problem into an equation. If x represents the number of glasses and y represents the number of plates that Nicholas bought, then 2x + 3.5y = 18. Both x and y must be integers. Therefore 18 - 2x must be a multiple of 3.5. The possible multiples of 3.5 for this problem are 3.5, 7, 10.5, 14 and 17.5. The only multiple of 3.5 that is equivalent to 18 - 2x as an integer is 14. Thus x = 2 (x = drinking glasses, y = number of plates...2x + 3.5y = 18)
A boxplot for a set of data is shown. Which of the following is true? A. The only outlier is 200 B. The only outlier is 1,000 C. The only outliers are 200, 700, and 1,000 D. All values greater than 500 or less than 300 are outliers
(B) The question requires an understanding of how to describe a set of data. The interquartile range is 500 - 300 = 200. Therefore any value less than 300 - (200 x 1.5) = 0 or greater than 500 + (200 x 1.5) = 800 is an outlier. The only outlier in the data set, ie., the data value less than 0 or greater than 800, is 1000.
9, 11, 1, 4, 7, 12, 10, 4, 9, 2, 5, 9, 8 The list shows the number of books each of the 13 students in a class read over the summer. The teacher wants to summarize the data using the boxplot shown. What is the value of C on this boxplot? A. 7 B. 8 C. 9
(B) The question requires an understanding of how to identify, construct, and complete a boxplot that correctly represents given data. Point C on the boxplot is the median of the data. To find the median, that data points must be reordered from least to greatest, which yields 1, 2, 4, 4, 5, 7, 8, 9, 9, 9, 10, 11, 12. The median is the 7th data point, which is 8.
-1/2x - 6 > 4 - 3x Which of the following is the graph of the solutions to the inequality shown?
(B) The question requires an understanding of how to interpret solutions of multi step one-variable linear equations and inequalities. The inequality can be solved by first adding 6 and 3x to each side of the inequality and then adding like terms, resulting in 5/2x > 10. Multiplying each side by 2/5 yields x > 4. On the number line, this inequality corresponds to a graph of the numbers greater than 4 but not including 4. (Note open bubble)
Two friends went out for lunch and decided to share the dessert. One of them ate 1/2 of the desert and the other ate 1/3 of the remaining part. What fraction of the desert was left over? A. 1/6 B. 1/3 C. 2/3 D. 5/6
(B) The question requires an understanding of how to solve multi step mathematical and real-world problems. The first friend ate 1/2 of the desert, while the second friend ate 1/3 of the remaining part; that is 1/3 (1 - 1/2), or 1/6. Altogether they ate 1/2 + 1/6 = 4/6, or 2/3 of the dessert. Therefore, the fraction left over is 1 - 2/3, or 1/3 of the dessert.
Which of the following graphs in the xy-plane could be used to solve graphically the inequality x - 2 < -2( x + 1) and shows the solution to the inequality?
(C) The question requires an understanding of inequalities and the ability to solve them graphically. The given inequality is equivalent to x - 2 < -2 x - 2, which is equivalent to 3 x < 0, which yields x < 0. A way to solve the given inequality graphically is to consider the equations y = x - 2 and y = -2 (x + 1). The graphs of the two equations in the xy-coordinate plane are lines. The solution to the inequality is the set of points (a, 0) on the xy-axis for which the point (a, a - 2) is "below" the point (a, -2 (a + 1)); that is, when the graph of the line y = x - 2 is "below" the graph of the line y = -2( x + 1).
Tasha has T books, and Aisha has A books. Aisha has twice as many books as Tasha, and altogether they have 30 books. Which of the following proportions can be used to find out how many books Aisha has? A. A : T = 2 : 3 B. T : A = 2 : 3 C. A : (T + A) = 2 : 3 D. T : (T + A) = 2 : 3
(C) The question requires an understanding of ratios and unit rates to describe relationships between quantities. Since Aisha has two as many books as Tasha, the ratio of T : A is equal to 1 : 2 that is, Tasha has one-third of the books, while Aisha has two-thirds. Therefore, the ratio of Aisha's books. A, and the total T + A is 2 : 3.
(2x + 5x -2) - (x + y - 3y - 5x + 2) Which of the following is equivalent to the expression shown? A. 11x + 2y - 4 B. 3x - 2y - 4 C. 11x - 2y
(A) The question requires an understanding of how to add and subtract linear algebraic expressions. Adding like terms in the given expression yields the equivalent expression (7x - 2) - (-4x - 2y + 2), which is equivalent to 7x - 2 + 4x + 2y - 2. Adding like terms again yields 11x + 2y - 4
In a set of number cubes, the length of the edge of each number cute is 2/3 inch. It takes 27 of these number cubes to completely fill a box in the shape of a cube. What is the volume, in cubic inches, of the box?
The correct answer is 8. The question requires an understanding of how to find the volume of the right rectangular prism. If 27 number cubes fit in the cubical box, there must be 3 layers of number cubes, with each layer consisting of 3 rows of 3 number cubes. With 3 number cubes, each having edge length 2/3 inch, along one edge of the box, each edge of the box is 3 x 2/3 = 2 inches in length. The volume of the box is thus 2^3 = 8 cubic inches. Another approach is to find the volume of each number cube first. The volume of each number cube is (2/3)^3, or 8/ 27 cubic inches. Since there are 27 number cubes in the box, the volume of the box is 8/ 27 x 27, or 8 cubic inches.
Which word describes each angle in an equilateral triangle? A. Acute B. Obtuse C. Right D. Straight
(A) The question requires an understanding of how to classify angles based on their measure. An equilateral triangle is also equiangular: that is, all its angles have the same measure. Therefore, each angle has a measure of 180 / 3, or 60*. An acute angle is an angle that measures less than 90*. Therefore, the angles of an equilateral triangle are all acute. Acute: Smaller than right angle (0-30 degrees) Obtuse: More than 90 degrees and less than 180 Congruent: Each side has the same degree
Cube A has a volume of 1 cubic inch. The length of each side of cube B is 1 inch greater than the length of each side of cube A. The volume of cube B is how many times the volume of cube A? A. 2 times B. 3 times C. 4 times D. 8 times
(D) The question requires an understanding of how changes to dimensions change area and volume of three-dimensional shapes. Since the volume of cube A is 1 cubic inch, its sides must have a length of 1 inch. The length of each side in cube B is 1 inch greater than the 1 inch of each side of cube A, so each side in cube B has a length of 2 inches. The volume of cube B is thus 8 cubic inches, which is 8 times the volume of cube A.
The perimeter of each of the seven regular hexagons in the figure shown is 18, what is the perimeter of the figures? A. 54 B. 63 C.108 D. 126
(A) The question requires an understanding of geometric reasoning. Since each of the small hexagons in the figure is regular, each hexagon's sides are equal in length. Therefore, the length of any one hexagon's side is 18 / 6, or 3. The perimeter of the large figure shown is found by multiplying the number of exposed hexagon's sides by the length of one hexagon side. There are 18 exposed sides, and each one is 3 units long. Therefore, the perimeter of the figure is 18 x 3, or 54.
Which of the following statements must be true about the two non-right angles of a right triangle? A. Both angles are acute B. One angle is acute and one is obtuse C. The angles are congruent D. Both angles are obtuse
(A) The question requires an understanding of how to classify angles based on their measure. The sum of the measures of the angles in a triangle is 180*. Since the right angle measures 90*, the otehr two angles must add up to the remaining 90*. Since both together make 90*, each individually must be less than 90* and therefore be acute. Acute: Smaller than right angle (0-30 degrees) Obtuse: More than 90 degrees and less than 180 Congruent: Each side has the same degree
The first three figures in a pattern are shown. The 1st figure is composed of two triangles and one square. Each figure after the 1st figure is composed of two triangles and one square more than the preceding figure. How many line segments are in the 10th figure of the pattern? A. 35 B. 38 C. 41
(A) The question requires an understanding of how to identify and extend a pattern. The first figure has 8 line segments. Adding a square to each figure is equivalent to adding 3 line segments. So the number of line segments of the nth figure can be described by the equation f(n) = 5 + 3n, with n = 1, 2, 3,...Therefore the 10th figure of the pattern has f(n) = 5 + 3 x 10, or 35 line segments.
The surface area of a cube is 54 in^2. What is the volume of the cube? A. 27 in^3 B. 54 in^3 C. 81 in^3 D. 108 in^3
(A) The question requires an understanding of how to solve problems involving elapsed time, money, length, volume, and mass. If the length of the side of the cube is in s in, then its surface area is 6s^2 in^2. Since the surface area is 54 in^2, then the length of the side of the cube, in inches, can be found by solving the equation 6s^2 = 54, which yields s = 3. The volume of the cube can then be found by solving the equation v = s^3, thus v =3^3. Therefore the volume is 27 in^3
A certain polygon has the following attributes. I. There are 2 pairs of parallel sides II. It is a quadrilateral III. One pair of parallel sides has length 2, and the other pair of parallel sides has length 4. Which one of the following types of polygons has all of the attributes listed? A. Parallelogram B. Rhombus C. Triangle D. Square
(A) The question requires an understanding of how to use attributes to classify or draw polygons and solids. A quadrilateral is a polygon with four sides. A parallelogram is a quadrilateral with two pairs of parallel sides. A rhombus is a parallelogram with all sides the same length. A square is a rhombus with at least one right angle. For attributes 1 and 2, the polygon is not a triangle. For attribute 3, the polygon is neither a rhombus nor a square. Therefore, the polygon must be a parallelogram.
The net shown above forms a right triangular prism. Which of the following represents the surface area of the right triangular prism? A. 84 B. 92 C. 96 D. 120
(A) The question requires an understanding of how to use nets to determine the surface area of three-dimensional figures. The surface area of the three-dimensional figure is the sum of the areas of its faces. The faces of the right triangular prism are 6 by 5 rectangle, a 6 by 4 rectangle, a 6 by 3 rectangle, and two congruent right triangles, each of which has sides of lengths 4 and 3. The areas of the rectangular faces are, respectively, 6 x 5, or 30; 6 x 4, or 24; and 6 x 3, or 18. The area of each of the triangular faces is 1/2 x 3 x 4, or 6. Therefore, the surface area of the figure is 30 + 24 + 18 + 2 x 6, or 84
In the xy-plane, point A has coordinates (2, 1) and the point B has coordinates (5,1). Which of the one following could be the coordinates for point C so that the area of triangle ABC is equal to 9 sq units? A. (5,7) B. (5,4) C. (5,2) D. (5,1)
(A) The question requires an understanding of how to solve problems by plotting points and drawing polygons in the coordinate plane. The formula for the area of a triangle is A = 1/2 x b x h where b is the length of a side of the triangle that is used as the base and h is the length of the height relative to the side chosen as a base. Since the coordinates of points a and b are given and the segment ab is parallel to the x-axis, the length of side ab can be easily computed, making ab the best choice for the base of the triangle. The length of segment ab is (5 - 2), or 3 units. Substituting the value of 9 for a and the value of 3 for b the formula yields 9 = 1/2 x 3 x h, that is, h = 6. Therefore, the length of the height of the triangle relative to side ab must be 6. Since ab is parallel to the x-axis, point c must be on the line with equation 7 = 7 or on the line with the equation y = -5; that is, it must have a y- coordinate of either 7, or -5.
1, 2, 4 The first three terms of a certain sequence are shown. Which of the following mathematical relationships could describe the terms of the sequence? A. The first term is 1. Each subsequent term of the sequence is obtained by multiplying the fist constant quantity by the preceding term. B. The first term is 1. Each subsequent term of the sequence is obtained by adding a constant quantity to the preceding term. C. The first term is 1. Each subsequent term of the sequence is obtained by adding the preceding term and a quantity that increases by 1 with each new term. D. The first term is 1. Each subsequent term of the sequence is obtained by adding 1 to the square of the preceding term. E. The first term is 1. Each subsequent term of the sequence is obtained by squaring a number that is one greater than the number that was squared to obtain the preceding term.
(A, C) The question requires an understanding of how to make conjectures, predictions, or generalizations based on patterns. In the pattern described in option (A), the first term is 1, and if the constant described in the pattern is 2, then the first 3 terms of the sequence are 1, 2, and 4. Hence, the pattern described in option (A) could produce the three terms 1, 2, and 4, and if extended, the fourth term would be 8. In the pattern described in option (C), the first term is 1, and if the quantity that is added is initially 1, than the first three terms of the sequence are 1, 2, and 4. Hence the pattern described in option (C) could produce the three terms 1, 2, and 4, and, if extended, the fourth term would be 7. In option (B), the quantity that would add to the first term, 1, to produce the second term, 2 must be 1. Hence the first three terms produced by the pattern are 1, 2, and 3. The third term is 3, not 4, as required. In option (D), the terms produced by the pattern would be 1, 2, and 5. The third term is 5, not 4, as required. In option (E), the first three terms are 1, 4, and 9. The second and third term values, respectively, are not 2 and 4, as required.
In the figure shown, quadrilateral ABFG is a square and quadrilateral FCDE is a rectangle. Which of the following statements must be true about the figure? Select al that apply... A. G lies on the x-axis B. C lies on the y-axis C. A is in the first quadrant D. D is in the fourth quadrant E. The area of the polygon ABCDEFG is 27 square units F. The perimeter of the polygon ABCDEFG is 25 units
(A, D, and E) The question requires an understanding of the coordinate plane and area of polygons. Since the x-axis is the horizontal axis, point G lies on the x-axis and has coordinates (-3, 0). The points in the fourth quadrant have a positive x-coordinate and a negative y-coordinate. Since point D has coordinates (4, -3), point D is in the fourth quadrant. The polygon is made of a square, a right triangle, and a rectangle. The length of the side of the square is 3 units; therefore, the area of the square is 9 square units. The lengths of the sides of the right triangle are 3 and 4 units, respectively; therefore the area of the rectangle is 12 square units. Finally, the area of the polygon is the sum of the areas of its parts; that is, 9 + 6 + 12, or 27 square units.
In the sequence of figures shown, the first term is made of a single square and each successive term is made by adding one cross and one square to the preceding term. To form each square, 4 line segments are needed, and to form each cross, 2 line segments are needed. Which of the following expressions can be used to find the number of line segments needed to form the nth term in the sequence of figures? A. n + 6 B. 6n - 2 C. n^2 + 3n
(B) The question requires an understanding of how to identify, extend, describe, or generate number and shape patterns. The nth figure includes n squares made up of 4 line segments and n-1 crosses made up of 2 line segments. Therefore, the number of line segments required for the nth figure is equal to 4n + 2(n - 1), which simplifies to 6n - 2.
Which of the following graphs represents the solution set to the inequality 18b - 5 < 20b + 11?
(B) The question requires an understanding of how to interpret solutions of multi step one-variable linear equations and inequalities. The first step to find the solution set of the inequality is to use the addition property of inequality to add -20b + 5 to both sides of he inequality. This yields 18b - 5 - 20b + 5 < 20b + 11 - 20b + 5. The second step is to add like terms. This yields -2b < 16. The third step is to use the multiplication property of inequality to multiply both sides by - 1/2. One must not forget to flip the direction of the inequality sign when multiplying by a negative number. This yields (-1/2)(-2b)> (-1/2)(16), which is equivalent to b > -8. The line graph in (B) represents all numbers greater than -8.
Mr. Scythe has asked his students to come up with function rules for the following table of data. Adam says that the function rule is y = -x. Belinda says that the function rule is y = -x^2, and Chandra says that the function rule is y = - x^2 - 2x. Which of the three equations could be the function rule for the table? A. Adam's only B. Adam's and Chandra's only C. Belinda's and Chandra's only D. Adam's, Belinda's, and Chandra's
(B) The question requires an understanding of the concept of function and its definition. A function is a rule that establishes a relationship between two quantities; the input and the output. To find out whether or not Adam's function could be the rule for the given table, it is necessary to substitute each pair of values. (0, 0) and (-1, 1), into the y = -x to verify whether or not the substitution gives the correct output. This process must be repeated for both Belenda's function and for Chandra's function. It is easy to see that (0, 0) satisfies all three rules. Since 1 = -(-1), the pair (-1, 1) satisfies Adam's function rule. Since 1 = -(-1)^2 - 2 (-1) = - 1 + 2 = 1, the pair (-1, 1) also satisfies Chandra's function rule. The pair(-1, 1) does not satisfy Belinda's rule, since 1 (does not equal) -(-1)^2 = -1.
A two-dimensional net of a certain three-dimensional figure includes five faces, nine edges, and six vertices. Which of the following three-dimensional figures is represented by the net? A. Triangular pyramid B. Triangular prism C. Rectangular pyramid D. Rectangular prism
(B) The question requires an understanding of three-dimensional geometry. Triangular pyramids and rectangular prisms have 4 and 6 faces, respectively. Triangular prisms have 5 faces; 2 are triangles, and 3 are quadrilaterals. Rectangular pyramids also have 5 faces; 1 is a rectangle, and 4 are triangles. Triangular prisms have 9 edges, while rectangular pyramids only have 8 edges.
In the figure shown, which two lines form a pair of perpendicular lines? A. Line a and line b B. Line b and line c C. Line c and line d
(C) The question requires an understanding of how to use definitions to identify lines, rays, line segments, parallel lines, and perpendicular lines. Two lines in the plane are perpendicular if they form at least one right angle. The figure indicates that lines c and d for a right angle; therefore, line c and line d are perpendicular.
A painter used 1 1/2 cans of paint to paint 2/3 of a room. At this rate , how much more paint does the painter need to paint the remainder of the room? A. 1/3 can B. 1/2 can C. 3/4 can D. 1 can
(C) The question requires an understanding of how to use proportional relationships to solve ration an percent problems. Since the painter has already painted 2/3 of the room, the painter still needs to paint 1- 2/3 or 1/3 of the room. To determine the amount of paint x needed to paint the rest of the room, one can set up the proportion 1 1/2: 2/3 = x : 1/3, which yields the equation 2/3 x = (1 1/2) x 1/3. Simplifying the right side of the equation yields 2/3 x = 1/2. Therefore, x = 3/2 x 1/2; that is, x = 3/4.
The Statue of Liberty casts a shadow that is 37 meters long at the same time that a nearby vertical 5-meter pole casts a shadow that is 2 meters long. Based on shadow height, the height, in meters, Statue of Liberty must be within which of the following ranges? A. 115 meters to 120 meters B. 105 meters to 110 meters C. 90 meters to 95 meters D. 60 meters to 65 meters
(C) The question requires an understanding of proportions. The ratio between the height of the Statue of Liberty and the length of its shadow is equal t the ratio between the height of the pole and the length of its shadow. The proportion will look like this (where L represents the height of the Statue of Liberty): L/37 = 5/2. Multiplying both sides by 37 and then simplifying both sides of the equation gives you L = 92.5 m. Note that other proportions can be set up, such as the height (L) divided by pole height (5 meters) equals state shadow length (37 meters) divided by pole shadow length (2 meters). This will also give the correct result.
A spinner is divided into 12 congruent sections. Each section is either red, blue, green, or yellow. The table shows the theoretical probability of landing on a color when spinning the spinner once. How many sections are blue? A. 2 B. 4 C. 5 D. 8
(C) The question requires an understanding of probability. The sample space consists of the possible outcomes of a spin which are red, blue, green or yellow. The sum of the probabilities of the possible outcomes in the sample space is 1; that is, 1/3 + x + 1/6 + 1/12 = 1, which is equivalent to 7/12 + x = 12/12. It follows that the probability x of landing on blue is 5/12. Therefore, 5 out of 12 sections of the spinner are blue.
The first two terms of the sequence shown are 2 and 3. Each subsequent term, beginning with third term is found by adding the two preceding terms and multiplying the sum by -2. What is the value of a5, the fifth term of the sequence? A. -14 B. -12 C. -10 D. -8
(D) The question requires an understanding of following a rule to continue a sequence of numbers. The first term is 2, the second term is 3, and the third term is found by adding 2 + 3 x -2. So the third term is -10. The fourth term is found by adding 3 + (-10) x -2 or -7 x -2 = 14. The fifth term is found by adding -10 + 14 x -2 or 4 x -2 = -8
Into how many equilateral triangles can a regular hexagon be decomposed? A. 3 B. 4 C. 5 D. 6
(D) The question requires an understanding of how to compose and decompose two-dimensional shapes. A regular hexagon can be inscribed in a circle. When segments are drawn from the center of he circle to the vertices of the hexagon, 6 triangles are formed. The triangles are congruent because of teh SSS Theorem. Moreover, the triangles are isosceles because, in each triangle, two of the sides are radi of the circle. In each triangle, teh angle that is opposed to the base measures 360*/ 6 =60*. Then, each base angle measures (180*-60*)/ 2 = 120*/ 2 = 60*. It follows that, in each triangle all angles have the same measurement; thus, the triangles are equilateral. A hexagon has 6 sides...
The table shows the cost of a membership to Gym B for the five possible membership lengths. Gym A has the same possible membership lengths, and the cost, y, in-dollars, of a membership to Gym A for x months is given by the equation 2y - 50x = 85. Which of the following is true about the cost, in dollars, of a membership to Gym A compared with the cost of a membership to Gym B? A. The cost of membership to Gym B is greater than the cost of a membership to Gym A for membership lengths of 6 months or less but is greater for membership lengths of greater than 6 months. B. The cost of a membership to Gym A includes the same initial membership fee as the cost of a membership t Gym B but a greater monthly fee. D. The cost of a membership to Gym B is greater than the cost of a membership to Gym A for any number of months.
(D) The question requires an understanding of how to use linear relationships represented by equations, tables, and graphs to solve problems.The table describes the costs of varying lengths of membership to Gym B and can be represented by the linear equation y = 25x + 50, where y is the cost of a membership lasting x months. The equation that describes the cost y of a membership to Gym A lasting for x months can be rewritten as y = 25x + 42.50. The monthly fees, represented by the slopes of the two linear equations, are equal for the two memberships. However, the y-intercept of the equation representing Gym B is greater than the y-intercept of the line representing Gym A. This can be interpreted to mean that the initial fee for Gym B is greater than the initial fee for Gym A. Since the monthly memberships are the same but Gym B has a greater initial fee, the membership cost for Gym B is always more expensive than the membership cost for Gym A for any number of months.
What is the area, in square inches, of the polygon shown? A. 21 B. 54 C. 1,008 D. 1,080
(D) The question requires an understanding of relative sizes of United States customary units. To find the area of the polygon in square inches, all measurements must first be converted to equivalent measurements in inches using the conversion that one foot is equivalent to 12 inches and 1 yard is equivalent to 3 feet. The lengths, in inches, of the labeled sides in the polygon are 36 inches, 18 inches, and 24 inches. The polygon can be decomposed into two rectangles; one having a length of 36 inches and a width of 18 inches, and the other having a length of 24 inches and a width of 18 inches. The first has an area of 36 x 18, or 648 square inches, and the second has an area of 24 x 18, or 432 square inches. The area of the polygon is the sum of the areas of these two rectangles, which equals 648 + 432, or 1,080 square inches.
Which of the following could be used to describe the polygon shown? A. Regular and convex B. Regular and concave C. Irregular and convex D. Irregular and concave
(D) The question requires and understnding of polygons and their properties. The polygon can be described as irregular because its sides are of different lengths, and the polygon can be described as concave because it is possible to connect two vertices of the polygon with a segment that is not contained in the interior of the polygon.