# MATH LEVEL 1 ARITHMETIC (ACCUPLACER)

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1 MATH LEVEL ARITHMETIC (ACCUPLACER) 7 Questions This test measures your ability to perform basic arithmetic operations and to solve problems that involve fundamental arithmetic concepts. There are 7 questions on the Arithmetic tests, divided into three types. Operations with whole numbers and fractions: Topics included in this category are addition, subtraction, multiplication, division, recognizing equivalent fractions and mixed numbers, and estimating. Operations with decimals and percents: Topics include addition, subtraction, multiplication, and division with decimals. Percent problems, recognition of decimals, fraction and percent equivalencies, and problems involving estimation are also given. Applications and problem solving: Topics include rate, percent, and measurement problems; simple geometry problems; and distribution of a quantity into its fractional parts. Arithmetic Sample Questions For each of the questions below, choose the best answer from the four choices given. You may use the paper you received as scratch paper Which of the following is the least? All of the following are ways to write percent of N EXCEPT 0. N N N N 00. Which of the following is closest to 7.8 x 9.? 80 00,800, A soccer team played 0 games and won percent of them. How many games did it win? Three people who work full-time are to work together on a project, but their total time on the project is to be equivalent to that of only one person working full-time. If one of the people is budgeted for one-half of his time to the project and a second person for one-third of her time, what part of the third worker s time should be budgeted to this project? 8 ELAC-Assessment Center Page

2 9. is 0 percent of what number? MATH Level Arithmetic ,. What is rounded to the nearest integer? 99. Three of four numbers have a sum of. If the average of the four numbers is 8, what is the fourth number? x If n, then n is between and and and 7 7 and 9. What is % of 0? A box in a college bookstore contains books, and each book in the box is a history book, an English book or a science book. If one-third of these books are history books and one-sixth are English books, what fraction of the books are science books? 8. The measures of two angles of a triangle are and. What is the measure of the third angle of the triangle? Erica bought yards of fabric. If she uses of the fabric to make a curtain, how much will she have left? yd. yd. yd. yd. 0. Jen wants to tile the floor of her kitchen. The floor is rectangular and measures feet by 8 feet. If it costs \$.0 per square foot for the materials, what is the total cost of the materials for tiling the kitchen floor? \$0 \$00 \$0 \$0 ARITHMETIC ANSWERS: (B); (A); (C); (C); (D); (A); 7(B); 8(C); 9(C); 0(C); (C); (C); (D); (B); (C); (B); 7(B); 8(B); 9(C); 0(D) ELAC-Assessment Center Page

3 MATH LEVEL ELEMENTARY ALGEBRA (ACCUPLACER) A total of questions of three types are administered in this test. Questions The first type involves operations with integers and rational numbers, and includes computation with integers and negative rationals, the use of absolute values, and ordering. The second type involves operations with algebraic expressions using evaluation of simple formulas and expressions, and adding and subtracting monomials and polynomials. Questions involve multiplying and dividing monomials and polynomials, the evaluation of positive rational roots and exponents, simplifying algebraic fractions, and factoring. The third type of question involves translating written phrases into algebraic expressions and solving equations, inequalities, word problems, linear equations and inequalities, quadratic equations (by factoring), and verbal problems presented in an algebraic context. Elementary Algebra Sample Questions For each of the questions below, choose the best answer from the four choices given. You may use the paper you received as scratch paper.. If A represents the number of apples purchased at cents each, and B represents the number of bananas purchased at 0 cents each, which of the following represents the total value of the purchases in cents? A + B (A + B) 0A + B A + 0B.? What is the value of the expression x + xy y when x and y? In the figure below, both circles have the same center, and the radius of the larger circle is R. If the radius of the smaller circle is units less than R, which of the following represents the area of the shaded region?. πr π(r ) πr π πr π(r - ) x y 9x y 9x y 9x y xy 9x y xy. If x >, then x x x x x x x x 7. ( ) x x, then x If ELAC-Assessment Center Page

4 MATH Level Elementary Algebra 9. ( ) ( ) Which of the following expressions is equivalent to 0 x? x x x ½ x ½. Which of the following lists of numbers is ordered from least to greatest?,,,,,,,,,,,,. If t, then t 8. For which of the following equations are x x x 0 x 0 x 0x 0 x 0 8 x and x both solutions? u u u. If x 0, then x x x 7x u u 7x 9u x u x The solution set of which of the following inequalities is graphed on the number line above? x x x x 7. x y x y How many solutions x, y are there to the system of equations above? None One Two More than two 7. Which of the following is a factor of both x x and x x? x x x x 0x 8x 8. x 9x x x x x x 9. A rectangular yard has area 9 square feet. If the width of the yard is feet less than the length, what is the perimeter, in feet, of the yard? 0 8 ELAC-Assessment Center Page

5 MATH Level Elementary Algebra 0. On Monday, it took Helen hours to do a page of science homework exercises. The next day she did the same number of exercises in hours. If her average rate on Monday was p exercises per hour, what was her average rate the next day, in terms of p? p exercises per hour p exercises per hour p exercises per hour p exercises per hour ELEMENTARY ALGEBRA ANSWERS: (D); (C); (A); (D); (D); (B); 7(D); 8(B); 9(B); 0(A); (B); (C); (D); (C); (C); (A); 7(A); 8(C); 9(A); 0(D) MATH LEVEL COLLEGE-LEVEL MATHEMATICS (ACCUPLACER) 0 Questions The College-Level Mathematics test measures your ability to solve problems that involve college-level mathematics concepts. There are six content areas measured on this test: (a) Algebraic Operations, (b) Solutions of Equations and Inequalities, (c) Coordinate Geometry, (d) Applications and other Algebra Topics, (e) Functions and (f) Trigonometry. The Algebraic Operations content area includes the simplification of rational algebraic expressions, factoring and expanding polynomials, and manipulating roots and exponents. The Solutions of Equations and Inequalities content area includes the solution of linear and quadratic equations and inequalities, systems of equations, and other algebraic equations. The Coordinate Geometry content area presents questions involving plane geometry, the coordinate plane, straight lines, conics, sets of points in the plane, and graphs of algebraic functions. The Functions content area includes questions involving polynomial, algebraic, exponential and logarithmic functions. The Trigonometry content area includes trigonometric functions. The Applications and other Algebra Topics content area contains complex numbers, series and sequences, determinants, permutations and combinations, factorials, and word problems. A total of 0 questions are administered on this test. College-Level Mathematics Sample Questions For each of the questions below, choose the best answer from the four choices given. You may use the paper you received as scratch paper.. E.. If a b and, then x x a b b a b a ab. If x x + 7 0, then (x ) ELAC-Assessment Center Page 7 9 ab a b E. 8 E. 9 ab a b 0 9

6 MATH Level College-Level Mathematics. The graph of which of the following equations is a straight line parallel to the graph of y x? x y x y x y x + y E. x y. An equation of the line that contains the origin and the point (, ) is y x y x y x y x + E. y x. An apartment building contains units consisting of one- and two-bedroom apartments that rent for \$0 and \$0 per month, respectively. When all units are rented, the total monthly rental is \$,90. What is the number of two-bedroom apartments? E If the two square regions in the figures below have the respective areas indicated in square yards, how many yards of fencing are needed to enclose the two regions? E If log 0 x, then x 0, E If f(x) x + and g(x) x x x x, then f(g(x)) x x x x x E. 0. If θ is an acute angle and sin θ, then cos θ - 0 E.. y (y - ) + (y - ) (y + ) (y + ) (y + ) (y - ) (y - ) (y + ) (y - ) (y - ) E. 0y (y - ). For what real numbers x is x x + 9 negative? - < x < x < - or x > x - or x 0 < x < E. For no real numbers x ELAC-Assessment Center Page

7 MATH Level College-Level Mathematics. A root of x x 0 is E. 9. In the x y - plane, the graph of y x and the circle with center (0,) and radius have how many points of intersection? None One Two Three E. More than three. (0,r) y 0 x (s, 0) If an equation of the linear function in the figure above is r s r s y mx b, then m rs r E. s. One ordering of the letters T, U, V and W from left to right is UTVW. What is the total number of orderings of these letters from left to right, including UTVW? 8 0 E. 7. If f(x) x and f - - is the inverse of f -, what is the value of f - - ()? 7 E. 8. The sequence { a n } is defined by a 0 and a n an + for n 0,,,. What is the value of a? E. 9. From employees at a company, a group of employees will be chosen to work on a project. How many different groups of employees can be chosen? 0 E. 0. If f(x) and a < b, which of the following must be true? f (a) + f (b) f (a) + f (b) f (a) f (b) f (a) < f (b) E. f (a) > f (b) COLLEGE-LEVEL MATHEMATICS ANSWERS: (C); (E); (E); (C); (A); (E); 7(C); 8(B); 9(A); 0(D); (B); (E); (E); (C); (A); (E); 7(E); 8(E); 9(D); 0(E) ELAC-Assessment Center Page 7

8 MATH SOLUTIONS MATH LEVEL ARITHMETIC (ACCUPLACER) For each of the questions below, choose the best answer from the four choices given. You may use the paper you received as scratch paper NOTE: Remember to line up the decimal points when you Add or Subtract decimals. 7.8 x decimal places decimal place decimal places 7. 0 NOTE: To convert fractions to decimals, divide the numerator by the denominator. 0.. denominator numerator. Which of the following is the least? least means smallest NOTE: Compare digits in the same place value from left to right. When digits are not equal, the number with the larger digit is the larger decimal.. All of the following are ways to write percent of N EXCEPT or % of N N 0. N or % N 00 or 0.N N N or % 0. % % or N ELAC-Assessment Center Page 8

9 MATH SOLUTIONS: Math Level Arithmetic Approximation. Which of the following is closest to 7.8 x 9.? ,800,000 8 x A soccer team played 0 games and won percent of them. Total: 0 games Won % How many games did it win? Number of games won 0 % (0.) 0 0 x Three people who work full-time are to work together on a project, but their total time on the project is to be equivalent to that of only one person working full-time. If one of the people is budgeted for one-half of his time to the project and a second person for one-third of her time, what part of the third worker s time should be budgeted to this project? + + x LCD: st person: + + x 8 nd person: rd person: x + + x + + x x x 9. is 0 percent of what number?.8 8 is equal of multiply what number unknown x: the number 0% x x 80 x 0. Convert to improper fraction:, LCD: ELAC-Assessment Center Page 9

10 MATH SOLUTIONS: Math Level Arithmetic ,. What is rounded to the nearest integer? 99 NOTE: We can easily eliminate and as the answer, but it s very hard to tell whether the answer is or x m. Three of four numbers have a sum of. If the average of the four numbers is 8, st # what is the fourth number? nd # Sum rd # th # x or greater st nd rd th # + # + # + # 8 x 8 + x Note: 0.0 x 0 Average.. OR: Move the decimal point x 0.0 to the left times. ELAC-Assessment Center 0. Page 0

11 MATH SOLUTIONS: Math Level Arithmetic. If n, then n is between NOTE: When dividing fractions, write the st fraction times the reciprocal of the nd and and fraction. Then multiply and simplify. and 7 7 and 9 Between and 7. What is % of 0? what unknown x is of multiply x : unknown x % 0 x 0. (0) x. 0 x A box in a college bookstore contains books, and each book in the box is a history book, an English book or a science book. If one-third of these books are history books and one-sixth are English books, what fraction of the books are science books? + + x (LCD: ) History + + x English Science x + + x + + x x 8. The measures of two angles of a triangle are and. What is the measure of the third angle of the triangle? st angle: 9 00 x nd angle: rd angle: x 0 0 NOTE: The sum of the measures of the three angles is x x 80 x x 00 ELAC-Assessment Center Page

12 MATH SOLUTIONS: Math Level Arithmetic 9. Erica bought yards of fabric. If she uses of the fabric to make a curtain, how much will she have left? Used: yd. yd. 7 7 of yards yd. yd. Left: yd. yards 0. Jen wants to tile the floor of her kitchen. The floor is rectangular and measures feet by 8 feet. If it costs \$.0 per square foot for the materials, what is the total cost of the materials for tiling the kitchen floor? \$0 \$00 \$0 \$0 Rectangle feet 8 feet Area length width Area (8) Area 9 square feet Total Cost \$.0 (9) Total \$0 ARITHMETIC ANSWERS: (B); (A); (C); (C); (D); (A); 7(B); 8(C); 9(C); 0(C); (C); (C); (D); (B); (C); (B); 7(B); 8(B); 9(C); 0(D) LEVEL ELEMENTARY ALGEBRA (ACCUPLACER) Elementary Algebra Sample Questions For each of the questions below, choose the best answer from the four choices given. You may use the paper you received as scratch paper.. If A represents the number of apples purchased at cents each, and B represents the number of bananas purchased at 0 cents each, which of the following represents the total value of the purchases in cents? A: # of apple ( each) A + B (A + B) B: # of bananas (0 each) 0A + B A + 0B Total Cost A + 0B ELAC-Assessment Center Page

13 MATH SOLUTIONS: Math Level Elementary Algebra.? x Property:. What is the value of the expression x xy y when x and y? x + xy y () + ()(-) (-) In the figure below, both circles have the same center, and the radius of the larger circle is R. If the radius of the smaller circle is units less than R, which of the following represents the area of the shaded region? R πr π (R ) Big circle: Radius R Small circle: Radius R R πr π πr π (R ) Area of shaded region Area of Big circle Area of Small circle πr π(r ). x y 9x y 9x y 9x y xy 9x y xy (x y) (x y)(x y) You may use the formula: (x y) 9x xy xy + y (A B) A AB + B (x y) 9x xy + y FOIL ELAC-Assessment Center Page

14 MATH SOLUTIONS: Math Level Elementary Algebra. If x >, then x x x x x x x x 7. ( ) ( ) 0 8. If x x, then x ( ) ( ) Which of the following expressions is equivalent to 0 x? x x x x x (x + ) x x xn x x x + x x 7 x x 7 ( ) ( ) (- ) (- ) x 0 x 0 x NOTE: *When multiplying or dividing by a negative number on both sides, the inequality sign reverses. x 0 < ELAC-Assessment Center Page

15 MATH SOLUTIONS: Math Level Elementary Algebra. Which of the following lists of numbers is ordered from least to greatest?,,,,,,,,,,,, Common denominator: Order: 9,, 9 0,,. If t, then t 8 8. For which of the following equations are x and x x x 0 x 0 x 0x 0 x 0,, t + + t + t + t x both solutions? u u u. If x 0, then x x x 7x u x (x ) is a factor x (x + ) is a factor equation: (x )(x + ) 0 x x + x 0 9u x u 7x u x x 0 NOTE: You may also solve each equation to get the correct answer. + + ELAC-Assessment Center Page

16 MATH SOLUTIONS: Math Level Elementary Algebra The solution set of which of the following inequalities is graphed on the number line above?. x y x x x x 7 Solve each: x y How many solutions x, y are there to the system of equations above? None One Two More than two 7. Which of the following is a factor of both x x and x x? 8. x x x 0x 8x x 9x x x x x x x x x x x + (x )(x + ) (x )(x ) Common factor: (x ) 0x 8x 0x 8x + x x x ELAC-Assessment Center Page x x x x - x + x x 7 x - + x x + x 7 + x x x x 8 x x < NOTE: The st equation is a multiple of the nd equation for the left side, but not the right side. That is, these two lines are parallel. OR x x We can try to solve it x + y x + y multiply by - - x + y Hence, there is no solution. -x y False No Solution Use: m a n a m - n a

17 MATH SOLUTIONS: Math Level Elementary Algebra 9. A rectangular yard has area 9 square feet. If the width of the yard is feet less than the length, what is the perimeter, in feet, of the yard? 0 8 Length L Length L Width L Width W Area Length Width 9 L (L ) 9 L L 0 L L 9 0 (L )(L + 8) L 0 or L L L 8 (not possible) So, Length ft Perimeter L + W m Width 8ft Peri mete r Perimeter 0 0. On Monday, it took Helen hours to do a page of science homework exercises. The next day she did the same number of exercises in hours. If her average rate on Monday was p exercises per hour, what was her average rate the next day, in terms of p? p exercises per hour p exercises per hour p p exercises per hour exercises per hour ELEMENTARY ALGEBRA ANSWERS: (D); (C); (A); (D); (D); (B); 7(D); 8(B); 9(B); 0(A); (B); (C); (D); (C); (C); (A); 7(A); 8(C); 9(A); 0(D) LEVEL COLLEGE-LEVEL MATHEMATICS (ACCUPLACER) College-Level Mathematics Sample Questions For each of the questions below, choose the best answer from the four choices given. You may use the paper you received as scratch paper.. Let x be the # of exercises Monday: Rate Tuesday: Rate E. ELAC-Assessment Center Page 7 x ( ) p Factor out GCF

18 MATH SOLUTIONS: Math Level College-Level Math. If a b and, then x x a b b a b a ab a b ab E. ab a b +, LCD: xab x(b a) -ab xab ( + ) xab x mab + xb xa xb xa -ab. If x x + 7 0, then (x ) x x x x -7 x x - x x + ( ) - + ( ) E (x (x ) ) 0 9 NOTE: Complete the square to get the answer. (x ) - +. The graph of which of the following equations is a straight line parallel to the graph of y x? x y x y x y x + y E. x y Parallel lines same slopes y x slope m x y x y x y x y x + x y x + x y x + y x y x y x m m m ELAC-Assessment Center Page 8

19 MATH SOLUTIONS: Math Level College-Level Math. An equation of the line that contains the origin and the point (, ) is y x y x y x y x + E. (0, 0) and (, ) - 0 m - 0 y mx + b y x y x. An apartment building contains units consisting of one- and two-bedroom apartments that rent for \$0 and \$0 per month, respectively. When all units are rented, the total monthly rental is \$,90. What is the number of two-bedroom apartments? E. 7 x: # of one-bedroom apartments 90y 0 y: # of two-bedroom apartments y 7 (two-bedroom) x + y multiply by -0 x + y 0x + 0y 90 x + 7-0x 0y x (one-bedroom) + -0x + 0y -90 So, there are 7 two-bedroom apartments. 7. If the two square regions in the figures below have the respective areas indicated in square yards, how many yards of fencing are needed to enclose the two regions? E. 0 Small Square: Larger Square: Perimeter: Let x be the length Let y be the length + x y + 0 x y yards ELAC-Assessment Center Page 9

20 8. If log 0 x, then x 0, sin θ m x θ 0 cos θ MATH SOLUTIONS: Math Level College-Level Math E. 0 x 9. If f (x) x + and g(x), then f (g(x)) x f (g(x)) f ( ) x x x x x x ( ) + x x x E. x + x 0. If θ is an acute angle and sin θ, then cos θ - 0. y (y - ) + (y - ) m x 000 (y + ) (y + ) (y + ) (y - ) (y - ) (y + ) (y - ) (y - ) E. 0y (y - ) a x b a E. y(y ) + (y ) *Factor out (y ) (y )(y + ). For what real numbers x is x x + 9 negative? < x < x < or x > x or x 0 < x < E. For no real numbers x (x x + 9 < 0 negative (x ) < 0 No solution as square cannot be negative. ELAC-Assessment Center Page 0

21 MATH SOLUTIONS: Math Level College-Level Math. A root of x x 0 is E. 9 x x 0 a, b -, c - x b b ac a ( ) ( ) ()( ) () 9. In the x y - plane, the graph of y x and the circle with center (0, ) and radius have how many points of intersection? None One Two Three E. More than three We can graph the two graphs. ) y x : vertex (0, 0), opens up ) circle : center (0, ), r y Intersect at points x - We can also solve it. Circle x + (y ) 9 y x x + (y ) 9 y + (y ) 9 y + y y + 9 y y 8 0 y x x y y y x ELAC-Assessment Center Page

22 MATH SOLUTIONS: Math Level College-Level Math y (0, r) (s, 0) 0 x. If an equation of the linear function in the figure above is y mx b, then m r r rs s s Points: (0, r), (s, 0) m m y y x x 0 r s 0 m s r r E. s. One ordering of the letters T, U, V and W from left to right is UTVW. What is the total number of orderings of these letters from left to right, including UTVW? 8 0 E. letters to choose from We can also compute by permutation letters, taken at a time P! 7. If f(x) x and f - - is the inverse of f -, what is the value of f - - ()? 7 E. f(x) f ( ) Hence, f - () m x x 7 x ELAC-Assessment Center Page

23 MATH SOLUTIONS: Math Level College-Level Math 8. The sequence { a n } is defined by a 0 and a n an + for n 0,,, What is the value of a? E. a0, an+ a n + a a0 + + a a a a From employees at a company, a group of employees will be chosen to work on a project. How many different groups of employees can be chosen? 0 E. Combination of. Choose at a time.! C C C!( )!!!! 0 ( ) 0. If f(x) x and a < b, which of the following must be true? f (a) + f (b) f (a) + f (b) f (a) f (b) f (a) < f (b) E. f (a) > f (b) f(x) ( ) x, decreasing. So if a < b, Then f (a) > f (b) COLLEGE-LEVEL MATHEMATICS ANSWERS: (C); (E); (E); (C); (A); (E); 7(C); 8(B); 9(A); 0(D); (B); (E); (E); (C); (A); (E); 7(E); 8(E); 9(D); 0(E) Rev. // ELAC-Assessment Center Page

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