Exhaustive Guide to ACCUPLACER Arithmetic, Elementary Algebra, and College-Level Mathematics

Arithmetic Overview

• Test Composition: The Arithmetic test consists of 17 questions.

• Content Areas:     * Operations with Whole Numbers and Fractions: This section covers basic arithmetic operations including addition, subtraction, multiplication, and division. Topics also include the recognition of equivalent fractions, mixed numbers, and estimation techniques.     * Operations with Decimals and Percents: Topics include addition, subtraction, multiplication, and division with decimals. It also encompasses percent problems, the recognition of decimals, percent equivalencies, and estimation.     * Applications and Problem Solving: This area focuses on practical math applications such as rate, percent, and measurement problems. It also includes simple geometry problems and the distribution of quantities into fractional parts.

Arithmetic Sample Questions

• Question 1: Solve for the sum: $2.75 + 0.003 + 0.158$     * A. $0.436$     * B. $2.911$     * C. $2.938$     * D. $4.36$

• Question 2: Solve for the product: $7.86 \times 4.6$     * A. $36.156$     * B. $36.216$     * C. $351.56$     * D. $361.56$

• Question 3: Find the decimal equivalent: $\frac{7}{20}$     * A. $0.035$     * B. $0.35$     * C. $0.858$     * D. $3.5$

• Question 4: Which of the following is the least value?     * A. $0.105$     * B. $0.501$     * C. $0.015$     * D. $0.15$

• Question 5: All of the following are ways to write 25 percent of $N$ EXCEPT:     * A. $(0.25)N$     * B. $\frac{25}{100}N$     * C. $\frac{1}{4}N$     * D. $25N$

• Question 6: Which of the following is closest to the product of $27.8 \times 9.6$?     * A. $280$     * B. $300$     * C. $2,800$     * D. $3,000$

• Question 7: A soccer team played $160$ games and won $65\%$ of them. How many games did the team win?     * A. $94$     * B. $104$     * C. $114$     * D. $124$

• Question 8: Relative to a project where the total work time must equal only one full-time person, three workers are assigned. One worker is budgeted for $\frac{1}{3}$ of his time (Note: transcript says 1/2, corrected via logic/options context as $\frac{1}{3}$) and a second for $\frac{1}{2}$ of her time. What part of the third worker's time should be budgeted to this project?     * A. $\frac{1}{3}$     * B. $\frac{1}{6}$     * C. $\frac{1}{8}$     * D. $\frac{3}{10}$

• Question 9: $32$ is $40\%$ of what number?     * A. $12.8$     * B. $128$     * C. $80$     * D. $800$

• Question 10: Solve the subtraction: $3\frac{1}{3} - 2\frac{2}{5}$     * A. $\frac{1}{15}$     * B. $\frac{14}{15}$     * C. $1\frac{1}{15}$     * D. $1\frac{1}{2}$

• Question 11: Solve the addition: $2\frac{1}{2} + 4\frac{2}{3}$     * A. $6\frac{1}{6}$     * B. $6\frac{5}{6}$     * C. $7\frac{1}{6}$     * D. $7\frac{5}{6}$

• Question 12: What is $\frac{1,345}{99}$ rounded to the nearest integer?     * A. $12$     * B. $13$     * C. $14$     * D. $15$

• Question 13: Three of four numbers have a sum of $22$. If the average of the four numbers is $8$, what is the fourth number?     * A. $4$     * B. $6$     * C. $8$     * D. $10$

• Question 14: Solve: $46.2 \times 10^{-2}$     * A. $0.0462$     * B. $0.462$     * C. $4.62$     * D. $462$

• Question 15: If $\frac{1}{2} + \frac{3}{4} + \frac{5}{8} = n$, then $n$ is between which two whole numbers?     * A. $1 \text{ and } 3$     * B. $3 \text{ and } 5$     * C. $5 \text{ and } 7$     * D. $7 \text{ and } 9$

• Question 16: What is $12\%$ of $120$?     * A. $10$     * B. $14.4$     * C. $18.4$     * D. $28.8$

• Question 17: A box contains history, English, and science books. If $\frac{1}{3}$ of these are history books and $\frac{1}{4}$ are English books, what fraction of the books are science books?     * A. $\frac{1}{3}$     * B. $\frac{5}{12}$     * C. $\frac{7}{12}$     * D. $\frac{3}{4}$

• Question 18: The measures of two angles of a triangle are $35^{\circ}$ and $45^{\circ}$. What is the measure of the third angle of the triangle?     * A. $95^{\circ}$     * Note: Interior angles of a triangle must sum to 180 degrees.     * B. $100^{\circ}$     * C. $105^{\circ}$     * D. $110^{\circ}$

• Question 19: Erica bought $3\frac{3}{4}$ yards of fabric. If she uses $\frac{2}{3}$ of the fabric to make a curtain, how much will she have left?     * A. $1 \text{ yard}$     * B. $1\frac{1}{4} \text{ yards}$     * C. $1\frac{1}{2} \text{ yards}$     * D. $2\frac{1}{2} \text{ yards}$

• Question 20: Jen wants to tile her rectangular kitchen floor ($12 \text{ feet} \times 8 \text{ feet}$). If materials cost $\$2.50$ per square foot, what is the total cost?     * A. $\$160$     * B. $\$200$     * C. $\$220$     * D. $\$240$

Elementary Algebra Overview

• Test Composition: The Elementary Algebra test consists of 12 questions.

• Content Areas:     * Numbers and Quantities: Covers integers and rational numbers, including computation with integers and negative rationals, absolute value, and ordering.     * Algebraic Expressions: Focuses on the evaluation of simple formulas and expressions, addition and subtraction of monomials and polynomials, and multiplication and division of the same. It also includes evaluating positive rational roots and exponents, simplifying algebraic fractions, and factoring.     * Problem Solving: Covers translating written phrases into algebraic expressions, solving linear equations, linear inequalities, and quadratic equations (via factoring). Verbal problems in algebraic contexts are included.

Elementary Algebra Sample Questions

• Question 1: If $A$ represents apples purchased at 15 cents each and $B$ represents bananas at 10 cents each, what represents the total value in cents?     * A. $A + B$     * B. $25(A + B)$     * C. $10A + 15B$     * D. $15A + 10B$

• Question 2: Evaluate: $\sqrt{2} \times \sqrt{15}$     * A. $\sqrt{17}$     * B. $\sqrt{30}$     * C. $17$     * D. $30$

• Question 3: Evaluate $2x^2 + 3xy - 4y^2$ when $x = 2$ and $y = -4$.     * A. $-80$     * B. $-32$     * C. $32$     * D. $80$

• Question 4: Shaded region area of two concentric circles: Radius of larger is $R$, radius of smaller is $3 \text{ units less than } R$.     * A. $\pi R^2$     * B. $\pi(R - 3)^2$     * C. $\pi R^2 - \pi \times 3^2$     * D. $\pi R^2 - \pi(R - 3)^2$

• Question 5: Expand: $(3x - 2y)^2$     * A. $9x^2 - 4y^2$     * B. $9x^2 + 4y^2$     * C. $9x^2 - 6xy + 4y^2$     * D. $9x^2 - 12xy + 4y^2$

• Question 6: If x > 2, simplify: $\frac{x^2 - x - 6}{x^2 - 4}$     * A. $\frac{x - 3}{2}$     * B. $\frac{x - 3}{x - 2}$     * C. $\frac{x - 3}{x + 2}$     * D. $\frac{3}{2}$

• Question 7: Solve: $\frac{4 - (-6)}{-5}$     * A. $-2$     * B. $-\frac{2}{5}$     * C. $1$     * D. $2$

• Question 8: Solve for $x$: $2x - 3(x + 4) = -5$     * A. $-17$     * B. $-7$     * C. $7$     * D. $17$

• Question 9: Solve: $-3(5 - 6) - 4(2 - 3)$     * A. $-7$     * B. $-1$     * C. $1$     * D. $7$

• Question 10: Which is equivalent to $20 - \frac{4}{5}x \geq 16$?     * A. $x \leq 5$     * B. $x \geq 5$     * C. $x \leq 6\frac{1}{2}$     * D. $x \geq 6\frac{1}{2}$

• Question 11: Order from least to greatest: $\left\{ -\frac{3}{5}, -\frac{2}{3}, -\frac{3}{3} \right\}$ or similar sets.     * A. Set starting with $\dots, -\frac{3}{5}, -\frac{2}{3}, \frac{3}{5}$     * B. Set starting with $\dots, -\frac{2}{3}, -\frac{3}{5}, \frac{3}{5}$     * C. Set starting with $\dots, -\frac{2}{3}, -\frac{3}{5}, \frac{3}{3}$     * D. Set starting with $\dots, -\frac{3}{5}, -\frac{2}{3}, \frac{3}{3}$

• Question 12: If $5t + 2 = 6$, then $t = \dots$     * A. $8$     * B. $\frac{5}{4}$     * C. $\frac{4}{5}$     * D. $-8$

• Question 13: For which equation are $x = 5$ and $x = -5$ both solutions?     * A. $x^2 + 25 = 0$     * B. $x^2 - 25 = 0$     * C. $x^2 + 10x - 25 = 0$     * D. $x^2 - 5x - 25 = 0$

• Question 14: If $x \neq 0$, then $\frac{5u}{x} + \frac{2u}{x} = \dots$     * A. $7x$     * B. $\frac{7u}{x}$     * C. $\frac{29u}{5x}$     * D. $\frac{31u}{5x}$

• Question 15: A number line shows a shaded region starting at $x = 2$ and extending to the right. Which inequality matches?     * A. $2x - 4 \geq -3$     * B. $2x + 5 \leq 6$     * C. $3x - 1 \leq 5$     * D. $4x - 1 \geq 7$

• Question 16: How many solutions $(x, y)$ exist for the system: $2x + 6y = 5$ and $x + 3y = 2$?     * A. None     * B. One     * C. Two     * D. More than two

• Question 17: Which is a factor of both $x^2 - x - 6$ and $x^2 - 5x + 6$?     * A. $x - 3$     * B. $x - 2$     * C. $x + 2$     * D. $x + 3$

• Question 18: Simplify: $\frac{10x^6 + 8x^4}{2x^2}$     * A. $9x^{12}$     * B. $14x^4$     * C. $5x^4 + 4x^2$     * D. $5x^3 + 2x^2$

• Question 19: A rectangular yard has an area of $96 \text{ square feet}$. If the width is $4 \text{ feet less than the length}$, what is the perimeter in feet?     * A. $40$     * B. $44$     * C. $48$     * D. $52$

• Question 20: Helen completed science exercises in $3 \text{ hours}$ on Monday at rate $p$ ($\text{exercises per hour}$). The next day she did the same number of exercises in $2 \text{ hours}$. What was her rate on Tuesday?     * A. $2(p + 1)$     * B. $3(p - 1)$     * C. $\frac{2}{3}p$     * D. $\frac{3}{2}p$

College-Level Mathematics Overview

• Test Composition: The College-Level Mathematics test consists of 20 questions.

• Content Areas:     * Algebraic Operations: Includes simplifying rational algebraic expressions, factoring/expanding polynomials, and manipulating roots/exponents.     * Solutions of Equations and Inequalities: Solving linear/quadratic equations and inequalities, systems of equations, and other algebraic equations.     * Coordinate Geometry: Plane geometry, the coordinate plane, straight lines, conics, sets of points, and algebraic function graphs.     * Functions: Covers polynomial, algebraic, exponential, and logarithmic functions.     * Trigonometry: Specific focus on trigonometric functions.     * Applications and Other Topics: Complex numbers, series/sequences, determinants, permutations, combinations, factorials, and word problems.

College-Level Mathematics Sample Questions

• Question 1: Evaluate: $2^{\frac{5}{2}} - 2^{\frac{3}{2}}$     * A. $2^{\frac{1}{2}}$     * B. $2$     * C. $2^{\frac{3}{2}}$     * D. $2^{\frac{5}{2}}$     * E. $2^2$

• Question 2: If $\frac{1}{a} + \frac{1}{b} = \frac{1}{x}$, then $x = \dots$     * A. $\frac{1}{a} + \frac{1}{b}$     * B. $\frac{b + a}{ab}$     * C. $\frac{ab}{a + b}$     * D. $a + b$     * E. $ab$

• Question 3: If $3x^2 - 2x + 7 = 0$, then $\left( x - \frac{1}{3} \right)^2 = \dots$     * A. $\frac{20}{9}$     * B. $\frac{7}{9}$     * C. $-\frac{7}{9}$     * D. $-\frac{8}{9}$     * E. $-\frac{20}{9}$

• Question 4: Which equation represents a line parallel to $y = 2x$?     * A. $4x - y = 4$     * B. $2x - 2y = 2$     * C. $2x - y = 4$     * D. $2x + y = 2$     * E. $x - 2y = 4$

• Question 5: Find the equation of the line containing the origin $(0, 0)$ and point $(1, 2)$.     * A. $y = 2x$     * B. $2y = x$     * C. $y = x - 1$     * D. $y = 2x + 1$     * E. $y = x + 1$

• Question 6: An apartment building has $12 \text{ units}$ (1-bedroom for $\$360$ and 2-bedroom for $\$450$). Total monthly rent is $\$4,950$. How many 2-bedroom units are there?     * A. $3$     * B. $4$     * C. $5$     * D. $6$     * E. $7$

• Question 7: Two square regions have areas of $125 \text{ square yards}$ and $5 \text{ square yards}$ respectively. How many yards of fencing are needed to enclose both separately?     * A. $4\sqrt{130}$     * B. $20\sqrt{10}$     * C. $24\sqrt{5}$     * D. $100$     * E. $104\sqrt{5}$

• Question 8: If $\log_{10}(x) = 3$, then $x = \dots$     * A. $3^{10}$     * B. $1,000$     * C. $30$     * D. $\frac{10}{3}$     * E. $\frac{3}{10}$

• Question 9: If $f(x) = 2x + 1$ and $g(x) = \frac{x - 1}{2}$, then $f(g(x)) = \dots$     * A. $x$     * B. $\frac{x - 1}{4x + 2}$     * C. $\frac{4x + 2}{x - 1}$     * D. $\frac{5x + 1}{2}$     * E. $\frac{(2x + 1)(x - 1)}{2}$

• Question 10: If $\theta$ is an acute angle and $\sin(\theta) = \frac{1}{2}$, then $\cos(\theta) = \dots$     * A. $-1$     * B. $0$     * C. $\frac{1}{2}$     * D. $\frac{\sqrt{3}}{2}$     * E. $2$

• Question 11: Factor: $5y(2y - 3) + (2y - 3)$     * A. $(5y + 1)(2y + 3)$     * B. $(5y + 1)(2y - 3)$     * C. $(5y - 1)(2y + 3)$     * D. $(5y - 1)(2y - 3)$     * E. $10y(2y - 3)$

• Question 12: For what real numbers $x$ is the value of $x^2 - 6x + 9$ negative?     * A. -3 < x < 3     * B. x < -3 \text{ or } x > 3     * C. $x = -3 \text{ or } x = 3$     * D. 0 < x < 6     * E. For no real numbers $x$

• Question 13: A root of $x^2 - 5x - 1 = 0$ is:     * A. $\frac{1 - \sqrt{29}}{2}$     * B. $\frac{5 - \sqrt{17}}{2}$     * C. $\frac{1 + \sqrt{29}}{2}$     * D. $\frac{5 + \sqrt{17}}{2}$     * E. $\frac{5 + \sqrt{29}}{2}$

• Question 14: In the $xy\text{-plane}$, how many points of intersection exist between the graph of $y = x^2$ and the circle with center $(0, 1)$ and radius $3$?     * A. None     * B. One     * C. Two     * D. Three     * E. More than three

• Question 15: If an equation of the linear function shown (passing through $(0, r)$ and $(s, 0)$) is $y = mx + b$, then $m = \dots$     * A. $r$     * B. $\frac{r}{s}$     * C. $rs$     * D. $-\frac{r}{s}$     * E. $-s$

• Question 16: What is the total number of orderings of the letters $T, U, V, \text{ and } W$ from left to right?     * A. $8$     * B. $12$     * C. $16$     * D. $20$     * E. $24$

• Question 17: If $f(x) = 3x - 1$ and $f^{-1}$ is the inverse, what is $f^{-1}(3)$?     * A. $\frac{1}{3}$     * B. $\frac{2}{3}$     * C. $1$     * D. $\frac{4}{3}$     * E. $2$

• Question 18: The sequence $(a_n)$ is defined by $a_1 = 1$ and $a_{n+1} = 2a_n + 2$. What is the value of $a_3$?     * A. $8$     * B. $10$     * C. $16$     * D. $20$     * E. $22$

• Question 19: From $5 \text{ employees}$, a group of $3$ will be chosen. How many different groups can be chosen?     * A. $3$     * B. $5$     * C. $6$     * D. $10$     * E. $15$

• Question 20: If $f(x) = \left( \frac{1}{3} \right)^x$ and a < b, which of the following is true?     * A. $f(a) + f(b) = 3$     * B. $f(a) + \frac{1}{3} = f(b)$     * C. $f(a) = f(b)$     * D. f(a) < f(b)     * E. f(a) > f(b)*