Points A and B lie on a circle with radius 1, and arc \(\overset{\frown}{AB}\) has length ^{π}⁄_{3}. What fraction of the circumference of the circle is the length of arc \(\overset{\frown}{AB}\)?

Solution
The correct answer is ^{1}⁄_{6}, .166, or .167. The circumference, C, of a circle is C = 2πr, where r is the radius of the circle. For the given circle with a radius of 1, the circumference is C = 2(π)(1), or C = 2π. To find what fraction of the circumference the length of arc AB is, divide the length of the arc by the circumference, which gives ^{π}⁄_{2} ÷ 2𝜋. This division can be represented by ^{𝜋}⁄_{2}∙\(\frac{1}{2\pi }\) = ^{1}⁄_{6}.The fraction ^{1}⁄_{6} can also be rewritten as .166 or .167.
How many liters of a 25% saline solution must be added to 3 liters of a 10% saline solution to obtain a 15% saline solution?

Solution
The correct answer is 1.5 or ^{3}⁄_{2}. The total amount, in liters, of a saline solution can be expressed as the liters of each type of saline solution multiplied by the percent of the saline solution. This gives 3(0.10), x(0.25), and (x + 3)(0.15), where x is the amount, in liters, of a 25% saline solution and 10%, 15%, and 25% are represented as 0.10, 0.15, and 0.25, respectively. Thus, the equation 3(0.10) + 0.25x = 0.15(x + 3) must be true. Multiplying 3 by 0.10 and distributing 0.15 to (x + 3) yields 0.30 + 0.25x = 0.15x + 0.45. Subtracting 0.15x and 0.30 from each side of the equation gives 0.10x = 0.15. Dividing each side of the equation by 0.10 yields x = 1.5, or x = ^{3}⁄_{2}.
In the figure above, \(\overline{BD}\) is parallel to \(\overline{AE}\). What is the length of \(\overline{CE}\) ?

Solution
The correct answer is 30. In the figure given, since \(\overline{BD}\) is parallel to \(\overline{AE}\) and both segments are intersected by \(\overline{CE}\), then angle BDC and angle AEC are corresponding angles and therefore congruent. Angle BCD and angle ACE are also congruent because they are the same angle. Triangle BCD and triangle ACE are similar because if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. Since triangle BCD and triangle ACE are similar, their corresponding sides are proportional. So in triangle BCD and triangle ACE,\(\overline{𝐵𝐷}\) corresponds to \(\overline{𝐵𝐷}\) and \(\overline{CD}\) corresponds to \(\overline{CE}\). Therefore,\(\frac{BD}{CD}=\frac{AE}{CE}\). Since triangle BCD is a right triangle, the Pythagorean theorem can be used to give the value of CD: 6^{2} + 8^{2} = CD^{2}. Taking the square root of each side gives CD = 10. Substituting the values in the proportion \(\frac{BD}{CD}=\frac{AE}{CE}\) yields \(\frac{6}{10}=\frac{18}{CE}\). Multiplying each side by CE, and then multiplying by \(\frac{10}{6}\) yields CE = 30. Therefore, the length of \(\overline{CE}\) is 30.
^{2}⁄_{3}t = ^{5}⁄_{2}
What value of t is the solution of the equation above?

Solution
The correct answer is The correct answer is \(\frac{15}{4}\) or 3.75. Multiplying both sides of the equation ^{2}⁄_{3}𝑡 =^{5}⁄_{2} by ^{3}⁄_{2} results in 𝑡 = \(\frac{15}{4}\), or t = 3.75.
If a^{b⁄4} = 16 for positive integers a and b, what is one possible value of b ?

Solution
The correct answers are 1, 2, 4, 8, or 16. Number 16 can be written in exponential form a^{b⁄4},where a and b are positive integers as follows: 2^{4}, 4^{2}, 16^{1}, (16^{2})^{1⁄2},(16^{4})^{1⁄4}. Hence, if a^{1⁄2}, where a and b are positive integers, then ^{b}⁄_{4} can be 4, 2, 1,,^{1}⁄_{2}, or ^{1}⁄_{4}. So the value of b can be 16, 8, 4, 2, or 1. Any of these values may be gridded as the correct answer.
Which of the following is equivalent to \(\left ( a+\frac{b^{2}}{2} \right )\)?

Solution
The expression can be rewritten as (𝑎 + ^{b}⁄_{2}) (𝑎 + ^{b}⁄_{2}). Using the distributive property, the expression yields (𝑎 + ^{b}⁄_{2}) (𝑎 + ^{b}⁄_{2}) = 𝑎^{2} + \(\frac{ab}{2}+\frac{ab}{2}+\frac{b^{2}}{4}\). Combining like terms gives 𝑎^{2} + 𝑎𝑏 + ^{b2}⁄_{4}.
Choices A, B, and C are incorrect and may result from errors using the distributive property on the given expression or combining like terms.
A laundry service is buying detergent and fabric softener from its supplier. The supplier will deliver no more than 300 pounds in a shipment. Each container of detergent weighs 7.35 pounds, and each container of fabric softener weighs 6.2 pounds. The service wants to buy at least twice as many containers of detergent as containers of fabric softener. Let d represent the number of containers of detergent, and let s represent the number of containers of fabric softener, where d and s are nonnegative integers. Which of the following systems of inequalities best represents this situation?

Solution
The number of containers in a shipment must have a weight less than 300 pounds. The total weight, in pounds, of detergent and fabric softener that the supplier delivers can be expressed as the weight of each container multiplied by the number of each type of container, which is 7.35d for detergent and 6.2s for fabric softener. Since this total cannot exceed 300 pounds, it follows that 7.35d + 6.2s ≤ 300. Also, since the laundry service wants to buy at least twice as many containers of detergent as containers of fabric softener, the number of containers of detergent should be greater than or equal to two times the number of containers of fabric softener. This can be expressed by the inequality d ≥ 2s.
Choice B is incorrect because it misrepresents the relationship between the numbers of each container that the laundry service wants to buy. Choice C is incorrect because the first inequality of the system incorrectly doubles the weight per container of detergent. The weight of each container of detergent is 7.35, not 14.7 pounds. Choice D is incorrect because it doubles the weight per container of detergent and transposes the relationship between the numbers of containers.
2x^{2} − 4x = t
In the equation above, t is a constant. If the equation has no real solutions, which of the following could be the value of t ?

Solution
The number of solutions to any quadratic equation in the form ax^{2} + bx + c = 0, where a, b, and c are constants, can be found by evaluating the expression b^{2} − 4ac, which is called the discriminant. If the value of b^{2} − 4ac is a positive number, then there will be exactly two real solutions to the equation. If the value of b^{2} − 4ac is zero, then there will be exactly one real solution to the equation. Finally, if the value of b^{2} − 4ac is negative, then there will be no real solutions to the equation.
The given equation 2x^{2}− 4x = t is a quadratic equation in one variable, where t is a constant. Subtracting t from both sides of the equation gives 2x^{2} − 4x − t = 0. In this form, a = 2, b = −4, and c = −t. The values of t for which the equation has no real solutions are the same values of t for which the discriminant of this equation is a negative value. The discriminant is equal to (−4)^{2} − 4(2)(−t); therefore, (−4)^{2} − 4(2)(−t) < 0. Simplifying the left side of the inequality gives 16 + 8t < 0. Subtracting 16 from both sides of the inequality and then dividing both sides by 8 gives t < −2. Of the values given in the options, −3 is the only value that is less than −2. Therefore, choice A must be the correct answer.
Choices B, C, and D are incorrect and may result from a misconception about how to use the discriminant to determine the number of solutions of a quadratic equation in one variable.
Which of the following is equivalent to \(\frac{4x^{2} + 6x}{4x + 2}\)?

Solution
The vertex of the parabola in the xyplane above is (0,c) . Which of the following is true about the parabola with the equation y = – a(x – b)^{2} + c?

Solution
Since the shown parabola opens upward, the coefficient of x^{2} in the equation y = ax^{2} + c must be positive. Given that a is positive, –a is negative, and therefore the graph of the equation y = −a(x − b)^{2} + c will be a parabola that opens downward. The vertex of this parabola is (b, c), because the maximum value of y, c, is reached when x = b. Therefore, the answer must be choice B.
Choices A and C are incorrect. The coefficient of x^{2} in the equation y = −a(x − b)^{2} + c is negative. Therefore, the parabola with this equation opens downward, not upward. Choice D is incorrect because the vertex of this parabola is (b, c), not (−b, c), because the maximum value of y, c, is reached when x = b.