The line with the equation ^{4}⁄_{5}x + ^{1}⁄_{3}y = 1 is graphed in the xy‑plane. What is the xcoordinate of the x‑intercept of the line?

Solution
The correct answer is 1.25 or ^{5}⁄_{4}. The ycoordinate of the xintercept is 0, so 0 can be substituted for y, giving ^{4}⁄_{5}x + ^{1}⁄_{3}(0) = 1. This simplifies to ^{4}⁄_{5} x = 1. Multiplying both sides of ^{4}⁄_{5}x = 1 by 5 gives 4x = 5. Dividing both sides of 4x = 5 by 4 gives x = ^{5}⁄_{4}, which is equivalent to 1.25. Either 5/4 or 1.25 may be gridded as the correct answer
In the xyplane, the graph of y = 3x^{2} − 14x intersects the graph of y = x at the points (0, 0) and (a, a). What is the value of a ?

Solution
The correct answer is 5. The intersection points of the graphs of y = 3x^{2}− 14x and y = x can be found by solving the system consisting of these two equations. To solve the system, substitute x for y in the first equation. This gives x = 3x^{2} − 14x. Subtracting x from both sides of the equation gives 0 = 3x^{2} − 15x. Factoring 3x out of each term on the lefthand side of the equation gives 0 = 3x(x − 5). Therefore, the possible values for x are 0 and 5. Since y = x, the two intersection points are (0, 0) and (5, 5). Therefore, a = 5.
A laboratory supply company produces graduated cylinders, each with an internal radius of 2 inches and an internal height between 7.75 inches and 8 inches. What is one possible volume, rounded to the nearest cubic inch, of a graduated cylinder produced by this company?

Solution
The possible correct answers are 97, 98, 99, 100, and 101. The volume of a cylinder can be found by using the formula V = πr^{2}h, where r is the radius of the circular base and h is the height of the cylinder. The smallest possible volume, in cubic inches, of a graduated cylinder produced by the laboratory supply company can be found by substituting 2 for r and 7.75 for h, giving V = π(2^{2})(7.75). This gives a volume of approximately 97.39 cubic inches, which rounds to 97 cubic inches. The largest possible volume, in cubic inches, can be found by substituting 2 for r and 8 for h, giving V = π(2^{2})(8). This gives a volume of approximately 100.53 cubic inches, which rounds to 101 cubic inches. Therefore, the possible volumes are all the integers greater than or equal to 97 and less than or equal to 101, which are 97, 98, 99, 100, and 101. Any of these numbers may be gridded as the correct answer.
2(5x – 20) – (15 + 8x) = 7
What value of x satisfies the equation above?

Solution
The correct answer is 31. The equation can be solved using the steps shown below.
2(5x − 20) − 15 − 8x = 7
2(5x) − 2(20) − 15 − 8x = 7 (Apply the distributive property.)
10x − 40 − 15 − 8x = 7 (Multiply.)
2x − 55 = 7 (Combine like terms.)
2x = 62 (Add 55 to both sides of the equation.)
x = 31 (Divide both sides of the equation by 2.)
A group of friends decided to divide the $800 cost of a trip equally among themselves. When two of the friends decided not to go on the trip, those remaining still divided the $800 cost equally, but each friend’s share of the cost increased by $20. How many friends were in the group originally?

Solution
The correct answer is 10.Let n be the number of friends originally in the group. Since the cost of the trip was $800, the share, in dollars, for each friend was originally \(\frac{800}{n}\). When two friends decided not to go on the trip, the number of friends who split the $800 cost became n − 2, and each friend’s cost became \(\frac{800}{n2}\). Since this share represented a $20 increase over the original share, the equation \(\frac{800}{n}+20=\frac{800}{n2}\) must be true. Multiplying each side of \(\frac{800}{n}+20=\frac{800}{n2}\) by n(n − 2) to clear all the denominators gives
800(n − 2) + 20n(n − 2) = 800n
This is a quadratic equation and can be rewritten in the standard form by expanding, simplifying, and then collecting like terms on one side, as shown below:
800n − 1600 + 20n^{2} − 40n = 800n
40n − 80 + n^{2}− 2n = 40n
n^{2} − 2n − 80 = 0
After factoring, this becomes (n + 8)(n − 10) = 0.
The solutions of this equation are −8 and 10. Since a negative solution makes no sense for the number of people in a group, the number of friends originally in the group was 10.
The scatterplot below shows the amount of electric energy generated, in millions of megawatthours, by nuclear sources over a 10‑year period.
Of the following equations, which best models the data in the scatterplot?

Solution
The data in the scatterplot roughly fall in the shape of a downwardopening parabola; therefore, the coefficient for the x^{2} term must be negative. Based on the location of the data points, the yintercept of the parabola should be somewhere between 740 and 760. Therefore, of the equations given, the best model is y = −1.674x^{2} + 19.76x + 745.73.
Choices A and C are incorrect. The positive coefficient of the x^{2} term means that these these equations each define upwardopening parabolas, whereas a parabola that fits the data in the scatterplot must open downward. Choice B is incorrect because it defines a parabola with a yintercept that has a negative ycoordinate, whereas a parabola that fits the data in the scatterplot must have a yintercept with a positive ycoordinate.
A motor powers a model car so that after starting from rest, the car travels s inches in t seconds, where s = 16t√t. Which of the following gives the average speed of the car, in inches per second, over the first t seconds after it starts?

Solution
The average speed of the model car is found by dividing the total distance traveled by the car by the total time the car traveled. In the first t seconds after the car starts, the time changes from 0 to t seconds. So the total distance the car traveled is the distance it traveled at t seconds minus the distance it traveled at 0 seconds. At 0 seconds, the car has traveled 16(0)√0 inches, which is equal to 0 inches. According to the equation given, after t seconds, the car has traveled 16t√t inches. In other words, after the car starts, it travels a total of 16t√t inches in t seconds. Dividing this total distance traveled by the total time shows the car’s average speed: \(\frac{16t\sqrt{t}}{t}\) = 16t√t inches per second.
Choices A, C, and D are incorrect and may result from misconceptions about how average speed is calculated.
Two different points on a number line are both 3 units from the point with coordinate −4. The solution to which of the following equations gives the coordinates of both points?

Solution
The coordinates of the points at a distance d units from the point with coordinate a on the number line are the solutions to the equation x − a = d. Therefore, the coordinates of the points at a distance of 3 units from the point with coordinate −4 on the number line are the solutions to the equation x − (−4) = 3, which is equivalent to x + 4 = 3.
Choice B is incorrect. The solutions of x − 4 = 3 are the coordinates of the points on the number line at a distance of 3 units from the point with coordinate 4. Choice C is incorrect. The solutions of x + 3 = 4 are the coordinates of the points on the number line at a distance of 4 units from the point with coordinate −3. Choice D is incorrect. The solutions of x − 3 = 4 are the coordinates of the points on the number line at a distance of 4 units from the point with coordinate 3.
In the xyplane, the graph of 2x^{2} − 6x + 2y^{2} + 2y = 45 is a circle. What is the radius of the circle?

Solution
One way to find the radius of the circle is to put the given equation in standard form, (x − h)^{2} + (y − k)^{2} = r^{2}, where (h, k) is the center of the circle and the radius of the circle is r. To do this, divide the original equation, 2x^{2} − 6x + 2y^{2} + 2y = 45, by 2 to make the leading coefficients of x^{2} and y^{2} each equal to 1: x^{2} − 3x + y^{2} + y = 22.5. Then complete the square to put the equation in standard form. To do so, first rewrite x^{2} − 3x + y^{2} + y = 22.5 as (x^{2} − 3x + 2.25) − 2.25 + (y^{2} + y + 0.25) − 0.25 = 22.5. Second, add 2.25 and 0.25 to both sides of the equation: (x^{2} − 3x + 2.25) + (y^{2} + y + 0.25) = 25. Since x^{2} − 3x + 2.25 = (x − 1.5)^{2}, y^{2} − x + 0.25 = (y − 0.5)^{2} , and 25 = 5^{2} , it follows that (x − 1.5)^{2} + (y − 0.5)^{2} = 5^{2}. Therefore, the radius of the circle is 5.
Choices B, C, and D are incorrect and may be the result of errors in manipulating the equation or of a misconception about the standard form of the equation of a circle in the xyplane.
A gear ratio r:s is the ratio of the number of teeth of two connected gears. The ratio of the number of revolutions per minute (rpm) of two gear wheels is s:r. In the diagram below, Gear A is turned by a motor. The turning of Gear A causes Gears B and C to turn as well.
If Gear A is rotated by the motor at a rate of 100 rpm, what is the number of revolutions per minute for Gear C?

Solution
Since Gear A has 20 teeth and Gear B has 60 teeth, the gear ratio for Gears A and B is 20:60. Thus the ratio of the number of revolutions per minute (rpm) for the two gears is 60:20, or 3:1. That is, when Gear A turns at 3 rpm, Gear B turns at 1 rpm. Similarly,since Gear B has 60 teeth and Gear C has 10 teeth, the gear ratio for Gears B and C is 60:10, and the ratio of the rpms for the two gears is 10:60. That is, when Gear B turns at 1 rpm, Gear C turns at 6 rpm. Therefore, if Gear A turns at 100 rpm, then Gear B turns at \(\frac{100}{3}\) rpm, and Gear C turns at \(\frac{100}{3}\) × 6 = 200 rpm.
Alternate approach: Gear A and Gear C can be considered as directly connected since their “contact” speeds are the same. Gear A has twice as many teeth as Gear C, and since the ratios of the number of teeth are equal to the reverse of the ratios of rotation speeds, in rpm, Gear C would be rotated at a rate that is twice the rate of Gear A. Therefore, Gear C will be rotated at a rate of 200 rpm since Gear A is rotated at 100 rpm.
Choice A is incorrect and may result from using the gear ratio instead of the ratio of the rpm when calculating the rotational speed of Gear C. Choice B is incorrect and may result from comparing the rpm of the gears using addition instead of multiplication. Choice D is incorrect and may be the result of multiplying the 100 rpm for Gear A by the number of teeth in Gear C.