The Downtown Business Association (DBA) in a certain city plans to increase its membership by a total of n businesses per year. There were b businesses in the DBA at the beginning of this year. Which function best models the total number of businesses, y, the DBA plans to have as members x years from now?

Solution
The DBA plans to increase its membership by n businesses each year, so x years from now, the association plans to have increased its membership by nx businesses. Since there are already b businesses at the beginning of this year, the total number of businesses, y,the DBA plans to have as members x years from now is modeled by y = nx + b.
Choice B is incorrect. The equation given in choice B correctly represents the increase in membership x years from now as nx. However, the number of businesses at the beginning of the year, b, has been subtracted from this amount of increase, not added to it. Choices C and D are incorrect because they use exponential models to represent the increase in membership. Since the membership increases by n businesses each year, this situation is correctly modeled by a linear relationship.
An online bookstore sells novels and magazines. Each novel sells for $4, and each magazine sells for $1. If Sadie purchased a total of 11 novels and magazines that have a combined selling price of $20, how many novels did she purchase?

Solution
Let n be the number of novels and m be the number of magazines that Sadie purchased. If Sadie purchased a total of 11 novels and magazines, then n + m = 11. It is given that the combined price of 11 novels and magazines is $20. Since each novel sells for $4 and each magazine sells for $1, it follows that 4n + m = 20. So the system of equations below must hold.
4n + m = 20
n + m = 11
Subtracting side by side the second equation from the first equation yields 3n = 9, so n = 3.Therefore, Sadie purchased 3 novels.
Choice A is incorrect. If 2 novels were purchased, then a total of $8 was spent on novels. That leaves $12 to be spent on magazines, which means that 12 magazines would have been purchased. However, Sadie purchased a total of 11 novels and magazines. Choices C and D are incorrect. If 4 novels were purchased, then a total of $16 was spent on novels. That leaves $4 to be spent on magazines, which means that 4 magazines would have been purchased. By the same logic, if Sadie purchased 5 novels, she would have no money at all ($0) to buy magazines. However, Sadie purchased a total of 11 novels and magazines.
The weight of an object on Venus is approximately \(\frac{9}{10}\) of its weight on Earth. The weight of an object on Jupiter is approximately \(\frac{9}{10}\) of its weight on Earth. If an object weighs 100 pounds on Earth, approximately how many more pounds does it weigh on Jupiter than it weighs on Venus?

Solution
The weight of an object on Venus is approximately \(\frac{9}{10}\) of its weight on Earth. If an object weighs 100 pounds on Earth, then the object’s weight on Venus is given by \(\frac{9}{10}\)(100)= 90 pounds. The same object’s weight on Jupiter is approximately \(\frac{23}{10}\) of its weight on Earth; therefore, the object weighs \(\frac{23}{10}\)(100)= 230 pounds on Jupiter. The difference between the object’s weight on Jupiter and the object’s weight on Venus is 230 − 90 = 140 pounds. Therefore, an object that weighs 100 pounds on Earth weighs 140 more pounds on Jupiter than it weighs on Venus.
Choice A is incorrect because it is the weight, in pounds, of the object on Venus. Choice B is incorrect because it is the weight, in pounds, of an object on Earth if it weighs 100 pounds on Venus. Choice D is incorrect because it is the weight, in pounds, of the object on Jupiter.
If 3(c + d) = 5, what is the value of c + d?

Solution
The value of c + d can be found by dividing both sides of the given equation by 3. This yields c + d =^{5}⁄_{3}.
Choice A is incorrect. If the value of c + d is ^{3}⁄_{5}, then 3 × ^{3}⁄_{5} = 5; however,^{9}⁄_{5} is not equal to 5.Choice C is incorrect. If the value of c + d is 3, then 3 × 3 = 5; however, 9 is not equal to 5.Choice D is incorrect. If the value of c + d is 5, then 3 × 5 = 5; however, 15 is not equal to 5.
To make a bakery’s signature chocolate muffins, a baker needs 2.5 ounces of chocolate for each muffin. How many pounds of chocolate are needed to make 48 signature chocolate muffins?
(1 pound = 16 ounces)

Solution
If 2.5 ounces of chocolate are needed for each muffin, then the number of ounces of chocolate needed to make 48 muffins is 48 × 2.5 = 120 ounces. Since 1 pound = 16 ounces, the number of pounds that is equivalent to 120 ounces is \(\frac{120}{16}\) = 7.5 pounds. Therefore, 7.5 pounds of chocolate are needed to make the 48 muffins.
Choice B is incorrect. If 10 pounds of chocolate were needed to make 48 muffins, then the total number of ounces of chocolate needed would be 10 × 16 = 160 ounces. The number of ounces of chocolate per muffin would then be \(\frac{160}{48}\) = 3.33 ounces per muffin, not 2.5 ounces per muffin. Choices C and D are also incorrect. Following the same procedures as used to test choice B gives 16.8 ounces per muffin for choice C and 40 ounces per muffin for choice D, not 2.5 ounces per muffin. Therefore, 50.5 and 120 pounds cannot be the number of pounds needed to make 48 signature chocolate muffins.
Some values of the linear function f are shown in the table above. Which of the following defines f ?

Solution
Because f is a linear function of x, the equation f(x) = mx + b, where m and b are constants, can be used to define the relationship between x and f(x). In this equation, m represents the increase in the value of f(x) for every increase in the value of x by 1. From the table, it can be determined that the value of f(x) increases by 8 for every increase in the value of x by 2. In other words, for the function f the value of m is ^{8}⁄_{2}, or 4. The value of b can be found by substituting the values of x and f(x) from any row of the table and the value of m into the equation f(x) = mx + b and solving for b. For example, using x = 1, f(x) = 5, and m = 4 yields 5 = 4(1) + b. Solving for b yields b = 1. Therefore, the equation defining the function f can be written in the form f(x) = 4x + 1.
Choices A, B, and D are incorrect. Any equation defining the linear function f must give values of f(x) for corresponding values of x, as shown in each row of the table. According to the table, if x = 3, f(x) = 13. However, substituting x = 3 into the equation given in choice A gives f(3) = 2(3) + 3, or f(3) = 9, not 13. Similarly, substituting x = 3 into the equation given in choice B gives f(3) = 3(3) + 2, or f(3) = 11, not 13. Lastly, substituting x = 3 into the equation given in choice D gives f(3) = 5(3), or f(3) = 15, not 13. Therefore, the equations in choices A, B, and D cannot define f
According to the line graph above, between which two consecutive years was there the greatest change in the number of 3‑D movies released?

Solution
The change in the number of 3D movies released between any two consecutive years can be found by first estimating the number of 3D movies released for each of the two years and then finding the positive difference between these two estimates.Between 2003 and 2004, this change is approximately 2 − 2 = 0 movies; between 2008 and 2009, this change is approximately 20 − 8 = 12 movies; between 2009 and 2010, this change is approximately 26 − 20 = 6 movies; and between 2010 and 2011, this change is approximately 46 − 26 = 20 movies. Therefore, of the pairs of consecutive years in the choices, the greatest increase in the number of 3D movies released occurred during the time period between 2010 and 2011.
Choices A, B, and C are incorrect. Between 2010 and 2011, approximately 20 more 3D movies were released. The change in the number of 3D movies released between any of the other pairs of consecutive years is significantly smaller than 20.
A school district is forming a committee to discuss plans for the construction of a new high school. Of those invited to join the committee, 15% are parents of students, 45% are teachers from the current high school, 25% are school and district administrators, and the remaining 6 individuals are students. How many more teachers were invited to join the committee than school and district administrators?

Solution
The correct answer is 8. The 6 students represent (100 − 15 − 45 − 25)% = 15% of those invited to join the committee. If x people were invited to join the committee, then 0.15x = 6. Thus, there were \(\frac{6}{0.15}\)= 40 people invited to join the committee. It follows that there were 0.45(40) = 18 teachers and 0.25(40) = 10 school and district administrators invited to join the committee. Therefore, there were 8 more teachers than school and district administrators invited to join the committee.
Jeremy deposited x dollars in his investment account on January 1, 2001. The amount of money in the account doubled each year until Jeremy had 480 dollars in his investment account on January 1, 2005. What is the value of x ?

Solution
The correct answer is 30. The situation can be represented by the equation x(2^{4}) = 480, where the 2 represents the fact that the amount of money in the account doubled each year and the 4 represents the fact that there are 4 years between January 1, 2001, and January 1, 2005. Simplifying x(2^{4}) = 480 gives 16x = 480. Therefore, x = 30.
Andrew and Maria each collected six rocks, and the masses of the rocks are shown in the table above. The mean of the masses of the rocks Maria collected is 0.1 kilogram greater than the mean of the masses of the rocks Andrew collected. What is the value of x ?

Solution
The correct answer is 2.6 or \(\frac{13}{5}\). Since the mean of a set of numbers can be found by adding the numbers together and dividing by how many numbers there are in the set, the mean mass, in kilograms, of the rocks Andrew collected is \(\frac{2.4 + 2.5 + 3.6 + 3.1 + 2.5 + 2.7}{6}=\frac{16.8}{6}=2.8\). Since the mean mass of the rocks Maria collected is 0.1 kilogram greater than the mean mass of rocks Andrew collected, the mean mass of the rocks Maria collected is 2.8 + 0.1 = 2.9 kilograms. The value of x can be found by using the algorithm for finding the mean:\(\frac{x+3.1+2.7+2.9+3.3+2.8+2.9}{6}=2.9\). Solving this equation gives x = 2.6, which is equivalent to \(\frac{13}{5}\). Either 2.6 or 13/5 may be gridded as the correct answer.