Shaun is developing a weight loss regimen, which includes both a workout plan and a calorierestriction plan. Shaun wants to work out for no less than 30 minutes and no more than 60 minutes a day and consume no less than 2,000 and no more than 2,500 calories. If each minute, m, of his workout time burns 50 calories, which of the following inequalities represents the number of minutes, m, that Shaun can work out each day to burn off as many calories as he consumes?

Solution
If each minute of his workout time burns 50 calories, and he wants to consume no fewer than 2,000 calories, Shaun must work out for a minimum of \(\frac{2,000}{50} = 40\) minutes.
If he wants to consume no more than 2,500 calories, Shaun must work out for a maximum of \(\frac{2,500}{50} = 50\) minutes.
Since the question asks for the inequality that represents the number of minutes for which Shaun will burn off as many calories as he consumes, (D) is correct, as it includes both the minimum (40 minutes) and maximum (50 minutes) amount of time that he can work out.
Choice (C) is incorrect because the answer should include 50 (he can work out for a "maximum" of 50 minutes, so he could work out for 50 minutes), but the lesser than sign ("<") excludes 50.
If 8x + 8y = 18 and x^{2} – y^{2} = –^{3}⁄_{8}, what is the value of 2x – 2y ?

Solution
Start by simplifying 8x + 8y = 18 by dividing each term by 8: x + y = ^{18}⁄_{8} or : x + y = ^{9}⁄_{4}.
The second equation provided in the question can be factored: x^{2}  y^{2} is the same as (x + y)(x  y), so the second equation can also be written (x + y)(x  y) = ^{3}⁄_{8}.
Since you know that x + y = ^{9}⁄_{4}, you can rewrite the second equation as ^{9}⁄_{4}(x  y) = ^{3}⁄_{8}.
Multiply both sides by : ^{4}⁄_{9} : x  y = ^{3}⁄_{8}(^{4}⁄_{9}) or x  y = ^{1}⁄_{6}.
Since the question asks for the value of 2x  2y, simply multiply everything by 2: 2(x  y) = 2(^{1}⁄_{6}) = ^{1}⁄_{3}, which is (A).
Each winter, Captain Dan’s Ski Lodge rents both pairs of skis and snowboards to its guests for a flat daily rate per pair of skis and a flat daily rate per snowboard. Five pairs of skis and two snowboards will cost a family $370. Three pairs of skis and four snowboards will cost a family $390. During a particularly slow season, Captain Dan announces a 10% discount on all skis and snowboards. What would be the cost of renting two pairs of skis and two snowboards if they were rented during this discount period?

Solution
In order to determine the normal cost for renting skis and snowboards, you need to write two equations and then manipulate and solve those equations.
If you call skis x and snowboards y, your two equations will be 5x + 2y = 370 and 3x + 4y = 390.
Look for a way to stack and add the equations to eliminate one of the variables.
For instance, multiply the first equation by 2 to get 10x + 4y = 740, and then stack and subtract the equations, as follows:
\(\frac{\begin{matrix} & 10x & + & 4y & = & 740\\  & 3x & + & 4y & = & 390 \end{matrix}}{ \begin{matrix} & \;\;\;\; 7x & & & \;\;\;\;\,\; \; \; = & 350 \end{matrix}}\)
So, 7x = 350 and x = 50, so the price of a pair of skis is $50.
Plug this number back into either equation to find the cost of a snowboard: 10(50) + 4y = 740, so 4y = 740  500 and 4y = 240.
Therefore, y = 60, the cost of a snowboard.
So, the cost of two pairs of skis and two snowboards would normally be 2(50) + 2(60) = 100 + 120 = 220.
Finally, remember that prices are discounted by 10%, so multiply the price of $220 by 10% to get $22, and subtract $22 from the price.
The final cost of two pairs of skis and two snowboards is 220  22 = 198, which is (C).
George and Joe both interview the same 20 fellow students regarding their interest in their school’s new Model UN Club. George asked the students to respond with Interested, Sort of Interested, and Not Interested. Joe asked the students to rate their interest on a scale of 1 to 5. The results of the polls are below.
George’s Poll
Response  Number of Students 
Interested  8 
Sort of Interested  5 
Not Interested  7 
Joe’s Poll
Rating  Number of Students 
1  5 
2  4 
3  3 
4  4 
5  4 
After reviewing the data, the Model UN advisors determine that Joe neglected to include whether a 1 or 5 was the best rating in his report. What additional piece of information would most help the advisor determine whether a 1 or 5 was the best rating?

Solution
The issue that needs clarification here is whether the students polled by Joe thought that a score of 1 or a score of 5 was good.
Since (A) and (C) deal with George's poll, they would do nothing to help clarify this ambiguity.
Choice (B) might help us to figure out which of the students Joe polled were interested in the Model UN Club; it would not help to determine whether 1 or 5 was the best rating.
Choice (D) is thus the best answer.
The graph, y = f(x), shown above models the performance of a certain crop, where x is the nutrients subtracted or added to the soil and y is the gain or loss of pieces of fruit added to the total harvest. A more powerful fertilizer that is used causes the graph y = f(x) to be reflected over the line y = x. Which of the following best describes the behavior of the crop with the new fertilizer?

Solution
Start by finding the slope of the line provided on the graph using the points (0,4) and (6,0) and the pointslope formula:\(\frac{y_{2}  y_{1}}{x_{2}  x_{1}} = \frac{0 \left (4 \right )}{6  0} = \frac{4}{6} = \frac{2}{3}\).
When this line is reflected across the line y = x, the x and y values switch, so the new slope would be the reciprocal of the original slope.
Since our original slope was ^{2}⁄_{3}, our new slope will be ^{3}⁄_{2}.
The numerator here reflects the gain or loss of pieces of fruit in the harvest, and the denominator reflects the nutrients subtracted or added.
This means that for every two nutrients added, there will be a harvest gain of three pieces of fruit, which is (D).
Three different chefs work together to prepare meals for 280 dinner guests. Each works at a different speed, and their combined output throughout the night is modeled by the equation 8x + 4x + 2x = 280. If x is a positive integer, which of the following could 8x represent in the equation?

Solution
Since work = rate × time, the 280 in the equation must represent the total number of meals (i.e. the "work").
All three chefs are working together, so they work for the same amount of time, and x must represent that time.
The coefficients 8, 4, and 2 must therefore represent the chefs' respective rates, or how many meals each prepares in a set amount of time.
Since 8 is the greatest of these three coefficients, 8x must be the meal output of the fastest chef, either (B) or (C).
Now you need to solve the equation: 8x + 4x + 2x = 280.
Combining like terms gives you 14x = 280.
Divide both sides by 14 to determine that x = 20.
This number represents the amount of time that the chefs worked, so the actual number of meals prepared by the fastest chef would be 8 × 20 = 160 meals, which is (B).
If \(a = \frac{4a^{2}}{16}\) and a is a nonzero integer, which of the following is equivalent to a ?

Solution
The first step here is to simplify the equation and solve for a.
Start by multiplying both sides by 16 to get 16a = 4a^{2}.
Divide both sides by 4 to get 4a = a^{2}.
Divide both sides by a to get 4 = a.
This is now your target answer.
Plug a = 4 into the values of a in the answer choices to determine which one matches 4.
Choice (D) is the answer, since 2√a = 2√4 = 4.
The number of eggs that Farmer Jones has in his chicken coop will grow exponentially as Farmer Jones buys more chickens to increase production. The number of eggs Farmer Jones has in the coop can be modeled by the equation y = 3^{x} beginning on Day 1, where x is given by x = 1, and y is the number of eggs currently in the coop. If the coop can support only 4,000 eggs, and Farmer Jones empties the coop every day, on which day will the chickens produce too many eggs for the coop to support?

Solution
The best way to deal with this question is to Plug in the Answers (PITA), starting with (A).
If x = 6, then y = 3^{6} = 729.
This is less than 4,000, so eliminate (A) and move to the next answer choice.
If x = 7, then y = 3^{7} = 2,187.
This is still less than 4,000, so eliminate (B).
If x = 8, then y = 3^{8} = 6,561.
This is greater than 4,000, so (C) must be the correct answer.
Two graphs, f(x) and h(x), are shown above. If f(x) = 3x + 4 and f(x) and h(x) are perpendicular, which of the following could be the equation of h(x) ?

Solution
Don't get too thrown off by the graph.
All you need to know to solve this question is that perpendicular lines have slopes that are the negative reciprocals of each other.
Since the standard equation for a line is y = mx + b, the slope of the f(x) line is 3.
The slope of the h(x) line must therefore be ^{1}⁄_{3}.
The only answer choice that matches is (B).
Given the scatterplot graph above, ten students at Welton Academy were polled at random about their usage of the school’s new physicscentered social media app, E = MC Shared. The app was developed to encourage students to discuss physics curricula and concepts in ways that mirrored social media trends in 2013. Students were asked how many times they logged into the app each day as well as how many posts they actually made using the app. With the given data, what conclusions can be drawn about this group of students?

Solution
The best way to approach this question is through POE.
Choice (A) states that the majority of students polled logged in more times than they posted.
The values along the xaxis of the graph are, for most of the data points, higher than the values along the yaxis of the graph, and thus (A) is true according to the data provided.
This same data contradicts (B) and (C).
You can eliminate (D) because the data does, in fact, allow you to draw a conclusion about the relationship between the variables.