⇒ \(\frac{2d^{2} – d – 10}{d^{2} + 7d + 10} = \frac{d^{2} – 4d + 3}{d^{2} + 2d – 15}\)
In the equation above, what is the value of d ?

Solution
Whenever the question includes variables and the answer choices are numbers, think Plugging In the Answers.
In (A), d = 4, and the equation becomes \(\frac{2\left (4 \right )^{2}  \left (4 \right )  10}{\left (4 \right )^{2} + 7\left ( 4 \right ) + 10} = \frac{\left ( 4 \right )^{2}  4\left ( 4 \right ) + 3}{\left ( 4 \right )^{2} + 2\left ( 4 \right )  15}\).
Solve both sides of the equation to get \(\frac{2\left (16 \right ) + \left (4 \right )  10}{16  28 + 10} = \frac{16 + 16 + 3}{16  8  15}, \;\; or \;\; \frac{26}{2} = \frac{35}{7}\).
Reduce both fractions to get 13 = 5. This is not true, so eliminate (A).
In (B), d = 2, and the equation becomes \(\frac{2\left (2 \right )^{2}  2  10}{\left (2 \right )^{2} + 7\left (2 \right ) + 10} = \frac{\left (2 \right )^{2}  4\left (2 \right ) + 3}{\left (2 \right )^{2} + 2\left (2 \right )  15}\).
Solve both sides of the equation to get \(\frac{2\left (4 \right )  \left (2 \right )  10}{4 + 14 + 10} = \frac{4  8 + 3}{4 + 4  15}, \;\; or \;\; \frac{4}{28} = \frac{1}{7}\).
Reduce both fractions to get \(\frac{1}{7} = \frac{1}{7}\).
Eliminate (B).
In (C), d = 4 and the equation becomes \(\frac{2\left (4 \right )^{2}  \left (4 \right )  10}{\left (4 \right )^{2} + 7\left ( 4 \right ) + 10} = \frac{\left ( 4 \right )^{2}  4\left ( 4 \right ) + 3}{\left ( 4 \right )^{2} + 2\left ( 4 \right )  15}\).
Solve both sides of the equation to get \(\frac{2\left (16 \right )  \left (4 \right )  10}{16 + 28 + 10} = \frac{16  16 + 3}{16 + 8  15}, \;\; or \;\; \frac{18}{54} = \frac{3}{9}\).
Reduce both fractions to get \(\frac{1}{3} = \frac{1}{3}\).
The correct answer is (C).
A dental hygiene company is creating a new 24ounce tube of toothpaste by combining its most popular toothpastes, Cavity Crusher and Bad Breath Obliterator. Cavity Crusher contains 0.25% of sodium fluoride as its active ingredient, and Bad Breath Obliterator contains 0.30% of triclosan as its active ingredient for a total of 0.069 ounces of active ingredients in both toothpastes. Solving which of the following systems of equations yields the number of ounces of Cavity Crusher, c, and the number of ounces of Bad Breath Obliterator, b, that are in the new toothpaste?

Solution
Start with the easier equation and use Process of Elimination.
The easier equation is related to the total number of ounces, c + b, in the tube.
According to the question, the tube has 24 ounces, so c + b = 24.
Eliminate (A), since it does not include this equation.
The other equation in the set is related to the amount of active ingredients.
According to the question, c includes 0.25% of sodium fluoride and b contains 0.30% triclosan.
0.25% = 0.0025 and 0.30% = 0.003.
Therefore, in the correct equation, c should be associated with 0.0025 and b should be associated with 0.003.
Eliminate (C) and (D) because both of these equations get the percentages wrong.
The correct answer is (B).
The rotation rate of a mixing blade, in rotations per second, slows as a liquid is being added to the mixer. The blade rotates at 1,000 rotations per second when the mixer is empty. The rate at which the blade slows is four rotations per second less than three times the square of the height of the liquid. If h is the height of liquid in the mixer, which of the following represents R(h), the rate of rotation?

Solution
Treat this question as a translation problem.
According to the question, R(h) = four rotations per second less than three times the square of the height of the liquid.
The height of the liquid is represented by h.
Therefore, three times the square of the height of the liquid = 3h^{2}.
Four less than this amount is 3h^{2}  4.
Since the original speed was 1,000 subtract this value from 1,000 to get the current rate of rotation.
The correct answer is (D).
If b is two more than onethird of c, which of the following expresses the value of c in terms of b ?

Solution
Whenever there are variables in the question and in the answers, think Plugging In.
Let c = 30. Therefore, b = 2 + ^{1}⁄_{3}(30) = 2 + 10 = 12.
Plug 12 in for b in the answers to see which answer equals the target number of 30.
Choice (A) becomes \(\frac{12  2}{3} = \frac{10}{3} = 3.\overline{3}\).
Eliminate (A), since it does not equal the target number.
Choice (B) becomes \(\frac{12 + 2}{3} = \frac{14}{3} = 4.\overline{6}\).Eliminate (B).
Choice (C) becomes 3(12  2) = 3(10) = 30. Keep (C), but check (D) just in case it also works.
Choice (D) becomes 3(12  6) = 3(6) = 18. Eliminate (D).
The correct answer is (C).
Jeff tests how the total volume occupied by a fluid contained in a graduated cylinder changes when round marbles of various sizes are added. He found that the total volume occupied by the fluid, V, in cubic centimeters, can be found using the equation below, where x equals the number of identical marbles Jeff added, one at a time, to the cylinder, and r is the radius of one of the marbles.
\(V = 24 \pi + x\left ( \frac{4}{3}\pi r^{3} \right )\)If the volume of the graduated cylinder is 96π cubic centimeters, then, what is the maximum number of marbles with a radius of 3 centimeters that Jeff can add without the volume of the fluid exceeding that of the graduated cylinder?

Solution
This is a good Plug In the Answers problem. Start with (B) and plug in 2 for x and 3 for r in the equation to get \(V = 24 \pi + 2\left ( \frac{4}{3}\pi 3^{3} \right )\), which is equal to the target amount of 96π, so (B) is correct.
The number of bonus points, B(p), that a credit card holder receives is given by the function B(p) = 4p + 7, where p represents the number of purchases made. If the number of purchases is increased by 3, by how much does the number of bonus points increase?

Solution
Whenever there are variables in the question and the answer choices, think Plugging In.
If 2 purchases were made, then p = 2, and the number of bonus points can be calculated as 4(2) + 7 = 8 + 7 = 15.
If the number of purchases were then increased by 3, the new p equals 5 and the number of bonus points can be calculated as 4(5) + 7 = 27.
The bonus points increased by 27  15 = 12. Therefore, the correct answer is (C).
What is the equation of a line that contains the point (1, 6) and has a yintercept of 4 ?

Solution
All of the answers are written in the slopeintercept form y = mx + b, where b is the yintercept and x and y are points on the line.
Eliminate (D) because the yintercept in that equation is 2.
For the remaining answer choices, plug in the x and yvalues to determine which equation works.
If x = 1 and y = 6, (A) becomes 6 = ^{1}⁄_{2}(1) + 4.
Solve both sides of the equation to get 6 = 4^{1}⁄_{2}. Eliminate (A).
Choice (B) becomes 6 = 1 + 4, so eliminate (B).
Choice (C) becomes 6 = 2(1) + 4, or 6 = 6.
Therefore, the correct answer is (C).
Syed took out a cash advance of d dollars from a financing company. The company deducts a fee of of the original advanced amount along with a wire transfer fee of $30.00. Which of the following represents the final advanced amount that Syed receives after all applied fees, in dollars?

Solution
Whenever there are variables in the question, plug in. Be sure to plug in a number that is divisible by 3. Let d = 300.
^{1}⁄_{3} of the original amount of $300 is $100, and that is deducted by the company, leaving Syed with $200.
Now, subtract the wire transfer fee to get $200  $30 = $170, which is the target number.
Plug in 300 for d in the answers to see which answer is equal to the target number of 170.
In (A), ^{1}⁄_{3}(300)  30 = 70.
This is not the target number, so eliminate (A).
Likewise in (B), ^{1}⁄_{3}(300  30) = 90, and in (C), ^{2}⁄_{3}(300  30) = 180.
Neither of these is the target number, so eliminate (B) and (C).
In (D), ^{2}⁄_{3}(300)  30 = 170, which is the target number.
The correct answer is (D).
A beverage store charges a base price of x dollars for one keg of root beer. A sales tax of a certain percentage is applied to the base price, and an untaxed deposit for the keg is added. If the total amount, in dollars, paid at the time of purchase for one keg is given by the expression 1.07x + 17, then what is the sales tax, expressed as a percentage of the base price?

Solution
You can plug in to make sense of this equation. Say that x = $100.
The amount of the keg would then be $107 + $17.
The $17 must be the untaxed deposit since it is a flat fee rather than percentage based.
Therefore, the tax is $7, which is 7% of the original $100 base price.
The answer is (C).
Which of the following equations has a vertex of (3, 3) ?

Solution
The vertex form of a parabola is y = a(x  h)^{2} + k, where (h, k) denotes the vertex.
Plug in the point (3, 3) into the vertex form to get y = a(x  3)^{2}  3.
The correct answer is (A).