QUESTION EXPLANATIONS

1. (B) — This can be written as 3x – 20 = 40, or 3x = 60, so x = 20.

2. (C) — We can set the equations equal to each other as 5n + 6 = – 2n – 24, then rearrange to get – + 7n + 30 = 0. Factoring the left side gives –(n – 10)(n + 3) = 0, meaning that the solutions are n = 10 and n = –3. Of these, only 10 is given as an option.

3. (C) — We can write (x - 5)/4 = 6 and multiply both sides by 4 to get x - 5 = 24. Adding 5 to both sides gives x = 29.

4. (B) — If x is positive, – 7x is the same as 3x – 7x, or –4x.

5. (A) — Slope is defined as rise (difference in y values) divided by run (difference in x values). The rise is –1, and the run is 2, so the slope is –½.

6. (D) — We can approach this problem by noticing that every time y increases by 1, x increases by 3, since the slope (1/3) is the rise (increase in y) divided by the run (increase in x). Since the line passes through (1, 1), we can see that only the point (7, 3) fits these criteria. Alternatively, we can write an equation in standard form: y = 1/3 x + b. Plugging in the given point, (1, 1), for x and y, we find that 1 = 1/3 + b, meaning that b = 2/3. We now have our equation: y = 1/3 x + 2/3. By checking the given points, we can again confirm that D) is the only point that satisfies this equation.

7. (C) — We can rewrite as x x x , or –1 x –1 x –1 x –1. This comes out to 1 x 1, or 1.

8. (B) — If the programmer wants to write 6 websites, she will need to code 6 pages of HTML. Since she can write 3 pages of HTML in 5 hours, it will take her 10 hours to code 6 pages. Similarly, to code the required 6 pages of JavaScript, she will need 9 hours since it takes 3 hours to code 2 pages of JavaScript. 10 + 9 = 19.

9. (C) — The formula for the volume of a cone is V = 1/3 . Since the radius is half the diameter, r = 8. We can now see that this is a 3-4-5 triangle (keep in mind that the figure is not drawn to scale), so the height is 6. We now have V = 1/3 , or 128.

10. (A) — To avoid mistakes in a problem like this, multiply one term at a time, and then add or subtract as needed: .

11. (D) — We can rewrite the first equation as x = 6y. Setting the equations equal to each other gives 6y = 8y + 16. Subtracting 8y from both sides gives –2y = 16, or y = –8.

12. (C) — The length of an arc is the radius times the central angle, or 6 x 30 degrees. Since the answers are in radians, we must convert 30 degrees to radians: 30 degrees x (2radians/360 degrees) = /6 radians, so 6 x /6 = .

13. (A) — We can represent the original price as p. Since there was a 15% discount, the discounted price can be written as 0.85p. With tax, the price is (1.06)(0.85)p = d. We can now find p in terms of d: p = d/(1.06)(0.85).

14. (C) — We want to find an equation in which e increases as d increases. Answer C) is the only equation that satisfies this; plotting a few points from the other equations confirms this. In Answer A), the term grows faster than d, so e decreases as d increases. In Answer B), grows faster than 100d, so again e decreases as d increases. In answer D), grows faster than , so e decreases as d increases.

15. (D) — The effect of doubling the distance is the same as saying that d2 becomes , or . This means that the denominator is four times larger, meaning that the force of gravity will decrease by 3/4.

16. (3/2) — For two lines to be perpendicular, their slopes must be reciprocal and have opposite signs. Since the slope of line k is -2/3, the slope of line j must be its reciprocal (-3/2), with the opposite sign (3/2)

17. (1) — We can model Amelie’s position as y = 5x and her brother’s position as y = 15(x – 2), where x is hours. Setting these equations equal to each other gives 5x = 15x – 30, or 10x = 30. So x = 3, meaning that Amelie has been running for 3 hours by the time her brother catches up with her. Since her brother left two hours later, it took him 3 – 2 = 1 hour to reach her.

18. (157) — g(3) = f(2 x 3) = f(6) = 63 – 10(6) + 1 = 216 – 60 + 1 = 157.

19. (100) — Charlie must travel 15 miles at 12 miles per hour, meaning it will take him 15 miles/12 miles/hour = 1.25 hours = 75 minutes to reach the top. Coming back, he must travel 15 miles at 36 miles per hour, taking 15 miles/36 miles/hour = 0.417 hours = 25 minutes. 75 + 25 = 100.

20. (108) — After 5 years, Sarah will have 30 pairs of shoes. After 10 years, she will have 60, and after 15, she will have 120. She donates 10%, or 12 pairs of shoes, leaving her with 120 – 12 = 108 pairs of shoes.