The Correct Answer is (B) — This can be written as 3x – 20 = 40, or 3x = 60, so x = 20.
The Correct Answer is (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.
The Correct Answer is (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.
The Correct Answer is (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.
The Correct Answer is (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.
The Correct Answer is (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).
The Correct Answer is (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.
The Correct Answer is (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)
The Correct Answer is (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.
The Correct Answer is (157) — g(3) = f(2 x 3) = f(6) = 63 – 10(6) + 1 = 216 – 60 + 1 = 157.
The Correct Answer is (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.
The Correct Answer is (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.