Even if something other than a number is given inside the parentheses of a function, the rule is the same: substitute the expression inside the parentheses for x in the function definition. For example, if g(x) = 2x+1, then g(cabbage) = 2·cabbage+1. For this problem, substitute 3a for x to get f(3a) = 2(3a)2 = 18a2. Since 2f(3a) = 144, then 36a2 = 144 so that a2 = 4 and a = ±2, making answer D correct. (Note that −2 is not listed as a choice: there can only be one answer.)
You should be able to factor the left-hand side of the equation as follows: x2 + 5x − 14 = (x + 7)(x − 2) = 0 so that either x = −7 or x = 2. Be careful to read the question: since the problem specified x > 0, the answer is not A; instead, the answer is C, or x = 2.
Hint: you need a statement that mentions a cabbage which isn’t red.
Strategy: work with the answers. Suppose the answer is A: 9 pounds of cabbage are needed. This means that 81 pounds of water are needed, since the ratio 81 : 9 is equal to the ratio 9 : 1. But this is only 81 + 9 = 90 pounds of soup, so answer A is incorrect. Using answer B, we have 18 pounds of cabbage and 9 · 18 = 162 pounds of water (to get the 9 : 1 ratio). This makes 162 + 18 = 180 pounds of soup, so answer B is correct. Math Teacher Solution: Let c be the number of pounds of cabbage, and w be the number of pounds of water. Then, c + w = 180 (cabbage and water add up to 180 pounds), and w/c = 9 (ratio of water to cabbage is 9 : 1). Solving the second equation for w and substituting into the first gives: c+9c = 180 so that 10c = 180, or c = 18.
Hint: for the SAT you should know the first few prime numbers (2, 3, 5, 7, 11, . . .) as well the meaning of “consecutive integers”.
he positive multiples of 4 less than 30 are 4, 8, 12, 16, 20, 24, and 28, and the positive multiples of 6 less than 30 are 6, 12, 18, and 24, so the intersection of these two sets is the set consisting of the numbers 12 and 24.
