(B) Pick numbers, say 40 for x and −50 for y. Then (A) is −50 − 40 = −90, and (B) is 40 − (−50) = 40 + 50 = 90. So (B) is greater. Quantity (B) will always be greater because it will be positive in value (since the double negatives cancel to make a positive), while Quantity (A) will always be negative.

(A) The probability of any event is between 0 and 1, inclusive; that is, 0 ≤ p ≤ 1, for any probability, p. The probability of two independent events both occurring is the product of their individual probabilities. If the probability of Event C occurring were as low as .09, the value in column (B), then the probability of both events occurring would be less than or equal to .09, since .09 times a number from 0 through 1 would be at most .09. But the given probability of both events occurring is .11, which is greater than .09, so the probability of Event C occurring cannot be as low as .09, and therefore it must be greater than .09, and (A) is correct.

(A) Let p = the full price, before any discounts. Then Mary, who received 25% marked off the price, paid p – .25p = .75p. Since she also paid $201, we know .75p = 201. Divide by .75 to obtain 1p = $268. Nan received a 15% discount, so she paid 1p – .15p = .85p, which is .85 × 268 = $227.80, a little more than $227, so (A) is correct

(C, D, E) Easiest is to pick numbers for the two odds, say p = 3, and m = 5. Then (A) is pm = 15, not an even number, and (B) is 3(3) – 2(5) = –1, also not even, so (A) and (B) are both incorrect. (C) is always even by definition, because it is written as 2 times another integer. Because 3 + 5 = 8, (D) is correct. 8 is even, and, in general, the sum of two odds is even. (E) is correct, because the value of the inner parentheses is even (since it is the sum of two odd numbers), and an even times any integer is even. Here, for example, (3 × 3 + 5 × 5) = 9 + 25 = 34. Answer is (C), (D), and (E).

(C, D) The triangle inequality states that in a triangle, the sum of any two of the side lengths is always greater than the length of any one. Equivalently, any one side length in a triangle must be less than the sum of the other two, but greater than the difference of the other two side lengths. Here, 17 – 12 < x < 17 + 12, or 5 < x < 29. Then the sum of the triangle’s three side length, its perimeter, will be greater than 5 + 12 + 17 but less than 29 + 12 + 17. This means 34 < perimeter < 58. Choices (C) and (D) satisfy this.

(B, C, D) The restaurant will spend more than 55 + 23 = 78% of its revenue but less than 70 + 28 = 98% of its revenue on food, supplies, and staff salaries combined. Then its remaining net profit, after these expenses are subtracted from revenue, will be between 2% and 22% of revenue, because 100 – 78 = 22, and 100 – 98 = 2. Then for a revenue of $80,000, the profit, p, will satisfy .02 × 80,000 < p < .22 × 80,000, which implies 1600 < p < 17600. (B), (C), and (D) satisfy this.

(2,640) Draw out four blanks to represent the four places in the word to be filled: ________ ________ ________ ________. In each blank, write the number of different ways that particular place can be filled. The total number of combinations will then equal the product of the numbers in the blanks. There are 12 consonants and five vowels in the language. Words fit the CVVC pattern, so we might expect the total number of possible words to equal 12 × 5 × 5 × 12. But since we are looking for words with no repeated letters, the number of possibilities for the second vowel and for the second consonant are each reduced by 1, so the answer becomes 12 × 5 × 4 × 11 = 2,640 possible words with four different letters.

($78,104) When Company D’s value decreased by 3%, it retained 97% of its value of $66,000, which is .97 × 66,000 = $64,020. In the following year, it increased in value by 22% of this, or .22 × $64,020 = $14,084.40. When this increase is added to 64,020, the final value becomes $78104.4, which rounds to $78,104.

*A nice shortcut is to think of the final value as 1.22(.97(66,000)), which is entered on the calculator as 1.22 × .97 × 66,000 = 78,104.4 ≈ 78,104. In this approach, we obtain 1.22 mentally by thinking of (value + 22% of value) as 1V + .22V = 1.22V. Similarly, value minus 3% of value is 1V – .03V = .97V, justifying the shortcut.

(E) We are informed of the percent change in the companies’ total sales, but we have no information on the actual sales amounts in dollars. A small percentage of a high sales total could be greater than a larger percentage of a low sales total. Therefore, (E), there is not enough information to determine.

(A, B) For (A), it is easiest to imagine that the company started with sales of $100. After a 10% increase, it had sales of 100 + 10 = $110. When the company next experienced a 10% decrease in sales, it lost .10 × 110 = $11 in sales, so it had only $110 – 11 = $99 in sales. (Or .9 × 110 = $99, as in 16* above.) This makes (A) correct, because 99 < 100, corresponding to a decrease in sales from the starting value. To compute Company B’s change in sales for the 2-year period, multiply 1.12 by 1.13 to represent the 12% increase “on top of” a 13% increase (again, see 16* above). This yields 1.12 × 1.13 ≈ 1.266, or about a 26.6% increase for the 2-year period. Inspection of the other numbers shows that none of the other companies increased by as much, so (B) is correct. You can use the percent change figures to compute Company C’s total sales ratio for 2011 to 2009. Assume the figure is $100 in 2009. After a 17% increase, the sales will equal $117. However, to decrease this number, $117, by 5% is not to simply subtract 5 from 117, to obtain 112, and so the ratio will not equal 112:100, and (C) is not correct. Company C’s sales in 2011 in this example would equal .95 × 117 = 111.15, and so the proper ratio of sales between the two years would be 111.15:100. Answer is (A) and (B).