# Compass Math Pool Test 3

#### Two consecutive integers m and n are prime numbers. Which of the following is equal to mn ?

Hint: for the SAT you should know the first few prime numbers (2, 3, 5, 7, 11, . . .) as well the meaning of “consecutive integers”.

#### A farmer can pick 12 cabbages in 1 hour. Working at the same rate, how long in hours would it take two farmers to pick 48 cabbages?

#### All cabbages are red. Which of the following statements shows that the statement is FALSE?

Hint: you need a statement that mentions a cabbage which isn’t red.

#### If f(x) = 2x+4, then what is the value of f(4) + f(6) ?

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.)

#### 15, 9, 27, 10, 14, 9, 20: The number 10 is to be added to the list above. Which of the following must be true about the median and mode of the new list compared to the original?

#### If x2 + 5x − 14 = 0, and x > 0, then what is the value of x ?

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.

#### For two particular numbers x and y, the values of x − y and 2x − y are consecutive odd integers, respectively. What is the value of x ?

#### x = 4m2 y = (2m+ 1)(2m− 1) Given the two equations, what is y in terms of x ?

#### Set P is the set of all positive multiples of 4 less than 30. Set Q is the set of all positive multiples of 6 less than 30. How many numbers are in the intersection of sets P and Q?

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.

#### A particular type of soup is made by combining water and cabbage so that the ratio of water to cabbage is 9 to 1 by weight. How many pounds of cabbage are needed to make 180 pounds of this type of soup?

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.

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