Math Pool Flashcards

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

Answer: x + 1

Given the equations x = 4m² and y = (2m + 1)(2m - 1), we first simplify the expression for y. Using the difference of squares formula (a+b)(a-b) = a² - b², we get y = (2m)² - 1² = 4m² - 1. Since we know that x = 4m², we can substitute x into the simplified equation for y, resulting in y = x - 1. Please note that the provided correct answer 'x + 1' is inconsistent with the given equations, as the correct derivation leads to y = x - 1.

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?

Answer: The number of modes will decrease and the median will decrease.

First, sort the original list: 9, 9, 10, 14, 15, 20, 27. The median is the middle number, 14. The mode is 9 (appears twice). When 10 is added, the new sorted list is: 9, 9, 10, 10, 14, 15, 20, 27. The new median is the average of the two middle numbers (10 and 14), which is (10+14)/2 = 12. The new modes are 9 and 10, as both appear twice. Thus, the median decreases (from 14 to 12) and the number of modes increases (from 1 to 2). Therefore, the provided correct answer (D) is inconsistent with the calculation, as the number of modes actually increases, not decreases.

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

Answer: f(18)

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.)

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

Answer: 2

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.

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?

Answer: 2

First, determine the combined work rate of two farmers. If one farmer picks 12 cabbages in 1 hour, two farmers working at the same rate will pick 2 * 12 = 24 cabbages in 1 hour. To find out how long it would take them to pick 48 cabbages, divide the total number of cabbages by their combined rate: 48 cabbages / 24 cabbages/hour = 2 hours. Therefore, it would take two farmers 2 hours to pick 48 cabbages.

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 ?

Answer: 2

Consecutive odd integers differ by 2. Since (x - y) and (2x - y) are consecutive odd integers, their difference must be 2. Therefore, we can set up the equation: (2x - y) - (x - y) = 2. Simplifying the equation, we get 2x - y - x + y = 2, which reduces to x = 2. The variable y cancels out, allowing for a direct solution for x.

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

Answer: Keisha is eating a green cabbage.

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

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?

Answer: 18

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.

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

Answer: 6

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

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?

Answer: 2

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.