What is the smallest possible value for the product of 2 real numbers that differ by 6?

Solution
If 2 numbers, x and y, differ by 6, that means that x − y = 6. Multiplying the two numbers, (x)(y), will yield the product. Solve the first equation for x.
x − y = 6
x = y + 6
Substitute the result for x in the second equation.
(y + 6)y
Since one of the answer choices must be the solution to that equation, plug in the answer choices, starting with the smallest value (note that the answer choices are in ascending order):
(y + 6)y = −9
y2 + 6y + 9 = 0
(y + 3)^{2} = 0
y = −3
Now, substitute −3 for y in the first equation and solve for x:
x − (−3) = 6
x = 3
Since (x)(y) = (3)(−3) = −9 is the smallest value given as an answer, answer choice F must be correct.
In a game, 84 marbles numbered 00 through 83 are placed in a box. A player draws 1 marble at random from the box. Without replacing the first marble, the player draws a second marble at random. If both marbles drawn have the same tens digit (that is, both marbles are numbered between 00 and 09, or 10 and 19, or 20 and 29, etc.), the player is a winner. If the first marble Dave draws is numbered 23, what is the probability that Dave will be a winner on the next draw?

Solution
The rules of the game state that a player is a winner if two marbles drawn have the same tens digit. The player has already drawn the marble numbered 23, which has a 2 in the tens digit. In order to win, the player must draw another marble with a 2 in the tens digit. The possible winning marbles are 20, 21, 22, 24, 25, 26, 27,28, and 29. Therefore, the player has nine chances to draw a winning marble. Since he has already drawn one of the 84 marbles and did not put it back, he has nine chances out of 83 to draw a winning marble. The probability is \(\frac{9}{83}\).
If f (x) = x^{2} + 3, then f (x + y) = ?

Solution
To answer this questions, substitute (x + y) for x in the equation and solve using the FOIL method, as follows:
(x + y)^{2} + 3
(x + y)(x + y) + 3
x^{2} + xy + xy + y^{2} + 3
x^{2} + 2xy + y^{2} + 3
Jason has been hired to build a circular wading pool in his neighbor’s backyard. The rectangular backyard measures 60 feet wide by 50 feet long. Jason’s neighbors want the pool to be as large as possible, with the edge of the pool at least 8 feet from the edge of the backyard all around. How long should the radius of the pool be, in feet?

Solution
First, draw the picture of the wading pool according to the information given in the problem, where the distance from the edge of the pool to the edge of the long side of the rectangular region is 8 feet. The distance from the edge of the pool to the edge of the short side of the rectangular region can be anything greater than 8, but it is not necessary to know this distance to solve the problem:
Now you can determine the diameter of the circular pool. The diameter is the maximum distance from one point on a circle to another (the dashed line) through the center of the circle. Since the short side of the rectangular region is 50 feet, and the distance from the edge of the circular pool to each edge of the long sides of the rectangular region is set at 8 feet, the diameter of the circle must be 50 feet − 2(8 feet), or 50 feet − 16 feet, or 34 feet. The question asks for the radius of the pool, which is ^{1}⁄_{2} of the diameter. 34 ÷ 2 = 17.
For every cent increase in price of a pound of apples, the grocery store sells 25 fewer pounds per day. The grocery store normally sells 800 pounds of apples per day at $1.09 per pound. Which of the following expressions represents the number of pounds of apples sold per day if the cost is increased by 3x cents per pound of apples?

Solution
The problem asks for an expression that represents the number of pounds of apples sold, not the cost of the pounds of apples sold. Therefore, answer choices F and H can be eliminated because they include reference to the cost per pound of apples ($1.09). The problem states that fewer pounds per day are sold when the price is increased, so the correct expression cannot be J, which shows an increase from the
800 pounds of apples that the grocery store normally sells. Since we are given that the store sells 25 less for every cent increased, then for 3x cents increased, the store will sell 25(3x), or 75x less apples.
The area of a rectangular kitchen is 80 square feet. If the length of the floor is 4 feet less than four times the width, what is the width of the floor in feet?

Solution
The area of a rectangle is calculated by multiplying the width by the length (w × l). You are given that the length is 4 feet less than four times the width. Set the width equal to w;the length is then 4w − 4. Plug these values into the equation for the area of a rectangle:
(w)(4w − 4) = 80
4w^{2} − 4w = 80
Put this equation into the quadratic form and factor to find the solutions for w:
4w^{2} − 4w − 80 = 0
4(w^{2} − w − 20) = 0
4(w + ___)(w − ___) = 0
4(w + 4)(w − 5) = 0
(w + 4) = 0; w = −4
(w − 5) = 0; w = 5
Since the width of a room cannot have a negative value, the width must be 5
The average of a set of six integers is 38. When a seventh number is included in the set, the average of the set increases to 47. What is the seventh number?

Solution
If the average of six integers is 38, then the total must be 6 × 38, or 228. If the average of seven integers is 47, then the total must be 7 × 47, or 329. Since you are adding a seventh integer to the set, the value of the seventh integer will be the difference between 329 and 228: 329 − 228 = 101.
In the standard (x,y) coordinate plane, the yintercept of the line 6x + 2y = 14 is?

Solution
The slopeintercept form of a line is y = mx + b, where m is the slope and b is the yintercept. Put the equation given in the problem in the slopeintercept form:
6x + 2y = 14
2y = −6x + 14
y = −3x + 7
The yintercept is 7.
If X, Y, and Z are real numbers, and XYZ = 1, then which of the following conditions must be true?

Solution
If XYZ = 1, then Z cannot equal 0. If Z (or X or Y, for that matter) were 0, then XYZ would equal 0. Both sides of the equation can be divided by Y, which gives you XZ = 1 Y , answer choice F. Answer choice G is incorrect because two of the values could be −1. Answer choice H is incorrect because two of the values could be fractions and the third value could be a whole number, that, when multiplied by the fractions give you 1. Answer choices J and K are incorrect because you have already determined that none of the values can be equal to zero.
Given the vertices of parallelogram FGHJ in the standard (x, y) coordinate plane below, what is the area of triangle GHJ, in square units?

Solution
The area of a triangle is ^{1}⁄_{2}(bh), where b is the base, and h is the height. You can determine the base by measuring the distance along the xaxis, and you can determine the height by measuring the distance along the yaxis: The distance between −1 and 3 on the xaxis is 4 units; likewise, the distance between −2 and 2 on the xaxis is 4 units. The length of the base is 4.
The distance between −8 and 3 on the yaxis is 11. The height is 11.
Now plug these values into the formula for the area of a triangle:
A = ^{1}⁄_{2}(bh)
A = ^{1}⁄_{2}(4 × 11)
A = ^{1}⁄_{2}(44) = 22.