1.
If (1/3)y + 9 = 0, then y =
Correct Answer
A. -27
Explanation
The equation (1/3)y + 9 = 0 can be solved by first subtracting 9 from both sides to isolate the variable. This gives us (1/3)y = -9. To solve for y, we can multiply both sides of the equation by 3 to get y = -27. Therefore, the value of y that satisfies the equation is -27.
2.
In the figure above, P, Q, and R lie on the same line. P is the center of the larger circle, and Q is the center of the smaller circle. If the radius of the larger circle is 4, what is the radius of the smaller circle?
Correct Answer
B. 2
Explanation
Since P, Q, and R lie on the same line, the distance between P and Q is equal to the sum of the radii of the two circles. Therefore, the radius of the smaller circle is 4 - 2 = 2.
3.
Roy planted corn on 1/5 of his land. If he planted 45 acres of corn, how many acres of land does he have?
Correct Answer
D. 225
Explanation
If Roy planted 45 acres of corn on 1/5 of his land, it means that 1/5 of his land is equal to 45 acres. To find the total number of acres of land he has, we can set up a proportion: 1/5 is to 45 as 1 is to x (unknown total acres). Cross multiplying, we get 1 * 45 = 5 * x, which simplifies to 45 = 5x. Dividing both sides by 5, we find that x = 9. Therefore, Roy has 9 times the amount of land he planted corn on, which is 9 * 45 = 225 acres.
4.
6, 10, 18, 34, 66
The first number in the list above is 6. Which of the following gives a rule for finding each successive number in the list?
Correct Answer
C. Double the preceding number and then subtract 2 from that result.
Explanation
Each number in the list is obtained by doubling the preceding number and then subtracting 2 from that result. This pattern can be observed by multiplying 6 by 2 and subtracting 2 to get 10, then multiplying 10 by 2 and subtracting 2 to get 18, and so on. This rule applies consistently to each number in the list.
5.
The two semicircles in the figure above have centers R and S, respectively. If RS = 12, what is the total length of the darkened curve?
Correct Answer
C. 12π
Explanation
The total length of the darkened curve can be found by adding the circumference of both semicircles. Since the radius of each semicircle is not given, we cannot determine the exact circumference. However, we know that RS = 12, which means the distance between the centers of the semicircles is 12. This implies that the sum of the radii of the two semicircles is 12. Therefore, the total length of the darkened curve is equal to the circumference of a full circle with a radius of 12. The formula for the circumference of a circle is 2πr, so the total length is 2π(12) = 24π. However, since we are dealing with semicircles, the total length would be half of this, which is 12π.
6.
If h and k are positive numbers and h + k = 7, then (7 - k)/h =
Correct Answer
A. 1
Explanation
If h and k are positive numbers and h + k = 7, then (7 - k)/h simplifies to (7 - k)/h = (7/h) - (k/h). Since h + k = 7, we can substitute h = 7 - k into the expression, giving us (7/(7 - k)) - (k/(7 - k)). Simplifying further, we get 1 - (k/(7 - k)). Since k is positive, (k/(7 - k)) is always less than 1. Therefore, 1 - (k/(7 - k)) will always be greater than 1. Hence, the answer is 1.
7.
The table above shows the populations of two
countries and their population densities. The
number of square miles in the area of Country B
is approximately how much greater than the
number of square miles in the area of Country A ?
Correct Answer
D. 8,000
8.
If x^{2} = x + 6, which of the following must be true?
Correct Answer
E. X^2 > x
Explanation
If x^2 is greater than x, it means that the square of x is larger than x itself. In the given equation x^2 = x + 6, we can see that x^2 is greater than x because there is a positive constant (6) added to x. Therefore, x^2 > x must be true.
9.
Let the functions f be defined by f(x) = 5x - 2a, where a is a constant. If f(10) + f(5) = 55, what is the value of a?
Correct Answer
C. 5
Explanation
Given that f(x) = 5x - 2a, we can substitute x = 10 and x = 5 to find f(10) and f(5).
f(10) = 5(10) - 2a = 50 - 2a
f(5) = 5(5) - 2a = 25 - 2a
We are given that f(10) + f(5) = 55, so we can substitute the expressions above into the equation:
50 - 2a + 25 - 2a = 55
Combining like terms, we get:
75 - 4a = 55
Subtracting 75 from both sides:
-4a = -20
Dividing both sides by -4:
a = 5
Therefore, the value of a is 5.
10.
A number is called “even-odd” if it is halfway between
an even integer and an odd integer. If x is an even-odd
number, which of the following must be true?
I. 2x is an integer.
II. 2x is even-odd.
III. x is halfway between two even integers.
Correct Answer
A. I only
Explanation
If a number x is even-odd, it means that it is halfway between an even integer and an odd integer. If we multiply x by 2, we will get an even integer because we are essentially doubling the distance between the even and odd integers. Therefore, statement I is true. Statement II is not necessarily true because multiplying x by 2 does not guarantee that the result will still be halfway between an even and odd integer. Statement III is not necessarily true because x being halfway between two even integers is not a requirement for it to be considered even-odd.
11.
If m is a positive integer, which of the following is NOT equal to (2^{4})^{m} ?
Correct Answer
D. 4^m(2^m)
Explanation
The expression (24)m can be simplified as 2^(4m).
Option 2^(4m) is equivalent to (2^4)^m, which simplifies to 16^m.
Option 4^(2m) is equivalent to (2^2)^(2m), which simplifies to 4^m.
Option 2^m(2^(3m)) is equivalent to 2^(m+3m), which simplifies to 2^(4m).
Option 4^m(2^m) is not equal to (24)m.
Therefore, the correct answer is 4^m(2^m).
12.
In the figure above, l || m. Which of the following must equal 180?
Correct Answer
B. K + p + s
Explanation
If l || m, it means that l and m are parallel lines. When two parallel lines are intersected by a transversal, the corresponding angles are congruent. In this case, the angles k, p, and s are corresponding angles. Since corresponding angles are congruent, the sum of k, p, and s must equal 180 degrees.
13.
How many different ordered pairs (x, y) are there such that x is an even integer, where 4 ≤ x ≤ 10, and y is an integer, where 4 < y < 10 ?
Correct Answer
C. 20
Explanation
There are 4 even integers between 4 and 10 (4, 6, 8, 10) and 5 integers between 4 and 10 (5, 6, 7, 8, 9). To find the number of different ordered pairs, we multiply the number of even integers by the number of integers, which gives us 4 * 5 = 20. Therefore, there are 20 different ordered pairs (x, y) that satisfy the given conditions.
14.
N(t) = 500(0.81)^{t}
The function above can be used to model the population of a certain endangered species of animal. If n(t) gives the number of the species living t decades after the year 1900, which of the following is true about the population of the species from 1900 to 1920 ?
Correct Answer
C. It decreased by about 180.
Explanation
The function n(t) = 500(0.81)t represents the population of the species t decades after the year 1900. To find the change in population from 1900 to 1920, we substitute t = 2 into the function. n(2) = 500(0.81)2 = 500(0.6561) ≈ 328.05. Therefore, the population decreased by about 328.05 - 500 ≈ -171.95, which is approximately 180. Hence, the correct answer is "It decreased by about 180."
15.
A sphere of radius r inside a cube touches each one
of the six sides of the cube. What is the volume of the
cube, in terms of r ?
Correct Answer
E. 8r^3
Explanation
The cube has six faces, and each face is a square. The sphere touches each face of the cube, which means that the diameter of the sphere is equal to the length of the side of the cube. Since the radius of the sphere is r, the side length of the cube is 2r. The volume of a cube is calculated by cubing the length of its side, so the volume of the cube in terms of r is (2r)^3, which simplifies to 8r^3.
16.
According to the graph above, in which year was the
ratio of the number of students enrolled at School B
to the number of students enrolled at School A the
greatest?
Correct Answer
E. 1994
Explanation
The graph shows the ratio of the number of students enrolled at School B to the number of students enrolled at School A over the years. To determine the year when the ratio was the greatest, we look for the highest point on the graph. In this case, the highest point is in the year 1994, indicating that the ratio of students enrolled at School B to students enrolled at School A was the greatest in that year.