1.
Tom is four years older than Kate. In two years, Kate will be twice as old as Marianne, who is four. How old is Tom?
Correct Answer
C. 14
Explanation
Set up a simple equation: T = K + 4 K + 2 = 2 * (M + 2) M = 4 Substitute 4 for M. K + 2 = 2 * (4 + 2) K + 2 = 2 * 6 K + 2 = 12 K = 10 Then substitute 10 for K. T = K + 4 T = 10 + 4 T = 14
2.
If 2x = 3(x-2), 6x + 3 = ?
Correct Answer
D. 39
Explanation
2x = 3(x-2)
2x = 3x - 6
0 = x - 6
x = 6
Substitute 6 for x.
6*6 + 3 = 36 + 3 = 39
3.
Which of the following is not a rational number?
Correct Answer
D. The square root of 2
Explanation
All non-perfect square roots are irrational. These include the square root of 2, 3, 5, 6, 7, 8 and so forth.
4.
If 2(x-3) = 14, what is x^2 - 6x + 9?
Correct Answer
C. 49
Explanation
2(x-3) = 14, so (x-3) = 7.
x^2 - 6x + 9 factors into (x-3)^2.
Substitute 7 for (x-3).
(x-3)^2 = 7^2 = 49
5.
Which of the following values of x will yield an integer value for 1 /x^2-3
Correct Answer
C. Square root of 7/2
Explanation
To solve this problem, plug each of the given values of x in. You will find that only the square root of 7/2 comes out to a clean 1/(integer) answer. 1 / (sqrt(7/2)^2 - 3) = 1 / (7/2 - 3) = 1 / (7/2 - 6/2) = 1 / (1/2) = 2
6.
Which of the following could not the product of an even number and an odd number?
Correct Answer
C. 15
Explanation
An even number is always divisible by two, meaning two is always a factor in an even number. Therefore, the product of an even and an odd still includes two as a factor, making the product necessarily even. The only non-even answer is 15.
7.
Given that there are X red marbles, X-4 blue marbles, and 3X+2 green marbles in a bag of exactly 18 combined red, blue, and green marbles, what is the chance of randomly selecting a red or blue marble?
Correct Answer
B. 2/9
Explanation
There are a total of X + (X-4) + (3X + 2) marbles in the bag, which equals a total of 5X - 2 marbles. Furthermore, you are given the total number of marbles to be 18. So:
5X - 2 = 18
5X = 20
X = 4.
The chance of pulling a red or blue marble is equal to:
Probability(Red) + Probability(Blue) =
X/(5X-2) + (X-4)/(5X-2)
= (2X-4)/(5X-2)
Substituting 4 for X, you have the result:
(2X-4)/(5X-2)
(2*4 - 4)/(18)
4/18 = 2/9
8.
Thomas has twice the amount of money that John does. Terry has 40 dollars less than Thomas. If the amount of money that John has is "J," how much money would Terry have in terms of J if she received a 20-dollar gift?
Correct Answer
D. 2*J-20
Explanation
Thomas = 2J and Terry = Thomas - 40 = 2J-40.
If she receives an extra 20 dollars, it becomes:
2J-40 + 20 =
2J-20
9.
If x^3 = 12, x^6 =
Correct Answer
E. 144
Explanation
Break up x^6 into components that you understand.
x^3 * x^3 =
12 * 12 = 144
10.
A - (4 - a) = 3a + 3. a = ?
Correct Answer
A. -7
Explanation
First, simplify the left-hand side. a - (4 - a) = 3*a + 3 a - 4 + a = 2*a + 3 2a - 4 = 3a + 3 Now, solve for a by combining like terms: -4 = a + 3 a = -7
11.
Two positive integers have a product of 96. The difference between the largest and the smallest of them is 20. Which of the following is the sum of the integers?
Correct Answer
D. 28
Explanation
Call the largest integer x. Then the other integer = (x-20). Also, the product between the two is equal to 96. So:
x*(x-20) = 96
x^2 - 20x - 96 = 0
Using the quadratic formula, you find that the positive root of this equation is 24. Thus, the large factor X = 24.
The other factor is equal to X - 20 = 24 - 20 = 4.
So, the sum of the factors is equal to 24 + 4 = 28
12.
At how many points do the lines given by the equations 2x + 2y = 3 and y = -x + 1 intersect?
Correct Answer
A. 0
Explanation
Two lines can intersect at 0, 1, or many (infinite) points. Parallel lines (lines with the same slope) never intersect while all other lines intersect at exactly one point. If the two lines described are identical, they are everywhere intersecting. To determine if the lines are parallel, we find both of their equations in slope-intercept form:
2x + 2y = 3
2y = -2x + 3
y = -x + 3/2
The lines are:
y = -x + 3/2 and
y = -x + 1
Since they have the same slope but different y-intercepts, they never intersect.
13.
John wishes to build a square fence with an area of 121 square yards. What is the perimeter of the fence, in yards?
Correct Answer
D. 44
Explanation
To determine the perimeter, we will first need the length of a single side. Since the area of the square is equal to the length of the side squared, we find:
s^2 = 121
s = 11
Then, since the perimeter of a square is 4 times the length of a side, we have the result:
4 * s =
4 * 11 =
44
14.
If F(x) = 2x + 2 and G(x) = x^2 - 1, what is F(G(3)) - G(F(3))?
Correct Answer
A. -45
Explanation
To find F(G(3))-G(F(3)), first find the values: F(3) = 2(3) + 2 = 6 + 2 = 8 G(3) = 3^2 - 1 = 8 Then, substitute 8 in each respective function: F(G(3)) - G(F(3)) = F(8) - G(8) = 2(8) + 2 - (8^2 - 1) = 16 + 2 - 65 = -45
15.
Which of the following numbers has the same digit in the thousands place as it does in the tenths place?
Correct Answer
C. 41220.122
Explanation
Only in C does the digit in the thousands place (1) equal the digit in the tenths place, or one place to the right of the decimal.
16.
If X% of A is 20, what is A% of X?
Correct Answer
C. 20
Explanation
X% of A means (X / 100) times A.
So,
X% of A = 20
A*(X/100) = 20
A = 20/(X/100)
A = 2000/X
A% of X means X(A/100)
X*(A/100) =
X*(2000/X)/100 =
20.
17.
If F(x-3) = x, what is F(14)?
Correct Answer
D. 17
Explanation
Since F(x-3) = x, F(14) = F(17-3) = 17.
18.
Two consecutive positive integers have a product of A. If the larger integer is called X, what is the smaller integer in terms of X and A?
Correct Answer
E. A/X
Explanation
If two integers are consecutive, and the larger is called X, then they can be said to be:
X and X-1
Since X(X-1) = A:
X-1 = A/X.
19.
How many prime numbers are less than 31?
Correct Answer
E. 10
Explanation
Exactly 10 prime numbers are less than 31. To compute this, ignore even numbers other than 2(obviously not prime) and the number 1, which is neither prime nor composite. The primes are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
20.
If 2x - 3 = 9, what is x^2 - 4x + 2?
Correct Answer
B. 14
Explanation
First, solve for x:
2x - 3 = 9
2x = 12
x = 6.
Next, substitute 6 for x.
x^2 - 4x + 2 =
6^2 - 4*6 + 2 =
36 - 24 + 2 =
12 + 2 =
14
21.
For how many integer x does the expression x^2 - 4 < 36 hold true?
Correct Answer
E. 13
Explanation
To determine for how many integer values of x the expression x^2 - 4 < 36 holds true, you can set up the inequality and solve it:
x^2 - 4 < 36
First, add 4 to both sides of the inequality:
x^2 - 4 + 4 < 36 + 4
x^2 < 40
Now, take the square root of both sides, but remember to consider both the positive and negative square roots:
√(x^2) < √40
x < ±√40
x < ±2√10
So, the inequality x^2 - 4 < 36 holds true for all integer values of x that are less than ±2√10. This means that x can be any integer between -6 and 6 (excluding -6 and 6), because ±2√10 is approximately ±6.32, and integers in the range -6 to 6 (excluding -6 and 6) satisfy the inequality. Therefore, there are 13 integer values of x that satisfy the inequality.
22.
Train A travels at 100 miles per hour. Train B travels at 150 miles per hour. Train B leaves station exactly 30 minutes after Train A. After how many minutes will Train B catch up to Train A?
Correct Answer
D. 60
Explanation
Set up a simple equation:
A = 100t; B = 150t
You want to know when the distance traveled by A = dist traveled by B, so set A + (head start) = B
A + t(A) = B
100t + .5(100) = 150t
100t + 50 = 150t
50t = 50
t = 1
One hour = 60 minutes.
23.
If five runners are present in a race, in how many ways can the top three runners place?
Correct Answer
C. 60
Explanation
To find the number of ways the top three runners can place in a race with five runners, you can use permutations. In this case, you want to calculate the number of permutations of 3 runners out of 5.
The formula for permutations is:
n P r = n! / (n - r)!
Where:
n is the total number of items to choose from (in this case, 5 runners).
r is the number of items to choose (in this case, 3 runners).
"!" denotes factorial, which means multiplying all positive integers from 1 to the given number.
So, for this problem:
5 P 3 = 5! / (5 - 3)! 5 P 3 = 5! / 2!
Calculating the factorials:
5! = 5 x 4 x 3 x 2 x 1 = 120 2! = 2 x 1 = 2
Now, divide 5! by 2!:
5 P 3 = 120 / 2 = 60
There are 60 different ways the top three runners can place in the race.