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The sum of two consecutive odd integers is 256. What is the value of the larger integer?
A) 103 B) 127 C) 129 D) 153 E) 155
Solution:
The correct answer is C.
If we let the first integer be represented by x, then the second integer will be x + 2, since they are both odd. Therefore, x + x + 2 = 256, which simplifies to 2x + 2 = 256, and x = 127. However, this is the value of the smaller of the pair; the larger is x + 2 = 129.
The sum of a number n and 4 is 10. What is the value of the sum of n and −1?
A) 2 B) 3 C) 5 D) 9 E) 11
Solution:
The correct answer is C.
If the sum of n and 4 is 10, then we can write n + 4 = 10, subtract 4 from either side to isolate n, and determine that n = 6. The sum of n − 1 then is (6) − 1 = 5.
The length of one leg of a right triangle is three times as large as the length of the other leg. If the hypotenuse has a length of 10 inches, which of the following is the perimeter of the triangle in inches?
A) √10 B) √410 C)10 D) 10+10 E) 10+410
Solution:
The correct answer is E.
If the length of the smaller leg is x, then the larger leg is 3x, and by the Pythagorean Theorem, the hypotenuse is x2+9x=10, the given length. Solving for x,x10√ = √10and the perimeter is 10√+310√+10.
A line in the (x, y) coordinate plane is defined such that each y-coordinate is found by multiplying the x-coordinate by three and adding two. What is the y-intercept of this line?
A) 0 B) 2 C) 5 D) 8 E) 10
Solution:
The correct answer is B.
The process described can be written as y = 3x + 2 for an x-coordinate of x. This equation is in slope-intercept form, so it indicates that the resulting line would have a y-intercept of 2.
The product of two numbers is 20, while their sum is 12. What is the difference of the two numbers?
A) 8 B) 17 C) 33 D) 188 E) 212
Solution:
The correct answer is A.
Let x and y represent the two numbers. Here we can write xy = 20, and x + y = 12, so y = 12 − x. Substituting this expression for y into the first equation, we find that x(12− x)= 20, which can be expressed as x2 – 12x + 20 = 0. This equation factors into (x − 2)(x − 10) = 0, which has solutions 2 and 10. Finally, the difference is 10 − 2 = 8.
The larger of two numbers exceeds three times the smaller number by 4. The sum of twice the larger number and 4 times the smaller number is 58. If x is the smaller number, which equation below determines the correct value of x?
A) 3(2x + 4) + 4x = 58 B) 3(2x − 4) + 3x = 58 C) 2(3x + 4) + 2x = 58 D) 2(3x + 4) + 4x = 58 E) 2(2x − 4) + 4x = 58
Solution:
The correct answer is D.
A simple way to solve this problem is to let the larger number be y. Therefore, you know that y = 3x +4, and that 2y + 4x = 58. Substitute 3x + 4 for y in the last equation to arrive at 2(3x + 4) + 4x = 58. This equation allows you to solve for x.
A total of f men went on a fishing trip. Each of the r boats that were used to carry the fishermen could accommodate a maximum number of m passengers. If one boat had 5 open spots and the remaining boats were filled to capacity, which of the following expresses the relationship among f, r, and m?
A) rm + 5 = f B) rm − 5 = f C) r + m + 5= f D) rf = m + 5 E) rf = m − 5
Solution:
The correct answer is B.
If each of the r boats that were used to carry the fishermen could accommodate a maximum number of m passengers, the total capacity of fisherman is rm. Since only one boat had 5 open spots and the remaining boats were filled to capacity, the number of fishermen was 5 less than the total capacity, or rm − 5. Thus f = rm − 5.
Gail made apple jelly and applesauce out of a bushel of apples. If the number of jars of jelly, j, is 3 less than twice the number of jars of applesauce, a, which expression shows the relationship of jars of jelly, j, to the jars of applesauce, a?
A) 2j = 2a − 3 B) j − 3 = 2a C) 2j = 3a D) j + 3 = 2a E) ja = 2a
Solution:
The correct answer is D.
You are given that the number of jars of jelly, j, is three less than twice the number of jars of applesauce, a. Put this into equation form: j = 2a − 3. Since none of the answers match this one, rearrange the equation: j + 3 = 2a.
Gary has turtles, cats, and birds for pets. The number of birds he has is 4 more than the number of turtles, and the number of cats is 2 times the number of birds. Of the following, which could be the total number of Gary’s pets?
A) 14 B) 18 C) 20 D) 22 E) 26
Solution:
The correct answer is C.
To solve this problem, realize that the number of pets that Gary has is determined through relationships between the quantities of the different types of pets: turtles (t), cats (c), and birds (b).
Because the number of birds, b, is 4 more than the number of turtles, t, this can be expressed as b = t + 4.
Since the number of cats, c, is 2 times the number of birds, this can be expressed as c = 2b.
Then, you might wish to use a table to show the numerical relationship between the numbers of each pet and the total number of pets, using the answer choices as a guideline:
Number of turtles (t) 0 1 2 3 4 Number of birds (b = t + 4) 4 5 6 7 8 Number of cats (c = 2b) 8 10 12 14 16 TOTAL 12 16 20 24 28 According to this table, Gary could have a total of 20 pets, but he could not have a total of 14, 18, 22, or 26 pets.
Last week, a student began collecting rare coins. Since then, he has doubled his collection by obtaining 36 more coins than he started with. How many coins are currently in his collection?
A) 18 B) 36 C) 72 D) 84 E) 116
Solution:
The correct answer is C.
If x represents the size of his original collection, then the statements give us the equation x + 36 = 2x. Therefore, x = 36, and his current collection has a size of 36 + 36 = 72.
What is the difference between 43 and the sum of the squares of −2, 1, and 6?
A) 23 B) 97 C) 53 D) 31 E) 39
Solution:
The correct answer is A.
( −2 )2 + 12 + 62 = 41
43 = 4 × 4 × 4 =64
64 − 41 = 23
Note 
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