Question 1: The numbers of products a store sold on 4 consecutive days were x, x + 5, x + 3 and x + 12. If the average of the products sold was 13, what is the value of x?
(a) 4
(b) 6
(c) 7
(d) 8
(e) 11
Solution: The average of the 4 numbers is: (x + x + 5 + x + 3 + x + 12)/4,
(x + x + 5 + x + 3 + x + 12)/4 = 13
(4x + 20)/4 = 13
x + 5 = 13
x = 8
Question #2: Is x > 5?
(1) x2 < 4
(2) |x – 5| > 0
(a) Statement (1) ALONE is sufficient, but statement (2) is not sufficient.
(b) Statement (2) ALONE is sufficient, but statement (1) is not sufficient.
(c) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(d) EACH statement ALONE is sufficient.
(e) Statements (1) and (2) TOGETHER are NOT sufficient.
Answer:
The first inequality, x2 < 4 can be written as:
(x – 2)(x + 2) < 0
x < 2 and x > -2. In conclusion the first inequality answers the question ‘Is x > 5?’.
The second inequality |x – 5| > 0 is true any real x except 0, so it cannot be determined if x is greater than 5.
In conclusion (a) is the correct answer.
Question #3: If x = mx + n and y = mx – n, then y – x =
(a) x – y
(b) -2n
(c) m + n
(d) 2m
(e) -n
Answer: We subtract the 2 equations:
y – x = mx – n – (mx + n) = -2n
(b) is the correct answer.
Question #4: What is the product x·y ?
(1) 3x·9y = 81
(2) x + y = 16
(a) Statement (1) ALONE is sufficient, but statement (2) is not sufficient.
(b) Statement (2) ALONE is sufficient, but statement (1) is not sufficient.
(c) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(d) EACH statement ALONE is sufficient.
(e) Statements (1) and (2) TOGETHER are NOT sufficient.
3x·9y = 81 can be written as:
3x·32y = 34
x + 2y = 4, and alone, this equation cannot determine the product x·y.
Both equation, together form a system of 2 equations with 2 real solutions x and y, so the correct answer is (c). Note that it is not necessary to solve the system, just to realize that you can find x and y and then multiply them.
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Question #5: In the infinite sequence a1,a2,…an,…,each term after the first one is equal to three times the previous term. If a3 + a4 = 252, what is the value of a1?
(a) 5
(b) 7
(c) 9
(d) 10
(e) 11
Answer: a3 + a4 = 252
a3 + 3a3 = 252
4a3 = 252
a3 = 63
a2 = 63/3
a2 = 21
a1 = 21/3
a1 = 7
Question 6: Blue marbles, green marbles and red marbles are in a ratio of 5 to 7 to 8 in a box. If the total number of marbles is 240, how many green marbles are in the box?
(a) 81
(b) 83
(c) 84
(d) 86
(e) 90
Answer: If the number of blue marbles, green marbles and red marbles are x, y and z,
x/y = 5/7, y/z = 7/8 and x + y + z = 240
x = (5/7)y and z = (8/7)y
(5/7)y + y + (8/7)y = 240
5y + 7y +8y = 240·7
20y = 240·7
y = 84
Question 7: A customer bought a shirt and a tie during a sale. Which item did he buy at the greater dollar value?
(1) He bought the shirt at a 5 percent discount.
(2) He bought the belt at a 15 percent discount.
(a) Statement (1) ALONE is sufficient, but statement (2) is not sufficient.
(b) Statement (2) ALONE is sufficient, but statement (1) is not sufficient.
(c) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(d) EACH statement ALONE is sufficient.
(e) Statements (1) and (2) TOGETHER are NOT sufficient.
Answer: All we can determine is that a greater percentage discount was obtained on the belt. We cannot determine if this translates into a greater dollar discount.
(e) is the correct answer
Question #8: In a class with 25 students, 13 learn French and 18 learn Spanish and 2 do not learn either. How many students learn both French and Spanish?
(a) 7
(b) 8
(c) 9
(d) 10
(e) 11
Answer: Of the 25 students, 25 – 2 = 23 students learn Spanish, or learn French or both.
23 – 13 = 10 students learn Spanish only.
23 – 18 = 5 students learn French only.
The number of students that learn both French and Spanish is 23 – 10 – 5 = 8.
Question #9: What is the slope of line l in the rectangular coordinate system?
(1) Line l intersects line y = x + 5 in more than one point.
(2) Line l is parallel with line y = x + 12.
(a) Statement (1) ALONE is sufficient, but statement (2) is not sufficient.
(b) Statement (2) ALONE is sufficient, but statement (1) is not sufficient.
(c) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(d) EACH statement ALONE is sufficient.
(e) Statements (1) and (2) TOGETHER are NOT sufficient.
Answer: If line l intersects the line y = x + 5 in more than one point, it means the two lines are represented by the same equation, y = x + 5. Statement (1) alone is sufficient.
Line l is parallel with line y = x + 12, so its slope is also equal to 1. Statement (2) alone is sufficient.
In conclusion (d) is the correct answer.
Question #10: What is the area of the triangle ABC?
(1) The edge AB of the triangle is 5 inches.
(2) Triangle ABC is equilateral
(a) Statement (1) ALONE is sufficient, but statement (2) is not sufficient.
(b) Statement (2) ALONE is sufficient, but statement (1) is not sufficient.
(c) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(d) EACH statement ALONE is sufficient.
(e) Statements (1) and (2) TOGETHER are NOT sufficient.
Answer:
Statement (1) alone is not sufficient. Different triangles with different areas can have AB = 5 inches.
Statement (2) alone is not sufficient. Equilateral triangles can have different areas.
BOTH statements TOGETHER are sufficient. We can calculate the altitude of the triangle using the Pythagorean theorem, a = 5√3.
The area of the triangle is 5·5√3 / 2 = 25·√3 / 2.
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