1.
y = x2 – 1
y = 8
| Column A | Column B |
| y/5 | x |
(a) The quantity in Column A is greater.
(b) The quantity in Column B is greater.
(c) The two quantities are equal.
(d) The relationship cannot be determined from the information given.
Solution: The quantity in column A is y/5 = 8/5.
We need to find x to evaluate the quantity in column B.
y = x2 – 1
8 = x2 – 1
9 = x2 and the solutions are x1 = 3 and x2 = -3
x1 > 8/5 but x2 < 8/5 and relationship cannot be determined from the information given. (d) is the correct answer.
2.
p > 1, p is a real number
| Column A | Column B |
| 1/p | 1/2 + 1/5 |
(a) The quantity in Column A is greater.
(b) The quantity in Column B is greater.
(c) The two quantities are equal.
(d) The relationship cannot be determined from the information given.
Answer:
The quantity in column B is 1/2 + 1/5 = .7.
p > 1, 1/p < 1. For some values of p, 1/p is greater then .7 (e.g. p = 1.1), for others 1/p is less then .7 (e.g. p = 5).
(d) is the correct answer.
3. If the ratio students to teachers on a committee of 21 members is 8 to 6, how many members of the committee are students?
(a) 8
(b) 9
(c) 12
(d) 14
(e) 16
Answer: s + t = 21
s/t = 8/6,
where s is the number of students and t is the number of teachers in the committee.
t = (3/4)s
(3/4)s + s = 21
(7/4)s = 21
s = 12
4. If a, b and c are odd integers, which of the following expressions must be an even integer?
(a) a + b + c
(b) ab + bc + ca
(c) a(b + c)
(d) a(b + c – 1)
(e) (a – 2)bc
Answer:
(a) a + b + c is: odd + odd + odd = odd
(b) ab + bc + ca is: odd + odd + odd = odd
(c) a(b + c) is: odd (odd + odd) = odd·even = even
(d) a(b + c – 1) is: odd·(odd + odd – 1) = odd·odd = odd
(e) (a – 2)bc is: (odd – 2)·odd·odd = odd
(c) is the correct answer.
5. The figure below shows two concentric circles each with center O. The radius of C1 is r1, the radius of C2 is r2, and r1 = √2·r2.
| Column A | Column B |
| Area of circle C2 | Shaded area |
(a) The quantity in Column A is greater.
(b) The quantity in Column B is greater.
(c) The two quantities are equal.
(d) The relationship cannot be determined from the information given.
Answer: The quantity in Column A is ¶r22
The shaded area is equal to the area of circle C1 minus the area of circle C2.
AreaC1 – AreaC2 = ¶r12 – ¶r22 = (√2·r2)2 – ¶r22 = ¶r22
In conclusion the two quantities are equal.
6. Find the solutions of the x2 + 3x – 18 = 0 equation.
(a) x1 = 3 and x2 = -5
(b) x1 = 3 and x2 = -6
(c) x1 = 3 and x2 = 6
(d) x1 = 2 and x2 = -6
(e) x1 = 2 and x2 = 6
Answer:
x2 + 3x – 18 = 0
x1,2 = [-3 +/-√(32 + 4·18)]/2
x1,2 = (-3 +/- 9)/2
(b) x1 = 3 and x2 = -6 is the correct answer.
Question 7: The height h of the trapezoid ABCD is equal to the average of its two parallel sides.
| Column A | Column B |
| Area of ABCD | h2 |
(a) The quantity in Column A is greater.
(b) The quantity in Column B is greater.
(c) The two quantities are equal.
(d) The relationship cannot be determined from the information given.
Answer: The area of the trapezoid is:
AABCD = h·(AD + BC)/2
We also know that h = (AD + BC)/2
(AD + BC) = 2h
AABCD = h·(2h)/2
AABCD = h2 so the two quantities are equal.
Question #8: In the x,y coordinate system, line l passes through point P(2, 1) and is perpendicular to line y = x + 1.
| Column A | Column B |
| y intercept of line l | x intercept of line l |
(a) The quantity in Column A is greater.
(b) The quantity in Column B is greater.
(c) The two quantities are equal.
(d) The relationship cannot be determined from the information given.
Answer: The slope of line y = x + 1 is m = 1.
If line l is perpendicular to y = x + 1, its slope must be -1.
The equation of line l can be written as y = -x + n.
We substitute the coordinates of P(2, 1) in this equation:
1 = -2 + n and n = 3.
y = -x + 3
The y intercept of l is 3.
The x intercept of l is also 3.
The two quantities are equal.
Question 9: In which of the following years was the difference between the revenues of company A and company B the lowest?
(a) 1971
(b) 1972
(c) 1974
(d) 1975
(e) 1976
Answer: A quick inspection of the two charts shows that the only year with a difference between the revenues of company A and company B less than $100,000 is 1972.
Question #10: What is the approximate percentage of all employees of Company A that are operators?
(a) 12%
(b) 29%
(c) 35%
(d) 44%
(e) 63%
Answer:
There are about 270 operators out of around 600 employees so the percentage is 44% and (d) is the correct answer.