1. COMPLEX
REAL IMAGINARY
IRRATIONAL RATIONAL
NON-INTEGER INTEGER
ZERONEGATIVE POSITIVE
ALGEBRA
Block Diagram of the Number System
Definitions:
Complex numbers – numbers which take the form x + yi where x and y are real numbers.
x is the real part
y is the imaginary part
i is the imaginary unit which is equal to 1
Examples: 2 + 3i, 3 – 4i, -2 + 5i
Rational number – a number which can be expressed as the ratio of two integers such as
¾, ⅝, –3, 0.25.
Irrational number - a number which can not be expressed as the ratio of two integers such
as 3 , 3
4 , π, e.
Integer – any whole number which can be either negative, zero or positive such as –3, 0, 7.
Natural numbers- are positive whole numbers including zero such as 0, 1, 2, 3, 4, . . . .
Properties of Real Numbers
1. Commutative property : Changing the order of terms/factors in addition/multiplication
will not change the sum/product, i.e.
a + b = b + a for addition & ab = ba for multiplication
2. Associative property: Changing the grouping of terms/factors to be added/multiplied will
not change the sum/product, i.e.
a + (b + c) = (a + b) + c for addition
a(bc) = (ab)c for multiplication
3. Distributive property: a(b + c) = ab + ac
4. Reflexive property: Any number is equal to itself, i.e. a = a.
5. Symmetric property: If a = b, then b = a.
6. Transitive property: If a = b and b = c, then a = c.

2. Factors and Special Products
1. Difference of two squares: ))((22
bababa 
2. Sum of two cubes: ))(( 2233
babababa 
3. Difference of two cubes: ))(( 2233
babababa 
4. Square of a binomial: 222
2)( bababa 
5. Cube of a binomial: 32233
33)( babbaaba 
Laws of Exponents: Laws of Radicals
1.
nmnm
aaa 
 1.
nn
aa /1
2.
nm
n
m
a
a
a 
 2.
mnn mnm
aaa )(/

3.
mnnm
aa )( 3. aaa n nnn
)(
4.
nnn
baab )( 4.
nnn
abba 
5. n
nn
b
a
b
a






5.
n
n
n
b
a
b
a

6.
mnm n
aa 
A surd is a radical expressing an irrational number such as 3 ( a quadratic surd), 3
5 (a
cubic surd), 4
2 (a quartic surd), etc.
The Quadratic Equation:
The general form of a quadratic equation is 02
 cbxax where a, b & c are real
constants. The roots of this equation are expressed by the quadratic formula shown below.
a
acbb
x
2
42


where the discriminant, acb 42
 determines the nature of the roots.
If acb 42
 > 0, the roots are real and unequal;
If acb 42
 = 0, the roots are real and equal;
If acb 42
 < 0, the roots are imaginary.
Properties of the roots:
Sum of the roots =
a
b
 Product of the roots =
a
c
Logarithms
Definition: The logarithm of a given number is the exponent to which the base must be
raised in order to yield the number. Thus, if Nbx
 , then Nx blog .

3. Types of Logarithms:
1. Common (or Briggsian) logarithm – logarithm having 10 as base..
2. Natural (or Naperian) logarithm having the number e as base.
Properties of Logarithms:
1. yxxy bbb logloglog  Characteristic – the integral (or whole-number)
2. yx
y
x
bbb logloglog  part of the logarithm of a number.
3. xnx b
n
b loglog  Mantissa – the decimal part of the logarithm of
4.
b
a
a
c
c
b
log
log
log  a number.
5.
b
a
a
b
log
1
log  Thus, if log 7890 ≈ 3.897, then 3 is the
6. xb xb
log
characteristic and 0.789 is the mantissa.
Graph of the logarithmic function xy blog
From the graph, it can be seen that
1. Negative numbers have no (real) logarithm.
2. Numbers between 0 and 1 have negative logarithms.
3. 0bLog
4. bLog
Antilogarithm – the number corresponding to a given logarithm.
Cologarithm of a number N – the logarithm of the reciprocal of N, i.e. colog N = log
N
1

4. Progressions
I. Arithmetic Progression: - a sequence of numbers called terms, each of which, after the
first, is obtained from the preceding term by adding to it a fixed number called the
common difference (d). The sequence a1, a2, a3, . . . . .,an is an arithmetic
progression if a2 – a1 = a3 – a2 = . . . . . . = an – an-1 = d.
Example: 17, 14, 11, 9, . . . . with d = –3
nth
term of an Arithmetic Progression:
an = a1 + (n – 1)d where an – the nth
term
a1 – the first term
d – the common difference
Sum of first n terms:
 nn aa
n
S  1
2
or   dna
n
Sn 12
2
1  where Sn – the sum of the first n terms
Arithmetic Mean: - the term between any two given terms of an arithmetic progression.
Between two numbers a and b, the arithmetic mean (AM) is
AM =
2
ba 
II. Geometric Progression: - a sequence of numbers called terms, each of which, after the
first, is obtained by multiplying the preceding term by a fixed number called the
common ratio (r). The sequence a1, a2, a3, . . . . .,an is a geometric progression if
r
a
a
a
a
a
a
n
n

12
3
1
2
......
Example: 27, 9, 3, 1, 1/3, . . . . . with common ratio, r = 1/3
nth
term of a Geometric Progression:
1
1

 n
n raa
Sum of first n terms:
1
1
1



r
r
aS
n
n or
r
r
aS
n
n



1
1
1
Sum of an infinite Geometric Progression:
r
a
S


1
1
where r <1 or its equivalent –1 < r < 1
Geometric mean: - the term between any two given terms of a geometric progression.
Between two numbers a and b, the geometric mean (GM) is
GM = ab
III. Harmonic Progression: - a sequence of numbers whose reciprocals form an arithmetic
progression. Example: 1, 1/3, 1/5, 1/7, . . . . .

5. The Binomial Theorem
If n is a positive integer,
nnnnnn
bba
nnn
ba
nn
ba
n
aba 



 
.....
!3
)2)(1(
!2
)1(
!1
)( 33221
The numerical coefficients in the binomial expansion of n
ba )(  can also be obtained using a
triangular array known as Pascal’s triangle.
0
)( ba  1
1
)( ba  1 1
2
)( ba  1 2 1
3
)( ba  1 3 3 1
4
)( ba  1 4 6 4 1
5
)( ba  1 5 10 10 5 1
6
)( ba  1 6 15 20 15 6 1
7
)( ba  1 7 21 35 35 21 7 1
. . . . . . . . . .
Properties of the binomial expansion n
ba )(  :
1. There are (n+1) terms in the expansion.
2. The exponent of a decreases by 1 in each succeeding term; b appears in the 2nd
term and its exponent increases by 1 in each succeeding term.
3. The sum of the exponents of a and b in any term is always equal to n.
In general,
Term involving r
b = termthr )1(  = rrn
ba
r
factorsrtoupnnn 
!
)2)(1(
= rrn
rn baC 
or
11
1
11
)!1(
)2()2)(1( 





 rrn
rn
rrn
baCba
r
rnnnn
termrth
Remainder Theorem:
If a polynomial f(x) is divided by a binomial of the form x – r, then the remainder R = f(r).
Example: Find the remainder if 3x3
– x2
– 3x + 1 is divided by x – 2.
Solution: Let f(x) = 3x3
– x2
– 3x + 1
x – r = x – 2, hence x = 2
R = f(2) = 3(2)3
– (2)2
– 3(2) + 1 = 15
Lowest Common Multiple (LCM) and Highest Common Factor (HCF):
A number is prime if it is greater than 1 and has no other factor except itself and 1.
Examples of prime numbers are 2, 3, 5, 7, 11, 13, etc.
The lowest common multiple (LCM) of several numbers is the smallest number of which
each of the given numbers is a factor. It is obtained by taking the product of all the different
prime factors raised to the highest power to which it occurs in any of those numbers.

6. Example: Find the lowest common multiple of 60 and 72.
Solution: Expressing each of the given numbers as the product of prime factors, we have
60 = 22
∙3∙5 72 = 23
∙32
LCM = 23
∙32
∙5 = 360
The highest common factor (HCF) of several numbers is the largest number which is a
factor of the given numbers. It is obtained by taking the product of all the different prime
factors that are common to the given numbers, each taken to the lowest power to which it
occurs in any of these numbers.
Example: Find the highest common factor of 60 and 72.
Solution: Expressing each of the given numbers as the product of prime factors, we have
60 = 22
∙3∙5 72 = 23
∙32
HCF = 22
∙3 = 12
Ratio and Proportion
The ratio of a number a to another number b is the fraction a/b usually written as a:b where
a is called the antecedent and b is called the consequent.
A proportion is a statement that two ratios are equal, e.g. a:b = c:d. Here, a and d are
called extremes while b and c are called the means, d is called the fourth proportional to
a, b and c. If the means are equal, e.g. a:x = x:b, then x is called the mean proportional to a
and b while b is called the third proportional to a and x. The mean proportional between a
and b is also equal to their geometric mean, i.e. abx  .
Example: Find the fourth proportional to 3, 5, and 21.
Solution: Let x be the fourth proportional.
x:215:3  
x
21
5
3
  35
3
)5(21
x
Equations
An equation is a statement that two quantities are equal. It has two members, the left
member and the right member. Numbers which, when substituted for the variables (or the
unknowns), make the two members of the equation equal are said to satisfy or be a
solution (or root) of the equation.
A conditional equation is an equation which is satisfied by some, but not all, of the values
of the variables for which the members of the equation are defined.
An identical equation (or identity) is an equation which is satisfied by all the values of the
variables for which the members of the equation are defined.
A system of equations having a unique solution is said to be consistent; a system of
equations having no solution is inconsistent; a system of equations having infinitely many
solutions is said to be dependent.
Word Problems:
Suggestions for attacking word problems:
1. Read and reread the problem until it is clear what is stated, getting well in mind the
given data and the unknowns.
2. Express each unknown in terms of a literal symbol.
3. Find the quantities, involving the given data and the unknowns, which are equal.
Then form an equation or a system of equations.
4. Solve the equation (or system of equations) and check the result.

7. Example: If you own a store, at what price will you mark a camera for sale that cost Php 6000
in order that you may offer 20% discount on the marked price and still make a profit
of 25% on the selling price?
a. Php 10,000 b. Php 12,000 c. Php 15,000 d. Php 8,000
Solution:
Let x – marked price
xxx 80.020.0  - selling price
)80.0(25.0600020.0 xxx 
xx 20.0600080.0 
600060.0 x
000,10Phpx 
Variation
Direct variation: If a variable y is equal to a constant times another variable x, then y is said
to vary directly as x, i.e.
kxy 
where k is called the constant of proportionality, or the constant of variation.
The expressions
y varies directly as x
y varies as x
y is directly proportional to x
y is proportional to x
are all used with the same meaning.
Inverse variation: If a variable y is always equal to a constant times the reciprocal of another
variable x, then y is said to vary inversely as x, or y is inversely proportional to x.
This relation is expressed by the equation
x
k
y 
Joint variation: If a variable z is equal to a constant times the product of variables x and y,
then z is said to vary jointly as x and y. This relation is expressed by the equation
kxyz 
Example: The time required for an elevator to lift a weight varies directly with the weight and
the distance through which it is to be lifted and inversely as the power of the motor. If
it takes 30 seconds for a 10 HP motor to lift 100 lbs through 50 ft., what size of motor
is required to lift 800 lbs in 40 seconds through 40 ft.?
Solution:
Let t – time in seconds
W – weight in lbs
d – distance in ft.
P – the power of the motor in HP
P
Wd
t  
P
Wd
kt  
10
50)100(
30 k 
50
3
k
48
40
40)800(
50
3

t
Wd
kP HP

8. REVIEW QUESTIONS IN ALGEBRA
1. The product of (xy – 1) and (xy + 8) is
a. x2
y2
– 8 b. x2
y2
– 9xy – 8 c. x2
y2
+ 7xy – 8 d. x2
y2
– 7xy – 8
2. The expression (x – 3)3
is identical to
a. x3
– 27 b. (x – 3)(x2
– 3x + 9)
c. (x + 3)(x2
– 6x + 9) d. x3
– 9x2
+ 27x – 27
3. The product of (5x + 7y) and (5x – 7y) is
a. 25x2
– 70xy – 49y2
b. 25x2
+ 49y2
c. 25x2
+ 70xy – 49y2
d. 25x2
– 49y2
4. 






2
5
2
a
a. 25
2
5
4
25
4
22
 a
aa
b. 25
2
5
4
2
 a
a
c. 255
4
25
4
22
 a
aa
d. 255
4
2
 a
a
5. The greatest common monomial factor in 3x2
– 6x is
a. 3 b. 3x c. 6x2
d. x – 2
6. The factored form of y2
(2x – 3) – 4(2x – 3) is
a. y2
– 4(2x – 3)2
b. (2x – 3)(y2
– 4)
c. (y + 2)(y – 2)(2x – 3) d. (y2
– 4) (2x – 3)2
7. The factored form of 1 – 0.49x2
is
a. (1 + 0.7x)(1 – 0.7x) b. (1 + 0.07x)(1 – 0.07x)
c. (1 + 0.49x)(1 – x) d. (! + 0.49x)(1 + x)
8. The factored form of 4a2
– 20ab + 9b2
is
a. (2a – b)(2a – 9b) b. (2a – b)(2a + 9b)
c. (2a + b)(2a – 9b) d. (2a + b)(2a + 9b)
9. The expression 81x2
– 90xy + 25y2
is an example of a
a. sum of 2 squares b. difference between 2 perfect squares
c. sum of 2 cubes d. perfect square trinomial
10.All numbers below are composite except
a. 42 b. 53 c. 105 d. 154
11.The length of a rectangle with an area of (x2
+ 10x) square units if the width is x units is
a. x2
b. x2
+ 10 c. x + 10 d. (x + 10)2

9. 12. 











 2
42
3
2
2
3
y
x
y
x
a. 4
6
2
3
y
x
b. 4
6
2
3
y
x
 c. 3
8
y
x
 d. 3
8
y
x
13. The reduced form of
3
5
4







is
a.
125
64
b.
64
125
c.
125
64
 d.
64
125

14. 2
1
28
36 yx
is identical to
a. 4
36
x
y
b. 4
36x
y
c.
y
x4
6
d. 4
6
x
y
15. If ,4
4
4 3
2

y
then y =
a. 1/5 b. – 1/5 c. 5 d. – 5
16. If 23y + 1
= 64, then y =
a. 7/3 b. 3/7 c. 5/3 d. 3/5
17. )7512233(  expressed in simplest form is
a. 7537  b. 312 c. 12238  d. 336
18. In the example 3
64 , – 64 is the
a. index b. exponent c. radicand d. coefficient
19. Of the following numbers, which is irrational?
a. 25 b. 500,2 c. 250 d. 0025.0
20. Solve the equation 56 x .
a. – 1 b. 11 c. 19 d. 25
21. The square root of the sum of a number and 10 is 7. The number is
a. 39 b. 49 c. 59 d. 17
22. What is the solution set of the equation x
x
xx




1
5
2
52
?
a. x = 1 b. x = – 5/11 c. x = – 1 d. x = 5/11
23. If 2.5 kg of dressed chicken cost Php 105.00, what will be the cost of 3.75 kg?
a. Php 393.75 b. Php 262.50 c. Php 157.50 d. Php 147.00

10. 24. The quadratic equation whose roots are 2 and – 3 is
a. x2
– x – 6 = 0 b. x2
– x + 6 = 0 c. x2
+ x – 6 = 0 d. x2
+ x + 6 = 0
25. In the equation kx2
– 6x + 3 = 0, determine the value of k so that the roots are equal.
a. 4 b. 1 c. 2 d. 3
26. In the equation x2
+ 9x + k = 0, find the value of k so that one root of the equation is
twice the other root.
a. 18 b. 16 c. 12 d. 24
27. The solution set of the equation (x – 3)2
– 49 = 0 is
a. 10, 4 b. – 10, 4 c. 10, – 4 d. – 10, – 4
28. What value of k yields a perfect square trinomial for x2
– 2kx + 49?
a. 7 b. – 7 c. 1/7 d. – 1/7
29. The discriminant of x2
+ 6x + 9 = 0 is
a. 0 b. 1 c. 6 d. 9
30. To complete the square in x2
– 5x + k, you would replace k with
a. – 5/2 b. 5/2 c. – 25/4 d. 25/4
31. If x is directly proportional to y, and k is the constant of proportionality, the relationship
may be expressed as
a. xy = k b. x = k/y c. x = ky d. y = k/x
32. When x = 5, y = 10 and when x = 2, y = 4. This is an example of
a. direct variation b. inverse variation
c. joint variation d. combined variation
33. The set






12
1
,
6
1
,
3
1
,
3
2
is
a. an arithmetic sequence b. a geometric sequence
c. a harmonic sequence d. not a sequence
34. The geometric sequence has a common
a. difference b. product c. sum d. ratio
35. The third term of the binomial expansion of (5p – q)2
is
a. – 5pq2
b. – 5pq c. – q2
d. q2
36. The sum of the first eleven terms of the arithmetic sequence beginning with 50, 40, … is
a. 0 b. 5 c. 25 d. 50
37. In an arithmetic sequence, the second term is 2 and the fourth term is -6. The common
difference is
a. 4 b. 8 c. -4 d. -6
38. The fourth term of the binomial expansion of (3s + 2t)5
is
a. 24st4
b. 720s2
t3
c. 648s4
t d. 24s4
t

11. 39. The first three terms of a harmonic sequence is 2,
11
6
,
7
6
. The fifth term is
a. 5/2 b. 3 c. 6/19 d. 2/5
40. The arithmetic mean is
a. the sum of the progression b. a term between the extremes
c. the last term of the sequence d. the first term of the progression
41. The absolute value of a nonzero number is
a. always negative b. always positive
c. sometimes zero and sometimes negative d. always zero
42. Any combination of symbols and numbers related by the fundamental operations of
algebra is called a/an
a. term b. algebraic expression c. equation d. algebraic sum
43. A number written with the decimal point placed just after the leading digit multiplied by a
power of 10 is said to be in
a. exponential form b. radical form
c. scientific notation d. logarithmic form
44. What do you call the first and fourth terms in the proportion of four quantities?
a. numerators b. extremes c. means d. denominators
45. An equation, which by some mathematical process, acquires an extra root is called a
a. linear equation b. defective equation
c. literal equation d. redundant equation
46. A mathematical expression consisting of two terms is called a
a. binomial b. monomial c. duomial d. polynomial
47. Which of the following can not be a base for a logarithm?
a. π b. 10 c. e d. 1
48. For a quadratic equation, two distinct roots are produced only if the discriminant is
a. equal to zero b. either less than or greater than zero
c. less than zero d. greater than zero
49. When the imaginary number is raised to an even exponent, it becomes
a. a negative imaginary number b. a real number
c. infinite d. a relatively small number
50. The prefix nano- is equivalent to
a.
3
10
b. 6
10
c. 9
10
d. 12
10
51. nn
xx /1
loglog  is also equal to
a. log nx b. log xn
c.
n
xlog
d. n log x

12. 52. The prefix tera- is opposite to the prefix
a. giga- b. pico- c. nano- d. deka-
53. Find the term involving x8
in the expansion of
16
2 1







x
x .
a. 12,870x8
b. 10,680x8
c. 11,480x8
d. 14,620x8
54. b to the (m/n)th power =
a. nth root of b to the mth power
b. b to the power (m + n)
c. 1/n times the square root of b to the mth power
d. b to the mth power over n
55. The logarithm of a negative number is
a. a rational number b. imaginary c. a real number d. nonexistent
56. Find x if x + 2, 3x – 4 and 2x + 7 are in arithmetic progression.
a. 17/3 b. 5 c. 3/17 d. 17
57. The sum of the terms in a geometric progression is 1820. How many terms are there if
the first term is 5, the second term is 15 and the third term is 45?
a. 7 b. 8 c. 5 d. 6
58. The sum of the 4th
and the 8th
terms of an arithmetic progression is 32. The sum of the
10th
and the 14th
terms is 68. Find the sum of the first 15 terms.
a. 315 b. 300 c. 345 d. 330
59. The arithmetic mean of 2a and 3b is
a. ab6 b. (2a – 3b)/2 c. 6ab/2 d. (2a + 3b)/2
60. What expression is equivalent to log x – log (y + z)?
a. log x + log y + log z b. log x – log y – log z
c. log [x/(y + z)] d. log y + log (x + z)
61. How many significant digits do 10.097 have?
a. 2 b. 3 c. 4 d. 5
62. The other form of log a N = b is
a. N = ba
b. N = a b
c. N = ab d. N = b/a
63. From
n
x
y 






1
, if x is a positive integer and n is large and positive, then the value of y is
a. large b. negative c. irrational d. small
64. A bed sheet ½ mm thick is to be folded over itself 10 times. Find the height of the folded
sheet in mm.
a. 256 b. 512 c. 1024 d. 128

13. 65. If log5.2 1000 = x, what is the value of x?
a. 4.19 b. 5.23 c. 3.12 d. 4.69
66. Find the value of a in the equation loga 2187 = 7/2.
a. 3 b. 6 c. 9 d. 12
67. The expression log xy
/ log yx
is equal to
a. xy
/yx
b. y log x – x log y c. (y log x) / (x log y) d. 1
68. Solve for x: x = (log b a)(log c d)(log d c)
a. log b a b. log a c c. log b c d. log d a
69. Given log 2 = x and log 3 = y, find the value of log 4
48 .
a.  yx 4
4
1
 b.  yx 4
4
1
c. 4(x + 4y) d. 4(4x + y)
70. Solve for x: log x2
– log 5x = log 20
a. 0 b. 50 c. 100 d. 150
71. Find the third proportional to 8 and 14.
a. 4.57 b. 10.58 c. 24.5 d. 42.5
72. A triangle has sides of lengths 12, 17, and 22 inches. If the length of the shortest side of a
similar triangle is 8 inches, find the length of the longest side in inches.
a.
3
2
14 b.
3
1
11 c. 18 d. 19
73. Find the remainder if 2x4
+ 5x3
– 8x2
– 7x – 9 is divided by x + 2.
a. 35 b. – 35 c. 18 d. – 18
74. The length of a rectangle exceeds its width by 2 feet. If each dimension were increased
by 3 feet, the area would be increased by 51 square feet. Find the original dimensions.
a. 9 by 11 ft b. 8 by 10 ft c. 7 by 9 ft d. 6 by 8 ft
75. The altitude of a triangle is ¾ the length of its base. If the altitude were increased by 3
feet and the base decreased by 3 feet, the area would be unchanged. Find the length of
the altitude.
a. 12 ft b. 9 ft c. 6 ft d. 15 ft
76. The sum of the three angles of a triangle is 180o
. The sum of two of the angles is equal to
the third angle and the difference of the two angles is equal to 2/3 the third angle. Find
the smallest angle.
a. 15o
b. 20o
c. 25o
d. 30o
77. The logarithm of 1 to any base is
a. indeterminate b. one c. infinity d. zero

14. 78. Carry out the following multiplication and express your answer in cubic meters: 8 cm x
5 mm x 2 m.
a. 8 x 10-2
b. 8 x 102
c. 8 x 10-3
d. 8 x 10-4
79. If n is any positive integer, then (n – 1)(n – 2)(n – 3). . . . . .(3)(2)(1) is equal to
a. en-1
b. (n – 1)! c. n! d. (n – 1)en
80. If a two-digit number has x for its units’ digit and y for its tens’ digit, what represents the
number?
a. x + y b. x – y c. 10y + x d. 10x + y
81. Terms that differ only in numeric coefficients are known as
a. unequal terms b. unlike terms c. like terms d. equal terms
82. The sum of three numbers is 138. The second is 5 more than the smallest and the third
is 10 more than the smallest. Find the smallest number.
a. 51 b. 46 c. 41 d. 36
83. Three numbers are so related that the second number is 2 more than the first number and
the third is 4 more than the first. Find the smallest of the numbers if the sum of their
squares is 56 more than three times the square of the smallest number.
a. 3 b. 5 c. 9 d. 15
84. The difference of two numbers is 14 and twice the smaller number is 5 less than the larger
number. Find the larger number.
a. 21 b. 23 c. 25 d. 27
85. If the numerator and denominator of a certain fraction are each decreased by 2, the value
of the new fraction is ½. But if the numerator of the original fraction is increased by 2 and
the denominator decreased by 2, the resulting fraction is equal to ¾. Find the original
fraction.
a. 5/9 b. 3/7 c. 10/18 d. 12/16
86. The sum of the reciprocals of two numbers is 11. Three times the reciprocal of one of the
numbers is 3 more than twice the reciprocal of the other no. Find one of the numbers.
a. 1/6 b. 5 c. 6 d. 1/3
87. Find the second of three numbers such that the sum of the first and second is 67, the sum
of the first and third is 80, and the sum of the second and third is 91.
a. 17 b. 28 c. 39 d. 52
88. The number 0.123123123123….. is
a. irrational b. surd c. rational d. transcendental
89. Henry’s father is now twice as old as he is. Sixteen years ago, he was four times as old.
How old is the father now?
a. 24 b. 48 c. 18 d. 36
90. In 11 years, Jonathan will be 5 times as old as he was 9 years ago. How old is he now?
a. 10 b. 12 c. 14 d. 16

15. 91. A girl is half as old as her brother and two years younger than her sister. The sum of the
ages of the three children is 34 years. How old is the girl?
a. 8 b. 7 c. 9 d. 10
92. Two or more equations are equivalent if and only if they have the same
a. solution set b. degree c. order d. variable set
93. A can paint a certain house in 10 days and B can paint the house in 12 days. How long
will it take to paint the house if both men work?
a.
11
7
6 days b.
11
5
5 days c. 5 days d. 6 days
94. A can do a certain job in 4 hours, B can do the job in 6 hours, and C can do the job in 8
hours. How long will it take to do the job if A and B work 1 hour and B and C finish the
job?
a. 2 hr b. 4 hr c. 3 ½ hr d. 3 hr
95. A tank can be filled by one pipe in 9 hours and by another pipe in 12 hours. Starting
empty, how long will it take to fill the tank if water is being taken out by a third pipe at a
rate per hour equal to one-sixth the capacity of the tank?
a. 20 hr b. 28 hr ```` c. 36 hr d. 32 hr
96. The ratio or product of two expressions in direct or inverse relation with each other is
called
a. ratio and proportion b. constant of proportionality
c. means d. extremes
97. The logarithm of a number to the base e is called
a. Naperian logarithm b. characteristic
c. mantissa d. Briggsian proportion
98. _____________ progression is a sequence of terms the reciprocals of which form an
arithmetic progression.
a. Geometric b. Harmonic c. Algebraic d. Binomial
99. Naperian logarithms have a base closest to which number?
a. 1.53 b. 1.62 c. 2.72 d. 10
100. An equation in which a variable appears under a radical sign is called
a. literal equation b. radical equation
c. irradical equation d. irrational equation
101. Which of the following has three significant digits?
a. 0.372 b. 0.00372 c. 0.0372 d. all of these

16. 102. A man drove 220 miles in
2
1
3 hours. Part of the trip was at 60 miles per hour and the
rest at 65 miles per hour. Find the time spent at the lower speed.
a. 1 hr b.
2
1
1 hr c. 2 hr d.
2
1
2 hr
103. Town A is 11 miles west of town B. A man walks from A to B at the rate of 3 miles per
hour, and another man, starting at the same time, walks from B to A at the rate of 4
miles per hour. Find the time after starting that the men are 2 miles apart.
a.
7
2
1 hr b.
7
3
1 hr c.
7
4
1 hr d.
7
5
1 hr
104. A motorboat goes 3 miles upstream in the same time required to go 5 miles downstream.
If the rate of flow of the river is 3 miles per hour, find the speed of the motorboat in still
water.
a. 10 mph b. 11 mph c. 12 mph d. 14 mph
105. An airplane travels 360 miles in two hours with the wind and flying back the same route,
it took
5
3
3 hours against the wind. Find the velocity of the wind.
a. 30 mph b. 40 mph c. 100 mph d. 160 mph
106. A statement of equality between two ratios.
a. valuation b. theorem c. identity d. proportion
107. A certain two-digit number is equal to 9 times the sum of its digits. If 63 were
subtracted from the number the digits would be reversed. Find the number.
a. 81 b. 94 c. 74 d. 69
108. The sum of the second and third digits of a three-digit number is equal to the first digit.
The sum of the first digit and the second digit is 2 more than the third digit. If the
second and third digits were interchanged, the new number would be 54 more than the
original number. Find the number.
a. 844 b. 835 c. 826 d. 817
109. If a = b and b = c, then a = c. This is the _______property of real numbers.
a. reflexive b. symmetric c. transitive d. addition
110. What time after 4 o’clock will the hands of a clock be together for the first time?
a. 4:21:49.09 b. 4:21:49.16 c. 4:21:49.02 d. 4:21:49.23
111. How many minutes after 7 o’clock will the hands of a clock be directly opposite each
other for the first time?
a. 7:05:07.58 b. 7:05:07.46 c.
11
5
5 min. d.
11
7
5 min.
112. If a = b, then b = a. This property of real numbers is known as
a. reflexive property b. symmetric property
c. transitive property d. substitution property

17. 113. How many ounces of pure silver must be added to 100 ounces, 40% pure, to make an
alloy which is 65% pure silver?
a.
7
3
67 oz. b.
7
3
73 oz. c.
7
3
69 oz. d.
7
3
71 oz.
114. A perfumer wishes to blend perfume valued at $4.10 an ounce with perfume worth $2.50
an ounce to obtain a mixture of 40 ounces worth $3.00 an ounce. How much of the
$4.10 perfume should he use?
a. 7.5 oz. b. 15 oz. c. 12.5 oz. d. 10 0z.
115. A gourmet chef blends a salad dressing by mixing 20 ounces of a solution containing
85% olive oil with pure corn oil, in order that the dressing be 50% olive oil. How much
corn oil should be used?
a. 13 oz. b. 14 oz. c. 15 oz. d. 16 oz.
116. What percentage of a mixture of sand, gravel and cement containing 30% cement
should be replaced by pure cement in order to produce a mixture that is 40% cement?
a. 14.28% b. 15.39% c. 13.72% d. 12.65%
117. Which of the following non-terminating decimals is rational?
a. 3.14149265… b. 1.141421356… c. 2.470470… d. 2.71828182…
118. Any number multiplied by _________________ equals unity.
a. infinity b. itself c. its reciprocal d. zero
119. 27, 9, 3, 1, 1/3, . . . is what type of progression?
a. arithmetic b. geometric c. harmonic d. power series
120. Solve the inequality:
3
84
2
2


xx
a. {x | x <
5
28
 } b. {x | x >
5
28
 }
c. {x | 5 < x < 28) d. {x | –5 < x < 28}

18. ANSWERS:
1. c 11. c 21. a 31. c 41. b 51. c
2. d 12. a 22. c 32. a 42. b 52. b
3. d 13. b 23. c 33. b 43. c 53. a
4. d 14. d 24. c 34. d 44. b 54. a
5. b 15. c 25. d 35. d 45. d 55. b
6. c 16. c 26. a 36. a 46. a 56. a
7. a 17. b 27. c 37. c 47. d 57. d
8. a 18. c 28. a 38. b 48. d 58. d
9. d 19. c 29. a 39. c 49. b 59. d
10. b 20. c 30. d 40. b 50. c 60. c
61. d 71. c 81. c 91. a 101. d 111. c
62. b 72. a 82. c 92. a 102. b 112. b
63. d 73. b 83. a 93. b 103. a 113. d
64. b 74. d 84. b 94. d 104. c 114. c
65. a 75. b 85. c 95. c 105. b 115. b
66. c 76. a 86. a 96. b 106. d 116. a
67. c 77. d 87. c 97. a 107. a 117. c
68. a 78. d 88. c 98. b 108. d 118. c
69. b 79. b 89. b 99. c 109. c 119. b
70. c 80. c 90. c 100. d 110. a 120. b