REVIEW – MATHEMATICS, SURVEYING AND TRANSPORTATION ENGINEERING

ALGEBRA 1 Scientific Notation and Prefixes

Kilo - 1000
Hecto - 100
NUMBERS Deca - 10
Deci - .1
Natural Numbers Centi - .01
-Any counting number is called a natural Milli - .001
number since counting can be done using one’s
natural fingers. In fact, the symbols 1, 2, 3,
4, 5, 6, 7, 8, 9, 0 are called digits, from a PRACTICE PROBLEMS
Latin word meaning finger.
1. Convert is the temperature in degree
-Any natural number of our decimal system can
be expressed using only these ten digits. Celsius of absolute zero?
Thus, the digits 1 and 7 are used to express 2. What temperature will the °C and °F
the natural or counting numbers 17 and 71. readings be equal?
-The successor of a natural number is the next 3. Multiply and express your answer in
greater natural number. Thus, the successor of cubic meters: 3cm x 5mm x 2m.
99 is 100. 4. The numbers of board feet in a plank 4
inches thick, 2 foot wide, and 20 feet long
Whole Numbers- Non Decimal or Fraction is:
Numbers/ Natural Numbers and 0 5. Find the absolute temperature of the
freezing point of water in degree Rankine?
Odd and Even Whole Numbers 6. Prove that 423 a Prime Number?
-The set of even numbers consists of 0, 2, 4,
7. Express 3763 in Roman numerals:
6, 8, and all whole numbers whose last digit 8. MCMXCIV is equivalent to:
is one of these.
-The set of odd numbers consists of 1,3, 5,
7,9, and all whole numbers whose last digit is RULES OF ARITHMETIC
one of these. A whole number is either an odd
number or an even number. The basic arithmetiacal operations of
-For example, 1352 is an even number and 2461 addition, subtraction, multiplication and
is an odd number. division are performed subject to the
Fundamental Rules of Arithmetic. For any
Rational Numbers- have integers and fractions three numbers
and decimals/ may have repeating decimals. a, b and c:
Ratio of two integers (3/2, ½…) (a1) the commutative law of addition
a + b = b + a
Irrational numbers- integers and fractions and (a2) the commutative law of multiplication
decimals/ have decimals that go on forever. a × b = b × a
Ratio that cannot be represented by fractions (b1) the associative law of addition
(√3, ∏…) (a + b) + c = a + (b + c)
(b2) the associative law of multiplication
Whole numbers- non fraction numbers (a × b) × c = a × (b × c)
(c1) the distributive law of multiplication
Fraction- portion of a whole over addition and subtraction
(a + b) × c = (a × c) + (b × c)
(a − b) × c = (a × c) − (b × c)
Proper Fraction- When the denominator is
(c2) the distributive law of division over
greater than the numerator. (1/2, 2/3…)
addition and subtraction
Improper Fraction- When the denominator is (a + b) ÷ c = (a ÷ c) + (b ÷ c)
less than the numerator (4/3, 5/2…) (a − b) ÷ c = (a ÷ c) − (b ÷ c)
Here the brackets indicate which operation is
Prime Number- is a natural number greater than performed first. These operations are called
1 that has no positive divisors other than 1 binary operations because they associate with
and itself. every two members of the set of real numbers
a unique third member; for example,
Real Numbers- include all of the above and 2 + 5 = 7 and 3 × 6 = 18
fractions and integers
BASIC ALGEBRA TERMS
Roman Numerals- Expressed by I = 1, X = 10, Constants
L=50, C = 100, D = 500 and M = 1000. A fixed quantity that does not change. For
example: 3, –6, π,

CONVERSION OF UNITS Variables

A variable is a symbol that we assign to an
Temperature unknown value. It is usually represented by
Celcius to Farenheight C = (F-32)x5/9 letters such as x, y, or t.
Celcius to Kelvin K = C + 273.15
Farenheight to Rankine R = F + 460 Coefficients
Absolute Zero 0 K The coefficient of a variable is the number
that is placed in front of a variable.
 REVIEW – MATHEMATICS, SURVEYING AND TRANSPORTATION ENGINEERING

Terms
A term can be any of the following: For relationships of ratios:
a constant: e.g. 3, 10, π,
the product of a number (coefficient) and a Ratio of lengths = ratio of sides = scale
variable: e.g. –3x, 11y, factor
the product of two or more variables: e.g. Ratio of surface areas = (ratio of sides)2 =
x2, xy, 2y2, 7xy (scale factor)2
Ratio of volume = (ratio of sides)3 = (scale
Expressions factor)3
An expression is made up of one or more
terms.
PRACTICE PROBLEMS
Equations 1. A line on a map was drawn at a scale of
An equation consists of two expressions 4:100,000. If a line in the map is 300 mm
separated by an equal sign. The expression on long, the actual length of the line is:
one side of the equal sign has the same value 2. When rounded-off to four significant
as the expression on the other side. figures, 103.68886 become:
3. The scale on the map is 1:x. A lot having
Quadratic Equations an area of 720 sqm is represented by an area
A Quadratic Equation is an equation of the of 30.6 cm on the map. What is the value of
form: x?
ax2 + bx + c = 0, where a, b and c are 5. If x<0 and y and z are not equal to 0,
numbers and a ≠0 what is the sign of x4y3z2?
6. If a negative number is divided by a
positive number, what is the sign of the
SIGNIFICANT FIGURES quotient?
Specifically, the rules for identifying 7. Which of the following numbers can be a
significant figures when writing or product of an even prime number and odd prime
interpreting numbers are as follows:[2] number?
a. 4 b.12 c.6 d.8
All non-zero digits are considered 8. What is sign of the product of 3 negative
significant. numbers?

For example, 91 has two significant EXPONENTS, RADICALS and

figures (9 and 1), while 123.45 has five LOGARITHMS
significant figures (1, 2, 3, 4 and 5).
LAWS OF EXPONENTS (INDEX LAW)
Zeros appearing anywhere between two non-zero xn x is base and n is exponent
digits are significant. xn = x*x*x to n factors
Example: 101.1203 have seven (xm)(xn) = x mn
significant figures: 1, 0, 1, 1, 2, 0 and 3.
xm
Leading zeros are not significant. For = x mn
example, 0.00052 has two significant figures: xn
5 and 2. mn
(xm)n = x
Trailing zeros in a number containing a (xyz)n = xnynxn
decimal point are significant. n
x xn
  =
For example, 12.2300 have six significant
 y yn
figures: 1, 2, 2, 3, 0 and 0. The number m
0.000122300 still has only six n
significant figures (the zeros before the 1 x =
n
xm
are not significant). 1
In addition, 120.00 have five x-m =
significant figures since it has three xm
trailing zeros. x0 = 1
xm = xn then m = n if x ≠ 0
SCALE If xm+1 = xn+2 then m+1 = n+2

Scale factor is the factor by which all PROPERTIES OF RADICALS

the components of an object are multiplied in 1
order to create a proportional ax  x a
enlargement or reduction.
x
 ax
y y
So to say 1:4 or scale/actual measurement,
a
that means that the scale is 1/4 of the
actual.
 a
x
x
a
When the scale factor is larger than 1 it x
a  b  x abx
means it gets bigger and smaller when it’s
less than 1. But they will always remain x
a a
proportional. So you could also use ratio
x
x
and proportion. b b
 REVIEW – MATHEMATICS, SURVEYING AND TRANSPORTATION ENGINEERING

Provided that b≠0 x ≥ a if and only if x≥a or x≤-a

PROPERTIES OF LOGARITHMS 1. If the absolute value of x is greater than

the absolute value of y, which of the
logaMN = logaM + logaN following is always true?
loga M = logaM - logaN a. x-y>0 b. x2>y2
N 2. Find the interval of real numbers which
loga Mn = nlogaM contains x, if x satisfies the condition |2x-
loga a = 1 5|<3.
loga aa = a 3. Find the absolute value of x if |4+4x| =
loga 1 = 0 12.
If loga M = N, then aN = M 4. Find the area of the curve enclosed by
If loga M = loga N, the M=N |x|+|y|=1
logeM = ln M
e = 2.71828…. (Naperian logarithm ) EQUALITY
log10M = log M (Common Logarithm)
logn M = log M / log n = ln M/ ln n Properties of Equality
logb x = a then x = antilogb a Reflexive x=x
ax = antiloga x Symmetric if x=y, then y=x
log10 4751 = log10 (1000 * 4.751) Transitive if x=y and y=z, then x=z
= log 1000 + log 4.751 Sum x=y and z=w, then x+z=y+w
=3 + 0.6768 Product x=y and z=w, then xz=yw
=3.6768
3 is the integral part or the characteristic, INEQUALITIES
0.6768, a non-negative decimal fraction part,
is called the mantissa An inequality is a relationship between two
quantities that are not equal.
1. Simplify (x-2 y3)2/(x2y-1). We can represent s linear inequality in one
2. Solve for C if C = √(1-√(1-√(1-... variable on a number line. We use the
4. The logarithm of negative number is: following symbol in representation
5. The logarithms of the quotient and the (<,>,≤,≥,≠)
product of two numbers are 0.362182518 and
1.79630250, respectively. Find the first
number Solving Inequalities
6. Solve for y: y= ln(ex/ex-2)
7. Factor the expression x2 + 6x + 8 To solve an inequality, we can use the same
completely. method we use in solving for equality.
8. Factor the expression(x4–y4) completely.
9. Log of the nth root of x equals log of x We can represent a linear inequality in one
to the 1/n power and also equate to: variable on a number line.
10. What expression is equivalent to log x – We can use the following symbols in the
log(y+z)? representation.
a. Log x + log y + log z b. log[x/(y+z)]
11. If 10x = 4 find the value of 102x+1 A small circle is used for < and > to
12. Rationalize the denominator and determine indicate that the number is not included
the transformed fraction (a1/2)/(a1/2–a1/2b1/2) A filled in circle is for ≤ and ≥ to
13. Find the value of x if (ex+e-x)/(ex–e-x) = indicate that the number is included.
2 A line with an arrow indicates that the
14. If log (9!) = 5.5598, what is the log of line continues to infinity in the direction
10!? of the arrow

ABSOLUTE VALUE

This is the distance of a number from 0

regardless of direction or its location in
the number line.

When solving an equation with absolute Properties of Inequality

values, it is necessary to split the equation < less than
into two equations, one resulting in a > Greater than
positive value and the other resulting in a ≤ less than or equal
negative value. We can then solve the two ≥ greater than or equal
equations to obtain two possible solutions.
Theorems
x>y only if -x<-y (3>2; -3<-2)
x = x if x ≤ 0 or -x if x<0 If x>0 then -x<0 (2>0; -2<0)
If x>y and z<0 then xz < yz (multiplied by a
xy  x y negative)
If x>y and z>w then x+z > y+w
x ≤ a if -a ≤ x ≤ a If x>y and z>w and x,y,z,w > 0 then xz > yw
If x>0, y>0, x>y then 1/x < 1/y
 REVIEW – MATHEMATICS, SURVEYING AND TRANSPORTATION ENGINEERING

1. If 1/x=a+b and 1/y=a-b then x-y is equal

to: Synthetic Division
2. If 3x=4y then (3x2)/(4y2) is equal to:
3. If 1/a:1/b:1/c = 2 : 3 : 4, then (a+b+c) : 1 0 -2 4 2
(b+c) is equal to: 2 4 4
4. Given the following equations: a*b = 8, 1 2 2 8
a*c=3 and b*c=6. What is the product of a, b
and c? x2+2x+2 remainder 8
5. If xyz = 8 and y2z=12, what is the value
of x/y? If no remainder then assumed number is a
6. If abc-de is positive, which of the factor.
following is always correct? a. abc-de>0
b. abc≥de When resulting numbers on the third are all
Which of the following is not in the range? positive, root is upper bound. When they are
7. If the domain of y = 2x + 1 is ( x| - 2 ≤ alternating from positive to negative, it is
x ≤ 3). Which of the following is not in the lower bound.
range?
A. -4 b) 0 c) -2 d) 7 Depressed Equation - equation formed after 1
synthetic division.
FACTORING AND EXPANSION
Descartes’ Rule - used to determined zeroes
Expanding Brackets - multiplying all terms of in a polynomial
each bracket by the other.
(x+y+z)(a+b) = ax + ay + az + bx + by + FACTOR THEOREM
bz
Given a function f(x). f(1) = 0 then x-1 is a
Factoring- opposite of expanding. Simplify an factor of f(x)
expression to a shorter expression by use of
brackets. REMAINDER THEOREM

Special Products and Factoring If a function f(x) is divided by (x-r) until

a remainder free of x is obtained, the
Common Factors remainder is f(r). If f(r) = 0 then x-r is a
x(a+b) + y(a+b) = (x+y)(a+b) factor of f(x).

Difference of two Square 1. If f(x) = x2+x+1, then f(x)-f(x-1) =

a2-b2 = (a+b)(a-b) 2. Find k in the equation 4x2+kx+1=0 so that
it will only have one real root.
Perfect Square Trinomial 3. When (x+3)(x-4)+4 is divided by x-k, the
(a+b)2 = a2 + 2ab + b2 remainder is k. Determine the value of k.
(a-b)2 = a2 - 2ab - b2 4. The quotient of (x2+32) by (x+2) is:
5. When the expression x4+ax3+5x2+bx+6 is
Sum of two cubes divided by (x-2), the remainder is 18. When
(a3+b3) = (a+b)(a2-ab+b2) it is divided by (x+1) the remainder is 14.
Find the value of constant a?
Difference of two cubes 6. If x4-2x³-3x²-4x-8 is divided by (x-2),
(a3-b3) = (a-b)(a2+ab+b2) the remainder is:
7. By synthetic division, compute the
Trinomials remainder if we divide 2x³+x²-18x+7 by x-2.
acx2 + (ad+bc) + bdy2 = (ax + by)(cx + dy)
BINOMIAL THEOREM
Factoring by Grouping
2x+2y+ax+ay = (2+a)(x+y) A binomial is a polynomial with two terms. We
raise it to an exponent and our goal is to
Division of Polynomials get the rth term of the binomial raised to
the nth power.
Long Division
You should be familiar with Pascal's
x3-2x+4 divided by x-2 Triangle. The rth term of a binomial is
simplified as:
x 2  2 x  2 Remainder 8
x  2 x3  0 x 2  2 x  4 n!
rth term of (a+b)n = a nr 1b r 1
(n  r  1)!(r  1)!
x3  2 x 2
2x2  2x n
For the middle term r  1
2
2x  4x
2
Or all the terms by:
2x  4
2x  4
8
 REVIEW – MATHEMATICS, SURVEYING AND TRANSPORTATION ENGINEERING

+ 15x² - 36x +20. How many are its rational

roots?
11. In the expansion of (2x-1/x)12
Find the term independent of x. Find the 6th
n
n
(a  b) n    (a n k )(b k )
term. Find the coefficient of the 9th term
12. In the binomial expansion (a+b)n .
k 0  k  Determine the value of “n” if the
Where k = r-1 coefficients of the 4th and the 13th terms are
equal to each other. Determine the
Number of Terms = n+1 coefficient of the 8th term of the expansion.
1st Term = an Last Term = bn Determine the 10th term of the expansion.
Exponents a descends from 0 and b ascends 13. Find the 8th term of the expression (4a-
from n. b²)10
14. Expand the expression (a/2 – 7/2)²
Coefficient of Next Term

C = (Cprevious Term)(e of x)/(e of y)-1

Refrences:
Sum of Coefficient of Variables -Schaum's Review of Elementary Mathematics
-Modern Engineering Mathematics 5ed [2015}
Substitute 1 in all variables but subtract -1001-Solved-Problems-in-Engineering-Mathematics
the constant term. -Schaum's Basic Mathematics

Problems: ALGEBRA 2
1. Find the 8th term of the expansion (1/2a -
3)12 QUADRATIC EQUATION
2. In the expansion of (a+4b)12, the
numerical coefficient of the 5th term is Is an equation whose highest power of any
3. The middle term of the expansion of (a2- variable is to the 2nd power.
3)8 is:
4. Find the sum of the coefficients in the Quadratic Equation
expansion of (a+4b-c)8. Ax2+Bx+C=0
5. For the expression of (6x-3)8
What is the value of the 4th term? Quadratic Formula

 B  B 2  4 AC
x
2A

Discriminant B2-4AC

TAKE HOME PROBLEMS If B2-4AC = 0 roots are equal

1. A car has a mass of 1200 kg. A model of a If B2-4AC > 0 root are real but unequal
car is made to a scale of 1:60. Determine the If B2-4AC < 0 roots are imaginary
mass of the model if the car and its model
are made of the same material. Properties of roots
2. Express 3239 in Roman numerals: Sum of roots
3. Determine all possible values of x that B
will satisfy the equation |x-1| = 5- 2x. x1+x2= 
4. Which of the following expressions is A
equal to |x-y| Product of Roots
for all real numbers x and y? a) |y-x| C
5. Find -6|d|; given that d is not equal to x1x2 =
0. A
6. The area of a lot on the map is 500 mm².
If the scale of the map is 1:40,000 determine Problems:
the true area of the lot in hectares.
7. If it is given that f(x) = |x| + 10, then 1. In a quadratic equation Ax2+Bx+C=0, the
which of the following values of x make f(x) product of the roots is:
equal to f(-x)? a) all real x 2. Find the value of k in x^2+3x+k-1=0 if the
8. An earthquake is usually measured quadratic equation has:
by the magnitude of M on the Richter scale. a) equal roots
The intensity I of an earthquake and the b) one root is -3
magnitude M are related by the formula: M = 3. Find k if the roots of the equation (k+3)x^2
log (I/Io) where Io is the intensity of an - 2(k+1)x - (k+1) = 0 are equal. If k is not
arbitrary chosen earthquake. The earthquake equal to -3.
that hit Kobe, Japan, measured 5.7 on the 4. Two students attempt to solve a problem
Richter scale. The earthquake that hit Texas that reduces to a quadratic equation. One of
measured 7.8. The earthquake that hit Texas the students made a mistake only in the
measured 7.8. How many times stronger is the constant term of the quadratic equation and
earthquake that hit Texas? gives and answer of 8 and 2 for the roots. The
9. Determine how many positive real roots are other student solving the same problem made an
there for the polynomial 7x2 + 5x5 +3x³ + x. error in the coefficient of the first degree
10. A polynomial has an equation x5 – 5x4 + term only and gives his answer as -9 and -1
5x³ for the roots, if you are to check their
 REVIEW – MATHEMATICS, SURVEYING AND TRANSPORTATION ENGINEERING

solutions, what would be I = Prt

the correct quadratic equations?
5. In the equation Ax² + 14x + 12 = 0, one P= Principal
root is 8 times the other root. Find the value I= Interest earned r= rate
of A? t = time
6. What is k so that the expression kx2-kx+9
is a perfect square? SAMPLE PROBLEMS
7. Find the values of k so that the equation 1. You know that to make 20 pancakes you have
(k-2)x^2 + 4x – 2k+1 =0 has two distinct real to use 2 eggs. How many eggs are needed to
roots. make 100 pancakes?
8. In the equation 4x2+3x+(3h-4)=0, find h if 2. Given that x varies directly as y and
the product of the roots is 6 inversely as z, and x = 12 when y = 8 and z =
9. If the roots of the quadratic equation 3, find x when z = 6 and y = 12.
ax2+bx+c=0 are 3 and 2 and a, b and c are all 3. The electrical resistance of a cable
whole numbers. Find a+b+c. varies directly as its length and inversely
10. The roots of a quadratic equation are 1/2 as the square of its diameter. If a cable 800
and 2/3. What is the equation? meters long and 20 mm in diameter has a
11. Determine the value of k so that the sum resistance of 0.1 ohm, find the length of the
and product of the roots are equal from the cable 75 mm in diameter with resistance of
given equation 8x²+ (3k-1)x – 2k + 1 = 0 1/6 ohm.
12. Find the value of k in the quadratic 4. Sparrows and pigeons sit on a fence. When
equation 4x² - kx + x – 6k = 0 if 3 is one of 5 sparrows leave, there remain 2pigeons for
the roots. every sparrow. Then 25 pigeons leave and the
ratio of sparrows to pigeons becomes 3:1.
RATIO AND PROPORTION What is the original number of birds? Ans. 50
5. A man sold 100 chickens. Eighty of them
Ratio problems are word problems that use were sold at a profit of 20% while the rest
ratios to relate the different items in the were sold at a loss of 25%. What is the
question. percentage gain or loss on the whole stock?
6. Kobe bought two sports cars, one for
・ Change the quantities to the same unit if P700,000.00 and the other for P600,000.00. He
necessary. sold the first a a gain of 10% and the second
at a loss of 12%. What was his total
・ Write the items in the ratio as a percentage gain or loss?
fraction. 7. A grocery item costs $4. If the price
increases 10%, what will be the new price in
dollars of the grocery?
・ Make sure that you have the same items in
8. For a gas at constant temperature, the
the numerator and denominator.
volume of a fixed mass of gas is inversely
proportional to its absolute pressure. If a
・ In any proportion, the product of the gas occupies a volume of 2 m³ at a pressure
means is equal to the product of the of 300x10³ Pascals, determine the pressure
extremes: when the volume is 1.4m³
a/b = c/d 9. What is the mean proportion of 4 and 36?
a and d = extremes c and b = means 10. An item is being sold for 240 dollars at
d is the 4th proportional to a,b and c a 36% discount, what is the original price of
x2 = ab x is the mean proportional to and b the item? Ans. 375
・ In the ratio a/b, a is called antecedent
and b is SETS AND VENN DIAGRAMS
the consequent
・ If a/b = c/d then a/c =b/d Sets are collections of things.
・ If a/b = c/d then a+b/b =c+d/d Elements- members of a set.
・ If a/b = c/d then a-b/b = c-d/d Bracket- Braces or boundaries of a set
・ If a/b = c/d then a+b/a-b = c+d/c-d
Finite Set- exact number of things in a set
when talking about sets, it is fairly
DIRECTLY PROPORTIONAL standard to use Capital Letters to represent
AND INVERSELY PROPORTIONAL the set, and lowercase letters to represent
an element in that set.
Directly proportional: as one amount
increases, another amount increases at the Equality- sets having the same members
same rate. Inversely proportional: as one Subset- set which elements are also a member
amount increases, another amount decreases at of a bigger set
the same rate
Proper subset- if all elements in a subset
∝ = symbol of proportionality are present in another set, but there is at
Constant of proportionality is the value least one element not in the subset
which proportionate the amounts or k.
y = kx proportional A is subset of B: A ⊆B
y = k/x inversely proportional A is not a subset of B: A ⊈B
Empty or Null Sets- sets that don't have
SIMPLE INTEREST
elements. The empty set is a subset of every
set, including the empty set itself.
 REVIEW – MATHEMATICS, SURVEYING AND TRANSPORTATION ENGINEERING

To solve them, decompose into partial

Order- size of the set or the number of fractions then either:
elements in the set
1. Substitute a value of x that will
Problems: eliminate some of the unknowns to form
1. In an election, 25 voted to have an outing equations
in the beach, 30 voted to have a hiking and
10 voted to do both. How many participants 2. Equate coefficients of similar variables
are there?
2. In a group of 70 students, 35 are taking and power.
Chemistry, 25 are taking Calculus and 15 are
taking both subjects. How many students are Problems:
taking neither subject?
3. A veterinarian surveys 26 of his patrons. 1. Convert (x+2)/(x2-7x+12) into partial
He discovers that 14 have dogs, 10 have cats fraction
and 5 have fishes. Four have dogs and cats, 3 2. Solve for A: (7x-3)/(x(x-1)) = A/X + B/X-1
have dogs and fishes and 1 has a cat and 3. Solve for B: (7x-3)/(x²(x-1)) = A/X + B/X²
fish. If no one has all three kinds of pets, + C/X-1
how many patrons have none of these pets? 4. Solve for a if (3x-2a)/(x²+x-6) = 2/a+3 +
4. A set has 5 items and it has range of 7. 1/a-2
The set is composed of the following: 5. Find the value of E in the following
{1,2,m,5,m²) with m >0. Determine the average equation. (2x4+3x³+7x²+10X+10)/(X-1)(X²+3)² =
number in the set. A/X-1 + (BX+C)/(x²+3) + (Dx+E)/(x²+3)²
5. A student is given a simple set which (Bx+C)/(x²-2x+2) + (Dx+E)/(x²-2x+2)²
contains only two integers, 15 and 16 and is
written as set PROGRESSION
{15,16}. The set is equivalent to:
a) { x | 15 < x ≤ 16, where x is an integer} Arithmetic
b) { x |15 < x < 16, where x is an integer} A sequence such that the common diffrence
c) { x | 14 ≤ x < 16 , where x is an between two consecutive terms are constant.
integer}
d) { x| 14 < x ≤ 16, where x is an integer nth term of a sequence an = a1 + (n-1)d

Sum of n terms
PARTIAL FRACTIONS Sn = (n/2)(a1 + an)

For problems in calculus, you may encounter Arithmetic Mean

problems involving fractions that could not Am = (a1 + an)/n
be integrated easily. What you can do
is the D E C O M P O S I T I O N i n t o Geometric
PA R T I A L FRACTIONS or simply creating
a group of fractions equivalent to the A sequence such that a term divided by the
given fraction. previous term is a constant or the common
ratio.
Partial-fraction decomposition is the process
of starting with the simplified answer and nth term of a sequence a = a rn-1
taking it back apart, of "decomposing" the
final expression into its initial polynomial Sum of n terms
fractions. Sn = a1[(1-rn)/(1-r)] when r<1
Sn = a1[(rn-1)/(r-1)] when r>1
To be able to break them apart you should be
able to classify them: Sum of Infinite Geometric Progression
SGP= a1/(1-r) -1<r<1 n = infinity
Distinct Linear
Geometric Mean
Term A B
  Gm = √(a1 x an)/n
(ax  b)(cx  d ) ax  b cx  d Gm = n√(a1 x a2 x a3x...an)/n

Harmonic
Linear with Repeated
Term A B C A sequence such that the reciprocal forms an
   Arithmetic Progression.
(ax  b)(cx  d ) 2
(ax  b) (cx  d ) (cx  d ) 2
Distinct Quadratic Use the Sigma Notation to get the sum of all
Term A Bx  C terms
  2
(ax  b)(cx  d ) ax  b cx  d
2
Harmonic Mean
HM = 2/[(1/x)+(1/y)]
Quadratic with Repeated GM2 = AM*HM
Term A Bx  C Dx  E
   Problems:
(ax  b)(cx  d )
2 2
(ax  b) (cx  d ) (cx 2  d ) 2
2
 REVIEW – MATHEMATICS, SURVEYING AND TRANSPORTATION ENGINEERING

1. Find a in the series get the sum of all terms from 1 to 4 we need
1,1/3,0.2,a.. to add them up. Sigma Notation will be
2. The first term of an arithmetic helpful for complicated sequences which will
progression (A.P.)is 6 and the 10th term is require summation of terms that is more than
3 times the second term. What is the common 3 or 4.
difference?
3. The positive value of a so that 4a, 5a+4,
3a²-1 will be in arithmetic progression is: Mathematical Sequence
4. The 10th term of the progression
6/5,4/3,3/2, … is: nth term = Sn+1 - Sn
5. The geometric mean of 4 and 64 is:
6. Find the sum of the infinite geometric Mathematical Induction
progression 6, -2, 2/3….
7. Find the sum of the first 10 terms of the 1. 1  2  3  4  ...n  n(n  1)
Geometric Progression 2,4,8,16, … 2
8. The 1st, 4th, and 8th terms of an A.P. are
themselves geometric progression (G.P.). What 2. 2  4  6  ...2n  n(n  1)
is the common ratio of the G.P.?
9. The sum of three numbers in arithmetical
progression is 45. If 2 is added to the first 3. 12  32  52  ...(2n  1) 2  n(2n  1)(2n  1)
number, 3 to the second, and 7 to the third, 3
the new numbers will be in geometrical
progression. Find the common difference in 4. 13  23  33  ...n3  n
2
(n  1) 2
A.P. 4
10. The geometric mean and the harmonic mean
of two numbers are 12 and 36/5 respectively.
5. 22  42  62  ...(2n) 2  2n(n  1)(2n  1)
What are the numbers? 3
11. A basketball is dropped from a height of
5 m. on each rebound, it rises 2/3 of the
6. 3
height from which it last fell. Find the 1  33  53  ...(2n)3  n 2 (2n 2  1)
distance travelled by the ball before it
comes to rest.
12. The nth term of an arithmetic progression 7. 1  1  1  ... 1  n
is an = 3n +5. What is the common difference? 1 2 2  3 3  4 n(n  1) n  1
13. A sequence of numbers is described by the 1 1 1 1
equation an = 3 x 1.53n. What is the common 8.    ...
1 3 3  5 5  7 (2n  1)(2n  1)
ratio?
14. What is the 15th term of the progression
12, 18.5, 25 …..? n
9. a  (a  d )  (a  2d )  ...a(n  1)d  (2a  (n  1)d )
2
15. A new Acu Contractor failing to complete
his first building contract worth P70,000 in n 1
a specified time is compelled to pay a 10.a  ar  ar 2  ...ar n1  a  ar
1 r
penalty of ½ of 1% per day for first 6 days
of extra time required and for each
Stacked Balls
additional day thereafter, the stipulated
penalty is increase by 10% or P35 each day. Equilateral Triangle
If he pays a total penalty of P7160.00, how n(n  1)(n  2)
many days did he overrun his contract? S
6
16. If 1/a, 1/b, 1/c are in A.P., what is the
value of y?
17. If 1/a, 1/b, 1/c are in GP, what is the
value of y? Rectangular Base
18. If 1/a, 1/b, 1/c are in HP, what is the n(n  1)(3m  n  1)
S
value of y? 6
19. Determine the harmonic mean between x and m=long side
y n= Shorter side
20. Find the harmonic mean between ½ and 1/8
21. There are 6 geometric means between 3 and Problems:
729. Find the sum of the G.P 1. Find x if x+3x+5x+7x+….+49x=625
2. Find the sum of all numbers between 0 and
MATHEMATICAL SEQUENCE AND MATHEMATICAL 10,000 which is exactly divisible by 77.
INDUCTION and SIGMA NOTATION 3. What is the sum of the following finite
Sigma ∑ means to sum up or to add so we just sequence of terms? 18,25,32,39, … 67.
add up terms in a series. Adding up rth terms 4. If equal balls are piled in the form of a
from the lower limit up to the upper limit. complete pyramid with an equilateral triangle
4 as its base, find the number of balls in a
 x 1
x 1
pile, if each side contains 4 balls.
5. Balls of the same radius are piled in the
form of a pyramid with a square base until
Means that the terms of the sequence is
there is just one sphere at the top layer. If
solved by x + 1 or the 1st term is 2, 2nd
term is 3 and 3rd is 4 and 4th is 5. So to
 REVIEW – MATHEMATICS, SURVEYING AND TRANSPORTATION ENGINEERING

there are 4 balls on The length divided by the width is 4. (l/w =

each side of the square, find the total 4)
number of balls in the pile John traveled 100 miles in 3 hours. (j =
6. What follows logically in these series of 100/3)
numbers 2,3,5,9,17…
7. In the given series of numbers AGE PROBLEMS
1,1,1/2,1/6,1/24…….What is the 6th term.
8. Find the value of x if 1+2+3+4+…x=36. To solve for Age Problems, remember that if
Using 1+2+3….n = [n(n+1)]/2 John's age moves back x years, Peter's age
9. Find the value of a in the sequence of also moves back x years.
numbers shown: a+2a+3a+4a+…..8a=72
10. Determine the sum to first 7 terms of the Problems:
series 0.25, 0.75, 2.25, 6.75.
11. What is the 12th term of the series: 1. A man is currently 26 years older than his
5,10,20,40? son. Twelve years from the current time, the
man will be twice as old as his son. What is
References: the current age of the father? Ans. M=40
-Schaum's Review of Elementary Mathematics
- https://www.algebra.com/ 2. A man has three children. The oldest child
- Basic Engineering Mathematics - 6th Edition is three times as old as the youngest. The
_https://www.mathplanet.com/education/algebra second child is six years older than the
-1/how-to-solve-linear-equations/ratios-and- youngest and six years younger than the
proportions-and-how-to-solve-them oldest. How old are the children? Ans. 6, 12
_https://www.algebra.com/algebra/homework/Per and 18
centage-and-ratio-word-problems/Percentage-
and-ratio-word- 3. Three years ago, the sum of the ages of the
problems.faq.hide_answers.1.html quintuplets and their older sister was 155. If
Sis is five years older than her siblings,
ALGEBRA 3 then how old are the quints today? Ans. 28

WORD PROBLEMS 4. When the population of group of bacteria at

any time “t” is given by An=Aoe0.584t, If Ao = 4,
Word problems require grammatical analysis in find t when An =2100. t = 10.72
order for us to successfully convert them into
equations. We should look for the following 5. A molecule moves according to the equation
key words: S = ut + 1/2at², where S is the distance in
meters, u is the initial velocity in m/s, a
Addition is the acceleration in m/s², and t is the
Sum, total, more than, greater than, time in seconds. Given that when t = 2 sec, S
consecutive increased by, plus, older than, = 98.21 m and when t=10 sec, S = 292.867 m,
farther than determine the following:
The value of u in kph
Example: The value of “a” in kph/sec. The value of S
The sum of the length and width is 20. (l+w=20) when t = 12 s.
The length is 2 more than the width. (l=w+2)
6. Determine the unit's digit in the
Subtraction expansion of 3855
Difference, diminished by, fewer than, less
than, decreased by, minus, subtracted from, 7. Find the 1987th digit in the decimal
younger than equivalent of 1785/9999 starting from the
decimal point.
Example:
The difference between Jan's age and Alice's 8. Mary is 24 years old. Mary is twice as old
age is 10. (J-A=10)
Joan has 10 fewer coins than Alice. (J=A-10) as Ana when Mary was as old as Anna now. How
old is Ana?
Multiplication
Product, twice, doubled, tripled, times,
multiplied by, of 9. A woman is three times as old as his
daughter. Four years ago, he was four times
as old as his daughter was at that time. How
old is the daughter?
Example:
The product of the length and the width is 40.
(lw = 40) AVERAGE PROBLEMS
The length is twice the width. (l = 2w)
John took half the number of marbles. (j=1/2m)
Average = Sum of all Terms/ Number of Terms
Division
WEIGHTED AVERAGE PROBLEMS
Quotient, divided by, divided into, quotient
of, in, per
Weighted Average = Sum (Term x Weight)/ Total
Weight
Example:
 REVIEW – MATHEMATICS, SURVEYING AND TRANSPORTATION ENGINEERING

SPEED PROBLEMS
INTEGER WORD PROBLEMS
Speed = Distance/ Time
Integer word problems are word problems that
involve integers or digits and their Flying Towards Wind, Velocity is X (your
relationships to each other. speed)+ Wind Velocity
Flying Against Wind, Velocity is X - W
Consecutive integers are integers that follow
in sequence. MIXTURE WORD PROBLEMS
For consecutive integers:
n, n + 1, n + 2 or n-1, n, n+1 Mixture problems are word problems where
For consecutive odd or even integers: items or quantities of different values are
n, n + 2, n + 4 or n -2, n, n + 2 mixed together.

DIGIT WORD PROBLEMS (Volume1 x Concentration) + (Volume2 x

Concentration) = Total Volume x Resulting
Same as the Integer Problems, but Concentration
the relationship is limited on the digits of
the given number. Some problems will require Problems:
treating the digits as individual numbers.
We could easily represent a 3 digit number 1. Four thousand (4000) kg of steel
by: containing 7% nickel is to be made by mixing
100x + 10y + z steel containing 12% nickel with steel
with x,y and z being the digits. containing 4% nickel. How much of the steel
containing 12% nickel is needed?
Relations could then be easily made
thru this equation. If we flip the digits, 2. To get rid of a nasty strain of root worm,
then we have: an insecticide mixture should have a
100z + 10y + x. concentration that’s 9% insecticide. A farmer
has two mixtures on hand — one with 5%
We could then just proceed to insecticide and the other with 15%
solving the equations. insecticide. What is the ratio of the 5% to
15% mixtures that should be combined to get a
9% mixture? If the farmer needs
Problems: 40 gallons of mixture, how many gallons of
each should she use? Ans. 24 gal and 16 gal
1. The difference between two positive
numbers is 20. If you square the larger 3. Two trains leave the same station
number and subtract ten times the smaller traveling in opposite directions. The first
number from the square, you get 575. What are train leaves at 2 p.m. The second train
the two numbers? Ans. 25 and 5 leaves a half-hour later and travels at a
speed averaging 15 miles per hour faster than
2. In a three-digit number, the sum of the the first train. By 8 o’clock that evening,
hundreds digit and the tens digit is 2 more they’re 600 miles apart. How fast are the two
than the units digit. While the tens digit is trains traveling? Ans. First 45mph 2nd 60mph
twice hundreds digit. When the digits are
reversed it is 396 more than the original 4. Alberto can bicycle 2 miles per hour less
number. What is the original number? than twice as fast as Ollie, so Alberto
didn’t leave for the rally until two hours
3. One integer is four smaller than another. after Ollie left. If the total distance they
The sum of their reciprocals is 24/5 . What travelled was 504 miles and if Ollie
are the numbers? Ans. 12 travelled for 10 hours, then how fast can
Alberto bicycle? Ans.38 mph
4. Find two consecutive even integers such
that the square of the larger integer is 36 5. A boat takes 2/3 as much time to travel
more than the square of the smaller integer. downstream from A to B, as to return,
if the rate of the river's current is
5. The denominator of a certain fraction is 1 9kph, what is the speed of the boat in still
more than twice its numerator. If 3 is added water?
to both the numerator and the denominator the
resulting fraction will be 3/5. Find the 6. Jon leaves Chicago at noon and heads south
original fraction. toward Bloomington traveling at 45 mph. Jane
leaves Bloomington heading north for Chicago
6. A certain copier machine produces 13 at 1 p.m., traveling at 55 mph. If Chicago
copies every 10 seconds. If the machine and Bloomington are 145 miles apart, then
operates continuously, how many copies will what time will they meet? Ans. 2:00 pm
it produce in an hour?
7. The gasoline tank of a car contains 60
7. The average of 14 numbers is 11, If one liters of gasoline and alcohol, the alcohol
number is removed, the new average is 8. What comprising 24%. How much of the mixture must
number was removed? be drawn off and replaced by alcohol so that
the tank contain a mixture of which 60% is
alcohol?
 REVIEW – MATHEMATICS, SURVEYING AND TRANSPORTATION ENGINEERING

with four more workers at the beginning of

8. How much water must be added to 8 ounces the 6th day. Find the total number of days it
of 40% alcohol to produce a mixture that’s 7% took them to finish the job.
alcohol? Ans. 38 ounces
4. In the tank of a water tower, the intake
9. A man fires a target with a bullet’s valve closes automatically when the tank is
velocity of 970m/s. After 3.5 seconds, he full and opens up again when 5/3 of the water
hears the sound of the bullet that strikes is drained off. The intake fills the tank in
the target. Assuming sound has a constant 4 hours, and the outlet drains the tank in 12
velocity of 340m/s, how far is the target hours. If the outlet is open continuously,
from the man? then how long a time is it between two
different instances when the tank is
WORK WORD PROBLEMS completely full? Ans. 10hrs and 48 mins

Work problems are similar to rate problems, 5. The population density x miles from the
where we need to treat the work done in center of a certain city is P(x) = 16 e-0.08x
relation to the rate on which it was done. thousand people per square mile.
The formula for “Work” Problems that involve What is the population density at the center
two persons is of the city? What is the population density
10 miles from the center of the city? What is
1/A+ 1/B = 1/C the population density at a distance of 9000
A= Time forAto finish miles of the city?
B= Time for B to finish
C= Time for the 2 of them to finish together 6. If 9 men can cut 26 trees in a day, how
many trees can 20 men cut in a day?
This formula can be extended for more than
two persons. It can also be used in problems CLOCK PROBLEMS
that involve pipes filling up a tank.
Solving clock problems requires familiarity
For work problems involving changes in Number with the movement of the clock hands.
of Workers, this can be solved using the
conept of: When the minute hand moves 60 spaces, the
hour hand moves 5 spaces, therefore the ratio
Work Done = Number of Worker x Rate of Work of the to is 5:60 or 1/12. This means if the
minute hand moves x minutes the hour hand
POPULATION WORD PROBLEMS moves x /12 minutes.

P =Poert When the second hand moves 60 spaces, the

P = Po + rt minute hand moves 1 space. This means their
ratio is 1/60, so when the second hand moves
P = Population at time t x seconds, the minute hand moves x/60
Po = Initial Population seconds. The the hour hand moves x/60/12 so
r = rate of increase the hour hand moves x/720 seconds.
t = time
Note also that 1 revolution is 360 degrees
Problems: and every 5 minutes is equal to 30 degrees.

1. A three-man crew can harvest the field in x = distance traveled by the minute hand (in
six hours, while a four-man crew can harvest minutes)
the field in four hours. If the three-man x/12= distance traveled by the hour hand (in
crew worked for one hour and then were joined minutes)
by the fourth man, how long will it take the
four-man crew to finish the job? How long Note: Distances Together = 0
does it take from start to finish to do the Perpendicular = 15 minutes
whole job? Ans. 4hrs and 20 mins Opposite = 30 minutes

2. Sarah, Sue, and Sybil are making Problems:

chocolate-chip cookies for the annual club
bake sale. Working alone, it would take Sarah 1. In three hours, the minute hand of the
8 hours to do all the baking. Sue could do clock rotates through an angle of:
the whole job in 10 hours, and it would take
Sybil 12 hours by herself. They all started 2. How many minutes after 4:00 PM will the
working early in the morning. But, after minute hand of the clock overtakes the hour
2 hours, Sybil said that she had to leave for
an appointment and wouldn’t be back. One hour hand?
after that, Sue got tired of listening to
Sarah’s griping and left. How long did it 3. At what time after 12:00 noon will the
take Sarah to finish up the job by herself, hour hand and the minute hand of a clock
after the other two left? Ans. 1hr and 16 min first form an angle of 90°?

3. A work order could be done by eleven

workers in 15 days. Five workers 4. Brandon Ingram left his home at pas 3:00
started the job. They were reinforced o'clock PM as indicated in his wall clock.
 REVIEW – MATHEMATICS, SURVEYING AND TRANSPORTATION ENGINEERING

Between two to three

hours after, he returned home and noticed
that the hands of the clock interchanged. At
what time did he left his home?

5. At how many minutes after 4 PM will the

hands of a clock be: Together for the first
time. Opposite each other for the first time.
Perpendicular to each other for the first
time.

6. It is now between 9 and 10 oclock. In 4

min. The hour hand of a clock will be
directly opposite the position occupied by
the minute hand 3 minutes ago. Find what time
is it.

7. What time between 2 and 3 oclock will the

angle between the hands of the clock be
bisected by the line connecting the center of
the clock and the 3 o’clock mark?

8. The second hand of a clock is 7 inches

long. Find the speed of the tip of the second
hand.

DIOPHANTINE EQUATION

A diophantine equation is an equation in

which the solutions are integers. The key
difference is that diophantine equations
present problems which lack at least 1
equation to complete the set of equations
needed to solve the problem. Solving this
kind of equation involves finding the set of
integers that will satisfy the equations.
Usually, the choices in the board
exam would be very useful as you could
substitute them directly and find out which
of them satisfies the equation

Problems:

1. Find the least positive integer value of x

in the following equation: 27x+14y=316.
a. 4 c. 10
b. 6 d. 7

2. A vendor has three items on sale: a DVD

for Php60, a remote for Php 20 and a bag
for Php 1. At the end of the day she sold
a total of 100 of the three items and has
taken exactly Php1717 of the total sales.
How many remotes did she sell?
a. 20 c. 38
b. 18 d. 23

References:
- Math-Word-Problems-for-Dummies
- Math_Word_Problems_Demystified