Module in Math in The Modern World - Revised

Romblon State University
San Fernando, Campus

Business Administration and Management Department
RSU-SFC BAM Form No.005 Module
“Shine and Serve with Honor and Excellence”

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GE 4: MATHEMATICS IN THE MODERN WORLD
Dear future managers and entrepreneurs,
WELCOME to our MATHEMATICS IN THE MODERN WORLD class!
“WE sense that ‘normal’ isn’t coming back, that we are being born into a new normal: a
new kind of society, a new relationship to the earth, a new experience of being human.”
— Charles Eisenstein
What has become ‘normal’? The lockdowns, quarantines, masks, washing hands,
taking vitamins and nutrient-rich food to nurture health, covering a coughing mouth and
sneezing nose, physical distancing, and the omnipresent support technology are a
“must” to continue for a lifetime. The temporary shift to distance schooling can be a
time for students , for both parents and teachers to reshape their understanding of what
learning can be and it does mean that learning is unstoppable.
In this course you will learn to deal with nature of mathematics, appreciation of its
practical, intellectual, and aesthetics dimension, and application of mathematical tools
in daily life. The course begins with an introduction to the nature of mathematics as an
exploration of patterns (in nature and the environment) and as an application of
inductive and deductive reasoning. By exploring the topics, you are encouraged to go
beyond the typical understanding of mathematics as merely a set of formulas but as a
source of aesthetics in patterns of nature, for example, and rich language in itself ( and
of science ) governed by logic and reasoning.
The course then proceeds to survey ways in which mathematics provides a tool for
understanding and dealing with various aspects of present day living, such as managing
personal finances, making social choices, appreciating geometric designs,
understanding codes used in data transmission and security, and dividing limited
resources fairly. These aspects will provide opportunities for actually doing mathematics
in a broad range of exercises that bring out the various dimensions of mathematics as a
way of knowing, and test the students’ understanding and capacity.
At the end of the term, you are expected to:

Knowledge
• Discuss and argue about the nature of mathematics, what it is, how it is
expressed, represented, and used.
• Use different types of reasoning to justify statements and arguments
made about mathematics and mathematical concepts.
• Discuss the language and symbols of mathematics.
Skills
• Use a variety of statistical tools to process and manage numerical data;
• Analyze codes and coding schemes used for identification, privacy, and
security purposes;
• Use mathematics in other areas such as finance, voting, health and
medicine, business, environment, arts and design, and recreation.
Values
• Appreciate the nature and uses of mathematics in everyday life;
• Affirm honesty and integrity in the application of mathematics to various
human endeavors.
This module that you will use in this course has the 5E format and based from the
course syllabus. It is your responsibility to ensure that you will accomplish all required
readings and activities to help you demonstrate the above cited course learning
outcomes.

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Table of Contents
• Module 1 : Nature of Mathematics and Speaking Mathematically

Unit 1: Nature of Mathematics
a. Patterns and Numbers in Nature and the World……………..………..4
b. Fibonacci Sequence………………………………………………….…….6
c. Symmetry……………………………………………………………………..8
d. Order of Rotation………………………………………………….….…...10
e. Packing problems…………………………………………………...……..11
Unit 2- Speaking Mathematically
a. Variables…………………………………………………………….…….…12
b. The Language of Sets………………………………………………….….13
c. The Language of Relations and Functions……………………………16
• Module 2: Problem Solving and Statistics
Unit 3: Problem Solving
a. Inductive and Deductive Reasoning………………………………….17

b. Problem Solving with Patterns…………………………………………..18
c. Problem-Solving Strategies………………………………………………19
Unit 4- Statistics
a. Measures of Central Tendency………………………………….………21

b. Measures of Dispersion……………………………………………………25
c. Measures of Relative Position……………………………………………27
• Module 3: Normal Distribution and Logic
Unit 4 : Normal Distribution
a. Linear Regression and Correlation………………………………………29
Unit 5: Logic
a. Logic Statements and Quantifiers……………………………………….31

b. Truth Tables, Equivalent Statements, and Tautologies………………32
• Module 4 : Mathematics of Finance

Unit 7: Mathematics of Finance
a. Simple Interest, Compound Interest, and Annuities………………..34

b. Practical Applications of Interest Theory……………………………..35

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Module 1
Course: Mathematics in the Modern World
Unit No. 1
Topic: Nature of Mathematics
Score:
Name:
Year & Section:
Date:
LESSON 1.1: PATTERNS AND NUMBERS IN NATURE AND THE WORLD
In this chapter, we will be looking at patterns and regularities in the world, and
how mathematics comes into play, both in nature and in human endeavors.
Learning Objectives:
At the end of the lesson, you should be able to:
1. Understand the meaning of pattern.
2. Learn to identify patterns in a given example.
3. Enumerate the different types of symmetry.
4. Appreciates the Fibonacci Sequence in the natural environment
5.Understands the essence of the variables
6. Learns how to identify sets
7. Appreciates Relation and Function in real life situations.
CONTENT
PATTERNS
❑ Regular, repeated, recurring forms or designs
❑ Repeats in a predictable manner
Example 1: What Comes Next?
Solution:
1. Looking at the lines, it seems to rotate at 90-degree intervals in a counter clockwise

direction, always parallel to one side of the square.
2. Checking the length of the lines inside the square it follows a decreasing trend
3. Finally, looking at the number of lines inside the box, each succeeding figure has the
number of lines increase by 1.

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Conclusion:
1. The next figure should have five lines inside.
2. Leads to option A as the correct choice.
Check your Progress:
What is the next figure in the given pattern?
A B
Example 2: What Number Comes Next?
What number comes next in 1,3,5,7,9,____?
Solution:
Looking at the given numbers, the sequence is increasing, with each term being two
more than the previous term: 3= 1 + 2; 5= 3+2; 7= 5 + 2; 9 = 7+2. Therefore, the next term
should be 11 = 9+2.
TRY THIS !
What number comes next in 1,4,9,16,25,___?
Final Thoughts:
Patterns indicate a sense of structure and organization that it seems only humans are
capable of producing these intricate, creative, and amazing formations. It is from this
perspective that some people see an “intelligent design” in the way that nature forms.

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LESSON 1.2: FIBONACCI SEQUENCE

As we have seen in the previous section, the human mind is hardwired to recognize
patterns. In mathematics, we can generate patterns by performing one or several
mathematical operations repeatedly. Suppose we choose the number 3 as the first
number in our pattern. We then choose to add 5 to our first number, resulting in 8, which
is our second number. Repeating this process, we obtain 13, 18, 23, 28,… as the
succeeding numbers that form our pattern. In mathematics, we call these ordered lists of
numbers a sequence.
1. define and identify fibonacci sequence;
2. solve for the next term in the sequence:
3. relate the Fibonacci sequence in real life.
SEQUENCE
❑ Is an ordered list of numbers, called terms, that may have repeated values. The
arrangement of this set is through a definite rule.
Example 1: Generating a Sequence
Analyze the given sequence for its rule and identify the next three terms.
a. 1,10,100,1000…
b. 2,5,9,14,20…
Solution:
a. Looking at the set of numbers, it can be observed that each term is in the power
of 10:
1= 100, 10 = 101, 100 = 102, and 1,000 = 103. Following this rule, the next three terms
are: 104 = 10,000, 105 = 100,000, and 106= 1,000,000
b. The difference between the first and second term (2 and 5) is 3. The difference
between the second and third terms (5 and 9) is 4. The difference between the
third and fourth terms (9 and 14) is 5. The difference between the fourth and the
fifth terms is 6. Following this rule, it can be deduced that to obtain the next three
terms, we should add 7,8,9, respectively, to the current term. Hence, the next
three terms are 20 + 7 = 27, 27 + 8 = 35, 35 + 9 = 44.
TRY THIS ! 1
Analyze the given sequence for its rule and identify the next three terms.
a. 16, 32, 64, 128

b. 1,1,2,3,5,8
The sequence in Check Your Progress Item B is a special sequence called the
Fibonacci Sequence.

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FIBONACCI SEQUENCE
❑ Named after the Italian Mathematician Leonardo Pisano or Leonardo of Pisa,
who was better known by his nickname Fibonacci
❑ Discovered through the observation of how rabbits bred and reproduced. The
problem involved having a single pair of rabbits and then finding out how many
pairs of rabbits will be born in a year, with the assumption that a new pair of
rabbits is born each month and this new pair, in turn, gives birth to additional pairs
of rabbits beginning at two months after they were born. He noted that the set of
numbers generated from this problem could be extended by getting the sum of
the two previous terms.
❑ Each number is the sum of two numbers
Starting with 0 and 1, the succeeding terms in a sequence can be generated by

adding the two numbers that came before the term:
0+1=1 0, 1, 1
1+1=2 0, 1, 1, 2
1+2=3 0, 1, 1, 2, 3
2+3=5 0, 1, 1, 2, 3, 5
3+5=8 0, 1, 1, 2, 3, 5, 8
5 + 8 = 13 0, 1, 1, 2, 3, 5, 8, 13
… 0, 1, 1, 2, 3, 5, 8, 13,…
Final thoughts:
While the sequence is widely known as Fibonacci sequence, this pattern is said to
have been discovered much earlier in India. According to some scholarly articles,
Fibonacci sequence is evident in the number of variations of a particular category of
Sanskrit and Prakrit poetry meters. In poetry, meter refers to the rhythmic pattern of
syllables.
Fibonacci sequence has many interesting properties. Among these is that this
pattern is very visible in nature. Some of nature’s most beautiful patterns, like the spiral
arrangement of sunflower seeds, the number of petals in a flower, and the shape of a
snail’s shell. It is also interesting to note that the ratios of successive Fibonacci numbers
approach the number Φ (Phi), also known as the Golden Ratio. This approximately equal
to 1.618

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LESSON 1.3: SYMMETRY

1. define symmetry;
2. identify the different types of symmetry;
3. give examples of symmetry that exist in nature and arts;
4. appreciates symmetry in nature and the world
SYMMETRY
❑ Greek word “measure together”
❑ One shape becomes exactly like another when you move it in some ways, turn,
flip or slide.
It indicates that you can draw an imaginary line across an object and the resulting parts
are mirror images of each other.
Examples:
Human face
Wings of a butterfly
Vitruvian Man
▪ Leonardo da Vinci
▪ Shows proportions and symmetry of the human body

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DIFFERENT TYPES OF SYMMETRY
1. BILATERAL SYMMETRY
❑ “bi” means two
2. THREE-FOLD SYMMETRY
❑ 3 sides or faces
Spiderwort
3. FIVE-FOLD SYMMETRY
Starfish
4. SIX-FOLD SYMMETRY
Snowflakes

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ROTATIONAL SYMMETRY
The appearance of the image is still the same even if you rotates it in several
degrees.
ANGLE OF ROTATION
Smallest angle that a figure can be rotated while preserving the original
formation
FORMULA:
𝟑𝟔𝟎°
𝑨𝒓 = 𝒏
𝑨𝒓 = angle of rotation
n= number of sides or faces
GOLDEN RATIO
❑ Phi Φ
❑ Ratio between two numbers
❑ Geometrically, it can also be visualized as a rectangle perfectly
formed by a square and another rectangle, which can be repeated
infinitely inside each section.
❑ approximately 1.618 (≈1.618)
1/1 = 1.0000 13/8 = 1.6250
2/1 = 2.0000 21/13 = 1.6154
3/2 = 1.5000 34/21 = 1.6190
5/3 = 1.6667 55/34 = 1.6177
8/5 = 1.600 89/55 = 1.6182
EXAMPLES:
Galaxy Parthenon
Sunflower seeds
Snail’s shell Mona Lisa

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Final Thoughts:
The Golden Ratio can also be expressed as the ratio between two numbers, if
the latter is also the ratio between the sum and the larger of the two numbers.
Geometrically, it can also be visualized as a rectangle perfectly
formed by a square and another rectangle, which can be repeated infinitely inside
each section.
Shapes and figures that bear this proportion are generally considered to be
aesthetically pleasing. As such, this ratio is visible in many works of art and architecture
such as in the Mona Lisa, the Notre Dame Cathedral, and the Parthenon. In fact, the
human DNA molecule also contains Fibonacci numbers, being 34 angstroms long by 21
angstroms wide for each full cycle of the double helix spiral. This approximates the
Golden Ratio at a value of about 1.619 ( 1 angstrom = 10 -10 meter or 0.1 nanometer).
ASSESSMENT:
Direction: Determine what comes next in the given pattern.
1.) A, C, E, G, I, ____________
2.) 15, 10, 14, 10, 13, 10 ______
3.) 3, 6, 12, 24, 48, 96, ________
4.) 27, 30, 33, 36, 39, _________
5.) 41, 39, 37, 35, 33, _________
DIRECTION: Analyse the given sequence for its rule and identify the next three
terms.
6.) 1, 10, 100, 1000, ______,______,______
7.) 2, 5, 9, 14, 20 _______,_______,_______
IDENTIFY:
8.) He is an Italian mathematician who is responsible for the discovery of a

special kind of sequence through his observation on how rabbits bred and
multiplied.
9.) It is a regular, repeated, or recurring forms or design.
10.) One rotation is equivalent to ____ degrees.
11.) What completes the following pattern?
CSD, ETF, GUH, _____, KWL
12.) What number should come next in this sequence?
22, 21, 25, 24, 28, 27, ______
13.) What number comes next in 1, 8, 27, 64, 125, _______?
14.) What letters comes next in this pattern?
A, SS, TT, E SS, TT, I, SS, _____, _____, ______

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UNIT 2. SPEAKING MATHEMATICALLY

This chapter will introduce you to some of the special language that is a
foundation for much mathematical thought, the language of variables, sets,
relations, and functions. Think of the chapter like the exercises you would do
before an important sporting event. Its goal is to warm up your mental muscles so
that you can do your best.
1. define variables;
2. differentiate variables from letters;
3.solve for the value of the variables;
LESSON 2.1: VARIABLES
❑ Is a symbol, usually letters, which represents a value or a number

❑ Mathematical “John Doe”- fictitious name used in legal documents for an
unknown male person
❑ Representation of an unknown
EXAMPLE:
In an equation 2x + 4y =10 if x=4 what will be the value of y?
x2 + 7x + 10 = 0 what are the value of x?
SOLUTION:
In the first equation you need to use the substitution method. Substitute
the value of x in the equation.
x=4
2(4) + 4y = 10
8 + 4y = 10 [then combine like terms]
4y = 10 -8
4y = 2 [divide both terms by 4 to eliminate the constant in the variable]
1
y=2
In the second equation you need to use the factoring method.
x2 + 7x + 10 =0 [find two numbers that when you multiply will equal to 10 and
when you add will equal to 7]
(x + 2) (x + 5) [ now that you have the factors. Equate both terms to zero and
simplify]
x+2=0 x+5=0
x = -2 x = -5
therefore, the values of x are -2 and -5

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LESSON 2.2: THE LANGUAGE OF SETS
1. understands sets;
2. name a set;
3. know the different methods in writing a set;
4. identify the different types of sets;
5. give a real life examples of sets.
SETS
❑ Collections of elements ( related and well-defined)

❑ George Cantor ( German Mathematician)
NAMING OF SETS
Always a capital letter
A = { 1, 2, 3 } [ order is not important]
In a set of vowels
V = { a,e,i,o,u}
lower case
EXAMPLE:
Write down all the letters in the word “rainbow”
A = { r,a,i,n,b,o,w}
Write down all the letters in the word “door”
B = {d,o,r} [ remember no elements shall be repeated]
METHOD IN WRITING A SET
1. Listing Method
Example: Set of first three counting numbers
{ 1, 2, 3}
2. Set-Builder Method
{x : x < 4, x € c } [ read as : set of all x such as x is less than 4 and x belongs to

counting numbers]

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3. Venn Diagram
1 2
TYPES OF SETS
1. Singleton set – only one element
Example : Set of even prime number
{2}
2. Null set / Empty set – valid set with no element
- symbols are {} , Ø
Example: Set of numbers which when multiplied by any number results to

zero
{} or Ø [ correct answer]
{0} [ not a null set, it is a singleton set]
{Ø} [also not a null set]
3. Infinite sets – no ending
- represented by 3 dots (…)
Example: Set of integers
{ … -4, -3, -2, -1, 0, 1, 2, 3,4 …}
4. Finite sets – with ending, exact
Example: Set of months in a year
{ jan, feb, march, april,may,june,july,aug, sept, oct, nov, dec}
5. Equivalent sets- same number of elements
A= { 1, 2, 3} B= { 3, 4, 5, 1}
C= {6, 7, 8} D = { 2, 5, 6, 9}
N(A) = 3
N(B) = 4
NI = 3
N(D) = 4
Set A is equivalent to set C
Set B is equivalent to set D

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6. Equal set – precisely the same elements
E = { 1, 2 ,3 } D= { 3,1,2}
E=D
G = { 9,8,7,6} H= { 6,8,7,9}
G=H
Remember: Equal always Equivalent
Equivalent may or may not Equal
7. Subset- part of the original set
Example: Set of Ice cream flavors, chocolates, butterscotch, vanilla
I = { c, b, v} [ original set]
Subsets:
None only one two three
{} {c} {c,b} {c,b,v}
{b} {c, v}
{v} {b,v}
8. Universal set- contains everything (suitable)
Example:
A = {x:x is the students of BSBASS 1A}
B = { x:x is the students of BSBA 1B}
C = {x:x is the students of first year BSBA}
ASSESSMENT
1. In an equation 5x + 8y =26 if x=2 what will be the value of y?

2. Find the value of y if the equation is 10x+4y=80 , and x=6.
3. In an equation 3x + 2y =21 if x=3 what will be the value of y?
4. x2 + 10x + 25 = 0 what are the value of x? Use factoring method.
5. x2 + 4x - 12 = 0 what are the value of x? Use factoring method.
6. x2 - 4x - 21 = 0 what are the value of x? Use factoring method.
7. x2 - 14x +45 = 0 what are the value of x? Use factoring method.

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LESSON 2.3: THE LANGUAGE OF FUNCTIONS AND RELATIONS

FUNCTION- is paired exactly with one y
Example: ( 1,6) (2,5) (6,5) (0,8) [Function]
( 5,6) ( 7,2) (5,8) (6,3) [ Not a Function]
x y
X
Y
-3
0
-9
5
5
10
12
Function
Look at the x-values, if they are all different, it is a function
RELATION- connection between any two items
( 2, 1)
Domain Range
Relation can be represented in multiple ways x y
x y 2 1
2 1
4 2
4 2
8 4
Table Graphing Mapping Diagram

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Module 2
Unit No. 3
Topic: PROBLEM SOLVING and STATISTICS
Score:
Name:
Year & Section:
Date:
Most occupations require good problem-solving skills. For instance, architects and
engineers must solve many complicated problems as they design and construct modern
buildings that are aesthetically pleasing, functional, and that meet stringent safety
requirements. Two goals of this chapter are to help you become a better problem solver
and to demonstrate that problem solving can be an enjoyable experience.
1. differentiate inductive from deductive reasoning;
2. came up with conclusions in every problem;
3. relate to real life situations.
LESSON 3.1: INDUCTIVE AND DEDUCTIVE REASONING
INDUCTIVE REASONING
❑ Specific to general (conclusion)
❑ General (conclusion) is called “conjecture” – means it may or may not be true.
Examples:
❑ 3,6,9,12,15, ____?
Conjecture: 18, true because the series of numbers are multiples of 3
❑ Symptoms of a female students: Pale, Vomits, Big belly, sudden change in body
size.
Conjecture: Pregnant, may or may not be true because there is possibility that the
female stud is malnourish and having an ulcer attack
COUNTER EXAMPLES
❑ one case for which a statement is not true
❑ we can verify that each statement is a false statement by finding a counterexample
for each
Examples:
a. I x I > 0 b. x2 > x c. √𝑥 2 = x
x=0 x=1 x = -3
𝑥 𝑥+3
a. = 1 b. =x+1 c. √𝑥 2 + 16 =x+4
𝑥 3
x=0 x =5 x=2

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DEDUCTIVE REASONING
❑ General to specific ( arguments)
❑ Conclusions are logically true, realistically true, always correct
Example:
Prove that QUAD is a parallelogram
U A
Q D
Conclusion:
QD II VA
QU II DA
PROBLEM SOLVING WITH PATTERNS

SEQUENCE- series of numbers that follows an order or a pattern
Example: 2,4,6,8,10,12,14…
Terms
a 1= 2 a5= 10
a 2= 4 a6=12
a 3= 6 a8= 14
a 4= 8 an= nth term of a sequence
What is the next term?
What is the formula/ rule can be used to generate terms?
We construct a “difference table”- shows differences between successive terms
Examples:
Sequence: 2, 4, 6, 8, 10
First differences: 2 2 2 2

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Find the next term using the difference table:
Sequence: 5, 14, 27, 44, 65, 90
First differences: 9 13 17 21 25
Second differences: 4 4 4 4
How did we get 90 as the next term?
Simply add the last term of the first differences and the last term of the given
sequence, the sum will be the next term of the sequence [ 25+ 65 = 90]
FORMULA:
an= a1 + ( n-1) d
a1= first term n= number of terms
d= common difference
EXAMPLES:
3, 8, 13, 18, 23, 28…
Find: a10
a202
Solutions:
an= a1 + ( n-1) d an= a1 + ( n-1) d
a10= 3 + (10-1) 5 a202= 3 + (202-1)5
= 3 + (9) 5 = 3 + (201)5
a10= 48 a202= 1008

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Examples:
a48=145
a14= 43
Find: a1= ?
d= ?
[a1+ 7d = 25]-1 - a1 - 7d = -25
a1+ 13d = 43 a1 + 13d = 43
6d = 18
6 6
d=3
a48=25
an= a1 + ( n-1) d
25= a1 + ( 8-1) 3
- a1 = 21-25
[-a1 = -4] -1
a 1= 4
a14= 43
an= a1 + ( n-1) d
43 = a1 + (14-1) d
43 = a1 + (13) 3
43 = a1 + 39
- a1 = 39-43
[-a1= -4] -1
a 1= 4

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Unit 4 – Statistics
Statistics involves the collection, organization, summarization, presentation, and
interpretation of data. The branch of statistics that involves the collection ,
organization, summarization, and presentation of data called descriptive
statistics. The branch that interprets and draws conclusions from the data is
called inferential statistics.
1. Learns the measures of central tendency.
2. Appreciates the Measures of Dispersion.
3. Solve for the Measures of relative Position
4.1 Measures of Central Tendency
The Arithmetic Mean
One of the most basic statistical concepts involves finding measures of central tendency
of a set of numerical data. It is often helpful to find numerical values that locate, in some
sense, the center of a set data.
The arithmetic mean is the most commonly used measure of central tendency. The
arithmetic of a set of numbers is often referred to as simply the mean. To find the mean
for a set of data, find the sum of the data values and divide by the number of data
values. The traditional symbol used to indicate a summation is the Greek letter sigma, ∑.
Thus the notation ∑x, called summation notation, denotes the sum of all the numbers in a
given set. We can define the mean using summation notation.
Mean
The mean of n numbers is the sum of the numbers divided by n.
∑𝑥
Mean=
𝑛
Statisticians often collect data from small portions of a large group in order to determine
information about the group. In such, situations the entire group under consideration is
known as the population , and any subset of the population is called a sample. It is
traditional to denote the mean of a sample by x ( which is read as “x bar”) and to
denote the mean of a population by the Greek letter µ (lowercase mu).
Example ( Find a mean )
Six friends in a biology class of 20 students received test grades of
92, 84, 65, 76, 88, and 90
Find the mean of these test scores.
Solution
The 6 friends are a sample of the population of 20 students . Use x to represent

the mean.
∑𝑥 92+84+65+76+88+90 495
x = = = = 82.5
𝑛 6 6
The mean of these test scores is 82.5.

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TRY THIS ! 2
A doctor ordered 4 separate blood tests to measure a patient’s total blood cholesterol
levels. The test results were
245, 235, 220, and 220
Find the mean of the blood cholesterol levels.
The Median
Another type of average is the median. Essentially, the median is the middle number or
the mean of the two middle numbers in a list of numbers that have been arranged in
numerical order from smallest to largest or largest to smallest. Any list of numbers that is
arranged in numerical order from smallest to largest or largest to smallest is a ranked list.
Example (Find a Median)
Find the median of the data in the following lists.
a. 4 , 8, 1, 14, 9, 21, 12 b. 46, 23, 89, 77, 108

Solution
a. The list 4, 8, 1, 14, 9, 21, 12 contains 7 numbers. The median of a list with an odd
number of entries is found by ranking the numbers and finding the middle
number. Ranking the numbers from smallest to largest gives
1, 4, 8, 9, 12, 14, 21
The middle number is 9. Thus, 9 is the median.

b. The list 46, 23, 92, 89, 77, 108 contains 6 numbers. The median of a list of data with
an even number of entries is found by ranking the numbers and computing the
mean of the middle numbers. Ranking the numbers from smallest to largest gives
23, 46, 77, 89, 92, 108
The two middle numbers are 77 and 89 . The mean of 77 and 89 is 83. Thus, 83
is the median of the data.
TRY THIS ! 3
Find the median of the data in the following lists.
a. 14, 27, 3, 82, 64, 34, 8, 51

b. 21, 3, 37, 4, 11.6 , 82.5 , 17. 2
The Mode
A third type of average is the mode. The mode of a list of numbers is the number that
occurs most frequently. Sometimes , a list of numerical data can have more than one
mode.
Example (Find a Mode)
Find the mode of the data in the following lists.
a. 18, 15, 21, 16, 15, 14, 15, 21 b. 2, 5, 8, 9, 11, 4, 7, 23

Solution
a. In the list 18, 15, 21, 16, 15, 14, 15, 21, the number 15 occurs more often than the
other numbers. Thus 15 is the mode.

P a g e | 22
b. Each number in the list 2, 5, 8, 9, 11, 4, 7, 23 occurs only once. Because no

number occurs more often than the others , there is no mode.
TRY THIS! 4
Find the mode of the data in the following lists.
a. 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 8 b. 12, 34, 12, 71, 48, 93, 71
The Weighted Mean
A value called the weighted mean is often used when some data values are more
important than other.
Find a Weighted Mean
Course Course Course The able shows Dillon’s fall semester course grades. Use
Grade units the weighted mean formula to find Dillon’s GPA for the
English B 4 fall semester.
History A 3 Where : A= 4 , B= 3, C= 2, D= 1, F= 0
Chemistry D 3
Algebra C 4 Solution
The B is worth 3 points, with a weight of 4; the A is worth 4 points with a weight
of 3; the D worth 1 point, with a weight of 3; and the C is worth 2 points , with a weight of
4. The sum of all the weights is 4+3+3+4, or 14.
(3𝑥4)+(4𝑥3)+(1𝑥3)+(2𝑥4)
Weighted mean =
14
35
= 14 = 2.5
Dillon’s GPA for the fall semester is 2.5

TRY THIS! 5
Course Course Course
The table shows Janet’s spring semester course
Grade units
Biology A 4 grades. Use the weighted mean formula to find
Statistics B 3
Janet’s GPA for the spring semester. Round to the
Business C 3
Psychology D 2 nearest hundredth.
CAD F 2
Data that have not been organized or manipulated in any manner are called raw data .
A large collection of raw data may not provide much readily observable information.

P a g e | 23
A frequency distribution , which is a table that lists observed events and the frequency
occurrence of each observed event, is often used to organize raw data. For instance,
consider the following table , which lists the number of laptop computers owned by
families in each of 40 homes in a subdivision.
Table A. Number of Laptop Computers per Household
2 0 3 1 2 1 0 4
2 1 1 7 2 0 1 1
0 2 2 1 3 2 2 1
1 4 2 5 2 3 1 2
2 1 2 1 5 0 2 5
The frequency distribution in the Table A on the next table was constructed using the
data from Table A. The first column of the frequency distribution consists of the numbers
0, 1, 2, 3, 4, 5, 6 and 7. The corresponding frequency of occurrence , f, of each of the
numbers in the first column is listed in the second column.
Table B. A frequency Distribution for Table A.
Observed event Frequency

Number of laptop Number of households, f,
computers , x with x laptop computers
0 5
1 12
2 14 This row indicates that there are 14
3 3 households with 2 laptop computers.
4 2
5 3
6 0
7 1_
40 total
The formula for a weighted mean can be used to find the mean of the data in a
frequency distribution. The only change is that the weights 𝑤1 , 𝑤2 , 𝑤3 , .. 𝑤𝑛 are replaced
with the frequencies 𝑓1 , 𝑓2 , 𝑓3 , . . . , 𝑓𝑛
Example. Find the Mean of Data Displayed in a Frequency Distribution
Find the mean of the data in Table B.
Solution
The numbers in the right-hand column of Table B are the frequencies f for the numbers in
the first column. The sum of all the frequencies is 40.
∑(𝑥.𝑓)
Mean = ∑𝑓
(0.5)+(1.12)+(2.14)+(3.3)+(4.2)+(5.3)+(6.0)+(7.1 )
=
40
79
= 40
= 1. 975 ( The mean number of laptop computers per household for the home in the
subdivision is 1. 975.
TRY THIS ! 6
A housing division consists of 45 homes. The following frequency distribution shows the
number of homes in the subdivision that are two-bedroom homes, the number that are
three-bedroom homes, the number that are four-bedroom homes, and the number that
are five-bedroom homes. Find the mean number of bedrooms for the 45 homes.

P a g e | 24
Observed event Frequency

Number of bedrooms , x Number of homes with x
bedrooms
2 5
3 25
4 10
5 5
45 total
4.2 Measures of Dispersion

The Range
The range of a set of data values is the difference between the greatest data value and
the least data value .
Example:
Find the range:
a) 150, 250, 825, 400, 18, 500
b) 2.2, 1.8, 5.1, 0.3
Solution:
a) The largest value is 825. The smallest value is 18
Range = largest value − smallest value = 825 − 18 = 807
b) The largest value is 5.1. The smallest value is 0.3
Range = largest value − smallest value = 5.1 − 0.3 = 4.8
The Standard Deviation
A measure of dispersion that is less sensitive to extreme values is the standard deviation.
The standard deviation of a set of numerical data makes use of the amount by which
each individual data value deviates from the mean. These deviations, represented by
( x – x), are positive when the data value x is greater than the mean x and are negative
when x is less than the mean x . The sum of all the deviations ( x – x )is 0 for all sets of
data. This is shown in Table D for the Machine 2 data of Table C .
Because the sum of all the deviations of the data values from the mean is always 0, we
cannot use the sum of the deviations as a measure of dispersion for a set of data.
Instead, the standard deviation uses the sum of the squares of the deviations.
Table C Table D
Machine 1 Machine 2 x x- x
9.52 8.01 8.01 8.01- 8 =

6.41 7.99 7.99 7.99 – 8 = -0.01
10.07 7.95
7.95 7.95- 8= -0.05
5.85 8.03
8.03 8.03- 8= 0.03
8.15 8.02
8. 02 8.02- 8= 0.02
x = 8.0 x = 8.0
Sum of deviations = 0

P a g e | 25
Standard Deviations for Populations and Samples

If x1, x2 , x3 , …xn is a population of n numbers with a mean of 𝜇 , then the
√Σ (𝑥−𝜇)²
standard deviation of the population is 𝜎 = (1)
𝑛
If x1, x2 , x3 , …xn is a sample of n numbers with a mean of x , then the
√Σ (𝑥−𝑥)²
standard deviation of the sample is 𝑠= (2).
𝑛−1
Take Note
You may question why a denominator of n- 1 is used instead of n when we compute a

sample standard deviation. The reason is that a sample standard deviation is often used
to estimate the population standard deviation, and it can be shown mathematically that
the use of n-1 tends to yield better estimates.
Procedure for Computing a Standard Deviation
8. Determine the mean of the n numbers.

9. For each number, calculate the deviation(difference) between
the number and the mean of the numbers.
10. Calculate the square of each deviation and find the sum of these
squared deviation.
11. If the data is population , then divide the sum by n. If the data is a
sample , then divide the sum by n – 1.
12. Find the square root of the quotient in Step 4.
Example
The following numbers were obtained by sampling a population.

2, 4, 7, 12, 15
Find the standard deviation of the sample.
Solution
Step 1: The mean of the numbers is
2+4+7+12+15 40
X= = =8
5 5
Step 2: For each number, calculate the deviation between the number and the
mean.
x x-x
2 2-8= -6
4 4-8=-4
7 7-8= -1
12 12- 8= 4
15 15- 8= 7
Step 3: Calculate the square of each deviation in Step 2, and find the sum of these
squared deviations.

P a g e | 26
x x-x (x- x) ²
2 2 - 8= -6 (-6)² = 36
4 4 - 8=-4 (-4)²= 16
7 7- 8= -1 (-1)²= 1
12 12 - 8= 4 4² = 16
15 15 - 8= 7 7²= 49
Sum of the squared deviations
118
Step 4: Because we have a sample of n = 5 values, divide the sum 118 by n – 1,
which is 4.
118
= 29.5
4
Step 5: The nearest standard deviation of the sample is s= √29.5. To the nearest
hundredth. The standard deviation is s = 5.43.
TRY THIS ! 7
A student has the following quiz scores: 5, 8, 16, 17, 18 , 20. Find the standard
deviation for this population of quiz scores.
4.3 Measures of Relative Position

Percentiles
Most standardized examinations provide scores in terms of percentiles , which
are defined as follows:
pth Percentile
A value x is called the pth percentile of a data set provided p% of the data
values are less than x.
Example
In a recent year, the median annual salary for a physical therapist was Php.
74,480. If the 90th percentile for the annual salary of a physical therapist was Php.
105, 900 , find the percent of physical therapists whose annual salary was
a. More than 74, 480

b. Less than 105, 900
c. Between 74, 480 and 105,900.
Solution
a. By definition, the median is the 50th percentile. Therefore, 50% of the

physical therapists earned more than 74, 480 per year.
b. Because, 105, 900 is the 90th percentile , 90% of all physical
therapists made less than 105,900
c. From parts a and b , 90% - 50% =40 % of the physical therapists
earned between 74, 480 and 105, 900 .
Percentile for a Given Data Value

Given a set of data and a data value x,
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 𝑥
Percentile of score x=
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠
. 100

P a g e | 27
Example
On a reading examination given to 900 students, Elaine’s score of 602 was
higher than the scores of 576 of the students who took the examination. What is
the percentile for Elaine’s score?
Solution
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 602
Percentile = . 100
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠
576
= .100
900
= 64
Elaine’s score of 602 places her at the 64 th percentile.
TRY THIS ! 8
On an examination given to 8600 students, Hal’s score of 405 was higher than the scores
of 3952 of the students who took the examination. What is the percentile for Hal’s score ?
QUARTILES
The three numbers Q1, Q2, and Q3 that partition a ranked data set into four
(approximately) equal groups are called the quartiles of the data. For instance, for the
data set below, the values Q1= 11 , Q2= 29, Q3= 104 are the quartiles of the data.
2, 5, 5, 8, 11, 12, 19, 22 , 23 , 29, 31 , 45, 83 , 91, 104, 159, 181 , 312, 354
Q1 Q2 Q3
The quartile Q1 is called the first quartile . The quartile Q2 is called the second quartile. It is
the median of the data. The quartile Q 3 is called the third quartile . The following method
of finding quartiles makes use of medians.
The Median Procedure for Finding Quartiles

1. Rank the data.
2. Find the median of the data. This is the second quartile, Q2.
3. The first quartile , Q1, is the median of the data values less than Q2. The third
quartile , Q3, is the median of the data values greater than Q2.
Example
Step 1: Rank the data as shown in the following table.
1) 26 2) 32 3) 36 4) 36 5) 37 6) 39 7) 39 8) 40 9) 40
10) 41 11) 42 12) 42 13) 43 14) 45 15) 45 16) 48 17) 48 18) 49
19) 50 20) 53 21) 53 22) 56 23) 58 24) 62 25) 73
Step 2: The median of these 25 data values has a rank of 13. Thus, the median is 43. The
second quartile Q2 is the median of the data, so Q2 =43.
Step 3: There are 12 data values less than the median and 12 data values greater than
the median. The first quartile is the median of the data value less than the median. Thus
Q1 is the mean of the data values with ranks of 6 and 7.
39+39
Q1 = = 39
2
The third quartile is the median of the data values greater than the median. Thus Q3 is the
mean of the data values with ranks of 19 and 20 .
50+53
Q3 = = 51. 5
2

P a g e | 28
TRY THIS ! 9
The following table lists the weights , in ounces, of 15 avocados in a random sample. Find
the quartiles for the data.
Weights, in ounces, of Avocados
12.4 10.8 14.2 7.5 10.2 11.4 12.6 12.8 13.1

15.6 9.8 11.4 12.2 16.4 14.5
4.4 Normal Distributions

Frequency Distributions
Large sets of data are often displayed using a grouped frequency distribution or
histogram.
Table E : A Grouped Frequency Distribution with 12 classess
Download time (in seconds) Number of subscribers

0-5 6
5-10 17
10-15 43
15-20 92
20-25 151
25-30 190
35-40 149
40-45 90
45-50 45
50-55 15
55-60 10
Table E is called grouped frequency distribution. It shows how often (frequently) certain
events occurred. Each interval, 0-5, 5-10, and so on, is called a class. This distribution has
12 classes. For the 10-15 class, 10 is the lower class boundary and 15 is the upper class
boundary. Any data value that lies on a common boundary is assigned to the higher
class. The graph of a frequency distribution is called a histogram. A histogram provides a
pictorial view of how the data are distributed.
The type of frequency distribution that lists the percent of data in each class is called a
relative frequency distribution.
TABLE F
Download time (in seconds) Percent of subscribers

0-5 0.6
5-10 1.7
10-15 4.3
15-20 9.2
20-25 15.1
25-30 19.0
35-40 14.9
40-45 9.0
45-50 4.5
50-55 1.5
55-60 1.0
Example
Use the relative frequency distribution in Table F to determine the
a. Percent of subscribers who required at least 25 s to download the file.

b. Probability that a subscriber chosen at random will require at least 5 s but less
than 20 s to download the file.

P a g e | 29
Solution
a. The percent of data in all the classes with a lower boundary of 25 s or more is the
sum of the percent printed in gray in Table G below. Thus the percent of
subscribers who required at least 25 s to download the file is 69. 1 %
Table G
Download time (in seconds) Percent of subscribers
0-5 0.6
5-10 1.7 Sum is 15.2%
10-15 4.3
15-20 9.2
20-25 15.1
25-30 19.2
30-35 19.0
35-40 14.9 Sum is 69.1%
40-45 9.0
45-50 4.5
50-55 1.5
55-60 1.0
b. the percent of data in all the classes with a lower boundary of 5 s and an upper
boundary of 20 s is the sum of the percent printed in orange in Table G above. Thus the
percent of subscribers who required at least 5 s but less than 20 s to download the file is
15.2%. The probability that a subscriber chosen at random will require at least 5s but less
than 20 s to download the file is 0.152.
TRY THIS ! 10
Use the relative frequency distribution in Table F to determine the
a. Percent of subscribers who required less than 25 s to download the file.

b. Probability that a subscriber chosen at random will require at least 10 s but less
than 30 s to download the file.

P a g e | 30
Module 3
Unit No. 4
Topic: Logic Score:
Name:
Year & Section:
Date:
One of the first mathematicians to make a serious study of symbolic logic was Gottfried
Wilhelm Leibniz (1646-1716). Leibniz tried to advance the study of logic from a merely
philosophical subject to a formal mathematical subject. Augustus De Morgan (1806-
1871) and George Boole (1815-1864), contributed to the advancement of symbolic logic
as a mathematical discipline.
1. Understands everything about logic.
2. Formulate Logical reasoning
3. Appreciates the importance of Logic in real life situation
Logic Statements
Every language contains different types of sentences, such as statements, questions, and
commands. For instance,
“Is the test today?” is a question.
“Go get the newspaper” is a command.
“This is a nice car” is an opinion.
“Denver is the capital of Colorado” is a statement of fact.
The symbolic logic that Boole was instrumental in creating applies only to sentences that
are statements as defined below.
A Statement
A statement is a declarative sentence that is either true or false , but not both true and
false .
Example : Identify Statements
Determine whether each sentence is a statement.
a. Florida is state in the United States.

b. How are you?
c. 9⁹ + 2 is a prime number.
d. X+1 = 5.
Solution
a. Florida is one of the 50 states in the United States, so this sentence is true and it is a
statement.
b. The sentence “ How are you?” is a question; it is not a declarative sentence. Thus
it is not a statement.
c. You may not know whether 9⁹+ is a prime number; however, you do know that it is
a whole number larger than 1, so it is either a prime number or it is not a prime
number. The sentence is either true or it is false, and it is not both true and false, so
it is a statement.

P a g e | 31
d. X + 1 = 5 is a statement. It is known as an open statement . It is true for x = 4, and it

is false for any other values of x. For any given value of x, it is true or false but not
both.
TRY THIS ! 11
Determine whether each sentence is a statement.
1. Open the door.

2. 7055 is a large number.
3. In the year 2024, the president of the United States will be a woman.
4. X > 3.
Truth Value and True Tables
The truth value of a simple statement is either true (T) or false (F).
The truth value of a compound statement depends on the truth values of its simple
statements and its connectives.
A truth table is a table that shows the truth value of a compound statement for all
possible truth values of its simple statements.
Truth Table for ~p p ~p

T F
F T
Example 2: Write the Negation of a Statement
1. Ellie Goulding is an opera singer.

2. The dog does not need to be fed.
Solution
1. Ellie Goulding is not an opera singer.

2. The dog needs to be fed.
TRY THIS ! 12
a. The Queen Mary 2 is the world’s largest cruise ship.

b. The fire engine is not red.
Table H Logic Connectives and Symbols
Statement Connective Symbolic Form Type of Statement

not p Not ~p Negation
p and q And p ^q Conjunctions
p or q Or pvq Disjunctions
if p, then q If….. then p→q Conditional
p if and only if q If and only if p↔q Biconditional
Example: Write Compound Statements in Symbolic Form
Consider the following simple statements.
p: Today is Friday
q: It is raining
r: I am going to a movie.
S: I am not going to the basketball game.

P a g e | 32
Write the following compound statements in symbolic form.
1. Today is Friday and it is raining.

2. It is not raining and I am going to a movie.
3. I am going to the basketball game or I am going to a movie.
4. If it is raining, then I am not going to the basketball game.
Solution:
1. p ^q 2. ~q ^ r 3. ~s v r 4. q → s
TRY THIS 13 !
a. Today is not Friday and I am going to a movie.

b. I am going to the basketball game and I am not going to a movie.
c. I am going to the movie if and only if it is raining.
d. If today is Friday , then I am not going to a movie.

P a g e | 33
Module 4
Unit No. 5 Score:
Topic: Mathematics of Finance
Name:
Year & Section:
Date:
This module covers introductory topics in financial mathematics particularly interest
earned on loans and investments. The use of scientific calculators or Excel application
will be very helpful in working through the examples and the exercises given at the end
of every sections.
1. Learn how to solve simple interest, compound interest and annuities.
2. Calculate involving practical problems.
Simple Interest
The simple interest is given by I = Prt
Where P is the principal, r is the interest rate per year, and t is the number of years. If
the number of days used is 360 days in a year, the interest is called ordinary simple
interest. If it uses 365 days in a year or 366 days for a leap year then the interest is called
exact simple interest.
The future value or maturity value is the total amount paid including the interest. This
amount is obtained by the formula
F=P+I
= P + Prt
= P (1+rt)
Example
1. An employee borrows ₱ 60, 000 for 7 months at an interest rate of 12% per year.
Find the interest earned and the total amount he has to pay.
Solution
The interest earned is
7
I= Prt= (60,000)(0.12) ( )= ₱ 4,200.
12
The total amount to be paid is
7
F= P (1+rt) = 60, 000 (1 + 0.12) ( )) = ₱ 64, 200.
12
Example
2. The ordinary simple interest charged after 130 days on a loan of ₱ 10, 200
is ₱ 575. Find the interest rate.
Solution
Use 360 days per year in computing for the ordinary simple interest. Thus,
I= Prt
130
575= 10, 200 (r) ( )
360
r= .156 or 15.6 %

P a g e | 34
Compound Interest
The compound interest is computed based on the principal amount and the
total accumulated interest earned. The total accumulated amount on the
principal P for n periods at an interest rate of i per period is given by
F = P (1 + i ) n
𝑗
Where n= mt and i =𝑚
The interest rate j is called the nominal rate and m is the number of compounding
periods in a year. For example, if the interest rate is 6 % compounded quarterly for 2
years, then the total number of periods is n= mt = (4) (2) = 8 periods and the interest rate
𝑗 0.06
per quarter is i = = 4 = 0.015 or 1.5 %
𝑚
Example
1. A mother invested ₱100,000 in a mutual fund on the date her first son was
born. If the money is worth 10% compounded semi-annually , how much
will the son receive on his 18th birthday ?
Solution
The total number of periods, n= mt = (2)(18) = 36. The interest rate per period is
𝑗 0.1
i= = = 0.05 . Hence, the total amount the son should receive is
𝑚 2
F = P (1 + i ) n
= (100, 000) (1+ 0.05) 36
=₱ 579,182
Example
2. What is the amount that should be deposited today at 6.5% compounded

quarterly in order to withdraw ₱ 200, 000 and leave nothing in the fund at the end
of 5 years?
Solution
𝑗
F= P (1 + )mt
𝑚
𝐹
F= P 𝑗
(1+𝑚)mt
0.065 4(5)
200, 000= P (1 + )
4
P= ₱ 144, 883. 46

P a g e | 35
Example
3. How long will it take to triple the sum of money invested at 12 %

compounded monthly?
Solution
Given that F= 3P, then
0.12 n
F = 3P= P (1 + )
12
3= (1.01)n
1n (3) = n 1n (1.01)
1𝑛 (3)
n= = 110.41
1𝑛 (1.01)
Thus, the number of years is
𝑛 110.41
t= = = 9.2 yrs.
𝑚 12
Effective interest rate, ER, is equivalent to interest rate compounded annually . It

can be used to compare two rates with different compounding periods.
𝑗
ER = (1 + )m – 1
𝑚
Example
1. If one bank advertises its rate as 6.2 % compounded annually , and

another bank advertises its rate as 6.3 % compounded annually, which
rate is better for an investment?
Solution
For bank 1, the effective rate is

0.062 12
ER = (1 + ) – 1 = 0.064 or 6.4 %
12
For bank 2, the interest rate is 6.3 %.
Hence, bank 1 has a better interest rate. A higher rate is good for financial
investment.

P a g e | 36
An annuity is a series of equal payments made at regular time intervals. It is also

known as a sinking fund which is used for future financial obligations such as
educational expenses for children, replacement of machines or equipment,
amortization of loans or assets.
An ordinary annuity is one whose payments are made at the end of each
interest period. The present value P and the future value F of an ordinary annuity
A are given by
𝐴
P= (1 − (1 + 𝑖 )-n)
𝑖
𝐴
F= ((1 + 𝑖 )n – 1 )
𝑖
Where A is the payment made at the end of each successive period, i is the
interest rate per period , and n is the total number of periods.
Example
1. A house and lot amounting to ₱ 3.5M is to be amortized by monthly

payments over 5 years at an annual interest rate of 10%. What is the
monthly payment?
Solution
0.1
Here, P= 3,500,000, i= = 0.008333, and n= (12)(5)= 60 .
12
Using the formula for ordinary annuity, then
0.1
1−(1+ 12 )−60
3,500,000= A ( 0.1 ) = 𝐴 (47.0654)
12
3,500,000
A= = ₱ 74, 364.61
47.0654
ASSESSMENT
Solve each problem completely.
1. How much will be the future worth of money after 12 months if the
sum of ₱ 35, 000 is invested today at a simple interest rate of 3% per
month ?
2. A man expects to receive ₱ 125, 000 in eight years. How much is
that money worth now considering an interest rate of 12%
compounded quarterly ?
3. How long will it take the money to triple itself if invested at 9.5%
compounded semi-annually?
4. By the conditions stated in a will, the sum of ₱ 2.5 M is left to a son to
be held in a trust fund by his guardian until it amounts to ₱4.5M.
When will the son receive the money f the fund is invested at 10%
compounded quarterly?
5. What is the effective rate corresponding to 18% compounded daily
using 360 days in one year ?

P a g e | 37
6. Suppose an engineer purchases a home and secures a loan of ₱

2.5M from a commercial bank for 20 years at an annual interest rate
of 9% . Find the monthly amortization of the loan.
7. How much is the total amount paid over the life of the loan in
problem 6 ?
8. How much is the total amount of interest paid over 20 years in
problem 6 ?
REFERENCES
Adam, John A. A Mathematical Nature Walk, Aufman, R. et al. Mathematical Excursions
(Chaps. 1, 2, 3, 4, 5,8,11 and 13) 3rd Ed ( International Edition) COMAP Inc. For All
Practical Purposes, Introduction to Contemporary Mathematics (2 nd ed.), Fisher, Carol
Burns, The Language of Mathematics ( from One Mathematical Cat, Please! By Carol
Burn Fisher), Fisher Carol Burns, The Language and Grammar of Mathematics., Hersh,
R. What is Mathematics Really? (Chaps 4 and 5) , Johnson and Mowry. Mathematics, A
Practical Odyssey (Chap. 12). Moser and Chen. A Student’s Guide to Coding and
Information Theory. Stewart, Ian. Nature’s Numbers,. Vistro-Yu, C. Geometry: Shapes,
Patterns and Designs ( A chapter for the new ed. Of the Math) Jamson R.E (2000),
Learning the language of mathematics. Language and Learning across the Disciplines
4(1), 45-54. Akiyama and Ruiz. A Day’s Adventure in Math Wonderland, Aufman et al.
Mathematics Excursions (Chap. 2) Averbach and Chein. Problem Solving Through
Recreational Mathematics, Enzensberger. The Number Devil. Johnson and Mowry.
Mathematics, A Practical Odyssey (Chaps. 1 and 4) Nocheda, Palaspas, Stewart, Ian.
Professor Stewart’s Cabinet of Mathematical Curiosities. Sobecki, Bluman, and Schirck-
Matthews.Math in Our World.

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Module in Math in The Modern World - Revised

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Module in Math in The Modern World - Revised

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Romblon State University

San Fernando, Campus

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GE 4: MATHEMATICS IN THE MODERN WORLD

Dear future managers and entrepreneurs,

WELCOME to our MATHEMATICS IN THE MODERN WORLD class!

At the end of the term, you are expected to:

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• Module 1 : Nature of Mathematics and Speaking Mathematically

a. Inductive and Deductive Reasoning………………………………….17

a. Measures of Central Tendency………………………………….………21

a. Linear Regression and Correlation………………………………………29

a. Logic Statements and Quantifiers……………………………………….31

• Module 4 : Mathematics of Finance

a. Simple Interest, Compound Interest, and Annuities………………..34

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LESSON 1.1: PATTERNS AND NUMBERS IN NATURE AND THE WORLD

❑ Regular, repeated, recurring forms or designs

❑ Repeats in a predictable manner

Example 1: What Comes Next?

1. Looking at the lines, it seems to rotate at 90-degree intervals in a counter clockwise

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1. The next figure should have five lines inside.

2. Leads to option A as the correct choice.

Check your Progress:

What is the next figure in the given pattern?

Example 2: What Number Comes Next?

What number comes next in 1,3,5,7,9,____?

What number comes next in 1,4,9,16,25,___?

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LESSON 1.2: FIBONACCI SEQUENCE

Example 1: Generating a Sequence

a. 16, 32, 64, 128

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❑ Each number is the sum of two numbers

Starting with 0 and 1, the succeeding terms in a sequence can be generated by

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LESSON 1.3: SYMMETRY

❑ Greek word “measure together”

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DIFFERENT TYPES OF SYMMETRY

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n= number of sides or faces

1/1 = 1.0000 13/8 = 1.6250

2/1 = 2.0000 21/13 = 1.6154

3/2 = 1.5000 34/21 = 1.6190

5/3 = 1.6667 55/34 = 1.6177

8/5 = 1.600 89/55 = 1.6182

Snail’s shell Mona Lisa

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Direction: Determine what comes next in the given pattern.

2.) 15, 10, 14, 10, 13, 10 ______

3.) 3, 6, 12, 24, 48, 96, ________

4.) 27, 30, 33, 36, 39, _________

5.) 41, 39, 37, 35, 33, _________

6.) 1, 10, 100, 1000, ______,______,______

7.) 2, 5, 9, 14, 20 _______,_______,_______

8.) He is an Italian mathematician who is responsible for the discovery of a

9.) It is a regular, repeated, or recurring forms or design.

10.) One rotation is equivalent to ____ degrees.

11.) What completes the following pattern?

CSD, ETF, GUH, _____, KWL

12.) What number should come next in this sequence?

22, 21, 25, 24, 28, 27, ______

13.) What number comes next in 1, 8, 27, 64, 125, _______?

14.) What letters comes next in this pattern?

A, SS, TT, E SS, TT, I, SS, _____, _____, ______

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UNIT 2. SPEAKING MATHEMATICALLY

LESSON 2.1: VARIABLES

❑ Is a symbol, usually letters, which represents a value or a number

In an equation 2x + 4y =10 if x=4 what will be the value of y?

x2 + 7x + 10 = 0 what are the value of x?

8 + 4y = 10 [then combine like terms]

4y = 2 [divide both terms by 4 to eliminate the constant in the variable]

In the second equation you need to use the factoring method.

6.) 1, 10, 100, 1000, ,,__

7.) 2, 5, 9, 14, 20 ___,_,_____

A, SS, TT, E SS, TT, I, SS, ___, _, ____