MATHEMATICS IN THE MODERN WORLD

APPORTIONMENT AND VOTING SYSTEM

1. APPORTIONMENT
 HAMILTON METHOD
 JEFFERSON METHOD
 WEBSTER METHOD
 HUNTINGTON-HILL METHOD
2. VOTING SYSTEM
 MAJORITY SYSTEM AND PLURALITY OF VOTING
 BORDA COUNT METHOD
 PLURALITY WITH ELIMINATION
 THE TOP TWO RUNOFF METHOD
 APPROVAL VOTING SYSTEM
 THE METHOD OF PAIRWISE COMPARISON

MATHEMATICAL LOGIC

 SIMPLE AND COMPOUND STATEMENT

 LOGICAL CONNECTIVE
 TRUTH TABLES

MATHEMATICS OF FINANCE

 SIMPLE INTEREST
 COMPOUND INTEREST

Prepared by:
Jovito U. Buscas
 MATHEMATICS IN THE MODERN WORLD
INTRODUCTION TO APPORTIONMENT
Apportionment is a method of distributing a number of items proportionally into several groups based on
population sizes.

1. HAMILTON METHOD
Proposed by Alexander Hamilton to assign voting seats in the House of representatives to each
represented states. This method is based on standard divisor and standard quota of the population.

where P is the total population and A is the total allocation or the number of available seats to be
assigned.

where G is the size of the group. The whole number portion of the standard quota is called lower quota
(LQ)=round down.

Example 1. Consider a country with 6 states and 40 seats in the House of Representatives with
populations distributed as follow.

STATE POPULATION
A 58,805
B 32,780
C 109,111
D 78,900
E 67,430
F 44,862
TOTAL 391,888
Determine the apportionment of 40 seats to the 6 states using the Hamilton method.

SOLUTION:

State A State B State C State D State E State F Total

P 58,805 32,780 109,111 78,900 67,430 44,862 391,888
SQ=G/SD
Lower Quota
Final
Apportionment
A 40
SD 9797.2

2. JEFFERSON METHOD
Jefferson method of apportionment uses a MODIFIED Standard Divisor (MSD) which uses
chosen by trial and error until the sum of the lower quota is equal to the required number of
allocation.

EXAMPLE 2. Solve example 1 using Jefferson method.

Solution:
Prepared by:
Jovito U. Buscas
 MATHEMATICS IN THE MODERN WORLD

State A State B State C State D State E State F Total

State A State B State C State D State E State F Total

P 58,805 32,780 109,111 78,900 67,430 44,862 391,888
SQ=G/SD
Lower Quota
Final
Apportionment
A 40
MSD

3. WEBSTER METHOD
A variation of the Jefferson plan. Instead of using the lower quota, use the regular rules of
rounding to determine the final apportionment.
Use a modified standard divisor (MSD) that yields the correct number of representatives by trial
and error.

EXAMPLE 3. Solve example 1 using Webster method.

Solution:

State A State B State C State D State E State F Total

P 58,805 32,780 109,111 78,900 67,430 44,862 391,888
SQ=G/SD
Lower Quota
Final
Apportionment
A 40
MSD

4. Huntington-Hill Apportionment method

Use a modified standard divisor (MSD) that yields the correct number of representatives by trial
and error.
Calculate the standard quota, lower quota and upper quota of each sub-group.
Calculate the geometric mean (GC √ (round to 2 decimal places) each sub-
group’s lower quota and upper quota.
If the SD is Less than the Geometric mean, ROUND THE QUOTA DOWN.
If the SD is Greater than or Equal to the Geometric mean, ROUND THE QUOTA UP.

EXAMPLE 4. Solve example 1 using Huntington-Hill method.

Solution:

Prepared by:
Jovito U. Buscas
 MATHEMATICS IN THE MODERN WORLD
P 58,805 32,780 109,111 78,900 67,430 44,862 391,888
SQ=G/SD
Upper Quota
Lower Quota
Geometric
Mean
Final
Apportionment
A 40
MSD

VOTING SYSTEM
VOTING is a tool used by groups of people in making a collective decision. It can be presented
conveniently in term of an election system in which one can select a particular candidate out of a set of
candidates on the basis of ballots cast by voters.

1. MAJORITY SYSTEM AND PLURALITY METHOD OF VOTING

 MAJORITY SYSTEM is the most common voting system applied to an election with
only two candidates. The winner in the majority system requires more than half of the
people voting for an issue or a candidate.
 PLURALITY METHOD
Each voter votes for one candidate, and the candidate with the most votes wins. The
winning candidate does not have to have a majority of the votes.
Example1. Determine the Winner Using Plurality Voting
Fifty people were asked to rank their preferences of five varieties of chocolate candy,
using 1 for their favorite and 5 for their least favorite. This type of ranking of choices is
called a preference schedule.

Prepared by:
Jovito U. Buscas
 MATHEMATICS IN THE MODERN WORLD
SOLUTION:
To answer the question, we will make a table showing the number of first-place votes for each
candy.

The winner is Toffee center

2. Borda Count Method of Voting

In this method, each candidate is assigned a weight according to their rank in the prference list. For n
number of candidates, the highest rank (the most favorite) will have n points and the lowest rank (the
least favorite) will have 1 point. The Borda Count is the sum of these weights given to a candidate.
The candidate who receives the highest Borda count will be declared the winner.

Example2: Using the Borda count method, determine the winner in the preference list.
The result of an election with 4 candidates and 50 voters are shown in the preference list below.
Using the Borda count method of voting, which candidate wins this election?

Candidate Ranking
A 1 4 2 4 2
B 3 3 1 3 4
C 2 1 4 1 3
D 4 2 3 2 1
No. of Voters 12 8 16 10 4

SOLUTION:

n=4 CANDIDATE A n=4 CANDIDATE B

Rank Rank No. of Total Rank Rank No. of Total
point votes point votes
1 4 12 48 3 2 12 24
4 1 8 8 3 2 8 16
2 3 16 48 1 4 16 64
4 1 10 10 3 2 10 20
2 3 4 12 4 1 4 4
Borda Count 126 Borda Count 128

The candidate with the

n=4 CANDIDATE C n=4 CANDIDATE D highest Borda count is
Rank Rank No. of Total Rank Rank No. of Total candidate C. Thus, candidate
C wins the election.
point votes point votes
2 3 12 36 4 1 12 12
1 4 8 32 2 3 8 24
4 1 16 16 3 2 16 32
1 4 10 40 2 3 10 30
3 2 4 8 1 4 4 16
Borda Count 132 Borda Count 114

Prepared by:
Jovito U. Buscas
 MATHEMATICS IN THE MODERN WORLD

3. PLURALITY WITH ELIMINATION

Plurality with elimination is a variation of the plurality method in which the alternative
choices of the voters are taken into consideration.
The candidate with the fewest number of first-place votes is first eliminated. In case there
are two alternatives with the same lowest votes, then both are to be eliminated. The remaining
candidates are re-ranked with the assumption that voters’ preferences do not change from round
to round.
EXAMPLE 3: Answer Example 2 using the method of plurality with elimination.
Candidate Ranking
A 1 4 2 4 2
B 3 3 1 3 4
C 2 1 4 1 3
D 4 2 3 2 1
No. of Voters 12 8 16 10 4
SOLUTION:
For round 1,
Candidate Total First-Place Votes
A 12
B 16
C 8+10=18
D 4
Candidate D should be eliminated.
Round 2, Re-rank
Candidate Ranking
A 1 3 2 3 1
B 3 2 1 2 3
C 2 1 3 1 2
No. of Votes 12 8 16 10 4

Candidate Total First-Place Votes

A 12+4=16
B 16
C 8+10=18
Candidate A and B have the same lowest first-place votes, so both should be eliminated. Thus,
the winner for this method is candidate C.

4. THE TOP TWO RUNOFF METHOD

The two candidates with the most number of first place votes are removed from the
preference list and are then re-rank for a new preference list. The one with the higher first-place
votes in the new preference list between these two candidates will be declared the winner.
EXAMPLE 4. Apply the top two runoff method to the preference list from example 2.

Candidate Ranking
A 1 4 2 4 2
B 3 3 1 3 4
C 2 1 4 1 3
D 4 2 3 2 1
No. of Voters 12 8 16 10 4
Prepared by:
Jovito U. Buscas
 MATHEMATICS IN THE MODERN WORLD

Candidate Ranking Total 1st –

place votes
A 1 4 2 4 2 12
B 3 3 1 3 4 16
C 2 1 4 1 3 18
D 4 2 3 2 1 4
No. of Voters 12 8 16 10 4
SOLUTION:
Counting the total first-place votes for each candidate, you will obtain the following.
The top two candidates with the most number of First-place votes are candidates B and C. Thus,
Candidate Ranking Total 1st –
place votes
B 2 2 1 2 2 16
C 1 1 2 1 1` 12+8+10+4=34
No. of Voters 12 8 16 10 4
remove candidates A and D form the list. This gives us the following preference list.
Thus, candidate C wins this election.
5. Approval Voting System
In this type of voting, a voter may choose more than one option or candidate. Each vote
coming from one voter will be counted as one vote, and the one with the most number of total
votes will be declared the winner. For large elections, approval voting may be considered a better
method because it measures the overall support for a candidate.
EXAMPLE 5.
The members of a scholarship council have picked their choices from the top four
applicants for a scholarship. The results are indicated in the table of preference below.
Candidate Number of votes
Harry, Liam, and Louis 5
Harry, Liam, and Niall 3
Niall and Louis 2
Harry and Niall 4
Who gets the scholarship using the approval voting system?
SOLUTION:
Tallying the votes, you will obtain the following:
Candidate Number of votes
Harry 5+3+4=12
Liam 5+3=8
Louis 5+2=7
Niall 3+2+4=9
Hence, Harry gets the scholarship with a total of 12 votes.
6. THE METHOD OF PAIRWISE COMPARISON
In this method, each candidate is compared head-to-head with each of the candidates. The
candidate with the most number of wins from these comparisons will be declared the winner.
EXAMPLE 6
The members of a mechanical engineering club consisting of 165 active members were
asked to decide in which sports event the club should participate in the coming intercollegiate
games. If the members marked their ballots as shown in the preference table below, which event
wins as the top choice using the method od pairwise comparison?

Prepared by:
Jovito U. Buscas
 MATHEMATICS IN THE MODERN WORLD

Number of 26 60 35 44
voters
1st choice Volleyball Basketball Football Volleyball
2nd choice Basketball Volleyball Basketball Basketball
3rd choice Swimming Swimming Volleyball Football
4th choice Football Football Swimming Swimming

SOLUTION:

Number of 26 60 35 44 Total 1st-

voters Place
Votes
Volleyball 1 2 3 1 44+26=70
(V)
Basketball 2 1 2 2 60
(B)
Football (F) 4 4 1 3 35
Swimming 3 3 4 4 0
(S)

Possible comparisons are highlighted in the table below.

V B F S
V
B
F
S

Number of 26 60 35 44 Total First-

Voters place votes
V 1 2 2 1 26+44=70
B (winner) 2 1 1 2 60+35=95

Number of 26 60 35 44 Total First-

Voters place votes
V (winner) 1 1 2 1 26+60+44=130
F 2 2 1 2 35

Number of 26 60 35 44 Total First-place

Voters votes
V (winner) 1 1 1 1 26+60+35+44=165
S 2 2 2 2 0

Number of 26 60 35 44 Total First-

Voters place votes
B (winner) 1 1 2 1 26+60+44=130
F 2 2 1 2 35

Number of 26 60 35 44 Total First-place

Voters votes
B (winner) 1 1 1 1 26+60+35+44=165
Prepared by:
Jovito U. Buscas
 MATHEMATICS IN THE MODERN WORLD
S 2 2 2 2 0

Number of 26 60 35 44 Total First-

Voters place votes
F 2 2 1 1 35+44=79
S (winner) 1 1 2 2 60+26=86
Below is the summary of the winner in each comparison.
V B F S
V B V V
B B B
F S
S
From the above table, basketball has the most number of wins. Therefore, this option should be
the top choice by pairwise comparison method.

EXERCISE 1
ANSWER EACH PROBLEM COMPLETELY.
1. The following is the preference table from a science club election where the candidates are
Arrow (A), Bennett (B), Candice (C), and Danny (D).
Number of voters 42 23 35 28 14
1st choice A B C A D
2nd choice D A D B A
rd
3 choice B C B D C
4th choice C D A C B
Determine the winner of the election using the indicated method.
a. Plurality method
b. Plurality with elimination method
c. Borda count
d. Top two runoff
e. Pairwise comparison.

EXERCISE 2

1. Use the Hamilton method to apportion 32 seats among the given states.
State A B C D E
Population 23,403 33,870 28,777 12,000 45,909
2. Use the Jefferson method for apportionment to answer problem 1.
3. Use the Webster method for apportionment to answer problem 1.
4. Use the Huntington-hill method for apportionment to answer problem 1.

Prepared by:
Jovito U. Buscas
 MATHEMATICS IN THE MODERN WORLD
MATHEMATICAL LOGIC
Symbolic logic is a powerful tool for analysis and communication in mathematics. It represents
the natural language and mathematical language with symbols and variables.

A STATEMENT is an assertion which can be regarded as true or false.

A SIMPLE STATEMENT is a single statement which does not contain other statements as parts.
A COMPOUND STATEMENT contains two or more statements.
A LOGICAL CONNECTIVE combines simple statements in compound statements.

The following table shows some basic propositional logic with their symbols.
Connectives Propositional Symbols Example Read
Logic
Negation p Not p (p is false)
NOT
Conjunction ˄ p ˄q p and q
And/But
(both p and q are
true)
Disjunction ˅ p˅q p or q
or
(either p is true
or q is true or
both are true)
Conditional → p→q If p then q
Implies
Biconditional ↔ p↔q P implies q and
If and only if
q implies p

p if and only if q
EXAMPLE
Consider the following statements.
h: Harry is not happy.
v: Harry is going to watch a volleyball game.
r: It is going to rain.
s: Today is Sunday.
Write the following compound statements in symbolic form.
a. Today is Sunday and Harry is not happy.
b. Today is Sunday and Harry is not going to watch a volleyball game.
c. If it is going to rain, then Harry is not going to watch a volleyball game.
d. Harry is going to watch a volleyball game if and only if he is happy.
e. Harry is going to watch a volleyball game or it is going to rain.

DERIVES FORMS OF A CONDITIONAL STATEMENT

There are three ways to restate a conditional statement-the CONVERSE, INVERSE, and
CONTRAPOSITIVE. However, restating the statement into one of these forms may change the
meaning of the original statement. The conditional statement, p → q, may be restated in the following
forms.
CONVERSE FORM: q → p
NVERSE FORM: p→ q
CONTR POS T VE: q → p

Prepared by:
Jovito U. Buscas
 MATHEMATICS IN THE MODERN WORLD

EXAMPLE:
Write the following converse, inverse, and contrapositive of the given sentence.
“She is allowed to join the volleyball team, only if she knows how to receive the ball.”

SOLUTION:
p → q : f she is allowed to join the volleyball team, then she knows how to receive the ball.

CONVERSE FORM: q → p
“if she knows how to receive the ball, then she is allowed to join the volleyball team.”
NVERSE FORM: p → q
“ f she is not allowed to join the volleyball team, then she does not know how to receive the ball.”
CONTRAPOSITIVE: q → p
“if she does not know how to receive the ball, then she is not allowed to join the volleyball team.”

TRUTH TABLES
A logical statement may either be true of false. If the statement is true, then the TRUTH VALUE corresponding
to the statement is true and is denoted by letter T. if it is false, the statement has a value denoted by F. the
TRUTH TABLE is a summary of all possible truth values of a statement. Here are some examples.

ASSERTION
p
T
F

COMPOUND STATEMENTS AND THEIR TRUTH TABLES

In mathematics, statements expressed in different ways are considered to be equivalent if they have the
same truth value. For example “Manila is the capital of the Philippines” is considered equivalent to the statement
written as “the capital of the Philippines is Manila.”
The truth table of a compound statement involving two or more statements can be constructed from the
tables of each of the simple statements.
EXAMPLE1:
Construct the truth table for the compound statement.
p˅q p

Prepared by:
Jovito U. Buscas
 MATHEMATICS IN THE MODERN WORLD
SOLUTION:
Apply the basic rules discussed previously to obtain the following table.
p q p˅q p (p ˅ q p
T T T F F
T F T F F
F T T T T
F F F T F

EXAMPLE2:
Find the truth table of p ˅ q → p
SOLUTION:
p q p q q→p p˅ q → p
T T F F T T
T F F T T T
F T T F T T
F F T T F T

If the truth value of a compound statement is always true regardless of the truth values of each of the component
statements, then the statement is said to be a TAUTOLOGY.

The compound statement in the previous example is a tautology. If the truth value of a compound statement is
always false, regardless of the truth value of each of the component statement, then the statement is a
CONTRADICTION.

LOGICAL EQUIVALENCE

Two mathematical statements are logically equivalent if the final output of their truth tables are exactly the
same.

EXAMPLE:

Verify if the statements p → q and p ˅ q are logically equivalent

SOLUTION:

p q p P→q p˅q
T T F T T
T F F F F
F T T T T
F F T T T

Since the last two columns are identical, the given statements are logically equivalent.

Prepared by:
Jovito U. Buscas
 MATHEMATICS IN THE MODERN WORLD
EXERCISE 3

Construct the truth tables for the given compound statements.

1. p ^ ( p)

2. [p ^ ( q ] ˅ [ p ˅ q]

3. p ˅ q ˅ p

If p is false and q is true, what are the truth values of the given statements?

4. p ˅ q

5. ( p q

EXERCISE 4

Write the converse, inverse, and contrapositive of the given statement.

1. If a quadrilateral is not a rectangle, then it is not a square.

Converse:______________________________________________________________________
Inverse:________________________________________________________________________
Contrapositive:__________________________________________________________________
2. If yesterday is not Wednesday, then tomorrow is not Friday.
Converse:______________________________________________________________________
Inverse:________________________________________________________________________
Contrapositive:__________________________________________________________________

Prepared by:
Jovito U. Buscas
 MATHEMATICS IN THE MODERN WORLD
MATHEMATICS OF FINANCE

When a person borrows money from a lending company or a bank, he or she usually pays a fixed rate of interest
on the principal for using that money. The amount paid by the borrower is called INTEREST while the amount
of money that is loaned is called the PRINCIPAL or the PRESENT VALUE.

ISMPLE 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 year. If the number of days
used is 360 days in a year, the interest is called ORDINARY SIMPLE INTEREST. If it used 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
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

I=Prt
I = (60,000)(0.12)(7/12) = 4,200

The total amount to be paid is

F=P+I
F= 60,000 + 4,200 = 64,200.
EXAMPLE 2:

The ordinary simple interest changed 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
575 = (10,200)(r)(130/360)
r = 0.156 or 15.6%

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
Where n = mt and i =
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 per quarter is i= = = 0.015 or 1.5%

Prepared by:
Jovito U. Buscas
 MATHEMATICS IN THE MODERN WORLD

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)(8) = 36. The interest rate per period is i= = = 0.05.
Hence, the total amount the son should receive is
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:
mt
F=

P=

200,000 = P(1 + )4(5)

P = 144,883.46

EFFECTIVE INTEREST RATE, ER, is equivalent to interest rate compounded annually. It can be
used to compare two rates with different compounding periods.

ER= m
–1

EXAMPLE 1
If one bank advertises is rate as 6.2% compounded monthly, 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
ER = 12
– 1 = 0.064 or 6.4%

For bank 2, the effective rate is

ER = 1
– 1 = 0.063 or 6.3%

Hence, bank 1 has a better interest rate. A higher rate is good for financial investment.

Prepared by:
Jovito U. Buscas
 MATHEMATICS IN THE MODERN WORLD
EXERCISE 1
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. By the conditions stated in a will, the sum of 2.5M 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 if the fund is invested at
10% compounded quarterly?
4. What is the effective rate corresponding to 18% compounded daily using 360 days in one year?
5. Which terms offer the best investment for 1 year?
a. 10% simple interest
b. 9.6% compounded monthly
c. 10% compounded daily

Prepared by:
Jovito U. Buscas