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Lesson 2.1
Percentage
Increases and
Decreases
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Note:
Often prices are
increased or decreased by
a percentage. In this
section we consider how
to increase or decrease
quantities by using
percentage.
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Formula:
Percentage Increase
(𝑃𝑖):𝑃𝑖 = 𝐵 + 𝑃
Percentage Decrease
𝑃𝑑 :𝑃𝑑 = 𝐵 − 𝑃
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Example 2.1.1:
Ms. Pam Pamintuan earns
P1,000.00 per week for
her part-time job. She is
to be given a 5% pay rise.
How much will she earn
per week after the pay?
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Example 2.1.2:
The prices of all the
television televisions in a
shop are to be increased
by 8%. Calculate the new
price of a television that
originally cost
P10,500.00.
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Example 2.1.3:
In a sale the cost of a
computer is reduced by
30%. The normal price of
the computer was
P15,650.00. Calculate the
sale price of the
computer.
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Lesson 2.2The Concept of
Simple Interest
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Interest or Simple Interest
This refers to the amount
(or price) paid for the use
of money. We will use I(majuscule letter I) to
denote the Interest.
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Categories of Interest
1.Simple Interest;
2.Simple Discount; and
3.Compound Interest.
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Something to think about…
What do you think
are the reasons why
people borrow
money?
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Lender vs. Lendee
The lender refers to a person
or an entity that lends money.
On the other hand, the person
or the entity who borrows
money is referred to as the
lendee.
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Something to think about…
The lower the interest
rate the more people
are over-optimistic to
borrow money.
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Something to think about…
The higher the
interest rate the lesser
people are optimistic
to borrow money.
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Something to think about…
How high interest
hurts seasonal
businesses?
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Something to think about…
How government
benefits from
interest?
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Something to think about…
How
businesses/foreign
investors benefit
from interest?
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Three Factors where Interest
(I) depends:
1.The Principal (P)
2.The Rate of Interest (r)
3.The Time (t)
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The Principal (P)
This refers to the sum of money
invested, deposited or
borrowed. We will use P(majuscule letter P) to denote
or represent the Principal.
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The Rate of Interest (r)This refers to the percentage of the
principal per year and is generally
expressed in terms of percent.
The rate of interest is usually
represented by r (minuscule letter
r).
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Reminder:
Before computing
rate in percent must
be expressed in
decimal form.
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The Time (t)
This refers to the length of
time between the date the loan
is made and the date the loan
becomes payable to the lender.
To denote time we will use t(minuscule letter t).
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The Amount of Interest Formula
𝑰 = 𝑷𝒓𝒕𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟐. 𝟏
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Equation 2.1
This shall be used in
computing for the amount
of interest I given the
principal P, the rate r,
and the time t.
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Reminder:
In simple interest, the
amount of interest
earned per annum is
constant.
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Example 2.1:Nimfa Bebe deposited P5,000.00 in a
bank paying 6% simple interest for 5
years. Compute the:
(a) amount of interest per annum;
and
(b) total amount of interest for the
entire period.
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How about if rate r is unknown?
𝒓 =𝑰
𝑷𝒕(𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟐. 𝟐)
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Equation 2.2This shall be used in
computing for the rate of
interest r given the
amount of interest I, the
principal P and the time
t.
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Example 2.3:
The simple interest received
over a period of 5 years and
3 months on a loan of
P22,000.00 is P11,300.00.
Compute the rate of
interest.
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How about when time t is unknown?
𝒕 =𝑰
𝑷𝒓(𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟐. 𝟑)
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Equation 2.3This shall be used in
computing the time tgiven the principal P, rate
r and the amount of
interest I.
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Example 2.5:
Lady Supladita borrowed P5,000.00
from a bank charging 12% simple
interest. If she paid the amount of
interest equivalent to P1,200.00,
for how long did she use the
money?
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How about when Principal P is unknown?
𝑷 =𝑰
𝒓𝒕(𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟐. 𝟒)
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Equation 2.4
This shall be used in
computing the Principal P
when amount of interest
I, rate of interest r and
time t are given.
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Example 2.7:
Boss Mapagmahal08 paid
an interest of P2,800.00
on a loan for 2 years at
9.5%. How much was the
original loan?
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Let’s Practice: Solve the following
problems.
1. Compute for the amount of simple interest on the
following loans:
a. P180,000.00 at 8% for 3 years
b. P106,200.00 at 8 ½% for 2 years
2. If money is worth 8.5%, then how much shall be the
principal if it will earn an interest of P5,500.00 in 3
years and 6 months?
3. Ms. Taylor Swift borrowed P15, 820.00 at 6% simple
interest. She was charged an interest of P1,260.00.
Compute for the duration of the loan on years.
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Assignment: Solve the following
problems.1. Mr. Harry Styles borrowed P25,300.00 for 3 years at 8%
simple interest. How much was the amount charged for
the use of money?
2. For how long (in years) will an amount of P25,000.00 be
invested at 10.75% to earn an interest of P8,200.00?
3. At what interest rate will an investment of P50,000.00
earn an interest of P10,062.50 in 1 year and 9 months?
4. Mr. Nick Jonas invested an amount of P18,360.00 at
9.75%. How much shall be his interest earnings after 3
years and 2 months?
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Lesson 2.2The Accumulated Value
and the Present Value
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Accumulated Value
This refers to the sum of
money at the end of the
period when a certain
amount of money is
deposited or borrowed.
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Accumulated Value
Accumulated Value is also
called the future value. To
denote accumulated value
we will use AV (majuscule
letters A and V).
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How to compute for AV?
𝑨𝑽 = 𝑷 + 𝑰𝒐𝒓 𝑨𝑽 = 𝑷 𝟏 + 𝒓𝒕(𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟐. 𝟓 𝒂𝒏𝒅 𝟐. 𝟔)
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Equation 2.5 and 2.6
This shall be used in
computing the Accumulated
Value AV given the amount
of Principal P and the
amount of Interest I.
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Example 2.9:Mr. Hagardo Versosa paid an
interest of P5,000.00 on a loan
for 3 years at 8% simple interest.
Compute the value of:
(a) the original loan; and
(b) the amount he paid at the
end of 3 years.
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Something to think about…
Therefore, based
on example 2.9
what is now the
Present Value?
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Present Value (PV)
The present value is the current
worth of a future sum of invested,
barrowed, or deposited money
given a specified rate of return.
We will denote this using PV(majuscule letters P and V).
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How to compute for PV?
𝑷𝑽 =𝑨𝑽
𝟏 + 𝒓𝒕(𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟐. 𝟕)
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Equation 2.7This shall be used in
computing for present
value PV given the
accumulated value AV,
rate r, and time t.
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Example 2.11The accumulated value paid by
Mr. Ely Pante on a loan is
P72,000.00. If the loan was for 3
years at 9% simple interest, (a)
how much was the original loan?
(b) Compute the total amount of
interest.
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Therefore, for computing total amount of
interest (TI)…
𝑻𝑰 = 𝑨𝑽 − 𝑷𝑽(𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟐. 𝟖)
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Equation 2.8
This shall be used in
computing for the total
amount of interest TIgiven the accumulated
value AV and the present
value PV.
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Let’s Practice: Determine the missing
items in each of the following:
Principal Rate Time Interest Accumulated
Amount
P12,000.00 3.5% 3.5 years
6.0% 9 months P650.00
Miss Mary Christmas Aguinaldo deposited an amount
of P32,600.00 in a bank that gives 4.5% on savings
account. How much shall be the accumulated value
of the investment after 3 years and 7 months? How
much shall be the interest earnings?
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Lesson 2.3Exact and Ordinary
Interests
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Something to think
about…When the term of
investment is expressed in
days, what would be the
approach in computing for
the amount of interest?
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Two Approaches
1.Exact Interest
2.Ordinary
Interest
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Exact InterestThis is used when interest is
computed on the basis of 365
days a year or 366 days in a leap
year. To denote Exact Interest
we will use the symbol Ie(majuscule letter I and subscript
minuscule letter e).
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Therefore, for Exact Interest:
𝑰𝒆 =𝑷𝒓𝒕
𝟑𝟔𝟓(Equation 2.9)
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Equation 2.9
This shall be used when
computing for the Exact
Interest, given the
Principal amount, the rate
of interest and the time
per days.
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Ordinary InterestThis is used when interest is
computed on the basis of an
assumed 30-day/month or
360-day/year. We shall use Io(majuscule letter I and
subscript minuscule letter o).
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While for, Ordinary Interest:
𝑰𝒐 =𝑷𝒓𝒕
𝟑𝟔𝟎(Equation 2.10)
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Equation 2.10
This shall be used when
computing for the Ordinary
Interest, given the
Principal amount, the rate
of interest and the time
per days.
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Example 2.12Mr. Benny Bilang invested an
amount of P28,100.00 at 7%
simple interest for 100days.
Compute the value of the:
(a)exact interest; and
(b)ordinary interest.
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Example 2.13
Ms. Honeygirl Pulot-Pukyutan deposited
an amount of P12,800.00 in a time
deposit account at 8% simple interest for
150 days. Compute the value of the:
(a)exact interest; and
(b)the accumulated value at the end of
the term.
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Let’s Practice: Solve the
following.1.An amount of P15,600.00 was invested at 8%
simple interest for 115 days. Compute the
amount of interest and the accumulated
value using: (a) exact interest and (b)
ordinary interest.
2.Using exact interest of 8% simple interest,
determine the present value of an
investment that accumulated to P21,700.00
after 285 days.
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Lesson 2.3Simple Discount
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Recall…How do we compute
for the amount of
simple interest?
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Recall…In simple interest,
when do we pay for
the amount of
interest?
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Simple DiscountThis refers to the type of
interest where the amount of
interest is deducted in advance
and is computed based on the
maturity/accumulated/future
value.
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Something to think about…
In business, what
do you know about
discount?
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In business…a deduction from the usual
cost of something,
reductions to the basic
prices of goods or services
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The Concept of Simple
Discount
Here, the term discount
refers to the amount deducted
from the maturity
value/amount due of an
obligation.
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Thus,
the price of using
money is deducted
in advance.
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Reflect…In simple discount, the
borrower pays the original
sum of money borrowed at
the end of the term at a
specified discount rate.
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Therefore…The amount due (AV) at
the end of the term (t) is
charged an interest in
advance at simple
interest rate (d).
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In Simple Discount…
Here, in simple
discount, we will call the
amount of interest as the
amount of discount.
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Amount of Discount…
𝑰 = 𝑨𝑽𝒅𝒕(𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟐. 𝟏𝟓)
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Equation 2.15This equation shall be used when
computing for the amount of
discount (I) given the amount due
(AV), discount rate (r) and the
term of the loan (t).
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Amount Due
𝑨𝑽 =𝑰
𝒅𝒕(𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟐. 𝟏𝟔)
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Equation 2.16
This equation shall be used when
computing for the amount due
(AV) given the amount of discount
(I), discount rate (d) and the term
of the loan (t).
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Discount Rate
𝒅 =𝑰
𝑨𝑽𝒕(𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟐. 𝟏𝟕)
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Equation 2.17
This equation shall be used when
computing for the discount rate
(d) given the amount of discount
(I), the amount of the loan (AV)
and the term of the loan (t).
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Term of the Loan
𝒕 =𝑰
𝑨𝑽𝒅(𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟐. 𝟏𝟖)
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Equation 2.17
This equation shall be used when
computing for the term of the
loan (t) given the amount of
discount (I), the amount due (AV)
and the discount rate (d).
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Example 2.16
How much interest will be
collected in advance from a
loan of P15,000.00 made by
Ms. Victoria Malihim for 2
years if the discount rate is
8%?
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Example 2.17
On June 11, 2007, Dayuhin Mo Ako
Resort borrowed P6,000.00 from a
credit union and was charged P410.00
for the loan. If interest was deducted
in advance, what was the discount
rate? Assume that the loan shall be
paid on December 20 of the same year.
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Example 2.18Ms. Lily Mangipin borrowed a
certain amount from a cooperative
and was charged P2,000.00
deducted in advance. If the
discount rate is 12% for 6 months,
how much is the amount of the
loan?
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Reflect…If in simple interest, the
amount of interest is deducted
in advance, what do we call to
the money received by the
borrower when he loans?
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Recall…From our previous
lesson, how do we
compute for the
accumulated value
(AV)?
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Accumulated Value
𝑨𝑽 = 𝑷 + 𝑰
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Proceeds (Pr)
It is the amount
received by the
borrower at the
beginning of the term.
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Moreover…The value of proceeds is
equivalent to the concept
of principal/original
loan/present value.
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However…
For purposes of clarity,
we shall use Pr to indicate
the present value at
simple discount.
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Remember…
The phrase, “to
discount” means to
get the value of the
proceeds.
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Proceeds
𝑷𝒓 = 𝑺 𝑰 − 𝒅𝒕(𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟐. 𝟏𝟖)
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Equation 2.18
This equation shall be used when
computing for the amount received by
the borrower at the beginning of the
term (Pr), given the amount due (AV),
discount rate (r), and term of the loan
(t).
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Example 2.19
Determine the proceeds of
P18,500.00 for 2 years and 6
months at 10% simple
discount.
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Example 2.20Find the proceeds and the
amount of discount on
P8,000.00 due at the end of
2 years at 8% simple
discount.
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Let’s Practice: Solve the
following problems.1. Determine what is missing value in each of the
following:
2. Ms. Pining Garcia borrowed P55,000.00 from a bank
that charges 12% simple discount. If Ms. Pining agreed to
pay her accounts 5 months from today, what sum of
money will she receive from the bank?
Amount Due
(AV)
Discount Rate
(d)
Term of the
Loan (t)
Amount of
Discount (I)
Proceeds
(Pr)
P8,560.00 7.5% 125 days
8.25% 100 days P28,250.00