BE-STEM 11 IS – BLOCK—B

General Mathematics

MATHEMATICS OF
INVESTEMENTS
Gen Math | MODULE 5
• Simple interest is charged only on the loan amount called the principal. Simple interest is
calculated by multiplying the principal by the rate of interest by the number of payment
periods in a year.
SIMPLE INTEREST FORMULA

I I = Prt
I I I
P = rt r = Pt t = Pr
P r t Where I = Interest, P = Principal, r = rate of
interest, and t = time or term in years

o The Principal P of a loan is also called the face value or the present value of the
loan.
• To find the maturity value, simply add interest to the principal

MATURITY VALUE FORMULA

A =P+I A = P + Prt A = P(1 + rt)

Where A = Maturity Value, P = Principal, I = Interest

• In solving Worded problems, we will use the acronym GUFSA

o Given – Unknown – Formula – Solution - Answer

• Examples:
1. Teresa borrowed ₱ 120 000.00 from her uncle. If Teresa agreed to pay an 8% annual
interest rate, calculate the amount of interest she must pay if the loan period is (a) 1
year, (b) 9 months, and (c) 18 months.

a. Given: P = ₱ 120 000.00, r = 8% at 0.08, and t = 1 year

Unknown: Interest
Formula: I = Prt
Solution: 𝐼 = (120 000)(0.08)(1)
Answer: 𝐼 = ₱ 9 600

9
b. Given: P = ₱ 120 000.00, r = 8% at 0.08, and t = 12 year
Unknown: Interest
Formula: I = Prt
9
Solution: 𝐼 = (120 000)(0.08) (12 )
Answer: 𝐼 = ₱ 7 200

18
c. Given: P = ₱ 120 000.00, r = 8% at 0.08, and t = 12 year
Unknown: Interest
Formula: I = Prt
18
Solution: 𝐼 = (120 000)(0.08) (12 )
Answer: 𝐼 = ₱ 14 400
 General Mathematics 1st Trimester, S.Y. 2021-2022
Governor Pack Road, Baguio City, Philippines 2600
Tel. Nos.: (+6374) 442-3316, 442-8220; 444-2786;
442-2564; 442-8219; 442-8256; Fax No.: 442-6268 Grade Level/Section: Grade 11 -
Email: email@uc-bcf.edu.ph; Website: www.uc-bcf.edu.ph

MODULE 5 – Gen Math Subject Teacher:

2. To buy the school supplies for the coming school year, you get a summer job at a
resort. Suppose you save ₱ 4 200.00 of your salary and deposit it into an account
that earns simple interest. After 9 months, the balance is ₱ 4 263.00. What is the
annual interest rate?
9 3
Given: A = ₱ 4 263.00, P = ₱ 4 200.00, t = 9 months or 12 or 4 year

Unknown: rate

Formula: A = P(1 + rt) Formula should be based on the given and unknown

3
Solution: 4 263 = 4 200 [1 + 𝑟 (4)] Substitute all the given in the formula

4 263 = 4 200 + 3 150𝑟 Apply Distributive Property

63 = 3 150𝑟 Subtract 4 200 from each side

0.02 = 𝑟 Divide each side by 3 150

Answer: r = 2%

3. If ₱ 10 000.00 is invested at 4.5% simple interest, how long will it take to grow to ₱ 11
800.00?
Given: A = ₱ 11 800.00, P = ₱ 10 000.00, r = 4.5% or 0.045
Unknown: time
Formula: A = P(1 + rt)
Solution: 11 800 = 10 000(1 + 0.045𝑡) Substitute all the given in the formula
11 800 = 10 000 + 450𝑡 Appy Distributive Property
1 800 = 450𝑡 Subtract 10 000 from each side
4=𝑡 Divide each side by 450

Answer: t = 4 years

4. Find the present value of ₱ 86 000.00 at 8% simple interest for 3 years.

Given: A = ₱ 86 000.00, r = 8% or 0.08, t = 3 years

Unknown: Principal
𝐴
Formula: 𝑃 = 1+𝑟𝑡 Derived formula from 𝐴 = 𝑃(𝐼 + 𝑟𝑡)

86 000
Solution: 𝑃 = 1+(0.08)(3)

Answer: P = ₱ 69 354.84

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• When the interest due at the end of a certain period is added to the principal and that sum
earns interest for the next period, the interest paid us called compound interest. In short,
compound interest is “interest on interest”.

COMPOUND INTEREST FORMULA

𝑟 𝐾𝑡 𝐴
𝐴 = 𝑃 (1 + ) 𝑃= 𝑟 𝐾𝑡
𝐼𝑐 = 𝐴 − 𝑃
𝐾 (1+ )
𝐾

𝑟
Where: K = number of compounding times per year; 𝐾 = periodic rate;
Kt = number of compounding period

• Number of Compounding times per year (K)

Terms K
annually 1
semi-annually 2
quarterly 4
monthly 12

• Examples:
1. If ₱ 320 000.00 is invested for 5 years at 8% compounded quarterly, find (a) the compound
amount and (b) the compound interest
a.
Given: P = ₱ 320 000.00, t = 5 years, r = 8% or 0.08, K = 4
(quarterly)
Unknown: Compound amount
Formula: 𝑟 𝐾𝑡
𝐴 = 𝑃 (1 + )
𝐾
Solution: 0.08 (4)(5)
𝐴 = 320 000 (1 + )
4
Answer: A = ₱ 475 503.17
b.
Given: P = ₱ 320 000.00, A = ₱ 475 503.17
Unknown: Compound Interest
Formula: 𝐼𝑐 = 𝐴 − 𝑃
Solution 𝐼𝑐 = 475 503.17 − 320 000
Answer: 𝐼𝑐 = ₱ 155 503.17

2. What amount must be invested in order to have ₱ 128 376.52 after 8 years if money is worth 6%
compounded semi-annually?

Given: A = ₱ 128 376.52, t = 8 years, r = 6% or 0.06, K = 2

Unknown: Principal
Formula: 𝐴
𝑃=
𝑟 𝐾𝑡
(1 + 𝐾 )
Solution: 128 376.52
𝑃=
0.06 (2)(8)
(1 + 2 )
Answer: P = ₱ 80 000.003 or ₱ 80 000.00
• The flowchart below gives the different kinds of annuities

ANNUITY

Types ANNUITY
CERTAIN
ANNUITY
UNCERTAIN

Kinds SIMPLE
ANNUITY
GENERAL
ANNUITIES

GENERAL GENERAL
Classifications ORDINARY
ANNUITY
ANNUITY
DUE
DEFERRED
ANUITY
ORDINARY
ANNUITY
ANNUITY
DUE
PERPETUTIES

Definition of Terms

• Annuity – a fixed sum of money paid to someone at regular intervals subject to a fixed
compound interest rate.
• Periodic Payment – each payment in an annuity.
• Payment interval – the time between the successive payment dates of an annuity.
• Term of the annuity – the interval between the beginning of the first payment period
and the end of the last payment period.
• Annuity Certain – payable for a definite duration. Begins and ends on a definite or fixed
date.
o Simple Annuity – interest conversion or compounding period is equal or the same
as the payment interval.
▪ Ordinary Annuity – annuity in which the periodic payment is made at the
end of each payment interval.
▪ Annuity Due – an annuity in which the periodic payment is made at the
beginning of each payment interval.
▪ Deferred Annuity – the periodic payment is not made at the beginning nor
at the end of each payment interval, but some later date.
o General Annuity – interest conversion or compounding period is unequal or not
the same as the payment interval.
▪ General Ordinary Annuity – first payment is made at the end of every
payment interval.
▪ General Annuity Due – first payment is made at the beginning of every
payment interval.
▪ Perpetuities – a series of periodic payments which are to run infinitely or
forever.
• Annuity Uncertain – annuity payable for an indefinite duration; it depends on some
certain event.
• Future Value of an annuity is the total accumulation of the payments and interest
earned.
• Present Value of an annuity is the principal that must be invested today to provide the
regular payment of an annuity.

• Examples
1. Determine if the given situations represent simple annuity or general annuity
a. Payments are made at the end of each month for a loan that charges 1.05%
interest compounded quarterly.
✓ Since the payment interval at the end of the month is not equal to the
compounding interval, quarterly, the situation represents a general
annuity.
 General Mathematics 1st Trimester, S.Y. 2021-2022
Governor Pack Road, Baguio City, Philippines 2600
Tel. Nos.: (+6374) 442-3316, 442-8220; 444-2786;
442-2564; 442-8219; 442-8256; Fax No.: 442-6268 Grade Level/Section: Grade 11 -
Email: email@uc-bcf.edu.ph; Website: www.uc-bcf.edu.ph

MODULE 5 – Gen Math Subject Teacher:

b. A deposit of ₱ 5 500.00 was made at the end of every three months to an

account that earns 5.6% interest compounded quarterly.
✓ Since the payment interval at the end of every three months (or
quarterly) is equal to the compounding interval, quarterly, the situation
represents a simple annuity.

2. Determine whether the situation describes an ordinary annuity or an annuity due.

a. Jun’s monthly mortgage payment is ₱ 35 148.05 at the end of each month.
✓ Because the payments are made at the end of each month, Jun’s
stream of monthly mortgage payments is an ordinary annuity.
b. The rent for the apartment is ₱ 7 000.00 and due at the beginning of each
month.
✓ Since the payments come at the beginning of each month, the
stream of rental payments is an annuity due.

Simple Ordinary Annuity

FUTURE VALUE PRESENT VALUE PERIODIC PAYMENT

(𝐹𝑉)𝑖
(1 + 𝑖)𝑛 − 1 𝑃[1 − (1 + 𝑖)−𝑛 ] P=
PV = (1 + 𝑖)𝑛 − 1
FV = P ∙ 𝑖
𝑖
(𝑃𝑉)𝑖
𝑃=
1 − (1 + 𝑖)−𝑛

Where: FV = Future Value P = Periodic Payment

PV = Present Value n = total number of conversion periods

i = interest rate per period 𝑛 = 𝑡𝐾

𝑟
𝑖=𝐾

Page 2 of 5
 General Mathematics 1st Trimester, S.Y. 2021-2022
Governor Pack Road, Baguio City, Philippines 2600
Tel. Nos.: (+6374) 442-3316, 442-8220; 444-2786;
442-2564; 442-8219; 442-8256; Fax No.: 442-6268 Grade Level/Section: Grade 11 -
Email: email@uc-bcf.edu.ph; Website: www.uc-bcf.edu.ph

MODULE 5 – Gen Math Subject Teacher:

Examples:

1. Alex and Tony are twins. After graduation and being finally able to get a good job, they
plan for retirement as follows.
a. Starting at age 24, Alex deposits ₱ 10 000.00 at the end of each year for 36
years.
b. Starting at age 42, Tony deposits ₱ 20 000.00 at the end of each year for 18
years.

Who will have the greater amount at retirement if both annuities earn 12% per year
compounded annually?

G-U-F-S-A Alex’s Plan Tony’s Plan

Given P = ₱ 10 000.00 P = 20 000
𝑟 0.12 𝑟 0.12
𝑖= = = 0.12 𝑖= = = 0.12
𝐾 1 𝐾 1
𝑛 = 𝑡𝐾 = (36)(1) = 36 𝑛 = 𝑡𝐾 = (18)(1) = 18
Unknown FV
Formula (1 + 𝑖)𝑛 − 1
FV = P ∙
𝑖

Solution (1 + 0.12)36 − 1 (1 + 0.12)18 − 1

𝐹𝑉 = 10 000 ∙ 𝐹𝑉 = 20 000 ∙
0.12 0.12
Answer FV = ₱ 4 844 631.16 FV = ₱ 1 114 994.30
Alex will have a greater amount at retirement. This example shows the value of time
and the advantage of saving early.

2. Rose works very hard because she wants to have enough money in her retirement
account when she reaches the age of 60. She wants to withdraw ₱ 36 000.00 every 3
months for 20 years starting 3 months after she retires. How much must Rose deposit
at retirement at 12% per year compounded quarterly for the annuity?
Given P = ₱ 36 000 𝑛 = 𝑡𝐾 = (20)(4) = 80
𝑟 0.12
𝑖 = 𝐾 = 4 = 0.03
Unknown PV
Formula 𝑃[1 − (1 + 𝑖)−𝑛 ]
PV =
𝑖
Solution 36 000[1 − (1 + 0.03)−80 ]
PV =
0.03
Answer PV = ₱ 1 087 227.48

3. Eva obtained a loan of ₱ 50 000 for the tuition fee of her son. She has to repay the
loan by equal payments at the end of every six months for 3 years at 10% interest
compounded semi-annually. Find the periodic payment.

Given PV = ₱ 50 000 𝑛 = 𝑡𝐾 = (3)(2) = 6

𝑟 0.10
𝑖 = 𝐾 = 2 = 0.05
Unknown Periodic Payment
Formula (𝑃𝑉)𝑖
𝑃 = 1−(1+𝑖)−𝑛 use the second formula since the given is PV and
not FV

Solution (50 000)(0.05)

𝑃=
1 − (1 + 0.05)−6
Answer P = ₱ 9 805.87

Page 3 of 5
 General Mathematics 1st Trimester, S.Y. 2021-2022
Governor Pack Road, Baguio City, Philippines 2600
Tel. Nos.: (+6374) 442-3316, 442-8220; 444-2786;
442-2564; 442-8219; 442-8256; Fax No.: 442-6268 Grade Level/Section: Grade 11 -
Email: email@uc-bcf.edu.ph; Website: www.uc-bcf.edu.ph

MODULE 5 – Gen Math Subject Teacher:

Simple Annuity Due

FUTURE VALUE

[(1 + 𝑖)𝑛 − 1](1 + 𝑖) Where:

𝐹𝑉 = 𝑃 ∙
𝑖 FV = Future Value

PV = Present Value
PRESENT VALUE P = Periodic Payment

[1 − (1 + 𝑖)−𝑛 ](1 + 𝑖) i = interest rate per period

𝑃𝑉 = 𝑃 ∙
𝑖 𝑖=𝐾
𝑟

n = total number of
PERIODIC PAYMENT conversion periods

(𝐹𝑉)𝑖 𝑛 = 𝑡𝐾
𝑃=
[(1 + 𝑖)𝑛 − 1](1 + 𝑖)
(𝑃𝑉)𝑖
𝑃=
[1 − (1 + 𝑖)−𝑛 ](1 + 𝑖)

Examples:

1. Suppose Mr. and Mrs. Mariano deposited ₱ 20 000.00 at the beginning of each year for 5
years un an investment that earns 10% per year compounded annually, what is the
amount or future value of the annuity?

Given P = ₱ 20 000.00 𝑛 = 𝑡𝐾 = (5)(1) = 5

𝑟 0.10
𝑖 = 𝐾 = 1 = 0.10
Unknown FV
Formula [(1 + 𝑖)𝑛 − 1](1 + 𝑖)
𝐹𝑉 = 𝑃 ∙
𝑖
Solution [(1 + 0.10)5 − 1](1 + 0.10)
𝐹𝑉 = 20 000 ∙
0.10
Answer FV = ₱ 134 312.20

2. Hope borrows money for the renovation of her house and repays by making yearly
payments of ₱ 50 000.00 at the beginning of each year for a period of 10 years at an
interest rate of 8% compounded annually. How much did Hope borrow?

Given P = ₱ 50 000.00 𝑛 = 𝑡𝐾 = (10)(1) = 10

𝑟 0.08
𝑖 = 𝐾 = 1 = 0.08
Unknown PV
Formula [1 − (1 + 𝑖)−𝑛 ](1 + 𝑖)
𝑃𝑉 = 𝑃 ∙
𝑖
Solution [1 − (1 + 0.08)−10 ](1 + 0.08)
𝑃𝑉 = 50 000 ∙
0.08
Answer PV = ₱ 362 344.40

Page 4 of 5
 General Mathematics 1st Trimester, S.Y. 2021-2022
Governor Pack Road, Baguio City, Philippines 2600
Tel. Nos.: (+6374) 442-3316, 442-8220; 444-2786;
442-2564; 442-8219; 442-8256; Fax No.: 442-6268 Grade Level/Section: Grade 11 -
Email: email@uc-bcf.edu.ph; Website: www.uc-bcf.edu.ph

MODULE 5 – Gen Math Subject Teacher:

3. Mary borrows ₱ 500 000 to buy a car. She has 24 monthly payments every beginning of
the month at 12% per year. If the interest is compounded monthly, how much will be her
monthly payment?

Given P = ₱ 500 000.00 𝑛 = 𝑡𝐾 = (2)(12) = 24

𝑟 0.12
𝑖= = = 0.01
𝐾 12
Unknown Periodic Payment
Formula (𝑃𝑉)𝑖
𝑃=
[1 − (1 + 𝑖)−𝑛 ](1 + 𝑖)
Solution (500 000)(0.01)
𝑃=
[1 − (1 + 0.01)−24 ](1 + 0.01)
Answer P = ₱ 23 303.70

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