GENERAL MATHEMATICS

QUARTER 2 WEEK 3 & 4

SELF-LEARNING MODULE
DEPARTMENT OF EDUCATION
DIVISION OF NORTHERN SAMAR
 Annuities

How to use and learn from this module

Below are the guidelines for you in going about the module.

1. Read and follow instructions very carefully.

2. Answer the pre-test to determine how much you already know about the lessons
in this module.
3. Check your answers in the given answer key at the end of this module.
4. Read each lesson and do the activities that are provided for you.
5. Ask for assistance from your parents or teacher if you have some confusion about
the lessons in this module.
6. Perform all the activities diligently to help you understand the topic.
7. Take the self-test after each lesson to determine how much you understand the
topic.
8. Answer the posttest to measure how much you have gained from the lessons.

Goodluck and have fun!

What is this Module all about?

In many aspects of modern life, Mathematics plays an important role. In the field of
business, mathematics is essential in analyzing markets, predicting stock market prices,
business decision making, forecasting production, financial analysis, and in business
operation in general. This module will introduce the students to the basic concepts of
business mathematics such as the simple and compound interests.
This module covers the content standards on the third and fourth week of the Second
Quarter.

Module 7: Simple Annuities

Content Standard: The learner demonstrates understanding of key concepts of

simple and compound interests, and simple and general annuities.
Performance Standard: The learner is able to investigate, analyze and solve
problems involving simple and compound interests and simple and general annuities
using appropriate business and financial instruments.
Learning Competencies:
• illustrates simple and general annuities.
• distinguishes between simple and general annuities.
• finds the future value and present value of both simple annuities and general
annuities.
• calculates the fair market value of a cash flow stream that includes an annuity.
• calculates the present value and period of deferral of a deferred annuity.

General Mathematics 1
 Annuities

Lesson 3: Simple Annuity

Objectives
At the end of this lesson, you will be able to

a. Illustrate simple annuities;

b. Distinguish between simple and general annuities; and
c. Computes the future value, present value and periodic payment of simple annuity

Pre-test

Read the questions carefully. Write the letter that corresponds to your answer in
your notebook.

1. It is an annuity where the payment interval is the same as the interest period.
a. Simple Annuity c. Annuity Certain
b. General Annuity d. Contingent Annuity

2. It is a sequence of payments made at equal (fixed) intervals or periods of time.

a. Future Value of an annuity c. Annuity
b. Present Value of an annuity d. Periodic Payment

3. The sum of future values of all the payments to be made during the entire term of
annuity.
a. Future Value of an annuity c. Annuity
b. Present Value of an annuity d. Periodic Payment

4. The sum of all present values of all the payments to be made during the entire term
of the annuity.
a. Future Value of an annuity c. Time of Annuity
b. Present Value of an annuity d. Periodic Payment

5. Find the future value of an ordinary annuity with a regular payment of P1,000 AT 5%
interest rate compounded quarterly for 3 years.
a. P12,806.63 c. P12,806.36
b. P12,860.63 d. P12,860.36

General Mathematics 2
 Annuities
Preliminary
You use money in everyday life. In order to buy what you need, you do
transactions involving money.
In the previous lessons, you learned the methods of solving the value of money
under compound and simple interest environment. You have learned to illustrate and
distinguish between simple and compound. You also learned how to compute for the
interest, present value and future value in a simple and compound interest environment.
As well as solve problems involving real life situations of simple and compound interest.

Lesson Proper
Ma’am Angel wants to start a business with an initial capital of P100,000. She
decided to put up a fund with deposits made at the end of each month. If she wants to
gain the initial capital after 4 years, how much monthly deposit must be made?
In most cases where house or cars are purchased, a series of payments is needed
at certain points in time. Such Transaction is called ANNUITY.

An annuity is a sequence of payments(or deposits), usually equal, made at regular

intervals.
ANNUITY
Simple Annuity – an General Annuity – an
According to payment annuity where the payment annuity where the payment
interval and interest interval is the same as the interval is not the same as
period interest period the interest period.

According to time of Ordinary Annuity (Annuity Immediate) – a type of annuity

payment in which the payments are made at the end of each payment
interval

According to duration Annuity Certain – an annuity in which payments begin and

end at definite times.

Definition of terms:
Terms of an Annuity (t) – The time between the first payment interval and the last
payment interval.
Regular or Periodic Payment (R) – The amount of each payment.
Amount (Future Value) of an Annuity (F) – The sum of future value of all the payments
to be made during the entire term of the annuity.
Present Value of an Annuity – The sum of present value of all the payments to be made
during the entire term of the annuity.
Annuities may be illustrated using a time diagram. The time diagram for an ordinary
annuity (i.e., payments are made at the end of the year) is given below.

General Mathematics 3
 Annuities
Formulas

Future Value Present Value

(1 + 𝑗)𝑛 − 1 1 − (1 + 𝑗)−𝑛
𝐹 = 𝑅[ ] 𝑃 = 𝑅[ ]
𝑗 𝑗
Note: Note:
𝑖 𝑖
𝑗= 𝑗=
𝑚 𝑚
𝑛 = 𝑚𝑡 𝑛 = 𝑚𝑡

Example 1: Suppose Mrs. Manda would like to deposit P3,000 every month in a fund that
gives 9%, compounded monthly. How much is the amount of future value of her savings
after 6 months?

Given: Periodic payment (R) = P3,000

Term (t) = 6 months
Interest rate per annum (annually) (i) = 9% or 0.09
Number of conversions per year (m) = 12
𝑖 0.09
Interest rate per period 𝑗 = 𝑚 = = 0.0075
12

General Mathematics 4
 Annuities
(1) Illustrate the cash flow in time diagram and Find the future value of all the payments
at the end of term (t = 6).

(2) Add all the future values obtained from the cash flow.

3,000 = 3,000
3,000 (1 + 0.0075) = 3,022.50
3,000 (1 + 0.0075)2 = 3,045.17
3,000 (1 + 0.0075)3 = 3,068.01
3,000 (1 + 0.0075)4 = 3,091.02
3,000 (1 + 0.0075)5 = 3,114.20

Thus, the amount of this annuity is P18,340.89.

(3) Solution using formula for FUTURE VALUE.

Given: 𝐴(𝑡) =? R = 3,000 𝑖 = 0.09 m = 12

6 𝑖 0.09 6
𝑡(𝑎𝑛𝑛𝑢𝑎𝑙𝑙𝑦) = 𝑗= = = 0.0075 𝑛 = 𝑚𝑡 = 12 ( ) = 6
12 𝑚 12 12

(1+𝑗)𝑛 −1
𝐹 = 𝑅[ ]
𝑗
(1+0.0075)6 −1
𝐹 = 3,000 [ ]
0.0075

(1.0075)6 −1
𝐹 = 3,000 [ ]
0.0075

1.045852235−1
𝐹 = 3,000 [ ]
0.0075

0.045852235
𝐹 = 3,000 [ ]
0.0075

General Mathematics 5
 Annuities
𝐹 = 3,000(6.113631347)
𝑭 = 𝟏𝟖, 𝟑𝟒𝟎. 𝟖𝟗

Example 2: Suppose Mrs. Manda would like to deposit P3,000 every month in a fund
that gives 9%, compounded monthly. How much is the amount of future value of her
savings after 6 months?

Given: Periodic payment (R) = P3,000

Term (t) = 6 months
Interest rate per annum (annually) (i) = 0.09 or 9%
Number of conversion per year (m) = 12
𝑖 0.09
Interest rate per period 𝑗 = 𝑚 = = 0.0075
12

1−(1+ 𝑗 )−𝑚𝑡
𝑃 = 𝑅 [[ ]
𝑗
1−(1+ 0.0075 )−6
𝑃 = 3000 [ ]
0.0075
1−(1.0075 )−6
𝑃 = 3000 [ ]
0.0075
1−0.9561580178
𝑃 = 3000 [ ]
0.0075
0.04384198223
𝑃 = 3000 [ ]
0.0075
𝑃 = 3,000 ( 5.84559763)
𝑷 = 𝟏𝟕, 𝟓𝟑𝟔. 𝟕𝟗

Therefore, the amount of Present value of Mrs. Manda’s savings after 6 months
is P17,536.79.

Example 3: A certain fund currently has P100,000 and is invested at 3% interest

compounded annually. How much withdrawal can be made at the end of each year so
that the fund will have zero balance at the end of 12 years?

SOLUTION: Since withdrawals are made every end of the year, then this ordinary
annuity.
Given: Periodic payment (R) = P100,000
Term (t) = 12 years
Interest rate per annum (annually) (i) = 0.03 or 3%
Number of conversion per year (m) = 1
𝑖 0.03
Interest rate per period 𝑗 = 𝑚 = = 0.03
1

𝑃
𝑅 =[ 1−(1+ 𝑗 )−𝑚𝑡
]
( )
𝑗

100,000
𝑅 =[ 1−(1+ 0.03)−(1)(12)
]
( )
0.03

General Mathematics 6
 Annuities

100,000
𝑅 =[ 1−0.7013798802 ]
( 0.03
)

100,000
𝑅 =[ ]
9.954003994

𝑅 = 10,046.21
Hence, the amount of yearly withdrawal is P10,046.21.

Practice!
Direction: Answer as indicated.

1. Find the future value of an ordinary annuity with a regular payment of P1,000 at 5%
compounded quarterly for 3 years.

2. Find the present value of an ordinary annuity with regular quarterly payments worth
P1,000 at 3% annual interest rate compounded quarterly at the end of 4 years.
(see answer key to check your understanding)

Assessment
1. Find the amount and present value of an annuity of 1500 payable for years if money
is worth 10% compounded semi-annually.

2. How much must be deposited every month in a fund in order to have P200,000 at the
end of 10 years, if money is worth 8% compounded monthly?

Lesson 4: General Annuity

Objectives

At the end of this lesson, you will be able to:

a. Illustrate general annuities;

b. Find the future and present values of general annuities and compute the periodic
payment of a general annuity; and
c. Calculate the fair market value of a cash flow stream that includes an annuity

General Mathematics 7
 Annuities
Pre-test

Read the questions carefully. Write the letter that corresponds to your answer in
your notebook.

1. It is an annuity where the payment interval is not the same as the interest period.
a. Simple Annuity c. Annuity Certain
b. General Annuity d. Contingent Annuity

2. it is the amount of each payment.

a. Future Value of an annuity c. Annuity
b. Present Value of an annuity d. Periodic Payment

3. The sum of future values of all the payments to be made during the entire term of
annuity.
a. Future Value of an annuity c. Annuity
b. Present Value of an annuity d. Periodic Payment

4. The sum of all present values of all the payments to be made during the entire term
of the annuity.
a. Future Value of an annuity c. Time of Annuity
b. Present Value of an annuity d. Periodic Payment

5. It is a term that refers to payments received (cash inflow).

a. General Annuity c. Cash flow
b. General Ordinary Annuity d. Annuity Certain

General Mathematics 8
 Annuities
Preliminary
In the previous lessons, you learned to illustrate a Simple Annuity and you solve
the present and future values of simple Annuity. You also compute for the periodic
payment of simple annuity. As well as solve problems involving real life situations on
simple Annuities.

Lesson Proper
A GENERAL ANNUITY is an annuity where the length of the payment interval is
not the same as the length of the interest compounding period.

A GENERAL ORDINARY ANNUITY is a general annuity in which the periodic

payment is made at the end of the payment interval.

Examples of General annuity:

1. Monthly installment payment of a car, lo or house with an interest rate that is
compounded annually.
2. Paying a debt semi-annually when the interest is compounded monthly.

Future and Present Value of a General Ordinary Annuity

The Future value F and present value P of a general ordinary annuity is

given by:
𝑖 𝑚𝑡 𝑖 −𝑚𝑡
(1+ ) −1 1− (1+ )
𝑚 𝑚
𝐹 = 𝑅 [ 𝑖 ] and 𝑃 = 𝑅 [ 𝑖 ]
𝑚 𝑚

Note: j = 𝑖 𝑚 , n = mt
Where: R = is the regular payment

j = is the equivalent interest rate per payment interval converted from the
interest rate per period

n = the number of payments

Example 1: Cris started to deposit P1,000 monthly in a fund that pays 6% compounded
quarterly. How much will be in the fund after 15 years?

GIVEN: R = 1,000, n = 12(15) = 180 payments 𝑖 4 = 0.06, m = 4

Find F

SOLUTION: The Cash Flow for this problem is shown in the diagram below.

General Mathematics 9
 Annuities
(1) Convert 6% compounded quarterly to its equivalent interest rate for monthly
payment interval.

F1 = F2
12𝑡 4𝑡
𝑖 12 𝑖4
𝑃 (1 + ) = 𝑃 (1 + )
12 4
12𝑡 4𝑡
𝑖 12 𝑖4
(1 + 12
) = (1 + )
4
12
𝑖 12 0.06 4
(1 + 12
) = (1 +
4
)

12
𝑖 12
(1 + ) = (1.015)4
12

1
𝑖 12
(1 + ) = (1.015)4(12)
12

1
𝑖 12
= (1.015)3 − 1
12

𝑖 12
= 0.00497521 = 𝑗
12

Thus, the interest rate per monthly payment interval is 0.00497521%.

(2) Apply the formula in finding the future value of an ordinary annuity using the computed
equivalent rate.
(1 + 𝑗)𝑛 − 1
𝐹 = 𝑅 [ ]
𝑗
(1 + 0.00497521)180 − 1
𝐹 = 1,000 [ ]
0.00497521
𝐹 = 290,082.51

Thus, Cris will have P290,082.51 in the fund after 20 years.

Example 2: Ken borrowed an amount of money from Kat. He agrees to pay the
principal plus interest by paying P38, 973.76 each year for 3 years. How much money
did he borrow if the interest is 8% compounded quarterly?

GIVEN: R = 38,973.76, 𝑖 4 = 0.08, m = 4, n = 3 payments

Find P, Present Value

SOLUTION
The Cash Flow for this problem is shown in the diagram below.

General Mathematics 10
 Annuities
F1 = F2
1𝑡 4𝑡
𝑖1 𝑖4
𝑃 (1 + ) = 𝑃 (1 + )
1 4
1𝑡 4𝑡
𝑖1 𝑖4
(1 + 1 ) = (1 + )
4
1
𝑖1 0.08 4
(1 + 1 ) = (1 + 4
)
1
𝑖1
(1 + 1 ) = (1.02)4

𝑖1
= (1.02)4 − 1
1

1
𝑖 12
= (1.015)3 − 1
12

𝑖1
= 0.082432 = j = 8.24%
1

Thus, the interest rate per payment interval is 0.082432 or 8.24%.

(2) Apply the formula in finding the present value of an ordinary annuity using the
computed equivalent rate j = 0.082432.

1− (1+𝑗)−𝑛
𝑃 = 𝑅[ ]
𝑗

1− (1+0.082432)−3
𝑃 = 38,973.76 [ ]
0.082432

1− 0.7284462444
𝑃 = 38,973.76 [ ]
0.082432

0.2715537556
𝑃 = 38,973.76 [ ]
0.082432

P = 38,973.76[2.565829711]
P = 100,000

Hence, Ken borrowed P100,000 from Kat.

A cash flow is a term that refers to payments received (cash inflows) or

payments or deposits made (cash outflows). Cash inflows can be represented by
positive numbers and cash outflows can be represented by negative numbers.
The fair market value or economic value of a cash flow (payment stream)
on a particular date refers to a single amount that is equivalent to the value of the
payment stream at that date. This particular date is called focal date.

General Mathematics 11
 Annuities
Example 3: Mr. Ribaya received two offers on a lot that he wants to sell. Mr. Ocampo
has offered P50,000 and a P1million lump sum payment 5 years from now. Mr. Cruz
has offered P50,000 plus P40,000 every quarter for five years. Compare the fair market
value of the two offers if money can earn 5% compounded annually. Which offer has a
higher market value?

Mr. Ocampo’s Offer Mr. Cruz Offer

P50,000 down payment P50,000 down payment
P1,000,000 after 5 years P40,000 every quarter for 5 years

Find the market of each offer.

Choose a focal date and determine the values of the two offers at that focal date.
For example, the focal date can be the date at the start of the term.
Since the focal date is at t = 0, compute for the present value of each offer.

Mr. Ocampo’s Offer: Since P50,000 is offered today, then its present value is still P50,000.
The present value of P1,000,000 offered 5 years from now is

𝑃 = 𝐹 (1 + 𝑗)−𝑛
𝑃 = 1,000,000 (1 + 0.05)−5
𝑃 = 𝑃783, 526.20

𝐹𝑎𝑖𝑟 𝑀𝑎𝑟𝑘𝑒𝑡 𝑣𝑎𝑙𝑢𝑒 (𝐹𝑀𝑉) = 𝐷𝑂𝑊𝑁𝑃𝐴𝑌𝑀𝐸𝑁𝑇 + 𝑃𝑅𝐸𝑆𝐸𝑁𝑇 𝑉𝐴𝐿𝑈𝐸

𝐹𝑀𝑉 = 50,000 + 783, 526.20 𝐹𝑀𝑉 = 𝑃833,526.20

Mr. Cruz’s Offer: We first compute for the present value of a general annuity with quarterly
payments but with annual compounding period at 5%. Solve the equivalent rate,
compounded quarterly of 5% compounded annually.

General Mathematics 12
 Annuities
F1 = F2

4(5) 1(5)
𝑖4 𝑖1
𝑃 (1 + ) = 𝑃 (1 + )
4 1
20 5
𝑖4 𝑖1
(1 + 4 ) = (1 + )
1
20
𝑖4 0.05 5
(1 + 4 ) = (1 +
1
)

20
𝑖4
(1 + 4 ) = (1.05)5

1
20( ) 1
𝑖4 20
(1 + 4 ) = (1.05)5(20)
1
𝑖4
= (1.05)4 − 1
4
𝑖4
= 0.012272234 = 𝑗 = 1.23%
4

The present value of an annuity is given by

1−(1+ 𝑗)−𝑛
𝑃=𝑅 [ ]
𝑗

1−(1+ 0.012272234)−20
𝑃 = 40,000 [ ]
0.012272234

𝑃 = 705,572.70

FAIR MARKET VALUE (FMV) = DOWNPAYMENT + PRESENT VALUE

FMV = 50,000 + 705,572.70

FMV = 755,572.70

Hence, Mr. ocampo’s Offer has a higher market value. The difference between the market
values of the two offers at the start of the term is

833,526.20 – 756,572.70 = 𝑃77,953.50

Practice!

1. Example 2: Sandra borrowed an amount of money from Rowel. He agrees to pay the
principal plus interest by paying P38, 973.76 each year for 3 years. How much money
did he borrow if the interest is 8% compounded quarterly?
(see answer key to check your understanding)

General Mathematics 13
 Annuities
Assessment
Complete the sentence below. Write your answers on a separate sheet of paper.
1. _______________________ is an annuity where length of the payment interval is not
the same as the length of the interest compounding period.
2. _______________________ is general annuity in which the periodic payment is made
at the end of the payment interval.
3. _______________________ is a term that refers to payments received or payments
or deposits made.
4. _______________________ of a cash flow on a particular date refers to a single
amount that is equivalent to the value of the payment stream at that date.
5. _______________________ installments payment of a car, lot or house with an
interest rate that is compounded annually.

Lesson 5: Deferred Annuity

Objectives

At the end of this lesson, you will be able to:

a. Illustrate a Deferred Annuity
b. Find the present value of a deferred annuity
c. Calculate the period of deferral of a deferred annuity

Pre-test
Read the questions carefully. Write the letter that corresponds to your answer in
your notebook.

1. it is a kind of annuity whose payments (or deposits) start in more than one period
from the present.
a. Simple Annuity c. Deferred Annuity
b. General Annuity d. Term of Annuity

2. it is the length of time for which there are no payments.

a. Future Value of an annuity c. Deferment Period
b. Present Value of an annuity d. Periodic Payment

3. Annual payments of P2,500 for 24 years that will start 12 years from now. What is the
period of deferral in the deferred annuity?
a. 10 years c. 12 years
b. 11 years d. 13 years

4. The sum of all present values of all the payments to be made during the entire term
of the annuity.
a. Future Value of an annuity c. Time of Annuity
b. Present Value of an annuity d. Periodic Payment

5. Find the future value of an ordinary annuity with a regular payment of P1,000 AT 5%
interest rate compounded quarterly for 3 years.
a. P12,806.63 c. P12,806.36
b. P12,860.63 d. P12,860.36

General Mathematics 14
 Annuities
Preliminary
In this section, you will explore annuities whose payments do not necessarily start
at the beginning or at the end of the next compounding period. For instance, for certain
employee who will retire in 20 years, his pension will only start after 20 years.

Lesson Proper
A DEFERRED ANNUITY is a kind of annuity whose payments (or deposits) start
in more than one period from the present.

The time between the purchase of an annuity and the start of the payments for the
deferred annuity is called deferment period or period of deferral.

In the time diagram the period of deferral is k because the regular payments of R
start at the time k+1.
The rotation R* represent k”artificial payments”, each equal to R but are not
actually paid during the period of deferral.

The formula in finding the present value of a deferred annuity is

1 − (1 + 𝑗)−(𝑘+𝑛) 1 − (1 + 𝑗)−𝑘
𝑃 = 𝑅[ ]−𝑅[ ]
𝑗 𝑗

Where P = present value of deferred annuity

R = periodic payment
n = total number of payments (m x t)
k = period of deferment
𝑖
j = interest rate per conversion (𝑚)

Example 1: A sequence of quarterly payments of 3,500 each, with the first one due at
the end of 3 years and the last at the end of 8 years. Find the present value of the
deferred annuity, if money is worth 16% compounded quarterly.

Solution:

Given: R = 3,500 𝑖 = 16% 𝑜𝑟 0.16 𝑚=4

𝑖 0.16
𝑗=𝑚= = 0.04 𝑘 = 3 × 4 = 12 − 1 = 11 𝑛 = 𝑚𝑡 = 4 × 7 = 28 + 1 = 29
4

General Mathematics 15
 Annuities
1−(1+𝑗)−(𝑘+𝑛) 1−(1+𝑗)−𝑘
𝑃 = 𝑅[ ]−𝑅[ ]
𝑗 𝑗
1−(1+0.04)−(11+29) 1−(1+0.04)−11
𝑃 = 3,500 [ ] − 3,500 [ ]
0.04 0.04
1−(1.04)−40 1−(1.04)−11
𝑃 = 3,500 [ 0.04 ] − 3,500 [ ]
0.04
𝑃 = 69,274.71 − 30,661.67
𝑃 = 38,613.04

Practice!

1. A sequence of quarterly payments of 5,000 each, with the first one due at the end of 3
years and the last at the end of 10 years. Find the present value of the deferred annuity,
if money is worth 8% compounded quarterly.

Assessment
Directions: Answer the questions briefly. Write your answers in a separate sheet
of paper.
1. Differentiate Deferred Annuity and Period of Deferrral.
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________

2. What is a Deferred Annuity?

______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________

3. What is a period of deferral?

______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________

4. What is the formula in finding the present value of a deferred annuity? Identify each
variable represents.
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________

References
General Mathematics, Orlando, O. A., 2016
GENERAL MATHEMATICS Quarter 2 - Module 7: Annuities
Mathematics of Investment, Altares,Arce, et.al, 2007

General Mathematics 16
 Annuities
Answers Key
Pretest pg. 2 Pretest pg. 6 Pretest pg. 14
1. a 1. b 1. b
2. d 2. d 2. d
3. a 3. a 3. a
4.b 4.b 4.b
5. d 5. c 5. c

Practice pg. 7 #1 Practice pg. 7 #2

Given: R = 1,000, 𝑖 = 0.05, Given: R = 1,000, 𝑖 = 0.03,
𝑚 = 4, 𝑡 = 3 𝑦𝑒𝑎𝑟𝑠, 𝑚 = 4, 𝑡 = 4 𝑦𝑒𝑎𝑟𝑠,
𝑖 0.05 𝑖 0.03
𝑗=𝑚= = 0.0125, 𝑗=𝑚= = 0.0075,
4 4
n = 4x3=12 n = 4x4=16
(1+𝑗)𝑛 −1 1−(1+𝑗)−𝑛
𝐹 = 𝑅[ ] 𝑃 = 𝑅[ ]
𝑗 𝑗
(1+0.0125)12 −1 1−(1+0.0075)−16
𝐹 = 1,000 [ ] 𝑃 = 1,000 [ ]
0.0125 0.0075
𝐹 = 1,000[12.860361417839] 𝑃 = 1,000[15.024312610126]
𝐹 = 12,860.36 𝑃 = 15,024.31

Practice pg. 13
GIVEN: R = 38,973.76, 𝑖 4 = 0.08, m = 4, n = 3 payments
Find P, Present Value

SOLUTION
The Cash Flow for this problem is shown in the diagram below.

F1 = F2
1𝑡 4𝑡
𝑖1 𝑖4
𝑃 (1 + ) = 𝑃 (1 + )
1 4
1𝑡 4𝑡
𝑖1 𝑖4
(1 + 1 ) = (1 + )
4
1
𝑖1 0.08 4
(1 + 1 ) = (1 + 4
)
1
𝑖1
(1 + 1 ) = (1.02)4

𝑖1
= (1.02)4 − 1
1

1
𝑖 12
= (1.015)3 − 1
12

𝑖1
= 0.082432 = j = 8.24%
1

General Mathematics 17
 Annuities

Practice pg. 13
(2) Apply the formula in finding the present value of an ordinary annuity using the
computed equivalent rate j = 0.082432.

1− (1+𝑗)−𝑛
𝑃 = 𝑅[ ]
𝑗

1− (1+0.082432)−3
𝑃 = 38,973.76 [ ]
0.082432

1− 0.7284462444
𝑃 = 38,973.76 [ ]
0.082432

0.2715537556
𝑃 = 38,973.76 [ ]
0.082432

P = 38,973.76[2.565829711]
P = 100,000

Hence, Sandra borrowed P100,000 from Rowel.

Practice pg. 16
Given: R = 5,000 𝑖 = 8% 𝑜𝑟 0.08 𝑚=4
𝑖 0.08
𝑗=𝑚= = 0.02 𝑘 = 3 × 4 = 12 − 1 = 11 𝑛 = 𝑚𝑡 = 4 × 5 = 20 + 1 = 21
4

1−(1+𝑗)−(𝑘+𝑛) 1−(1+𝑗)−𝑘
𝑃 = 𝑅[ ]−𝑅[ ]
𝑗 𝑗
1−(1+0.02)−(11+21) 1−(1+0.02)−11
𝑃 = 5,000 [ ] − 5,000 [ ]
0.02 0.02
1−(1.02)−32 1−(1.02)−11
𝑃 = 5,000 [ 0.02 ] − 5,000 [ ]
0.02
𝑃 = 117,341.67 − 48,934.24
𝑃 = 68,407.43

General Mathematics 18