ENRICHMENT ACTIVITY
Make 2 examples of a situation where simple interest is illustrated.
Make 2 examples of a situation where compound interest is
illustrated.
Quarter 2 Week 1– A
(M11GM-IIa-1)
The learner illustrates simple and compound interests.
Republic Act 8293, section 176 states that: No copyright shall subsist in any
work of the Government of the Philippines. However, prior approval of the government
agency or office wherein the work is created shall be necessary for exploitation of such
work for profit. Such agency or office may, among other things, impose as a condition the
payment of royalties.
This MathPACKS is published to be utilized by the Schools Division of Iloilo.
ALL RIGHTS RESERVED. No part of this learning resource maybe reproduced
or transmitted in any form or by any means electronic or mechanical without written
permission from the Schools Division of Iloilo.
Writer:
Graphic Artist:
Layout Artist:
Division Quality Assurance:
EPS I - Mathematics:
Management Team:
JOHNLEX L. DIVINAGRACIA
PATRICK T. LOMIGO
KRISHEA MAE P. JARUDA
DR. KIM S. ARCEÑA
ROGER A. ALAVATA
DR. KIM S. ARCEÑA
Dr. Roel F. Bermejo, Dr. Nordy D. Siason, Jr.
Dr. Lilibeth T. Estoque, Dr. Azucena T. Falales
Ruben S. Libutaque, Lilibeth E. Larupay,
Dr. Kim S. Arceña
GENERALIZATIONS
Illustrating Simple and Compound Interests
Definition of Terms:
Lender or creditor – person (or institution) who invests the money or makes
fund available
Borrower or debtor - person (or institution) who owes the money or avails the
funds from the lender
Origin or load date – date on which money is received by the borrower
Repayment date or maturity date – date on which the money borrowed or
loaned is to be completely repaid
Time or term (𝒕) – amount of time in years the money is borrowed or invested;
length of time between the origin and maturity dates
Principal (𝑷) – amount of money borrowed or invested on the origin date
Rate (𝒓) – annual rate, usually in percent, charged by the lender, or rate of
increase of the investment
Interest 𝑰 – amount paid or earned for the use of money
Simple Interest (𝑰𝒔 ) – interest that is computed on the principal and then added
to it
Compound Interest (𝑰𝒄 ) – interest is computed on the principal and also on the
accumulated past interests
Maturity value or future value (𝑭) – amount after 𝑡 years that the lender
receives from the borrower on the maturity date .
EXAMPLE
Create situations that illustrate Simple and Compound interests.
SOLUTION
2. Bianca just graduated from college and is ready to buy her first car. She has
enough for a down payment but needs to borrow Php 20,000 to make the
purchase. The loan she gets has an annual interest rate (assessed using simple
interest) of 7 percent and a term of five years.
3. Ayesha has Php 19,000 in savings and decides to deposit it in a savings account
that offers a simple interest rate of 2 percent per year.
Compound Interest
1. An investor is happy with a modest 3% annual rate of return on her portfolio.
Her present Php 100,000 portfolio would, therefore, grow to Php 180,611 after 20
years. In contrast, a risk-tolerant investor who expects an annual return of 6% on
her portfolio would see Php 100,000 grow to Php 320,714 after 20 years.
2. Sam makes an initial investment of Php 10,000 for a period of 5 years. He
wants to know the amount of investment which he will get after the 5 years if the
investment earns the return of 6 % per annum compounded weekly.
3. Mr. Z makes an initial investment of Php 5,000 for a period of 3 years. Find
Simple interest
the value of the investment after the 3 years if the investment earns the return of
1. For example, assume you have a car loan of Php 20,000 with simple interest
10% compounded monthly.
at 4%. The loan is repayable over a five-year period in equal installments. Your
payment would work out to be Php 368.33 per month over 60 months. In the first
EXERCISE
month, the interest payable on the full Php 20,000 loan amount is Php 66.67
([20,000 x .04] / 12), which means that principal repayment is Php 301.66
(368.33 – 66.67). At the end of the first month, the principal amount is Php
19,698.34, on which interest payable is Php 65.66. The principal repayment in the
second month is Php 302.67, and so on. By the end of the 60th month, the loan
amount outstanding will be zero.
Create 3 situations that illustrate simple and compound interests.
General
Math
ENRICHMENT ACTIVITY
Determine whether the given situation exemplifies a simple
interest or compound interest. Answer the questions asked.
1. A planters corporation is lending its planters Php5, 000.00
for an interest of 10% per annum. How much
interest will be paid by the planters after two and
a half years?
2. A consumers cooperative is giving their member a
chance to invest in their Sari-sari Store business
an promises a 3% interest compounded annually.
How much interest would an investor earns if he
invested Php20, 000.00?
Quarter 2 Week 1– B
(M11GM-IIa-2)
The learner distinguishes between simple and compound interests.
Republic Act 8293, section 176 states that: No copyright shall subsist in any
work of the Government of the Philippines. However, prior approval of the government
agency or office wherein the work is created shall be necessary for exploitation of such
work for profit. Such agency or office may, among other things, impose as a condition the
payment of royalties.
This MathPACKS is published to be utilized by the Schools Division of Iloilo.
ALL RIGHTS RESERVED. No part of this learning resource maybe reproduced
or transmitted in any form or by any means electronic or mechanical without written
permission from the Schools Division of Iloilo.
Writer:
Graphic Artist:
Layout Artist:
Division
Quality Assurance:
Writer:
JOHNLEX L. DIVINAGRACIA
PATRICK T. LOMIGO
KRISHEA MAE P. JARUDA
DR. KIM S. ARCEÑA
ROGER A. ALAVATA
JOHNLEX L. DIVINAGRACIA
EPS I - Mathematics:
DR. KIM S. ARCEÑA
Layout Team:
Artist:
Management
KRISHEA MAE P. JARUDA
Dr. Roel F. Bermejo, Dr. Nordy D. Siason, Jr.
Dr. Lilibeth T. Estoque, Dr. Azucena T. Falales
EPS I - Mathematics
Ruben S. Libutaque, Lilibeth E. Larupay,
KIM S. ARCEÑA, EdD
Dr. Kim S. Arceña
Investment 2: Compound Interest (Bank)
GENERALIZATIONS
Simple Interest vs. Compound Interest
Simple Interest (𝑰𝒔 ) – interest that is computed on the principal and then
added to it
Compound Interest (𝑰𝒄) – interest is computed on the principal and also on
the accumulated past interests
Simple interest remain constant throughout the investment term. In compound
interest, the interest from the previous year also earns interest. Thus, the interest
grows every year.
Some differences between simple and compound interest:
1. The procedure in finding simple and compound interest;
2. The interest earned throughout the investment term, and
3. The formula used in solving the interest earned in each case.
Amount at
the start of
the year
𝑡
Rate
(𝑟)
1
10, 000.00
2
Compound Interest
Solution
Answer
Amount at the end of year 𝑡
(Maturity Value)
2%
(10, 000.00)(0.02)(1)
200.00
10, 000 + 200 = 10, 200.00
10, 200.00
2%
(10, 200.00)(0.02)(1)
204.00
10, 200 + 204 = 10, 404.00
3
10, 404.00
2%
(10, 404.00)(0.02)(1)
208.08
4
10, 612.08
2%
(10, 612.08)(0.02)(1)
212.24
5
10, 824.32
2%
(10, 824.32)(0.02)(1)
216.49
Time (𝑡)
10, 404 + 208.08
= 10, 612.08
10, 612.08
+ 212.24
= 10, 824.32
10, 824.32
+ 216.49
= 11, 040.81
EXAMPLE
Suppose Ressel Joy received her 13th month pay amounting to Php10, 000.00
and she plan to invest it for 5 years. A cooperative group offers 2% simple
interest per year. A bank offers 2% compounded annually. Which will she choose
and why?
Simple interest: 11, 000- 10, 000= 1, 000
Compound interest: 11, 040.81- 10, 000=1, 040.81
Since the interest incurred in investing Php 10, 000.00 for 5 years at 2%
compounded annually is higher than 2% simple interest per year, Ressel Joy
SOLUTION
would likely choose to invest in bank.
Investment 1: Simple Interest (Cooperative group)
EXERCISE
Amount after 𝑡
years
(Maturity Value)
Interest
rate
(𝑟)
Solution
Answer
1
2%
(10, 000.00)(0.02)(1)
200.00
10, 000 + 200 = 10,
200.00
2
2%
(10, 000.00)(0.02)(2)
400.00
10, 000 + 400 = 10,
400.00
3
2%
(10, 000.00)(0.02)(3)
600.00
10, 000 + 600 =10,
600.00
Time 𝑡
Principal
(𝑃)
10, 000.00
Simple Interest
4
2%
(10, 000.00)(0.02)(4)
800.00
10, 000 + 800 = 10,
800.00
5
2%
(10, 000.00)(0.02)(5)
1, 000.00
10, 000 + 1, 000 =
11, 000.00
Determine whether the given situation illustrates a simple interest or compound
interest.
1. How long will 1 million pesos earn a compound interest of Php200, 000.00
at 1% rate?
2. When invested at an annual interest rate of 7%, the amount earned
Php11, 200.00 of simple interest in two years. How much money
was originally invested?
EXAMPLE 3
General
Math
Solve for the ordinary and exact interest on Php1, 480.00 for 110 days at 5%.
SOLUTION
Given: 𝑃 = 1, 480
𝑟 = 5%
110
𝑡 = 360 𝑓𝑜𝑟 𝑜𝑟𝑑𝑖𝑛𝑎𝑟𝑦 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡
110
𝑡 = 365 𝑓𝑜𝑟 𝑒𝑥𝑎𝑐𝑡 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡
For Ordinary Interest:
110
𝐼𝑂 = 𝑃𝑟𝑡 = (1, 480)(0.05)(360)= 22.61
For Exact Interest:
𝐼𝐸 = 𝑃𝑟𝑡 = 1,480 0.05
110
= 22.30
365
EXERCISE 2
Solve for the exact and ordinary interest on Php12,000 for 90 days at 4%. Write
your answer in your answer sheet.
Quarter 2 Week 1– C
(M11GM-IIa-b-1)
The learner computes maturity value, and present value in simple
interest environment.
Republic Act 8293, section 176 states that: No copyright shall subsist in any
work of the Government of the Philippines. However, prior approval of the government
agency or office wherein the work is created shall be necessary for exploitation of such
work for profit. Such agency or office may, among other things, impose as a condition the
payment of royalties.
This MathPACKS is published to be utilized by the Schools Division of Iloilo.
ALL RIGHTS RESERVED. No part of this learning resource maybe reproduced
or transmitted in any form or by any means electronic or mechanical without written
permission from the Schools Division of Iloilo.
Writer:
Graphic Artist:
Layout Artist:
Division
Quality Assurance:
Writer:
JOHNLEX L. DIVINAGRACIA
PATRICK T. LOMIGO
KRISHEA MAE P. JARUDA
DR. KIM S. ARCEÑA
ROGER A. ALAVATA
JOHNLEX L. DIVINAGRACIA
EPS I - Mathematics:
DR. KIM S. ARCEÑA
Layout Artist:
Management
Team:
Dr. Roel F. Bermejo, Dr. Nordy D. Siason, Jr.
KRISHEA MAE P. JARUDA
EPS I - MathematicsDr. Lilibeth T. Estoque, Dr. Azucena T. Falales
Ruben S. Libutaque, Lilibeth E. Larupay,
KIM S. ARCEÑA, EdDDr. Kim S. Arceña
GENERALIZATION
Simple Interest
𝐼𝑠
𝑟𝑡
(a)𝑃=
Annual Simple Interest (𝑰𝒔)
𝑰𝒔 = 𝑷𝒓𝒕, where 𝑃 is the principal, or the amount invested or borrowed; 𝑟
is the simple interest rate; and 𝑡 is the term or time in years. Note: When the
𝑀
term is expressed in months 𝑀, it should be converted in years by 𝑡 = 12.
EXAMPLE 1
1,500
= (0.025)(4) = 15, 000
𝐼
4,860
( b ) 𝑟 = 𝑃𝑡𝑠 = (36,000)(1.5) = 0.09 = 9%
𝐼
275
( c ) 𝑡 = 𝑃𝑟𝑠 = (250,000)(0.005) = .22 𝑦𝑒𝑎𝑟
( d ) 𝐼𝑠 = 𝑃𝑟𝑡 = 500,000 0.125 10 = 625,000
How much interest is charged when Php50, 000.00 is borrowed for 9 month at an
annual interest rate of 10%?
SOLUTION
9
Given: 𝑃 = 50, 000 𝑟 = 10% = 0.10 𝑡 = 12 𝑦𝑒𝑎𝑟 = 0.75 𝑦𝑒𝑎𝑟
Find: 𝐼𝑠
𝐼𝑠 = 𝑃𝑟𝑡 = (50, 0000)(0.10)(0.75) = Php3, 750.00
EXERCISE 1
Complete the table below by finding the unknown. Write your answer in your
answer sheet.
EXAMPLE 2
Complete the table below by finding the unknown.
𝑃
Principal
𝑟
Rate
𝑡
Term or time in
years
𝐼𝑠
Interest
(a)
36, 000.00
250, 000.00
500, 000.00
2.5%
(b)
0.5%
12.5%
4
1.5
(c)
10
1, 5000.00
4, 860.00.
275.00
(d)
𝑃
Principal
𝑟
Rate
4,500.00
3,000.00
950.00
(d)
1%
3%
(c)
5.5%
𝑡
Term or time in
years
(a)
3 years
2 years
2.5 years
𝐼𝑠
Interest
8.00
(b)
75.00
270.00
SOLUTION
To find the unknown, use the triangle.
To use the triangle, we first determine the unknown.
If the unknown is rate, we cover ―Rate‖ in the triangle.
The remaining variables will be our formula. So our
𝐼
formula for Rate is 𝑟 = 𝑃𝑡𝑠 . If the ―Interest‖ is the
unknown value, we cover ―Interest‖ in the
triangle. Hence our formula in finding the
Interest is 𝐼𝑠 = 𝑃𝑟𝑡. The same rule applies in
finding the ―Principal‖ and the ―Time‖.
Principal
Ordinary and Exact Interest
Ordinary interest 𝐼𝑂 is interest computed for a given number of days, divided by
360.
𝑑𝑎𝑦𝑠
𝐼𝑂 = 𝑃𝑟 360
Exact interest 𝐼𝐸 is interest computed for a given number of days, divided by 365.
𝑑𝑎𝑦𝑠
𝐼𝐸 = 𝑃𝑟 365
Interest
Rate
Time
EXERCISE 2
General
Math
Complete the table by finding the unknown. Write your answers in your answer
sheet.
Present
value
𝑃
Rate
𝑟
Time
𝑡
Interest
𝐼𝑠
Maturity
value
𝐹
8,000.00
8%
6 years
(1)
(2)
5,000.00
(3)
1 month
(4)
6,000.00
(5)
12%
5 years and 3
months
(6)
400,000.00
Quarter 2 Week 1– D
(M11GM-IIa-b-1)
The learner computes maturity value, and present value in simple interest
environment.
Republic Act 8293, section 176 states that: No copyright shall subsist in any
work of the Government of the Philippines. However, prior approval of the government
agency or office wherein the work is created shall be necessary for exploitation of such
work for profit. Such agency or office may, among other things, impose as a condition the
payment of royalties.
This MathPACKS is published to be utilized by the Schools Division of Iloilo.
ALL RIGHTS RESERVED. No part of this learning resource maybe reproduced
or transmitted in any form or by any means electronic or mechanical without written
permission from the Schools Division of Iloilo.
Writer:
Graphic Artist:
Layout Artist:
Writer:
Division
Quality Assurance:
JOHNLEX L. DIVINAGRACIA
PATRICK T. LOMIGO
KRISHEA MAE P. JARUDA
DR. KIM S. ARCEÑA
ROGER
A. ALAVATA
JOHNLEX L. DIVINAGRACIA
EPS I - Mathematics:
DR. KIM S. ARCEÑA
Layout Artist:
Management Team:
KRISHEA MAE P. JARUDA
Dr. Roel F. Bermejo, Dr. Nordy D. Siason, Jr.
EPS I - Mathematics Dr. Lilibeth T. Estoque, Dr. Azucena T. Falales
KIM S. ARCEÑA, EdD Ruben S. Libutaque, Lilibeth E. Larupay,
Dr. Kim S. Arceña
GENERALIZATION
Find the actual and approximate days from Nov. 16, 2005 to June 13, 2008. Write
your answer in your answer sheet.
Actual and Approximate Time
EXAMPLE 1
Solve for (a) actual and (b) approximate number of days from March 18, 2006 to
November 30, 2006.
SOLUTION
Actual number of days
Mar (18-31)
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Total
EXERCISE 1
14
30
31
30
31
31
30
31
30
258 actual no. of days
Maturity/Future Value 𝑭 and Present Value 𝑷 of Simple Interest
Maturity/Future Value 𝑭
𝐹 = 𝑃 + 𝐼𝑠 𝑜𝑟 𝐹 = 𝑃(1 + 𝑟𝑡)
Where
𝐹 is the maturity/future value
𝑃 is the principal amount
𝑟 is the rate
𝑡 is the term/time in years
𝐼𝑠 is the simple interest
Present Value 𝑷
𝑃=
Where
𝐹
1 + 𝑟𝑡
𝑃 is the present value
𝐹 is the maturity/future value
𝑟 is the rate
𝑡 is the term/time in years
EXAMPLE 2
Find the maturity value if 1 million pesos is deposited in a bank at an annual
simple interest rate 0f 0.25% after 5 years.
Approximated number of days
SOLUTION
Year
Month
Day
2006
11
30
2006
-3
-18
8
12
Since all months are assumed to have only 30 days, then
8 𝑚𝑜𝑛𝑡ℎ𝑠 • 30 𝑑𝑎𝑦𝑠 = 240 + 12 = 252 𝑎𝑝𝑝𝑟𝑜𝑥. 𝑛𝑜. 𝑜𝑓 𝑑𝑎𝑦𝑠
Given: 𝑃 = 1,000,000 𝑟 = 0.25% = 0.0025
𝐹 = 𝑃 1 + 𝑟𝑡 = 1,000,000 [ 1 + 0.0025 5 = 1,012,500
The maturity value after 5 years is Php1, 012, 500.00
EXAMPLE 3
How much would you invest if the simple interest is 7.5% in order to have
Php300, 000.00 in 2 years?
SOLUTION
Given: 𝑟 = 7.5% = 0.075
𝐹 = 300,000
𝑡=2
𝐹
300,000
𝑃 = 1:𝑟𝑡 = 1:(0.075)(2) = 260,869.57
You should invest Php260, 869.57 to P300, 000.00 after 2 years
with 7.5% interest.
General
Math
OPTION B: Interest is compounded semi-annually, or every 6 months. Under this
option, the interest rate every six months is 1% (2% divided by 2).
Time (t) in
years
½
1
1½
2
2½
3
3½
4
4½
5
Principal = P10,000
Annual Int. rate = 2%, compounded semi-annually
Amount at the end of the year
10,000 * 1.01 = 10,100
10,100* 1.01 = 10,201
10,201* 1.01 = 10,303.01
10,303.01 * 1.01 = 10,406.04
10,406.04 * 1.01 = 10,510.10
10,510.10 * 1.01 = 10,615.20
10,615.20 * 1.01 = 10,721.35
10,721.35 * 1.01 = 10,828.56
10,828.56* 1.01 = 10,936.85
10,936.85 * 1.01 = 11,046.22
Answer: Option B will give the higher interest after 5 years
because a more frequent compounding will result in a higher
interest.
Quarter 2 Week 1– E
(M11GM-IIa-b-1)
The learner computes maturity value, and present value in compound
interest environment.
Republic Act 8293, section 176 states that: No copyright shall subsist in any
work of the Government of the Philippines. However, prior approval of the government
agency or office wherein the work is created shall be necessary for exploitation of such
work for profit. Such agency or office may, among other things, impose as a condition the
payment of royalties.
This MathPACKS is published to be utilized by the Schools Division of Iloilo.
ALL RIGHTS RESERVED. No part of this learning resource maybe reproduced
or transmitted in any form or by any means electronic or mechanical without written
permission from the Schools Division of Iloilo.
Writer:
Graphic Artist:
Layout Artist:
Division Quality Assurance:
JOHNLEX L. DIVINAGRACIA
PATRICK T. LOMIGO
KRISHEA MAE P. JARUDA
DR. KIM S. ARCEÑA
Writer:
ROGER A. ALAVATA
L. DIVINAGRACIA
EPS IJOHNLEX
- Mathematics:
DR. KIM S. ARCEÑA
Management
Team:
Layout Artist:
Dr. Roel F. Bermejo, Dr. Nordy D. Siason, Jr.
KRISHEA MAE P. JARUDA
Dr. Lilibeth T. Estoque, Dr. Azucena T. Falales
EPS I - Mathematics
Ruben S. Libutaque, Lilibeth E. Larupay,
Dr. Kim S. Arceña
KIM S. ARCEÑA, EdD
GENERALIZATIONS
Compound Interest
𝐼𝑐 = 𝐹 − 𝑃, where 𝐼𝑐 is the compound interest; 𝐹 is the maturity value; and 𝑃 is the
SOLUTION
Given: 𝐹 = 50,000
𝑟 = 10% = 0.10
𝑡 = 7 𝑦𝑒𝑎𝑟𝑠
𝐹
50,000
𝑃=
=
= 25, 657.91
(1 + 𝑟)𝑡 (1 + 0.10)7
The present value is Php25, 657.91
principal amount.
Maturity/Future Value and Compound Interest
𝐹 = 𝑃(1 + 𝑟)𝑡 , where 𝑃 is the principal amount; 𝐹 is the future value at the end of
the term; 𝑟 is the interest rate; and 𝑡 is term/time in years.
EXAMPLE 1
Find the (a) maturity value and (b) compound interest if Php10,000.00 is
compounded annually at an interest rate of 2% in 5 years.
SOLUTION
Given: 𝑃 = 10,000
𝑟 = 2% = 0.02
EXERCISE 2
How much money should a student place in time deposit in a bank that pays 1.1%
compounded annually so that he will have Php200,000.00 after 6 years? Write
your answer in your answer sheet.
Compounding More than Once a Year
𝑡 = 5 𝑦𝑒𝑎𝑟𝑠
𝐹 = 𝑃(1 + 𝑟)𝑡 = 10,000 1 + 0.02
5
= 11, 041.81
𝐼𝑐 = 𝐹 − 𝑃 = 11, 040.81 − 10,000.00 = 1, 040.81
The future value 𝐹 is P11,040.81 and the compound interest is Php1,040.81.
EXERCISE 1
Find the maturity value and interest if Php50,000 is invested at 5% compounded
annually for 8 years. Write your answer in your answer sheet.
Present value 𝑷 at Compound Interest
𝑭
𝑷=
(𝟏 + 𝒓)𝒕
EXAMPLE 3
Given a principal of Php10,000, which of the following options will yield greater
interest after 5 years:
OPTION A: Earn an annual interest rate of 2% at the end of the year, or
OPTION B: Earn an annual interest rate of 2% in two portions—1% after 6
months, and 1% after another 6 months?
SOLUTION
OPTION A: Interest is compounded annually
Time (t) in years
Where, 𝑃 is the principal amount; 𝐹 is the maturity value; 𝑟 is the interest rate;
and 𝑡 is the term/time in year.
Principal = P10,000
Annual Int. rate = 2%, compounded annually
Amount at the end of the year
1
10,000 * 1.02 = 10,200
EXAMPLE 2
2
10,200 * 1.02 = 10,404
3
10,404 * 1.02 = 10,612.08
What is the present value of Php50,000.00 due in 7 years if money is worth 10%
4
10,612.08 * 1.02 = 10,824.32
5
10,824.32 * 1.02 = 11,040.81
compounded annually?
General
Math
EXAMPLE 3
Find the present value of Php50,000 due in 4 years if money is invested at 12%
compounded semi-annually.
SOLUTION
Given: 𝐹 = 50,000
𝑡 = 4 𝑖 (2) = 0.12
First, compute for the interest rate per conversion period given
by
𝑖 (𝑚) 1(2)
=
= 0.06
𝑚
2
The total number of conversion periods is n = tm = (4)(2) = 8.
The present value can be computed by substituting these
values in the formula
𝐹
50,000
𝑃=
=
= 𝑃31, 370.62
𝑚
(1 + 0.06)8
𝑖
(1 + 𝑚 )𝑚𝑡
𝑗=
EXERCISE 2
What is the present value of Php25,000 due in 2 years and 6 months if
money is worth 10% compounded quarterly? Write your answer in your
answer sheet.
Quarter 2 Week 2 - A
(M11GM-IIb-2)
The learner solves problems involving simple and compound interests.
.
Republic Act 8293, section 176 states that: No copyright shall subsist in any
work of the Government of the Philippines. However, prior approval of the government
agency or office wherein the work is created shall be necessary for exploitation of such
work for profit. Such agency or office may, among other things, impose as a condition the
payment of royalties.
This MathPACKS is published to be utilized by the Schools Division of Iloilo.
ALL RIGHTS RESERVED. No part of this learning resource maybe reproduced
or transmitted in any form or by any means electronic or mechanical without written
permission from the Schools Division of Iloilo.
Writer:
JOHNLEX L. DIVINAGRACIA
Graphic Artist:
PATRICK T. LOMIGO
Writer:
Layout Artist:
KRISHEA MAE P. JARUDA
Division
Quality
Assurance:
DR. KIM S. ARCEÑA
JOHNLEX L. DIVINAGRACIA
ROGER A. ALAVATA
Layout Artist:
EPS I - Mathematics:
DR. KIM S. ARCEÑA
KRISHEA
MAE
P. JARUDA
Management
Team:
EPS I - Mathematics Dr. Roel F. Bermejo, Dr. Nordy D. Siason, Jr.
KIM S. ARCEÑA, EdD Dr. Lilibeth T. Estoque, Dr. Azucena T. Falales
Ruben S. Libutaque, Lilibeth E. Larupay,
Dr. Kim S. Arceña
GENERALIZATION
Maturity Value, Compounding m times a year
Definition of Terms: Frequency of conversion (m) – number of conversion
periods in one year
Conversion or interest period– time between successive conversions of
interest
Total number of conversion periods 𝒏
𝑛 = 𝑚𝑡 = 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑐𝑜𝑛𝑣𝑒𝑟𝑠𝑖𝑜𝑛 𝑋(𝑡𝑖𝑚𝑒 𝑖𝑛 𝑦𝑒𝑎𝑟𝑠)
Nominal rate(𝒊(𝒎) )– annual rate of interest
Rate (j) of interest for each conversion period
𝑖 (𝑚)
𝑎𝑛𝑛𝑢𝑎𝑙 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡
𝑗=
=
𝑚
𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑐𝑜𝑛𝑣𝑒𝑟𝑠𝑖𝑜𝑛
EXAMPLE 1
𝑚
𝑖
𝑚
)𝑚𝑡
Where, 𝐹 is the future/maturity value; 𝑃 is the principal amount; 𝑖
𝑚
is the
nominal rate of interest(annual rate); 𝑚 is the frequency of conversion; and 𝑡 is
the term/time in years.
EXAMPLE 2
Find the maturity value and interest if P10,000 is deposited in a bank at 2%
compounded quarterly for 5 years.
SOLUTION
Examples of nominal rates and the corresponding frequencies of conversion and
interest rate for each period:
𝑖𝑚
Nominal
rate(annual
rate of
interest)
2%
compounded
annually
𝑖 1 = 0.02
2%
compounded
semi-annually
𝑖 2 = 0.02
%
compounded
quarterly
𝑖 3 = 0.02
2%
compounded
monthly
12
i
= 0.02
2%
compounded
daily
365
i
= 0.02
𝐹 = 𝑃(1 +
𝑚
𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑐𝑜𝑛𝑣𝑒𝑟𝑠𝑖𝑜𝑛
1
𝑗
Interest rate
per
conversion
period
0.02
= 0.02
1
= 2%
Given: 𝑃 = 10,000
𝑖 (4) = 0.02
4
12
One
conversion
period
𝑗=
𝑖 (𝑚)
𝑚
=
𝑖 (4)
4
365
= 0.005
Compute for the total number of conversion periods given by
𝑛 = 𝑚𝑡 = 4 5 = 20 𝑐𝑜𝑛𝑣𝑒𝑟𝑠𝑖𝑜𝑛 𝑝𝑒𝑟𝑖𝑜𝑑𝑠
1 year
0.02
= 0.01
2
= 1%
6 months
0.02
4
= 0.005
= 0.5%
3 months
0.02
12
= 0.0016
= 0.16%
1 month
0.02
365
𝑚=4
Compute for the interest rate in a conversion period by
Compute for the maturity value using
𝐹 = 10,000
2
𝑡 = 5 𝑦𝑒𝑎𝑟𝑠
1+
𝑖4
4
4 5
= 11,048.96
The compound interest is given by 𝐼𝑐 = 𝐹 − 𝑃 = 11,048.96 −
10,000 = 𝑃1,048.96
EXERCISE 1
Find the maturity value and interest if P10,000 is deposited in a bank at 2%
compounded monthly for 5 years. Write your answer in your answer sheet.
Present Value 𝑷 at Compound Interest
𝑃=
1 day
𝐹
𝑖 𝑚 𝑚𝑡
(1:
)
𝑚
Where, 𝐹 is the future value; 𝑃 is the principal amount; 𝑖 𝑚 is the nominal rate of
interest(annual rate); 𝑚 is the frequency of conversion; and 𝑡 is the term/time in
years.
EXERCISE 2
General
Math
At what interest rate compounded quarterly will Php12, 000.00
double itself in 10 years? Write your answer in your answer
sheet.
Quarter 2 Week 2 - B
(M11GM-IIb-2)
The learner solves problems involving simple and compound interests.
.
Republic Act 8293, section 176 states that: No copyright shall subsist in any
work of the Government of the Philippines. However, prior approval of the government
agency or office wherein the work is created shall be necessary for exploitation of such
work for profit. Such agency or office may, among other things, impose as a condition the
payment of royalties.
This MathPACKS is published to be utilized by the Schools Division of Iloilo.
ALL RIGHTS RESERVED. No part of this learning resource maybe reproduced
or transmitted in any form or by any means electronic or mechanical without written
permission from the Schools Division of Iloilo.
Writer:
JOHNLEX L. DIVINAGRACIA
Graphic Artist:
PATRICK T. LOMIGO
Writer:
Layout Artist:
KRISHEA MAE P. JARUDA
Division
Quality
Assurance:
DR. KIM S. ARCEÑA
JOHNLEX L. DIVINAGRACIA
ROGER A. ALAVATA
Layout Artist:
EPS I - Mathematics:
DR. KIM S. ARCEÑA
KRISHEA
MAE
P. JARUDA
Management
Team:
EPS I - Mathematics Dr. Roel F. Bermejo, Dr. Nordy D. Siason, Jr.
KIM S. ARCEÑA, EdD Dr. Lilibeth T. Estoque, Dr. Azucena T. Falales
Ruben S. Libutaque, Lilibeth E. Larupay,
Dr. Kim S. Arceña
GENERALIZATION
Finding Interest Rate and Time in Compound Interest
Finding the Number of Periods n, for Compounded Interest:
Using the formula for maturity value F, present value P, and interest rate j,
𝐹 = 𝑃 1 + 𝑗 𝑛,
then log 𝐹 = log 𝑃 1 + 𝑗 𝑛 = 𝑛 log(1 + 𝑗)
log 𝐹
Thus, 𝑛 = log 𝑃(1:𝑗)
Finding the Interest Rate, j, per Conversion Period
Using the formula maturity value F, present value P, and interest rate j,
𝐹 =𝑃 1+𝑗 𝑛
Then
𝑛
𝐹
𝑃
Thus, 𝑗 =
Using 𝑗 =
=1+𝑗
𝑛
𝐹
−
𝑃
𝑖 (𝑚)
,
𝑚
1
then 𝑖 (𝑚) = 𝑚𝑗
EXAMPLE 1
How long will it take Php3,000 pesos to accumulate to Php3,500 in a bank savings
account at 0.25% compounded monthly?
EXAMPLE 2
At what nominal rate compounded semi-annually will Php10,000 accumulate to
Php15,000 in 10 years?
SOLUTION
Given: 𝑃 = 3,000
𝑗=
𝐹 = 3,500
𝑖 (12)
𝑚
=
𝑖 (12) = 0.25% = 0.0025 𝑚 = 12`
SOLUTION
0.0025
12
Substituting the given values in the maturity value formula 𝐹 = 1 + 𝑗 𝑛 results to
0.0025 𝑛
3500 = 3000(1 +
)
12
3500
0.0025 𝑛
= (1 +
)
3000
12
To solve for n, take the logarithms of both sides.
3500
0.0025 𝑛
log
= log(1 +
)
3000
12
3500
0.0025
log
= 𝑛 log(1 +
)
3000
12
3500
log 3000
𝑛=
= 740.00
0.0025
log(1 + 12 )
𝑛
Thus, payments must be made for 740 months, or 𝑡 = 𝑚 =
740
12
= 61.67 𝑦𝑒𝑎𝑟𝑠.
EXERCISE 1
How long will it take Php1,000 to earn Php300 if the interest is 12% compounded
semi-annually? Write your answer in your answer sheet.
Given: 𝐹 = 15,000
𝑃 = 10,000
𝑡 = 10 𝑚 = 2 𝑛 = 𝑚𝑡 = 2 10 = 20
𝐹 = 1+𝑗 𝑛
15,000 = 10,000 1 + 𝑗 20
15,000
= 1 + 𝑗 20
10,000
1.5 = 1 + 𝑗 20
1
(1.5)20 = (1 + 𝑗)
1
(1.5)20 −1 = 𝑗
𝑗 = 0.0205
The interest rate per conversion period is 2.05%.
The nominal rate (annual rate of interest) can be computed by
𝑖 (𝑚) 𝑖 (2)
𝑗=
=
𝑚
2
𝑖 (2)
0.0205 =
2
𝑖 (2) = 0.0205 2 = 0.0410 = 4.10%
Hence, the nominal rate is 4.10%.
E. Peter borrowed Php100,000 at 8% compounded annually?
How
much will he be paying after 2 years?
General
Math
F. What amount must be deposited by a student in a bank that
pays 2 % compounded annually so that after 12 years he will
have Php100,000?
G. Ressel Joy wants to compare the simple interest to compound
interest on a Php60,000 investment.
a. Find the simple interest if funds earn 8% simple
interest for 5 year.
b. Find the interest if funds earn 8% compounded
annually for 5 year
Quarter 2 Week 2 - C
(M11GM-IIb-2)
The learner solves problems involving simple and compound interests.
.
Republic Act 8293, section 176 states that: No copyright shall subsist in any
work of the Government of the Philippines. However, prior approval of the government
agency or office wherein the work is created shall be necessary for exploitation of such
work for profit. Such agency or office may, among other things, impose as a condition the
payment of royalties.
This MathPACKS is published to be utilized by the Schools Division of Iloilo.
ALL RIGHTS RESERVED. No part of this learning resource maybe reproduced
or transmitted in any form or by any means electronic or mechanical without written
permission from the Schools Division of Iloilo.
Writer:
Graphic Artist:
Layout Artist:
Division Quality Assurance:
EPS I - Mathematics:
Management Team:
JOHNLEX L. DIVINAGRACIA
PATRICK T. LOMIGO
KRISHEA MAE P. JARUDA
DR. KIM S. ARCEÑA
ROGER A. ALAVATA
DR. KIM S. ARCEÑA
Dr. Roel F. Bermejo, Dr. Nordy D. Siason, Jr.
Dr. Lilibeth T. Estoque, Dr. Azucena T. Falales
Ruben S. Libutaque, Lilibeth E. Larupay,
Dr. Kim S. Arceña
GENERALIZATIONS
ENRICHMENT ACTIVITY
Definition of terms:
Equivalent rates – two annual rates with different conversion periods
A. Complete the table below by finding the unknown.
that will earn the same maturity value for the same time/term
Nominal rate – annual interest rate (may be compounded more than
once a year)
Effective rate– rate when compounded annually will give the same
compound each year with the nominal rate; denoted by 𝑖 1 .
EXAMPLE
What effective rate is equivalent to 10% compounded quarterly?
SOLUTION
Given: 𝑖 (4) = 0.10
𝑚=4
Since the equivalent rates yield the same maturity value, then 𝐹1 = 𝐹2
𝑖 4 𝑚𝑡
𝑡
𝑃 1 + 𝑖 (1) = 𝑃(1 +
)
𝑚
𝑡
𝑖4
Dividing both sides by P results to 1 + 𝑖 (1) = (1 + 𝑚 )𝑚𝑡
Raise both sides to 1/t to obtain 1 + 𝑖
4
= (1 +
𝑖4 𝑚
)
𝑚
)4
4
0.10 4
1 + 𝑖 1 = (1 +
)
4
1
4
𝑖 = (1 + 0.025) −1
𝑖 1 = (1.025)4 −1
𝑖 1 = 1.103813 − 1
1+𝑖
1
𝑖
1
= (1 +
𝑖 (1) = 0.103813 0𝑟 10.3813%
EXERCISE
Complete the table by computing for the rates equivalent to the following nominal
rates. Round off your answer to six decimal places.
Given Interest Rates
Equivalent Interest Rates
12% compounded monthly
__________ compounded annually
8% compounded semi-annually
__________ compounded quarterly
12% compounded monthly
__________ compounded semi-annually
to
𝑃
Principal
𝑟
Rate
𝑡
Term or time in
years
𝐼𝑠
Interest
2,750.00
a
2 years
75.00
12, 500.00
8%
b
35, 800.60
4.25%
2 years and 3
months
c
61.20
d
3.60%
270 days
50.00
B. Php150,000.00 was invested at a local bank from June 7, 2002
January 20, 2006 at 4.5%. find the exact and ordinary interest
using:
a. Exact time
b. Approximate time
C. What are the amounts of interest and maturity value of a loan
for Php25,000 at 12% simple interest for 5 years?
D. How much money should you deposit in a bank so that it will
accumulate to Php100,000 at 1% simple annual interest for 10
years?
ANSWER KEY
QUARTER 2 WEEK 1-A
EXERCISE 1: ANSWERS MAY VARY
ENRICHMENT ACTIVITY: ANSWERS MAY VARY
QUARTER 2 WEEK 1-B
EXERCISE 1:
1. Compound interest
2. Simple interest
ENRICHMENT ACTIVITY:
1. Simple interest. Php1, 250.00
2. Compound interest. Php600.00
Quarter 2 Week 1 – C
EXERCISE 1:
a. 0.1778
b. 270.00
c. 3.947%
d. 1963.63
EXERCISE 2:
Exact interest: Php118.356
Ordinary interest: Php120.00
Quarter 2 Week 1 – D
EXERCISE 1:
Actual Days: 941
Approximate Days: 927
EXERCISE 2:
1. Php3,840.00
2. Php11,840.00
3. 240%
4. Php1, 000.00
5. Php245,398.773
6. Ph154,601.227
Quarter 2 Week 1 – E
EXERCISE 1:
Maturity value: Php73, 872.77
Interest: Php23, 872.77
EXERCISE 2:
Present value: Php187, 293.65
Quarter 2 Week 2– A
EXERCISE 1:
Maturity value: Php11,050.79
Interest: Php1,050.79
EXERCISE 2:
Present value: Php19, 529.96
Quarter 2 Week 2–B
EXERCISE 1:
Time: 8.88 mos or .74 yr
EXERCISE 2:
Interest rate: 0.07 or 7%
Quarter 2 Week 2–C
EXERCISE 1:
1. 1%
2. 7.92%
3. 12.30%
ENRICHMENT ACTIVITY
A.
a. 1.36%
b. Php2,250.00
c. 0.04 yrs
d. Php1,851.00
B. EXACT INTEREST
Using:
Exact/Actual time
Interest=Php24,484.93
Maturity=Php174,484.93
Approximate time
Interest=Php24,096.56
Maturity=Php174,096.56
ORDINARY INTEREST
Using:
Exact/Actual time
Interest=Php24,825.00
Maturity=Php174,825.00
Approximate time
Interest=Php24,431.25
Maturity=Php174,431.25
Quarter 2 Week 2–C
Enrichment Activity
C.
Interest = Php15,000.00
Maturity value= Php40,000.00
D. Php90,909.09
E. Php116,640.00
F. Php78,849.32
G.
a. Php4,800.00
b. Php4,800.00