General Mathematics Senior High School PDF 2021

General Mathematics Quarter 2 – Module 2: Interest, Maturity, Future, and Present Values in Simple and Compound Interests CO_Q2_General Mathematics SHS Module 2 General Mathematics – Senior High School Alternative Delivery Mode Quarter 2 – Modu...

General Mathematics Quarter 2 – Module 2: Interest, Maturity, Future, and Present Values in Simple and Compound Interests CO_Q2_General Mathematics SHS Module 2 General Mathematics – Senior High School Alternative Delivery Mode Quarter 2 – Module 2: Interest, Maturity, Future, and Present Values in Simple First Edition, 2021 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. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them. Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio Development Team of the Module Writer: Ann Michelle M. Jolo Editors: Elizabeth D. Lalunio, Elizabeth B. Dizon Anicia J. Villaruel, Roy O. Natividad Reviewers: Fritz A. Caturay, Necitas F. Constante, Dexter M. Valle, Jerome A. Chavez, Mary Grace D. Constantino and Jasmin J. Flores Illustrator: Hanna Lorraine Luna and Diane C. Jupiter Layout Artist: Roy O. Natividad, Sayre M. Dialola, Noel Rey T. Estuita and Argie L. Ty Management Team: Francis Cesar B. Bringas Job S. Zape, Jr. Ramonito Elumbaring Reicon C. Condes Elaine T. Balaogan Fe M. Ong-ongowan Hermogenes M. Panganiban Phillip B. Gallendez Josephine T. Natividad Anicia J. Villaruel Dexter M. Valle Printed in the Philippines by ________________________ Department of Education – Region 4A CALABARZON Office Address: Gate 2 Karangalan Village, Brgy. San Isidro, Cainta, Rizal Telefax: 02-8682-5773/8684-4914/8647-7487 E-mail Address: [email protected] General Mathematics Quarter 2 – Module 2: Interest, Maturity, Future, and Present Values in Simple and Compound Interests Introductory Message This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson. Each SLM is composed of different parts. Each part shall guide you step-by- step as you discover and understand the lesson prepared for you. Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these. In addition to the material in the main text, Notes to the Teacher are also provided to our facilitators and parents for strategies and reminders on how they can best help you on your home-based learning. Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task. If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator. Thank you.  What I Need to Know On your previous module, you have already illustrated and distinguished the difference between simple and compound interests. A deeper analysis of this topic will be the focus of this module wherein the relationships among the variables in solving the simple and compound interest will be explored. You will realize the importance of deriving a certain formula to compute the required variable involved in interest. The connection among interest, principal or present value, rate, time and maturity value will be reiterated. This topic will also revolve around money and since it is talking about money you will learn more on how to make decisions that concerns about it. Are you now ready for the new lesson, if so you may proceed to this module and have fun while learning. The module is composed of two lessons, namely: Lesson 1 – Interest, Maturity, Future, and Present Values in Simple Interest Lesson 2 – Interest, Maturity, Future, and Present Values in Compound Interest After going through this module, you are expected to: 1. compute interest, maturity value, future value, and present value in simple interest environment; 2. compute interest, maturity value, future value, and present value in compound interest environment; and 3. derive the formula of simple and compound interest to compute the maturity, future, and present value. 1 CO_Q2_General Mathematics SHS Module 2  What I Know Let’s find out how far you might already know about this topic by answering the assessment below. Choose the letter of the best answer. Write the chosen letter on a separate sheet of paper. 1. Which of the following is the formula to find the simple interest? a. 𝐼𝐼𝑠𝑠 = 𝑃𝑃𝑃𝑃𝑃𝑃 b. 𝐼𝐼𝑠𝑠 = 𝑃𝑃(1 + 𝑟𝑟)𝑡𝑡 c. 𝐼𝐼𝑠𝑠 = 𝐹𝐹 − 𝑃𝑃 𝐹𝐹 d. 𝐼𝐼𝑠𝑠 = (1+𝑟𝑟)𝑡𝑡 2. What formula will be used to find the present value of simple interest? a. 𝑃𝑃 = 𝐼𝐼𝑠𝑠 𝑟𝑟𝑟𝑟 𝐼𝐼𝑠𝑠 b. 𝑃𝑃 = 𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟 c. 𝑃𝑃 = 𝐼𝐼𝑠𝑠 𝐼𝐼 𝑟𝑟 d. 𝑃𝑃 = 𝑠𝑠 𝑡𝑡 3. If the investment amounting to ₱35,000 earned an interest of ₱2,500 how much will be the maturity value? a. ₱32,500 b. ₱37,500 c. ₱30,000 d. ₱40,000 4. Given that P = ₱5,250, r = 1.25% and t = 5 years, find the simple interest. a. ₱32,812.50 b. ₱3,281.25 c. ₱328.13 d. ₱32.82 1 5. Given that P = ₱10,500, r = 4 % and t = 8 months, find the simple interest. 2 a. ₱315 b. ₱3,150 c. ₱3,780 d. ₱31,500 6. Which of the following is the formula to find the compound interest? a. 𝐼𝐼𝑐𝑐 = 𝑃𝑃(1 + 𝑟𝑟)𝑡𝑡 b. 𝐼𝐼𝑐𝑐 = 𝐹𝐹 − 𝑃𝑃 𝐹𝐹 c. 𝐼𝐼𝑐𝑐 = (1+𝑟𝑟)𝑡𝑡 𝑖𝑖 𝑚𝑚 𝑚𝑚𝑚𝑚 d. 𝐼𝐼𝑐𝑐 = 𝑃𝑃(1 + ) 𝑚𝑚 2 CO_Q2_General Mathematics SHS Module 2 7. What is the frequency of conversion if the annual rate is compounded quarterly? a. 2 c. 4 b. 3 d. 12 8. If the interest rate is 5% compounded monthly, what is the interest rate per conversion period? a. 2.5% c. 1.25% b. 1.67% d. 0.42% 9. Given that 𝐼𝐼𝑠𝑠 = 750, r = 5% and t = 3 years, what is the principal or present value? a. ₱50 b. ₱500 c. ₱5.000 d. ₱50,000 10. If F = ₱25,000 and P = ₱20,000, how much is the compound interest? a. ₱45,000 b. ₱25,000 c. ₱20,000 d. ₱5,000 For numbers 11 – 12, use the following values: F = 40,000, t = 3 ½ years, 𝑖𝑖 4 = 0.10 and m= 4 11. What is the present value? a. ₱30, 000.25 b. ₱28, 309.09 c. ₱25, 307.15 d. ₱20, 432.75 12. How much is the compound interest? a. ₱9,999.75 b. ₱11,690.91 c. ₱14,692.85 d. ₱19,567.25 13. If the maturity value is ₱23,000 and the compound interest is ₱3,500, how much is the present value? a. ₱26,500 b. ₱23,000 c. ₱19,500 d. ₱15,000 For numbers 14 – 15, use the following values: P = ₱15,000, 𝑖𝑖12 = 5%, t = 4 years, m = 12. 14. How much is the maturity value? a. ₱18, 313.43 b. ₱17, 413.37 c. ₱16, 313.37 d. ₱15, 413.43 15. How much is the compound interest? a. ₱413.43 b. ₱1,313.37 c. ₱2,413.37 d. ₱3, 313.43 3 CO_Q2_General Mathematics SHS Module 2  Lesson Interest, Maturity, Future, 1 and Present Values in Simple Interest Business transactions are common events we experienced in our daily lives like banking, financing, marketing and pawning activities which involves withdrawals, deposits, used of credit cards and others. When these transactions occur there are two parties involved the lender and the borrower hence there involves a particular amount which we call interest. Everything that we have right now is just borrowed, our talents, jobs, and even our lives. God is the sole person who does not charge interest from what He lent. However, the money that we borrowed or loaned from others once paid earned an interest which will be tackled in this module. What’s In For you to begin consider the previous lesson essential in obtaining success in this module. In the last module, you differentiate simple and compound interest. Simple interest is computed by multiplying the principal (P) by the rate (r) and the length of time (t) (I=Prt) while compound interest is computed on the principal plus the accumulated past interests. Different terms related to simple and compound interest were also given emphasis such as the lender or creditor which refers to the person who invests or makes funds available and the borrower or debtor, which refers to the person who owes the money. Moreover, different terms essential in the interest formula were also explained such as time or term, principal or present value, rate, and maturity value. However, additional terms for compound interest were also given importance such as frequency of conversion, nominal rate, and rate of interest for each conversion period. These terms will be useful for you to have a better grasp of this module. 4 CO_Q2_General Mathematics SHS Module 2 Other skills such as expressing percent to decimals is also necessary for example: Express the following as decimal: 1. 12% - 0.12 2. 10.5% - 0.105 3. 300% - 3 1 4. 8 % - 0.85 2 Notes to the Teacher Use of calculator in this module is allowed because it will help them to easily compute what is asked, however reiterate to the learners that to ensure the accuracy and precision of the solution the use of correct formula is necessary. Also, inform them that in other books different variables were used to represent the components of interest formula, it will not affect the result as long as the relationship with other variables is the same with the relationship to be presented in this module. What’s New Read and analyze the situation below. Let’s Save Janice and Jamaica are both senior high school students. After class the two had a conversation: Janice: Wow, your cellphone is so cool! Is that new? Jamaica: Thank you! And yes, I just bought it yesterday. Janice: Did your mother give you money? What will you do with your old cellphone? 5 CO_Q2_General Mathematics SHS Module 2 Jamaica: No, I saved up my allowance to accumulate enough money to buy this phone. I am still using my old phone. I will use my new phone for my social media accounts. Janice: How much is that? Jamaica: It is only ₱2,300 but it is already android so I can download different applications like Tiktok, Snapchat and games. Therefore, I will not be bored! Janice: Oh! We are almost the same, I also saved ₱2,300 (already) from my allowance Jamaica: What (do you plan) will you buy from your savings? Janice: My old cellphone is still working, so I think there is no need for me to buy a new one. It will be better if I save this money. Jamaica: But your money might get lost if you will not spend it? Janice: I saw from an advertisement in Cooperative Bank that the minimum amount to open an account for students is ₱2,000, and they call it Kid Savers. Jamaica: Is there an advantage if you put the money on the bank? Janice: According to the advertisement there will be 2.5% interest in a year. I will use it in my college education. Questions: 1. What is the dialogue all about? ___________________________________________________________ 2. Do you think saving money is important? Explain your answer. ________________________________________________________________ 2. Do you think it is wrong for Jamaica to buy a new cellphone? ___________________________________________________________ 3. What can you say about Janice’s attitude towards money? ___________________________________________________________ 4. How much will Janice save after two years? ___________________________________________________________ 5. How do you value education? ___________________________________________________________ 6 CO_Q2_General Mathematics SHS Module 2  What is It From the previous conversation, you can say that allowance is one of the sources of funds of ordinary students. A person may decide on what they want to do with their money. In the scenario, there are two kinds of students. One who saves to buy the things they wanted and the other one saves to invest for her future. In doing so investing money in the bank earns interest. In computing the simple interest and other related components, the formula is 𝐼𝐼𝑠𝑠 = 𝑃𝑃𝑃𝑃𝑃𝑃 where: Is = simple interest P = principal or the amount invested or borrowed or present value r = simple interest rate t = time or term in years The formula can be manipulated to obtain the following relationships: The formula for finding the principal amount 𝐼𝐼𝑠𝑠 𝑃𝑃 = 𝑟𝑟𝑟𝑟 The formula for finding the rate 𝐼𝐼𝑠𝑠 𝑟𝑟 = 𝑃𝑃𝑃𝑃 The formula for finding the time 𝐼𝐼𝑠𝑠 𝑡𝑡 = 𝑃𝑃𝑃𝑃 7 CO_Q2_General Mathematics SHS Module 2 To find the maturity (future) value, you can use either of the following: 𝐹𝐹 = 𝑃𝑃(1 + 𝑟𝑟𝑟𝑟) or 𝐹𝐹 = 𝑃𝑃 + 𝐼𝐼𝑠𝑠 where: F = maturity (future) value Is = simple interest P = principal or the amount invested or borrowed or present value r = simple interest rate t = time or term in years Let us take the following for example: Example 1: Given: 𝑃𝑃 = ₱18, 500, 𝑟𝑟 = 0.03, 𝑡𝑡 = 5. Find simple interest (𝐼𝐼𝑠𝑠 ) Solution: Use the formula of Simple Interest 𝐼𝐼𝑠𝑠 = 𝑃𝑃𝑃𝑃𝑃𝑃 Substitute the given to the formula 𝐼𝐼𝑠𝑠 = 18,500(0.03)(5) Performing the operation 𝐼𝐼𝑠𝑠 = ₱2,775 Therefore, the simple interest is ₱2,775 Example 2: Given: 𝑃𝑃 = ₱20,000, 𝐼𝐼𝑠𝑠 = ₱4,000, 𝑡𝑡 = 4. Find the rate (𝑟𝑟) Solution: 𝐼𝐼 Use the formula in finding the rate 𝑟𝑟 = 𝑃𝑃𝑃𝑃𝑠𝑠 4000 Substitute the given to the formula 𝑟𝑟 = (20,000)(4) Performing the operations r = 0.05 Express your answer in percent r = 5% Therefore, the rate of interest is 5%. Example 3: Given: 𝑃𝑃 = ₱40,000., 𝐼𝐼𝑠𝑠 = ₱700, 𝑟𝑟 = 7%. Find time (𝑡𝑡). Solution: 𝐼𝐼 Use the formula in finding the time 𝑡𝑡 = 𝑃𝑃𝑃𝑃𝑠𝑠 700 Substitute the given to the formula 𝑡𝑡 = (40000)(0.07) Performing the operations t = 0.25 Express your answer as unit of time t= ¼ year or 3 mos. Therefore, the term or time in years is ¼ year or 3 months. 8 CO_Q2_General Mathematics SHS Module 2 Example 4: Given: 𝑃𝑃 = ₱15,000, 𝑡𝑡 = 4 months, 𝑟𝑟 = 2%. Find maturity (future) value (𝐹𝐹). Solution: Use the formula of maturity value 𝐹𝐹 = 𝑃𝑃(1 + 𝑟𝑟𝑟𝑟) 1 Substitute the given to the formula 𝐹𝐹 = 15,000(1 + (0.02) ) 3 4𝑚𝑚𝑚𝑚𝑚𝑚. 1 Hint: time must be expressed in years so 4 months will become = 𝑦𝑦𝑦𝑦𝑎𝑎𝑎𝑎 12𝑚𝑚𝑚𝑚𝑚𝑚 3 Performing the operations F = ₱15,100 Alternative Solution: 1 Solve first the simple interest 𝐼𝐼𝑠𝑠 = (15,000)(0.02)( ) 3 Is = 100 Use the formula 𝐹𝐹 = 𝑃𝑃 + 𝐼𝐼𝑠𝑠 F = 15,000 +100 Therefore, the maturity value is ₱15,100 What’s More Activity 1.1 1. If P = ₱4,500, r = 1.25% and t = 5 years, find the simple interest. a. What formula will be used? _______________________ b. How are you going to express the rate in decimal? ______________ c. How much is the simple interest? ___________________________ 2. If P = ₱5,000, = 2% and t = 8 mos., find the maturity value. a. Which formula will you use? __________________________ b. How are you going to express the time in years? __________________ c. How much is the maturity value? _________________________ Activity 1.2 9 CO_Q2_General Mathematics SHS Module 2 Find the value of the required components in each item. 1. Find the simple interest and maturity value if P = ₱13,000, r = 0.8% and t= 2 ½ years. 2. If Is = ₱625, r = 2% and t =3 years, find the present value Activity 1.3 Complete the table below by finding the unknown. Principal Rate Time Interest Maturity Value ₱45,000 2% 2.5 years 1.2% 4 years ₱4,560 ₱105,000 3% ₱114,450 Guide Questions: For the first set of values a. What unknown variable will you solve first? b. What formula will you use to find the missing values? For the second set of values a. Which among the missing components can be solved using the given? b. How will you solve the next missing component? For the third set of values a. What formula will you use to find the missing value? b. Which component can be solved first using the given? What I Have Learned 10 CO_Q2_General Mathematics SHS Module 2 Read the paragraph carefully and supply the missing term/s in the blanks that will make it logical. In doing a business transaction one of the essential things to consider is the interest because it will be the basis whether you have gained or lost. When the interest is computed based on the principal it is called ______________ its formula is ______________. There are other components included in the simple interest formula such as principal or present value, rate, time, and maturity value. In finding the principal the formula ______________ will be used. Meanwhile the rate must be expressed in ________ and in finding its value given the simple interest, principal and time you will use the formula__________. Moreover, time should be expressed in ________. In a simple interest environment, the formula to be used in finding the time is _____________. Lastly maturity value can be obtained by adding the __________ and _________ or if the simple interest is not given you can use the formula ___________________. What I Can Do Money Matters Read and analyze the situation below, then answer the question that follows. You are a new accounts clerk in Lucena Metropolitan Bank where you met Mr. and Mrs. Smith who are planning for the education of their children in the future. You introduce to them the advantage of time deposit having the following features: Option A: 1.10% interest annually in 3 years, Option B: 1.25% interest annually in 5 years Option C: 1.75% interest annually in 8 years The couple has an initial amount of ₱50,000 to be deposited. To help the couple in deciding the terms of their investment make a proposal by completing the table below. PROPOSAL PRINCIPAL TIME RATE INTEREST AMOUNT ₱50,000 ₱50,000 11 CO_Q2_General Mathematics SHS Module 2  ₱50,000 Aside from the amount to be yielded at the end of each term what are the other factors that you can consider? List down the possible questions that you will ask to the couple to determine those factors. ______________________________________________________________________________ ______________________________________________________________________________ Is there any disadvantage in choosing each term? What are those? ______________________________________________________________________ ______________________________________________________________________ What advice or tips can be given to the couple in choosing the term of interest? _______________________________________________________________________ _______________________________________________________________________ Rubrics for the Task Categories Excellent Fair Poor 3 2 1 Relevance of the Excellent Some Little to no advice understanding in understanding in understanding in creating advice for creating advice for creating advice for choosing a plan choosing a plan choosing a plan Interview Skills Excellent set of A fair set of A poor set of questions that questions that questions that elicit the answer to elicit the answer to elicit the answer to help in the help in the help in the decision making decision making decision making process is given process is given process is given Accuracy of The computations The computations There is no Solution made are all made have flaws attempt in making correct computation 12 CO_Q2_General Mathematics SHS Module 2  Lesson Interest, Maturity, Future, 2 and Present Values in Compound Interest The previous lesson reiterates the importance of simple interest in a certain transaction. Interest is a natural event in a business transaction, however not all interest is classified as simple interest some of them are considered compound interest. This lesson will delve into compound interest and the different components involved in it such as present value and maturity value. What’s In For you to begin considering the lesson on the previous module which is essential in obtaining success in this lesson. Compound interest is not like a simple interest wherein only the principal is considered in the computation this type of interest considers the principal and the accumulated past interest. The frequency of conversion, nominal rate, and rate of interest for each conversion period will also play an important role in this lesson. In the previous lesson the maturity value is computed using the formula 𝐹𝐹 = 𝐼𝐼𝑠𝑠 + 𝑃𝑃 where F is the future value , Is is the simple interest and P the present value or the Principal and to find the principal or present value given the interest and maturity value the formula 𝑃𝑃 = 𝐹𝐹 − 𝐼𝐼𝑠𝑠 can be employed. Let us take the following example. Example 1: Given: P = ₱35,000 and Is = ₱ 4,000, find F. Solution: F F = ₱35,000 + ₱4,000 = ₱39,000 Example 2: Given: F = ₱50,000 and P = ₱45,000, find Is. Solution: Is = ₱50,000 – ₱45,000= ₱5,000 Example 3: Given Is = ₱2,000 and F = ₱23,000., fnd P. Solution: P = ₱23,000 - ₱2,000 b= ₱21,000 13 CO_Q2_General Mathematics SHS Module 2  What’s New Read and analyze the situation below. Let’s Save Michael is planning to apply for a loan in a Cooperative Bank, and he is already aware of the terms and conditions of payment for his loan. When he is about to pass his application form and compare his computation with the bank’s payment terms he notices some discrepancies. Michael’s Computation Computation from the bank Amount of Loan: ₱100,000 Amount of Loan: ₱100,000 Interest rate: 3% Interest rate: 3% Due Date: After 3 years Computation: Year 1 Year 2 Year 3 I = (100,000)(0.03(3) Int 3000 6090 9272.70 I = ₱9,000 Amt 103,000 106,090 109,272.70 Amount to be paid after 3 years ₱109,000 To be enlightened, he asked some explanations why they have different computations and the bank gave him the detailed computation: Initially at t = 0 ₱100,000 at t = 1 ₱100,000 (1.03) = ₱103,000 at t = 2 ₱103,000 (1.03) = ₱106,090 at t = 3 ₱106,090 (1.03) = ₱109,272.70 14 CO_Q2_General Mathematics SHS Module 2 Questions 1. Is Michael’s computation correct? ____________________________________________________________ 2. Is the bank’s computation fair? Why? _____________________________________________________________ 3. How much is the difference in the total amount to be paid between Michael’s computation and the bank’s computation? ______________________________________________________________ 4. Why do you think the bank’s computation yielded more interest? ______________________________________________________________ 5. Do you think the bank committed an error in the computation of the amount to be paid? ______________________________________________________________ 6. If the term of payment will be longer what do you think will happen between the difference of the amount to be paid in Michael’s computation and the bank’s computation? _________________________________________________________________ 7. If you are Michael and you follow the computation made by the bank do you think there is a way to lessen the amount to be paid at the end of 3 years? How? ________________________________________________________________ ________________________________________________________________ 15 CO_Q2_General Mathematics SHS Module 2  What is It Notice that there is a difference between the computation of Michael and the bank concerning the amount to be paid for the loan. Michael used simple interest to find the amount to be paid for the loan for three years. While the bank computed first the interest for the first year and added it to the loan amount, then the resulting amount becomes the basis for computing the total amount to be paid for the second year, and it follows the same pattern for the third year. Interest plays a major role in computation because it became one of the factors in determining the amount to be paid for the succeeding years. In such a case, we call that compound interest. To better understand of what compound interest is, the following formulas will be considered. To find the compound interest, which is compounded annually the formula to find the maturity value is: 𝐹𝐹 = 𝑃𝑃(1 + 𝑟𝑟)𝑡𝑡 where: F = maturity (future) value P = principal or present value r = interest rate t = term or time in years To find the compound interest use the formula: 𝐼𝐼𝑐𝑐 = 𝐹𝐹 − 𝑃𝑃 where: 𝐼𝐼𝑐𝑐 = compound interest P = principal or present value F = maturity (future) value To find the present value or principal of the maturity value F due in t years the formulas are: 𝐹𝐹 𝑃𝑃 = 𝑃𝑃 = 𝐹𝐹(1 + 𝑟𝑟)−𝑡𝑡 (1 + 𝑟𝑟)𝑡𝑡 or 16 CO_Q2_General Mathematics SHS Module 2 Example 1: Given: P= ₱18,500, r = 3% and compounded annually for 3 years, find the maturity value (F) and the compound interest (Ic ). Solution: Use the formula of maturity value 𝐹𝐹 = 𝑃𝑃(1 + 𝑟𝑟)𝑡𝑡 Substitute the given to the formula 𝐹𝐹 = 18,500(1 + 0.03)3 Performing the operations F = ₱20,215.45 Apply the formula of compound interest 𝐼𝐼𝑐𝑐 = 𝐹𝐹 − 𝑃𝑃 Substitute the value of F that you get and P 𝐼𝐼𝑐𝑐 = 20,215.45 − 18500 Performing the operations 𝐼𝐼𝑐𝑐 = ₱1,715.45 Therefore, the maturity value is ₱20,215.45 and the compound interest ₱1,715.45 Example 2: Given F = ₱15,000, r = 2% compounded annually for 4 years, find the present value (P). Solution: 𝐹𝐹 Use the formula in finding the present value 𝑃𝑃 = (1+𝑟𝑟)𝑡𝑡 15000 Substitute the given to the formula 𝑃𝑃 = (1+0.02)4 Performing the operations P= ₱13,857.68 Therefore, the present value is ₱13,857.68 Compounding More Than Once a Year In the examples above the interest are compounded annually, however, there are cases that interest are compounded more than once a year so in this case additional terms must be clarified such as: Frequency of conversion (m) - number of conversion period in one year Conversion or interest period – time between successive conversions of interest Total number of conversion periods (n) n = mt = (frequency of conversion) 𝑥𝑥 (time in years) Nominal rate (𝒊𝒊𝒎𝒎 ) - annual rate of interest or interest rate per year 𝑖𝑖 (𝑚𝑚) 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑜𝑜𝑜𝑜 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 Rate (j) of interest for each conversion period 𝑗𝑗 = = 𝑚𝑚 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑜𝑜𝑜𝑜 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 17 CO_Q2_General Mathematics SHS Module 2 Study the table below. Situations m N 𝑖𝑖 𝑚𝑚 j 2% compounded Annually m =1, t = 3 𝑖𝑖1 = 0.02 𝑖𝑖 𝑚𝑚 𝑗𝑗 = annually for 3 𝑚𝑚 m=1 n = (1)(3)=3 years 0.02 𝑗𝑗 = = 0.02 1 2% compounded Semi – m = 2, t = 3 𝑖𝑖 2 = 0.02 0.02 𝑗𝑗 = = 0.01 2 semi – annually annually n = (2)(3)=6 for 3 years m=2 2% compounded Quarterly m = 4, t = 3 𝑖𝑖 4 = 0.02 0.02 𝑗𝑗 = = 0.005 4 quarterly for 3 m=4 n =(4) (3)=12 years 2% compounded Monthly m = 12, t = 3 𝑖𝑖12 = 0.02 0.02 𝑗𝑗 = 12 monthly for 3 = 0.0016 m = 12 n = (12)(3) = 36 years 2% compounded Daily m = 365, t = 3 𝑖𝑖 365 = 0.02 0.02 𝑗𝑗 = 365 daily for 3 years m = 365 n = (365(3) =1095 Since the rate for each conversion period is represented by j, then in t years, interest is compounded mt times. Thus, the formula of Maturity Value for interest compounding m times a year is: 𝐹𝐹 = 𝑃𝑃(1 + 𝑗𝑗)𝑛𝑛 where: F = maturity value P = present value 𝑖𝑖 (𝑚𝑚) j= 𝑚𝑚 n = mt Meanwhile, the formula in finding the present value given the maturity value is: 𝐹𝐹 𝑃𝑃 = (1 + 𝑗𝑗)𝑛𝑛 18 CO_Q2_General Mathematics SHS Module 2 Let us take the following for example: Example 3: Given 𝑃𝑃 = ₱50,000.00, 𝑖𝑖 4 = 0.03, 𝑚𝑚 = 4, 𝑡𝑡 = 4, find F and Ic. Solution: Use the formula of maturity value 𝐹𝐹 = 𝑃𝑃(1 + 𝑗𝑗)𝑛𝑛 Solve for n and j 𝑛𝑛 = 𝑚𝑚𝑚𝑚; 𝑛𝑛 = 4(4); 𝑛𝑛 = 16 𝑖𝑖 (4) 0.03 𝑗𝑗 = ; 𝑗𝑗 = ; 𝑗𝑗 = 0.0075 𝑚𝑚 4 Substitute the values of the known variables 𝐹𝐹 = 50,000(1 + 0.0075)16 Performing the operations 𝐹𝐹 = 56,349.61 Use the formula of compound interest 𝐼𝐼𝑐𝑐 = 𝐹𝐹 − 𝑃𝑃 Substitute the values of F and P 𝐼𝐼𝑐𝑐 = 56,349.61 − 50,000 Performing the operation 𝐼𝐼𝑐𝑐 = 6,349.61 Therefore, the maturity value is ₱56,349.61 and the compound interest is ₱6,349.61 Example 4: Given = ₱45,000.00, 𝑖𝑖 2 = 0.02, 𝑚𝑚 = 2, 𝑡𝑡 = 4, find Ic. Solution: 𝐹𝐹 Use the formula for Present value 𝑃𝑃 = (1+𝑗𝑗)𝑛𝑛 Solve for n and j 𝑛𝑛 = 𝑚𝑚𝑚𝑚; 𝑛𝑛 = 2(4); 𝑛𝑛 = 8 𝑖𝑖 (2) 0.02 𝑗𝑗 = ; 𝑗𝑗 = ; 𝑗𝑗 = 0.01 𝑚𝑚 2 45000 Substitute the values of the known variables 𝑃𝑃 = (1+0.01)8 Performing the operations 𝑃𝑃 = 41,556.75 Use the formula for compound interest 𝐼𝐼𝑐𝑐 = 𝐹𝐹 − 𝑃𝑃 Substitute the values of F and P 𝐼𝐼𝑐𝑐 = 45,000 − 41,556.75 Performing the operation 𝐼𝐼𝑐𝑐 = 3,443.25 Therefore, the present value is ₱41,556.75 and the compound interest is ₱3443.25 19 CO_Q2_General Mathematics SHS Module 2  What’s More Activity 2.1 Read each statement and answer the question that follows. 1. If P = ₱85,500, and r = 1.25% compounded monthly for 1 year, find the compound interest. What is the first component that should be computed? ____________________ What formula will be used? _______________________ How much is the maturity value? _____________________________ How will you find the simple interest? ______________ How much is the simple interest? ___________________________ 2. If F = ₱50,000 with the rate 1.5% compounded quarterly for 5 years find the present value and compound interest Which should be find first present value or compound interest? __________________________ How are you going to express the rate in decimal? ___________________ What formula will you use in finding the present value? ____________ How much is the present value? _________________________ How will you find the compound interest? _________________ How much is the compound interest? ______________________________ 20 CO_Q2_General Mathematics SHS Module 2 Activity 2.2 Find the value of the required components in each item. 1. Find the compound interest and maturity value if P = ₱43,000, with a rate of 5% is compounded semi-annually for 6 years. 2. Find the compound interest and present value if F = ₱105,000 with a rate of 2.5% is compounded quarterly for 3 years Activity 2.3 Complete the table below by finding the unknown. Present Nominal Interest Interest Time in Total Compound Maturity Value rate (i(m)) compounded per years number of Interest Value conversion conversions period (j) (n) 60,000 3% quarterly 3 6% semi- 4 40,000 annually Guide Questions: For the first set of values a. What unknown variable will you solve first? b. How are you going to find j? c. What will be n? d. How much is the maturity value? e. How much is the compound interest? For the second set of values a. What unknown variable will you solve first? b. How are you going to find j? c. What will be n? d. How much is the maturity value? e. How much is the compound interest? Activity 2.4 Complete the table below by finding the unknown. Present Nominal Interest Interest Time in Total Compound Maturity Value rate (i(m)) compounded per years number of Interest Value conversion conversions period (j) (n) 78,000 4% quarterly 6 12% monthly 3 50,000 21 CO_Q2_General Mathematics SHS Module 2  What I Have Learned I. Fill in the blanks to make the statements true. 1. In finding the total number of conversion period you will multiply __________ by _____________. 2. To get the rate of interest for each conversion period the annual rate of interest will be ________ by the frequency of conversion. 2. The formula in getting the compound interest given the present value and maturity value is ____________________ 3. 4. The formula in getting the present value in a compound interest environment is ___________________ 5. The formula __________________ is used to get the maturity value in a compound interest environment. II. In not more than three sentences explain how to compute the compound interest. ______________________________________________________________________________ 22 CO_Q2_General Mathematics SHS Module 2  What I Can Do Money Matters Read and analyze the situation below then answer the questions that follow. Suppose you are a regular employee in a finance company that offers a loan payable through salary deduction. One of its privileges is availing a loan that offers a 5% interest compounded annually for 1 to 5 years. You are thinking to apply for a loan but you wanted to analyze if you can shoulder the monthly payment. Make a loan schedule and fill up an application form that will help you decide the amount of loan that you will make Loan Schedule Loan Amount Maturity Value Monthly t=1 t=2 t=3 t=4 t=5 Payment ₱10,000 ₱20,000 ₱30,000 ₱40,000 ₱50,000 ₱100,000 ₱150,000 ₱200,000 ₱250,000 23 CO_Q2_General Mathematics SHS Module 2  Loan Application Form Name: ________________________________________________________ Age:________________________ Date of Birth: ________________ Occupation: __________________________________________________ Monthly Income:______________________________________________ Sources of Funds:_____________________________________________ Loan Amount:_________________________________________________ Other Sources of Funds: ______________________________________ Monthly Payment of Loan:_____________________________________ Please answer the following questions truthfully: 1. Do you have an existing loans from other banks? _________________________________________________________ 2. What is the purpose of the loan? _________________________________________________________ 3. Can your salary cover the monthly payment of the loan? If not do you have other resources to pay the monthly payment? ______________________________________________________ Rubrics for Scoring Categories Excellent Fair Poor 3 2 1 Relevance of the Excellent Some Little to no advice understanding in understanding in understanding in creating advice for creating advice for creating advice for choosing a plan choosing a plan choosing a plan Planning The goal set is The goal set is The goal set is not achievable and hard to achieve achievable and realistic not realistic Accuracy of The computations The computations There is no Solution made are all made have flaws attempt in making correct computation 24 CO_Q2_General Mathematics SHS Module 2  Assessment Multiple Choice. Choose the letter of the best answer. Write the chosen letter on a separate sheet of paper. 1. Which of the following described to formula 𝐼𝐼𝑠𝑠 = 𝑃𝑃𝑃𝑃𝑃𝑃 ? a. simple interest b. compound interest c. present value d. maturity value 2. What will you get if you divide the simple interest by the product of the rate and time? a. present value of compound interest b. present value of simple interest c. maturity value of compound interest d. maturity value of simple interest 3. If the investment amounting to ₱45,000 earned an interest of ₱3,500 how much will be the maturity value? a. ₱41,500 b. ₱48,500 c. ₱40,000 d. ₱25,000 4. Given that P = ₱15,250, r = 3.25% and t= 3 years, find the simple interest. a. ₱24,868.80 b. ₱14,868.80 c. ₱1,486.88 d. ₱148.69 1 5. Given that P = ₱20,820, r = 2 % and t = 9 months, find the simple interest. 4 a. ₱ 351.34 b. ₱ 3513.38 c. ₱ 4,216.05 d. ₱ 42160.50 6. Which of the following is the formula to find the maturity value of the compound interest? a. 𝐹𝐹 = 𝑃𝑃(1 + 𝑟𝑟)𝑡𝑡 b. 𝐹𝐹 = 𝐹𝐹 − 𝑃𝑃 𝐹𝐹 c. 𝐹𝐹 = (1+𝑟𝑟)𝑡𝑡 𝑖𝑖 (𝑚𝑚) 𝑚𝑚𝑚𝑚 d. 𝐹𝐹 = 𝑃𝑃(1 + ) 𝑗𝑗 25 CO_Q2_General Mathematics SHS Module 2 7. What is the frequency of conversion if the annual rate is compounded monthly? a. 2 b. 3 c. 4 d. 12 8. If the interest rate is 12% compounded quarterly, what is the interest rate per conversion period? a. 3% b. 2.5% c. 1.25% d. 0.42% 9. Given that 𝐼𝐼𝑠𝑠 = 3,750, r = 2 ½ % and t = 3 years, what is the principal or present value? a. ₱500 b. ₱5,000 c. ₱50,000 d. ₱500,000 10. Given that F = ₱50,000 and P = ₱35,000 how much is the compound interest? a. ₱15, 000 b. ₱25,000 c. ₱35,000 d. ₱45,000 For numbers 11 – 12, use the following values: F = 150,000, t = 6 years, 𝑖𝑖 2 = 5% and m= 2 11. What is the present value? a. ₱100,353.83 b. ₱111,533.38 c. ₱123,153.83 d. ₱132,531.38 12. How much is the compound interest? a. ₱49646.17 b. ₱38466.62 c. ₱26846.17 d. ₱17468.62 13. If the maturity value is ₱50,000 and the compound interest is ₱3,500, how much is the present value? 26 CO_Q2_General Mathematics SHS Module 2  a. ₱53,500 b. ₱46,500 c. ₱46,000 d. ₱3,000 For numbers 14 – 15, use the following values: P=₱150,000, 𝑖𝑖 2 = 6%, t=4 years, m=2 14. How much is the maturity value? a. ₱190, 015.51 b. ₱179, 413.35 c. ₱167, 313.51 d. ₱159, 413.35 15. How much is the compound interest? a. ₱9,413.35 b. ₱17,313.51 c. ₱29,413.35 d. ₱40,015.51 Additional Activities 27 CO_Q2_General Mathematics SHS Module 2 Complete the table. A. Principal Rate Time Interest Maturity Value ₱60,000 4% 3.5 years 2.2% 5 years ₱345 ₱125,000 2% ₱135,000 B. Present Nominal Interest Interest Time Total Compound Maturity Value rate compounded per in number of Interest Value (i(m)) conversion years conversions period (j) (n) ₱80,000 2.1% quarterly 4 2% monthly 6 ₱70,000 Answer Key 28 CO_Q2_General Mathematics SHS Module 2  Module 2 CO_Q2_General Mathematics SHS 29 General Mathematics Teacher's Guide. Pasig City: Department of Education, 2016. 2016. General Mathematics Learner's Material. Pasig City: Department of Education, References What's More Activity 1.1 1. ₱281.25 2. ₱5066.67 Activity 1.2 1. I =₱260, F = ₱13,260 2. ₱10,416.67 Activity 1.3 P r t Is F 45,000 2% 2 2250 47250 1/2 95,000 1.2% 4 4560 99560 105000 3% 3 9450 114450 Activity 1.4 P r t Is F 60000 5% 3 9000 69000 40000 1 2 1200 41200 ½% 20000 0.5% 9mos 75 21075 Assessment Activity 2.1 What I 1. ₱1074.89 1. A 2. P=₱46393.68 Know Ic = 3606.32 2. B Activity 2.2 3. B 1. A 4. C 1. F = ₱57,830.22 2. B I = ₱14,830.22 5. A 3. B 2. P=₱97,435.81 6. A 4. C I=₱7564.19 7. D 5. A Activity 2.3 8. A 6. A P i(m m j t n Ic F ) 9. C 7. C 60000 3% 4 0- 3 12 5628. 65628.41 10.A 8. A 0075 41 31576. 6% 2 0.03 4 8 8423. 40000 11.B 9. C 37 63 12.B 10.D Activity 2.4 13.B 11.B P i(m m j t n Ic F ) 14.A 12.B 78000 4% 4 0.0 6 24 21039 99039.3 15.D 13.C 1.30 0 34946. 12 12 0.0 3 36 15053 50000 14.A 25 % 1.75 15.D Orines, Fernando B. Next Century Mathematics. Quezon, City: Phoenix Publishing House, 2016. Oronce, Orlando A. General Mathematics. Sampaloc, Manila: Re Bookstore, Inc., 2017. Santos, Durawin C.,Biason, Ma. Garnet P. Math Activated:Engage Yourself and Our World General Mathematics. Makati, City: Salesiana Books by Don Bosco Press, Inc., 2016. 30 CO_Q2_General Mathematics SHS Module 2 For inquiries or feedback, please write or call: Department of Education - Bureau of Learning Resources (DepEd-BLR) Ground Floor, Bonifacio Bldg., DepEd Complex Meralco Avenue, Pasig City, Philippines 1600 Telefax: (632) 8634-1072; 8634-1054; 8631-4985 Email Address: [email protected] * [email protected]