Compound Interest: Definition, Formulas, and Examples, Cheat Sheet of Mathematics

Download Compound Interest: Definition, Formulas, and Examples and more Cheat Sheet Mathematics in PDF only on Docsity! 1 COMPOUND INTEREST Definition of terms: 1___________ – also called maturity value, it is an accumulated amount obtained by adding the principal and the compound interest. 2___________ – the number of times in a year the interest will be compounded. The following are the common conversion periods in a year: annually : m = 3 semi-annually : m = 3 quarterly : m = monthly : m = 3 3___________ – the total number of times interest is calculated for the entire term of the investment or loan. 4___________ – the stated rate of interest per year. 5___________ – the interest rate per conversion period. 6___________ – this is the principal P, that will accumulate to F if there is an interest at periodic rate i for n conversion periods. What is It Compound interest (Ic) is usually used by banks in calculating interest for long-term investments and loans such as savings account and time deposits. In this type of interest, the interest due at stipulated interval is added to the principal and earns interest thereafter. It implies that the principal increases over a period of time, resulting to an increase in interest earned at every compounding period. Thus, compound interest is an interest resulting from the periodic addition of simple interest to the principal amount or simply the difference between the compound amount and the original principal. The problem below is an example of compound interest. 2 Example: ₱50,000.00 was loaned for a period of 3 years with 5% interest compounded annually. What amount of money will be needed to repay the loan? Principal at the start of the year Interest Amount at the end of the year First Year ₱50,000.00 ₱50,000 × 0.05 × 1 = ₱2,500.00 ₱50,000 + 2 500 = ₱52,500.00 Second Year ₱52,500.00 ₱52,500 × 0.05 × 1 = ₱2,625.00 ₱52,500 + 2625 = ₱55,125.00 Third Year ₱55,125.00 ₱55,125 × 0.05 × 1 = ₱2,756.25 ₱55,125 + 2 756.25 = ₱57,881.25 The required answer to the problem is ₱57,881.25. As shown in the table, the amount at the end of the year is equal to the sum of the principal and the interest for that year. Thus, Amount for First Year : A = 50000 + (50000 × 0.05) = 50000 (1 + 0.05) Amount for Second Year: A = 50000 (1 + 0.05) + (50000 (1 + 0.05)(0.05)) = 50000 (1 + 0.05) (1 + 0.05) = 50000 ( 1 + 0.05)2 Amount for Third Year: A = 50000 (1 + 0.05)2 + (50000 ( 1 + 0.05)2(0.05)) = 50000 (1 + 0.05)2 (1 + 0.05) = 50000 (1 + 0.05)3 Generally, when interest is compounded annually for n years, the amount A = P( 1 + i) n. 5 4.) How many years will it take for ₱13,000.00 to become ₱20,000.00 at 12.5% compounded annually? Solution: Given: P = ₱13,000.00 m = 1 F = ₱20,000.00 r = 12.5% or 0.125 𝑖 = 𝑟 = 0. = 0.125 𝑚 𝐹 log ( 𝑡 = 𝑃) ??? 𝑚[log(1 + 𝑖)] What I Have Learned Problems Involving Compound Interest 1. Joseph borrows ₱50,000.00 and promise to pay the principal and interest at 12% compounded monthly. How much must he repay after 6 years? Solution: Given: P = ₱50,000.00 m = 12 r = 12% or 0.12 t = 6 years 𝑟 𝑖 = = = 0.01 𝑛 = 𝑚𝑡 = 12(6) = 72 𝑚 𝐹 = 𝑃(1 + 𝑖)𝑛 ??? 2. A loan ₱125,000.00 at 8% compounded quarterly was paid back with an amount of ₱176,000.00 at the end of the period. For how long was the money borrowed? Solution: Given: P = ₱125,000.00 r = 8% or 0.08 F = ₱176,000.00 m = 4 0. 𝑖 = 𝑟 = = 0.02 𝑚 𝐹 log ( ) 𝑡 = 𝑃 ??? 𝑚[log(1 + 𝑖)] 6 3. How much must be invested today in a savings account in order to have ₱50,800.00 in 6 years and 9 months if money earns 5.4% compounded semi-annually? Solution: Given: F = ₱50,800.00 t = 6 years or 6.75 years r = 5.4% or 0.054 m = 2 𝑟 0. 𝑖 = = = 0.027 𝑛 = 𝑚𝑡 = 2(6.75) = 13.5 𝑚 𝑃 = 𝐹(1 + 𝑖)−𝑛 ??? Additional Activities Arrange the jumbled letters to form a word/s related to business mathematics. 1.) T M T U R I Y A A E D T 2.) N E E T T R I S E R T A 3.) I I P P C N L R A 4.) O O U D C M N P N U T M A O 5.) I P S E M L T E N T S R E I 6.) E V M N T T N E I S 7.) T E U Y A Q R L R 8.) C M C U E T U L A A 9.) R W O R B O R E 10.) M O C U D O N P T N T R E E S I 7 (Post-test) 1.) Date on which money is received by the borrower. A. Conversion period C. Maturity date B. Loan date D. Repayment date 2.) 3 % is equivalent to A. 0.0032 C. 0.32 B. 0.032 D. 3.2 3.) This refers to the interest charged on the principal alone for the entire duration or period of the loan or investment. A. Compound interest C. Interest rate B. Future value D. Simple interest 4.) This refers to the number of years for which the money is borrowed or invested. A. Conversion period C. Principal B. Interest rate D. Time 5.) An interest resulting from the periodic addition of simple interest to the principal amount. A. Compound amount C. Interest rate B. Compound interest D. Simple interest 6.) What is the formula in computing the present value of F in a financial transaction involving compound interest? A. 𝑃 = 𝐹(1 + 𝑖)−𝑛 C. 𝑃 = 𝐹(1 − 𝑖)−𝑛 B. 𝑃 = 𝐹(1 + 𝑖)𝑛 D. 𝑃 = 𝐹(1 − 𝑖)𝑛