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Business math CH 1

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Economía y finanzas

Empresariales

Sindisiwe SibulisiweSeguir

student at the university of JohannesburgPublicidad

- 2013/04/12 1 WORKING WITH INTEREST X-Kit Textbook Chapter 4 CONTENT Simple Interest Compound Interest Time Lines TERMINOLOGY Term Explanation Investment When you save money Loan When you borrow money Debt When you owe money Interest • The price you pay for borrowing money. • Earning for saving money. • Interest is added to the original loan or investment. SIMPLEINTEREST Jim wants to study graphic design at a college and needs R10 000 for his studies. He goes to a bank to borrow the money. The bank ask 5% interest on study loans. At the bank the following is discussed: TERMINOLOGY Term Explanation Present Value (PV) • The amount of money we borrow or save • The value today/now • The principal part of the loan • Amount before adding interest • Example R10 000 Interest Rate (𝒊) • The rate is used to calculate the amount of interest • Given as percentages • Change to decimal fractions • Example 𝟓% = 𝟎, 𝟎𝟓 Interest (I) • The amount we pay (borrow) or get (save) TERMINOLOGY Term Explanation Term of period (𝒏) • The length of time over which we borrow or save money • Due date for paying back the a loan • Maturity date for an investment • Measure term in years, half-years, months, weeks or days • The longer the term, the greater the amount of interest Future Value (FV) • The value of money at the end of the term • The sum of the principal and interest
- 2013/04/12 2 𝑰 = 𝑷𝑽 × 𝒊 × 𝒏 • Simple interest is interest that is calculated on the principal amount for the length of time for which it is borrowed. • Simple interest is due at the end of the term • We calculate simple interest by multiplying the present value by the interest rate by the term • Always use the same length of time to measure the rate of interest and the term, for example: If the interest rate is per year and the term is for 8 months, show the term as a fraction of a year ( 𝟖 𝟏𝟐) Example Calculate the simple interest that Jim has to pay the bank if he borrows R10 000 for 1 year. The interest rate is 5% per annum. 𝑰 = 𝑷𝑽 × 𝒊 × 𝒏 • 𝐼 = the interest (in rands) • 𝑃𝑉= the present value (the amount borrowed/saved) • 𝑖 = rate of interest (a percentage) • 𝑛 = the term or time Example What if Jim borrows the money for 2 years? CALCULATINGFUTUREVALUE(FV) Future Value = Present Value + Interest 𝑭𝑽 = 𝑷𝑽 + 𝑰 𝑭𝑽 = 𝑷𝑽 + 𝑷𝑽 × 𝒊 × 𝒏 𝑭𝑽 = 𝑷𝑽 𝟏 + 𝒊𝒏 Example Jim wants to know how much he will have to pay in interest on a 3-year loan of R10 000. Jim also wants to know how much money he will have paid in total by the end of the loan period? CALCULATINGPRESENTVALUE(PV) 𝑭𝑽 = 𝑷𝑽 𝟏 + 𝒊𝒏 𝑷𝑽 = 𝑭𝑽 𝟏 + 𝒊𝒏
- 2013/04/12 3 Example Jim is still at school and can make some money as a waiter during the holidays. Instead of borrowing the money, Jim wants to save for his studies. He wants to know how much money he must save now, at 5% simple interest to have R10 000 in 3 years’ time? CALCULATINGINTERESTRATE (𝒊) 𝑭𝑽 = 𝑷𝑽 𝟏 + 𝒊𝒏 Jim made a lot of money in tips working as a waiter over the summer holidays. He wants to put R8 500 in the bank to save for his post- matric studies. He has calculated the future value. He will need R10 000 for his studies in 3 years’ time. How much interest will he have to earn on his principal of R8 500? What is the required interest rate? CALCULATINGTHE TERM(𝒏) 𝑭𝑽 = 𝑷𝑽 𝟏 + 𝒊𝒏 Jim has R8 000 and he can bank it at an interest rate of 5%. He still thinks that he would need R10 000 for his studies. Jim wants to know how long it will take the present value of R8 000 to grow to a future value of R10 000 at a simple interest rate of 5%? CALCULATINGSIMPLEINTERESTFOR A FRACTIONOF THETERM Jim invested R8 000 for 7 months before he plans to begin college. What interest will Jim receive if he banks R8 000 for 7 months at 4.5% simple interest per annum? You are asked to calculate interest (amount in rands), not interest rate (a percentage) Example Grace receives R750 on her birthday. She decides to invest the money for 8 months at 6% per annum. 1. Find the simple interest. 2. Find the future value. Example To have a future value of R1 050 after 7 months, how much money must Jim save at 8% simple interest per annum? (Remember to change the time to the same unit as the interest rate.)
- 2013/04/12 4 Example Mary banked R500. After 180 days the amount in her account is R507. What was the interest rate that she received? Example Joe banked R500 at a bank that gave him 3% interest per annum. When he looked at his balance (the amount in the bank) it was R520. How long had Joe’s money been in the bank? TIMELINES Beginning of Term Time Period Interest Rate End of Term Example When Jim started high school, Jim’s mother began to save for his post-matric studies. She calculated that 5 years from then, when Jim leaves school, he will need at least R20 000 to study for 2 years. Jim’s mother has a savings account at a bank that pays 4,5% interest per year. How much would she have to save today in order to have R20 000 in 5 years’ time? Show your calculations on a time line. Example A man is offered R200 000 cash now for his house or R202 000 after 6 months. Which is the better offer if the current interest rate is 5% per year? Present all information on a time line. COMPOUNDINTEREST– Interestupon Interest •Compound interest is interest paid on the original investment as well as on the interest that you have earned previously. •Simple interest is only earned on the original principal.
- 2013/04/12 5 R100investedat 10% annually Year Simple Interest Compound Interest 1 R100 + R10 = R110 R100 + R10 = R110 2 R110 + R10 = R120 R110 + R11 = R121 3 R120 + R10 = R130 R121 + R12,10 = R133,10 4 R130 + R10 = R140 R133,10 + R13,31 = R146,41 5 R140 + R10 = R150 R146,41 + R14,64 = R161,05 10 R190 + R10 = R200 R235,79 + R23,58 = R259,37 50 R590 + R10 = R600 R10 671,90 + R1 067,19 = R11 739,09 WORKINGWITH COMPOUNDINTEREST 𝑭𝑽 = 𝑷𝑽 𝟏 + 𝒊 𝒏 • 𝐹𝑉 = Future Value • 𝑃𝑉 = Present Value (Principal) • 𝑖 = Annual Interest Rate • 𝑛 = Number of years/period •We can change the subject of the formula CALCULATINGFUTUREVALUE Jim receives R1 000 on his birthday and decides to save it. He can get an interest rate of 4% at the bank. Interest is compounded annually (yearly). Jim wants to know how much his investment will be worth at the end of 3 years. EXAMPLE In 10 years’ time, you want to have your own business. You believe that it is better to start small than not to start saving at all. You invest R5 000 now and leave it for 10 years. Interest is compounded at 8% annually. CALCULATINGPRESENTVALUE Before you start your own business, you’d like to travel. In 7 years’ time, you think you would need at least R85 000 to see some of the world. How much money would you need to invest now? The interest rate is 18% p.a. compounded annually. CALCULATINGINTERESTRATE A friend wants to borrow R3 000 from you. She says that she will give you R4 000 back after 3 years. But, you know about compound interest. You know that if you put your R3 000 in a bank for 3 years, you will earn compound interest at a rate of 8% p.a. What is the interest rate if you lend your friend the money?
- 2013/04/12 6 CALCULATINGTHE TERM/PERIOD Your younger brother earns R500 working in your family’s shop during the school holidays. He wants to buy a bicycle so that he can make more money by delivering pizzas. His dream bicycle cost R700. How long will he have to save if he can earn 8% interest compounded annually? DIFFERENTTIMEPERIODS • Interest compounded annually means that the interest is added to the capital amount ones a year at the end of the year. • To handle different time periods the compound interest formula is changed: 𝑭𝑽 = 𝑷𝑽 𝟏 + 𝒊 𝒎 𝒏×𝒎 • 𝐹𝑉 = Future Value after 𝑛 × 𝑚 periods of compounding • 𝑃𝑉 = Present Value • 𝑖 = Compound Interest Rate • 𝑚 = Number of compounding periods during one year • 𝑛 = Number of years that the investment is held EXAMPLE Let’s see how much R1 000 would be worth if we invested it for 10 years at 5% interest, and interest is compounded: 1. Annually 2. Semi-annually 3. Quarterly 4. Monthly 5. Daily The Future Value increases as the compounding periods increase. MORE EXAMPLES Find the future value of R200 invested for 7 years at 7,5% per annum and compounded annually. MORE EXAMPLES You have won a scratch-and-win competition held by your bank. You can choose one of the following prizes: •R10 000 now, or •R18 000 at the end of 5 years The interest rate is 12% compounded annually. Which prize do you choose?
- 2013/04/12 7 MORE EXAMPLES For R2 000 to become R3 000 in 10 years, work out what the annual compound interest rate should be. MORE EXAMPLES How long will it take R2 000 to grow to R3 000 at 8% interest compounded annually? MORE EXAMPLES Jim saves R500 at an interest rate of 6%. What is his investment worth after 5 years if interest is compounded: 1. Yearly 2. Quarterly 3. Monthly 4. Weekly 5. Daily

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