FINANCIAL MATHEMATICS (2)

Transcription

1 Lesson 27 FINANCIAL MATHEMATICS (2) Learning Outcomes and Assessment Standards Learning Outcome 1: Number and Number Relationships When solving problems, the learner is able to recognise, describe, represent and work confidently with numbers and their relationships to estimate, calculate and check solutions. Assessment Standards We know this when the learner is able to: Use simple and compound growth formulae to solve problems, including interest, hirepurchase, inflation, population growth and other real-life problems. Demonstrate an understanding of the implications of fluctuating foreign exchange rates (e.g. on the petrol price, imports, exports, overseas travel). Overview In this lesson you will: Learn about compound interest. Learn about inflation. DVD Lesson Compound interest Consider an amount of R1 000 invested at 10% per annum compound interest: Accumulated amount after one year: A 1 = , = R1 100 (Interest received is R100) Accumulated amount after two years: A 2 = , = R1 210 (Interest received is R110) Accumulated amount after three years: A 3 = , = R1 331 (Interest received is R121) Accumulated amount after four years: A 4 = , = R1 44,10 Accumulated amount after five years: A 5 = 1 44,10 + 0, ,10 = R1 10,51 The formula for compound interest (Interest received is R133,10) (Interest received is R14,41) The formula which helps us to calculate the accumulated amount (future value) if an original amount is invested or loaned for n years at a rate of r% compound interest is given by: where: P = present value of the investment or loan (original amount at the beginning) 09 LC G10 MATHS LWB.indb 2008/09/09 12:3:18 PM

2 A = accumulated amount or future value of the investment or loan after n periods n = time period in years i = r_ for the simplest rate r% (i is in decimal form) 100 Consider the previous example. The accumulated amount after 5 years can be calculated by using this formula: A = 1 000(1 + 0,10) 5 A = R1 10,51 Example 1 Malibongwe invests R7 000 for 7 years at 15% p.a. compounded annually. Find the future value of his investment after 7 years and the interest he receives. P A ,14 T 0 T 1 T 2 T 3 T 4 T 5 T T 7 A = 7 000(1 + 0,15) 7 A = 7 000(1,15) 7 A = R18 20,14 Interest received = R18 20,14 R7 000 = R11 20,14 Example 2 Kathy takes out a loan in order to start her small business. Three years after the granting of the loan, she agreed to repay the loan by means of a once off payment of R The bank charges her an interest rate of 24% p.a. compounded annually. How much money did she borrow from the bank (what is the present value of the loan)? P ,90 A T 0 T 1 T 3 T = P(1 + 0,24) = (1,24) LC G10 MATHS LWB.indb /09/09 12:3:19 PM

3 400 _ 000 (1,24) = P 3 P = R ,90 Example 3 R4 250 is invested for years and grows to R Find the interest rate if interest is compounded annually = 4 250(1 + i) _ = (1 + i) Raise both sides to the power of _ 1 : (_ ) 1_ = [(1 + i) ] 1_ (_ ) 1_ = 1 + i (_ ) 1_ 1 = i i = 0, r = 23,03% INDIVIDUAL SUMMATIVE ASSESSMENT Activity 1 Individual activity Summative assessment 1. Johan invests R in an account paying 18% p.a. compounded annually. Calculate the future value of his investment after 15 years. 2. Portia borrows money from a bank in order to finance a new business. The bank charges her an interest rate of 14% p.a. compounded annually. Calculate the present value of the loan (the amount she originally borrowed), if she pays off the loan years from now with a payment of R R 000 is invested for 4 years and grows to R Find the interest rate if interest is compounded annually. 4. Find the annual compound interest rate that makes R2 500 double in 5 years. 5. R is invested at 10% p.a. simple interest for 3 years. Thereafter, the total amount is reinvested in a different financial institution at 25% p.a. compound interest for 2 more years. What is the future value of the investment after the five-year period? 8 09 LC G10 MATHS LWB.indb /09/09 12:3:20 PM

4 Lesson Inflation Inflation is the steady compounded increase in prices over time throughout the economy. The effect of inflation is to erode the buying power of money over time. DVD Example A motorcar currently costs R If the rate of inflation is 12% p.a. compounded annually, how much will this car cost in 5 years time? A = (1,12) 5 A = R ,35 Activity 2 Individual activity Summative assessment 1. A home presently costs R Calculate the future value of the home in 12 years time taking into account an inflation rate of 14% p.a. 2. You want to travel overseas in 5 years from now. Inflation is 12% p.a. The trip is expected to cost R then. How much must you invest now in order to afford the trip? 3. If salaries double every 7 years in order to keep up with inflation, what rate of inflation does this imply? Lesson Exchange rates Different countries have different money systems. Currency is the term used to describe the particular money system of a country. Here are the currencies of a few countries. You might like to find out the currencies of other countries. INDIVIDUAL SUMMATIVE ASSESSMENT Country Currency Symbol South Africa Rand R United States of America US dollar $ United Kingdom British Pound Several European countries Euro Example 1 Simon wants to buy the new Playstation3, which has been advertised in a US catalogue for $00. He wants to order and pay online. The current rand/dollar exchange rate is R10,43 to the US dollar. Calculate the cost in rands of the Playstation3. $1 = R10, LC G10 MATHS LWB.indb /09/09 12:3:21 PM

5 $1 00 = R10,43 00 $00 = R 258 The Playstation3 will cost R 258. Example 2 Zinhle is on a trip to England to visit her family. The current rand/pound exchange rate is R12,50 to the British pound. She has R to spend in England. How many pounds does she have to spend in England? 1 = R12,50 1_ 12,50 = _ R12,50 12,50 0,08 = R1 0, = R = R Zinhle has 1 00 to spend in England. INDIVIDUAL SUMMATIVE ASSESSMENT Activity 3 Individual activity Summative assessment 1. A DVD player costs $25 in California. What would it cost in South Africa if the rand/dollar exchange rate is R10,45 to the US dollar? 2. You want to buy a book online costing 35 in London. How much will it cost you in rands? The rand/pound exchange rate is R13 to the pound. 3. You want to buy a box of Japanese sweets costing 50 yen ( ). If the rand/ yen exchange rate is one rand to 1,78 yen, calculate the cost of the sweets in rands. 4. Peter is visiting a friend in New York for a week. He has R2 500 to spend and will exchange the money for US dollars. How many dollars will he have to spend if the rand/dollar exchange rate is R10,38 to the US dollar? 5. Colin wants to purchase an electric guitar from a music dealership in London. The guitar costs and Vinesh has saved R The rand/ pound exchange rate is R13,24 per one pound. Will he have enough money, in rands, to buy this guitar? Show your reasoning.. One yen costs R0,22. How much yen can you spend in Japan if you have R3 000 available to you? LC G10 MATHS LWB.indb /09/09 12:3:22 PM

6 ANSWERS AND ASSESSMENT Lesson 2 Activity 1 1. (a) A = 4 000(1 + 0,09 8) A = R 880 (b) Simple Interest = R 880 R4 000 = R (c) _ = R = P(1 + 0,12 4) = P(1,48) P = R20 270, = 2 900(1 + 0,11 n) = n = 319n n =, years = 5 800(1 + 0,12 n) = n = 9n n = 8, 3 years = 3 000(1 + i 5) 000 = i = i i = 0,2 r = 20% = 4 000(1 + i 2) = i = 8 000i i = 0,125 r = 12,5% LC G10 MATHS LWB.indb /09/09 12:3:23 PM

7 Activity 2 1. (a) A = 9 800(1 + 0,15 5) (b) Simple Interest A = R = R R9 800 = R (c) _ = R285,83 + R12 = R297,83 per month 0 2. Deposit = 0, = R1 200 Balance = R A = (1 + 0,1 2) A = R14 25 He will pay R R14 25 = R Monthly payments = _ = R Cost = R Deposit (plus trade in) = R R2 000 = R3 500 Balance = R Monthly payments = R1 120,50 24 = R2 892 Interest paid = R2 892 R = R = (1 + 2i) = i = i i = 0,21712 r = 21,7% 4. Payments = 24 R500 = R = P(1 + 0,25 2) P = R8 000 Lesson 27 Activity 1 1. A = (1 + 0,18) = P(1 + 0,14) A = R ,79 _ (1,14) = P P = R , = 000(1 + i) = 2 500(1 + i) 5 7 _ = (1 + i)4 5 _ 000 = (1 + i) (_ 7 ) 1_ 4 = 1 + i ( 5 _ ) 1_ 5 = 1 + i (_ = i ( 5 _ ) 1_ 5 1 = i ) 1_ 09 LC G10 MATHS LWB.indb /09/09 12:3:24 PM

8 i = 0, i = 0, r = 3,9% r = 14,9% 5. A = (1 + 0,10 3) A = R5 000 A = 5 000(1 + 0,25) 2 A = R101 52,50 Activity 2 2 = (1 + i) 7 2 1_ 7 = 1 + i 2 1_ 7 1 = i i = 0,104 r = 10,4% 1. A = (1 + 0,14) x = x(1 + i) 7 A = R , = P(1 + 0,12) _ (1,12) = P 5 P = R51 08,42 Activity 3 1. $1 = R10, = R13 $1 25 = R10, = R13 35 $25 = R2 75,20 35 = R R1 = 1,78 4. $1 = 10,38 _ R1 1,78 = _ 1,78 $1_ 1,78 10,38 = _ R10,38 10,38 _ R1 1,78 50 = 50 $1_ = R ,38 R2, = 50 $240, = R2 500 = R2,98 = 50 $240,85 = R = R13,24. 1 = R0, = R13, = R He will afford it with R280 to spare. _ 1 0,22 = _ R0,22 0,22 _ = R , ,3 = R LC G10 MATHS LWB.indb /09/09 12:3:25 PM

9 TIPS FOR TEACHERS Lesson 2 It is important for learners to know how simple interest works as well as the formula for doing simple interest calculations. Learners need to understand that hire purchase problems involve simple interest. The idea of paying an initial deposit is important in real world situations. Lesson 27 It is important for learners to know how compound interest works as well as the formula for doing compound interest calculations. Inflation is calculated using the compound interest formula. Emphasise to learners that it is important to save money using an interest rate that is higher than the inflation rate. Otherwise, the buying power of money will be eroded. Exchange rates are important for learners to discuss, especially given that South Africa has become part of the international arena LC G10 MATHS LWB.indb /09/09 12:3:25 PM