Gr12_Mathematics_Learner_Eng (1)-144-150.pdf

What can I do to protect myself? You are your own best protection. It is your responsibility to make sure that you never give your money to any company or person that is not registered as a deposit-taking institution in terms of the Banks Act. Why is handing my money over so risky? When you hand ove...

What can I do to protect myself? You are your own best protection. It is your responsibility to make sure that you never give your money to any company or person that is not registered as a deposit-taking institution in terms of the Banks Act. Why is handing my money over so risky? When you hand over your money (notes and coins) to another person who then loses the money, steals it or goes bankrupt, you only have an unsecured claim against that person or their estate and you might not get all your money back. Why is it safer to hand my money to a bank? Banks and investment companies have to be registered so that they can be regulated and supervised, to make sure that your money is safe. Unregulated and unsupervised persons and groups don’t follow these rules and your money is at great risk with them. 3.6 Summary EMCG8 Always keep the rate of interest per time unit and the time period in the same units. Simple interest: A = P (1 + in) Compound interest: A = P (1 + i)n Simple depreciation: A = P (1 − in) Compound depreciation: A = P (1 − i)n m  (m) Nominal and effective annual interest rates: 1 + i = 1 + i m Future value of payments: x [(1 + i)n − 1] i F = Payment amount: x= F ×i [(1 + i)n − 1] Present value of a series of payments: P = x [1 − (1 + i)−n ] i Payment amount: x= P ×i [1 − (1 + i)−n ] Chapter 3. Finance 133 Exercise 3 – 6: End of chapter exercises 1. Mpumelelo deposits R 500 into a savings account, which earns interest at 6,81% p.a. compounded quarterly. How long will it take for the savings account to have a balance of R 749,77? 2. How much interest will Gavin pay on a loan of R 360 000 for 5 years at 10,3% per annum compounded monthly? 3. Wingfield school will need to replace a number of old classroom desks in 6 years’ time. The principal has calculated that the new desks will cost R 44 500. The school establishes a sinking fund to pay for the new desks and immediately deposits an amount of R 6300 into the fund, which accrues interest at a rate of 6,85% p.a. compounded monthly. a) How much money should the school save every month so that the sinking fund will have enough money to cover the cost of the desks? b) How much interest does the fund earn over the period of 6 years? 4. Determine how many years (to the nearest integer) it will take for the value of a motor vehicle to decrease to 25% of its original value if the rate of depreciation, based on the reducing-balance method, is 21% per annum. 5. Angela has just started a new job, and wants to save money for her retirement. She decides to deposit R 1300 into a savings account once each month. Her money goes into an account at Pinelands Mutual Bank, and the account receives 6,01% interest p.a. compounded once each month. a) How much money will Angela have in her account after 30 years? b) How much money did Angela deposit into her account after 30 years? 6. a) Nicky has been working at Meyer and Associates for 5 years and gets an increase in her salary. She opens a savings account at Langebaan Bank and begins making deposits of R 350 every month. The account earns 5,53% interest p.a., compounded monthly. Her plan is to continue saving on a monthly schedule until she retires. However, after 8 years she stops making the monthly payments and leaves the account to continue growing. How much money will Nicky have in her account 29 years after she first opened it? b) Calculate the difference between the total deposits made into the account and the amount of interest paid by the bank. 7. a) Every three months Louis puts R 500 into an annuity. His account earns an interest rate of 7,51% p.a. compounded quarterly. How long will it take Louis’s account to reach a balance of R 13 465,87? b) How much interest will Louis receive from his investment? 8. A dairy farmer named Kayla needs to buy new equipment for her dairy farm which costs R 200 450. She bought her old equipment 12 years ago for R 167 000. The value of the old equipment depreciates at a rate of 12,2% per year on a reducing balance. Kayla will need to arrange a bond for the remaining cost of the new equipment. An agency which supports farmers offers bonds at a special interest rate of 10,01% p.a. compounded monthly for any loan up to R 175 000 and 9,61% p.a. compounded monthly for a loan above that amount. Kayla arranges a bond such that she will not need to make any payments on the loan in the first six months (called a ’grace period’) and she must pay the loan back over 20 years. a) Determine the monthly payment. b) What is the total amount of interest Kayla will pay for the bond? 134 3.6. Summary c) By what factor is the interest she pays greater than the value of the loan? Give the answer correct to one decimal place. 9. Thabo invests R 8500 in a special banking product which will pay 1% per annum for 1 month, then 2% per annum for the next 2 months, then 3% per annum for the next 3 months, 4% per annum for the next 4 months, and 0% for the rest of the year. If the bank charges him R 75 to open the account, how much can he expect to get back at the end of the year? 10. Thabani and Lungelo are both using Harper Bank for their savings. Lungelo makes a deposit of x at an interest rate of i for six years. Three years after Lungelo made his first deposit, Thabani makes a deposit of 3x at an interest rate of 8% per annum. If after 6 years their investments are equal, calculate the value of i (correct to three decimal places). If the sum of their investment is R 20 000, determine how much Thabani earned in 6 years. 11. More questions. Sign in at Everything Maths online and click ’Practise Maths’. Check answers online with the exercise code below or click on ’show me the answer’. 1. 28KQ 7. 28KX 2. 28KR 8. 28KY 3. 28KS 9. 28KZ 4. 28KT 10. 28M2 www.everythingmaths.co.za 5. 28KV 6. 28KW m.everythingmaths.co.za Chapter 3. Finance 135 CHAPTER 4 Trigonometry 4.1 Revision 138 4.2 Compound angle identities 144 4.3 Double angle identities 151 4.4 Solving equations 154 4.5 Applications of trigonometric functions 161 4.6 Summary 171 4 4.1 Trigonometry Revision EMCG9 Trigonometric ratios We defined the basic trigonometric ratios using the lengths of the sides of a right-angled triangle. A c B b a C sin  = opposite = hypotenuse a c sin B̂ = opposite = hypotenuse b c cos  = adjacent = hypotenuse b c cos B̂ = adjacent = hypotenuse a c tan  = opposite adjacent a b tan B̂ = opposite adjacent b a = = Trigonometric ratios in the Cartesian plane We also defined the trigonometric ratios with respect to any point in the Cartesian plane in terms of x, y and r. Using the theorem of Pythagoras, r2 = x2 + y 2. y P (x; y) r α x O sin α = 138 y r cos α = 4.1. Revision x r tan α = y x CAST diagram The sign of a trigonometric ratio depends on the signs of x and y: y 90◦ Quadrant II Quadrant I S A sin θ all 180◦ 0 T 0◦ x 360◦ C cos θ tan θ Quadrant IV Quadrant III 270◦ Reduction formulae and co-functions: 1. The reduction formulae hold for any angle θ. For convenience, we assume θ is an acute angle (0◦ < θ < 90◦ ). 2. When determining function values of (180◦ ± θ), (360◦ ± θ) and (−θ) the function does not change. 3. When determining function values of (90◦ ±θ) and (θ ±90◦ ) the function changes to its co-function. y Second Quadrant: First Quadrant: sine function is positive all functions are positive sin (180◦ − θ) = sin θ sin (360◦ + θ) = sin θ cos (180◦ − θ) = − cos θ cos (360◦ + θ) = cos θ tan (180◦ − θ) = − tan θ tan (360◦ + θ) = tan θ sin (90◦ + θ) = cos θ sin (90◦ − θ) = cos θ cos (90◦ + θ) = − sin θ cos (90◦ − θ) = sin θ x O Third Quadrant: Fourth Quadrant: tangent function is positive cosine function is positive sin (180◦ + θ) = − sin θ sin (360◦ − θ) = − sin θ cos (180◦ + θ) = − cos θ cos (360◦ − θ) = cos θ tan (180◦ + θ) = tan θ tan (360◦ − θ) = − tan θ Chapter 4. Trigonometry 139