(b) (i) How much is to be paid as a deposit under this option? (1) Find the cost of the loan under Friendly Credit Terms.

Transcription

1 1. Angela needs $4000 to pay for a car. She was given two options by the car seller. Option A: Outright Loan A loan of $4000 at a rate of 12% per annum compounded monthly. Find (i) (ii) the cost of this loan for one year; the equivalent annual simple interest rate. Option B: Friendly Credit Terms A 25% deposit, followed by 12 equal monthly payments of $ (i) How much is to be paid as a deposit under this option? (1) (ii) Find the cost of the loan under Friendly Credit Terms. (c) Give a reason why Angela might choose (i) (ii) Option A Option B To help Angela, her employer agrees to give her an interest free loan of $4000 to buy the car. The employer is to recover the money by making the following deductions from Angela s salary: $x in the first month, $y every subsequent month. The total deductions after 20 months is $1540 and after 30 months it is $2140. (d) Find x and y. (4) (e) How many months will it take for Angela to completely pay off the $4000 loan? (Total 15 marks) IB Questionbank Mathematical Studies 3rd edition 1

2 2. Tony invested CHF 500 in a bank account at a constant rate of interest. The bank calculates his balance at the end of each year, rounded to two decimal places, as shown in the table below. Year Value at beginning of year 1st CHF 500 CHF 540 Value at end of year 2nd CHF 540 CHF rd CHF CHF th CHF CHF th CHF th What is the rate and type of interest? Complete the table for the fifth and sixth year of investment. (Total 4 marks) 3. At what interest rate, compounded annually, would you need to invest $100 in order to have $125 in 2 years? (Total 4 marks) 4. Hassan invested CHF at the end of The interest rate was 5% per annum. How much interest in total would Hassan have earned at the end of the year 1999 if he had removed the interest from his account at the end of each year; he had not removed the interest from his account at the end of each year. (Total 4 marks) 5. The table shows part of a currency conversion chart. For example GBP 1 is equivalent to FFR GBP USD FFR GBP 1 p 8.33 USD q FFR l For all calculations in this question give your answers correct to two decimal places. Calculate the value of (i) p; (ii) q. (4) IB Questionbank Mathematical Studies 3rd edition 2

3 Joe has USD 1500 to exchange at a bank. (i) Assuming no commission is charged, how much in GBP will Joe receive from the bank? (ii) Assuming the bank charges 1.5% commission, how much in GBP does Joe pay in commission? (1) how much in GBP does Joe actually receive for his USD 1500? (1) (c) Joe decides to invest GBP 700 of his money in a savings account which pays interest at 5%, compounded annually. (i) How much interest will the GBP 700 earn after 4 years? (ii) For how many years must Joe invest his GBP 700 in order to earn at least GBP 200 in interest? (d) After 4 years Joe has a total of GBP 900 in his savings account on an investment at 5% interest compounded annually. How much did he invest? Give your answer to the nearest one GBP. (Total 14 marks) 6. The table below gives the monthly repayments for a loan of 1000 Australian dollars (AUD). The interest rates are 18% per annum and 18.5% per annum, respectively. Number of years of the loan From the table, Monthly repayments on AUD % p.a. 18.5% p.a find the monthly repayment on a loan of AUD 1000 at 18.5% per annum taken over 1 year; IB Questionbank Mathematical Studies 3rd edition 3

4 calculate the total amount to be repaid on a loan of AUD 8000 taken over 5 years, if the interest rate is 18% per annum. (Total 4 marks) 7. In city Y, house prices have increased by 3% each year for the last three years. If a house cost USD three years ago, calculate, to the nearest dollar, its value today. In city Z, a house worth USD three years ago is now valued at USD Calculate the yearly percentage increase in the value of this house. 8. The following is part of a display on the notice board of a bank in the United Kingdom. It shows the exchange rate between one British pound (GBP) and other currencies. EXCHANGE RATES Bank buys foreign currency Bank sells foreign currency Denmark (KR) Finland (MKK) France (FFR) Germany (DM) Greece (DR) NO COMMISSION CHARGED Geraldine eats a meal in a restaurant while on holiday in Greece. The meal costs 4256 drachma (DR). (Total 4 marks) Use the bank-selling price to calculate the cost of the meal in British pounds. The Williams family go to Germany. Before leaving, they change 600 GBP into German marks. Calculate the number of German marks they receive for 600 GBP, giving your answer correct to two decimal places. They spend 824 DM in Germany, and on returning to the United Kingdom, they change their remaining German marks into British pounds. (c) Calculate the number of British pounds they receive, correct to two decimal places. (Total 4 marks) IB Questionbank Mathematical Studies 3rd edition 4

5 9. Takaya invested 1000 JPY at 6.3% simple interest for 15 years. Morimi invested 900 JPY at 6.3% interest compounded annually for 15 years. Who had more money at the end of the 15th year? Justify your answer clearly. (Total 5 marks) 10. The following is a currency conversion table: FFR USD JPY GBP French Francs (FFR) 1 p q US Dollars (USD) Japanese Yen (JPY) British Pounds (GBP) For example, from the table 1 USD = GBP. Use the table to answer the following questions. Find the values of p and q. Mireille wants to change money at a bank in London. (i) (ii) How many French Francs (FFR) will she have to change to receive 140 British Pounds (GBP)? The bank charges a 2.4% commission on all transactions. If she makes this transaction, how many British Pounds will Mireille actually receive from the bank? (4) (c) Jean invested 5000 FFR in Paris at 8% simple interest per annum. Paul invested 800 GBP in London at 6% simple interest per annum. (i) (ii) (iii) (iv) How much interest in FFR did Jean earn after 4 years? How much interest in US Dollars did Paul earn after 4 years? Who had earned more interest after 4 years? Explain your reasoning in part (c) (iii). (7) (Total 13 marks) 11. The rate of inflation from the beginning of 1995 has been 4.5% per year. A loaf of bread cost $1.70 on January 1, What did it cost on January 1, 1999? A car cost $ on January 1, What did it cost on January 1, 1997? (Give your answer to the nearest thousand dollars.) (Total 4 marks) IB Questionbank Mathematical Studies 3rd edition 5

6 12. On Vera s 18 th birthday she was given an allowance from her parents. She was given the following choices. Choice A Choice B Choice C Choice D $100 every month of the year. A fixed amount of $1100 at the beginning of the year, to be invested at an interest rate of 12% per annum, compounded monthly. $75 the first month and an increase of $5 every month thereafter. $80 the first month and an increase of 5% every month. Assuming that Vera does not spend any of her allowance during the year, calculate, for each of the choices, how much money she would have at the end of the year. (8) Which of the choices do you think that Vera should choose? Give a reason for your answer. (c) On her 19 th birthday Vera invests $1200 in a bank that pays interest at r% per annum compounded annually. Vera would like to buy a scooter costing $1452 on her 21 st birthday. What rate will the bank have to offer her to enable her to buy the scooter? (4) (Total 14 marks) 13. Frederick had to change British pounds (GBP) into Swiss francs (CHF) in a bank. The exchange rate is 1 GBP = 2.5 CHF. There is also a bank charge of 3 GBP for each transaction. How many Swiss francs would Frederick buy with 133 GBP? Let s be the number of Swiss francs received in exchange for b GBP. Express s in terms of b. (c) Frederick received 430 CHF. How many British pounds did he exchange? (Total 8 marks) 14. Keisha had USD to invest. She invested m USD at the Midland Bank, which gave her 8% annual interest. She invested f USD at the First National Bank, which gave 6% annual interest. She received a total of 640 USD in interest at the end of the year. Write two equations that represent this information. Find the amount of money Keisha invested at each bank. IB Questionbank Mathematical Studies 3rd edition 6 (Total 8 marks)

7 15. The exchange rate from US dollars (USD) to French francs (FFR) is given by 1 USD = 7.5 FFR. Give the answers to the following correct to two decimal places. Convert 115 US dollars to French francs. Roger receives 600 Australian dollars (AUD) for 2430 FFR. Calculate the value of the US dollar in Australian dollars. (Total 8 marks) 16. A Swiss bank shows currency conversion rates in a table. Part of the table is shown below, which gives the exchange rate between British pounds (GBP), US dollars (USD) and Swiss francs (CHF). Buy Sell GBP USD This means that the bank will sell its British pounds to a client at an exchange rate of 1 GBP = CHF. What will be the selling price for 1 USD? Andrew is going to travel from Europe to the USA. He plans to exchange 1000 CHF into dollars. The bank sells him the dollars and charges 2% commission. How many dollars will he receive? Give your answer to the nearest dollar. (Total 8 marks) 17. David invests 6000 Australian dollars (AUD) in a bank offering 6% interest compounded annually. Calculate the amount of money he has after 10 years. David then withdraws 5000 AUD to invest in another bank offering 8% interest compounded annually. Calculate the total amount he will have in both banks at the end of one more year. Give your answer correct to the nearest Australian dollar. (Total 8 marks) IB Questionbank Mathematical Studies 3rd edition 7

8 1. (i) Cost of loan = 4000(1.01) = $ (A1) Note: Accept $ and then $4450 for part (i) (ii) (100) Equivalent S.I. Rate = 4000 = = 12.7% (3 s.f.) (A1) 4 (i) Deposit = 25% of $4000 = $1000 (A1) (ii) Cost of the loan = (12) 3000 (or equivalent) = $ (A1) 3 (c) (i) Option A. Because she doesn t need a deposit (or equivalent appropriate explanation). (R1) (ii) Option B. Because it is cheaper by $( ) = $57.30 (or equivalent appropriate explanation). (R1) 2 (d) x + 19y = 1540 x + 29y = 2140 x = 400 (A1) y = 60 (A1) 4 (e) (n 1)60 = 4000 n = 61 Therefore, 61 months. (A1) 2 [15] 2. Compound interest (A1) 8% per year (A1) IB Questionbank Mathematical Studies 3rd edition 1

9 Year Value at beginning of year 1st CHF 500 CHF 540 Value at end of year 2nd CHF 540 CHF rd CHF CHF th CHF CHF th CHF CHF (A1) 6th CHF CHF (A1) [4] r 3. A = C n 2 r 125 = r 1.25 = r = 100 r = 11.8% (3 s.f.) (A1) (C4) [4] I = 100 = CHF (A1) I = (1.05) = CHF (or CHF to 3 s.f.) (A1) [4] IB Questionbank Mathematical Studies 3rd edition 2

10 5. Notes: If no method is shown, award (A1) if and only if answer is correct, otherwise award zero marks. However, award if correct method is shown; even if final answer is wrong. (i) p = (ii) q = = 1.56 (2 d.p.) (A1) = 5.26 (2 d.p.) (A1) 4 Notes: For parts (i) and (ii) accept and follow through with conversions routed via candidate s home currency. For example: USD 1 = GBP 0.64 GBP 1 = FFR 8.33 USD 1 = FFR (0.64) (8.33) q = 5.33 instead of 5.26 (i) GBP ( ) = GBP (A1) 2 Note: Accept ( (or candidate s p)) = GBP (ii) ( ) = GBP (A1) 1 Note: Follow through from part (i) above. ( ) = GBP (A1) 1 Note: Follow through from parts (i) and (ii). (c) (i) 700(1.05) 4 = GBP Therefore interest = GBP (A1) 2 (ii) 700(1.05) 5 = = (2 d.p.) 700(1.05) 6 = = (2 d.p.) therefore after 6 years (A1) 2 Note: Accept other correct methods. (d) C(1.05) 4 = C = 4 (1.05) Notes: Award the at the point where C has been correctly isolated 900 Accept C = = GBP = GBP 740 (nearest GBP) (A1) 2 [14] 6. Monthly repayment = AUD (A1) (C1) Monthly repayment on AUD 8000 taken over 5 years at IB Questionbank Mathematical Studies 3rd edition 3

11 18% per annum = = AUD Note: Award for multiplying by 8 Total repaid = = AUD (A1) (C3) Notes: Award for multiplying by (5 12) or 60. No accuracy specified, but the answer is exact. [4] 7. Price today = USD = USD = USD or US$ (to the nearest dollar) (A1) (C2) Note: Accept equivalent method = r = r 3 r = Rate = 6.00% per annum (3 s.f.) (accept 6%) (A1) (C2) [4] = 16 GBP (A1) Note: Penalize only once in parts and (c) if the buying and selling price are consistently confused throughout the question. Allow (ft) marks = DM (A1) (c) = DM = ( ) GBP = GBP (2 d.p.) (A1) Note: Penalize only once for answer not given to 2 d.p. [4] IB Questionbank Mathematical Studies 3rd edition 4

12 9. Takaya: (0.063)(15) = 1945 JPY (A1) Morimi: 900(1.063) 15 = 2250 JPY (A1) Morimi had more (A1) 5 Note: Award (M0)(A0) for computing interest only for Takaya. Award follow through, (A1) follow through for follow through with only interest for Morimi. Award (A1) for Morimi had more for comparing the two interests. Award (A1) for Morimi had more for any reasoning that shows understanding of difference between compound and simple interest. Award (A0) for Morimi had more after computing interest for Takaya, but interest + principal for Morimi. Award (A1) for Morimi had more with no work shown at all. [5] 10. p = (A1) q = 17.5 (A1) 2 (i) 140(9.901) = = 1390 FFR (3 s.f.) (A1) OR = = 1390 FFR (3 s.f.) (A1) 2 Also accept 1387 FFR (you must go up to get 140 GBP) (ii) 140(0.024) = 3.36 GBP Commission = or 137 (3 s.f.) (A1) 2 Note: Commission can be calculated in FFR also for (Ml). (c) (i) I = 5000(0.08)(4) = 1600 FFR (A1) 2 (ii) I = 800 (0.06)(4) = 192 GBP 192(1.585) = USD or 304 USD (3 s.f.) (A1) 2 IB Questionbank Mathematical Studies 3rd edition 5

13 (iii) Paul (A1) 1 (iv) Jean: 1600 FFR = or 254 (3 s.f.) (R1) Paul: or 304 (3 s.f.) (R1) 2 Note: Accept any same currency comparison. [13] A = = $ = $1.94 (3 s.f.) (A1) = C = C $ = C $37000 = C (to the nearest thousand dollars) (A1) Note: Accuracy is specified in the question, therefore this is not a paper accuracy penalty. [4] 12. Choice A = = $1200 (A1) 12 Choice B = = $ (A1) Choice C = = 12 { } $ (A1) Choice D = (1.05 1) = $ (1.05 1) (A1) Note: Award method marks if candidate works out each amount. But the answer has to be accurate to receive the mark of (A1). 8 IB Questionbank Mathematical Studies 3rd edition 6

14 Option D because the total allowance is the highest (C1) (R1) 2 (c) r = 1452 (A1) r = 1452 = 1.21 (A1) r = = 1.1 (A1) 100 r = r = 10% (A1) 4 [14] 13. S = 2.5(133 3) S = 325 CHF (A1) (C2) S = 2.5(b 3) or 2.5(b 3) (A2) (C2) (c) 430 = 2.5(b 3) (M2) 172 = b = b 175 GBP (A2) (C4) [8] m f = 640 (A2) (C2) m + f = (A2) (C2) 8 m + 6 f = m 8 f = f = (C2) f = 8000 or $8000 at First National Bank (A1) m = 2000 or $2000 at Midland Bank (A1) (C2) Note: Allow ft from part. [8] = (A1) USD (A1) (C3) IB Questionbank Mathematical Studies 3rd edition 7

15 METHOD FFR 600 x AUD 1 FFR = AUD (A1) 1 USD = 7.5 FFR = AUD = (A1) 1 USD = 1.85 AUD (2 d.p.) (A1) (C5) METHOD FFR = 600 AUD FFR = AUD (= AUD) USD = 7.5 FFR = 7.5 = (A1) USD = 1.85 AUD (2 d.p.) (A1) (C5) [8] CHF (A2) (C2) Method 1 2% of 1000 CHF = 20 (A1) Amount = = 980 CHF (A1) 980 = USD = USD (A1) = 554 USD (to nearest dollar) (A1) (C6) Method CHF = 1 USD CHF = 1.77 = (A1) (A1) = (A1) = 554 USD (to the nearest dollar) (ft from answer in ) (A1) (C6) [8] 17. A = 6000(1.06) 10 (A1) = (AUD) (A1) (C3) IB Questionbank Mathematical Studies 3rd edition 8

16 = 5745 (A1) = (A1) = (to the nearest AUD) (A1) (C5) [8] IB Questionbank Mathematical Studies 3rd edition 9