Ch 13 FM YR 12 Page 607 Monday, November 13, 2000 3:28 PM 13 Simple interest VCE coverage Area of study Units 3 & 4 • Business related mathematics In this cha chapter pter 13A Simple interest 13B Finding P, r and T 13C Bonds, debentures and term deposits 13D Bank savings accounts 13E Hire-purchase 13F Effective rate of interest Ch 13 FM YR 12 Page 608 Monday, November 13, 2000 3:28 PM 608 Further Mathematics Simple interest People often wish to buy goods and services that they cannot afford to pay for at the time, but which they are confident they can pay for over a period of time. The options open to these people include paying by credit card (usually at a very high interest rate), lay-by (where the goods are paid off over a period of time with no interest charged but no access to or use of the goods until the last payment is made), hire-purchase, or a loan from the bank. The last two options usually attract what is called simple interest. This is the amount of money charged by the financial institution for the use of its money. It is calculated as a percentage of the money borrowed multiplied by the number of periods (usually years) over which the money is borrowed. As an example, Monica wished to purchase a television for $550, but did not have the ready cash to pay for it. She made an agreement to borrow the money from a bank at 12% p.a. (per year) simple interest and pay it back over a period of 5 years. The amount of interest Monica would be charged on top of the $550 is $550 × 12% × 5 years which is $330. Therefore, Monica is really paying $550 + $330 = $880 for the television. Ch 13 FM YR 12 Page 609 Monday, November 13, 2000 3:28 PM Chapter 13 Simple interest 609 Total amount of loan or investment = Initial amount or Principal + Interest (charged or earned) A=P+I It would have been more economical for Monica to buy the television for cash at the time. However, by borrowing the money she has use of the television while she is paying it off. Also, by using this method she would be paying a small amount each month which is easy to budget for. The down-side is that she must pay the extra interest. Simple interest is the percentage of the amount borrowed or invested multiplied by the number of time periods (usually years). The amount is added to the principal either as payment for the use of the money borrowed or as return on money invested. PrT I = ---------100 I = Simple interest charged or earned ($) P = Principal (money invested or loaned) ($) r = Rate of interest per period (% per period) T = Time, the number of periods (years, months, days) over which the agreement operates Hint: The interest rate, r, and time period, T, must be stated and calculated in the same time terms, for example: 4% per annum for 18 months must be calculated over 1 1--2- years, as the interest rate period is stated in years (per annum); 1% per month for 2 years must be calculated over 24 months, as the interest rate period is stated in months. WORKED Example 1 Find the simple interest charged on borrowing $325 for 5 years at 3% p.a. (per annum or per year) interest. THINK WRITE 1 Write the simple interest formula. PrT I = ---------100 2 List the values of P, r and T. Check that r and T are in the same time terms. P = $325 r = 3% per year T = 5 years 3 Substitute into the formula. 325 × 3 × 5 I = --------------------------100 4 Use a calculator to evaluate. I = 48.75 5 Write your answer. The interest charged for borrowing $325 over 5 years is $48.75. Ch 13 FM YR 12 Page 610 Monday, November 13, 2000 3:28 PM 610 Further Mathematics Graphics Calculator tip! Simple interest calculations 1. Transpose the formula for simple interest so that it PrT equals zero. (0 = I – ---------- ). 100 2. Press MATH , choose 0: Solver then press the up arrow key and enter the equation. 3. Press ENTER , then enter the values of the known variables and move the cursor to the variable to be solved for. The given values for worked example 1 are shown in the screen at right. 4. Press ALPHA [SOLVE]. WORKED Example 2 Jan invested $210 with a building society in a fixed deposit account that paid 8% p.a. simple interest for 18 months. How much did she receive after the 18 months? THINK 1 Write the simple interest formula. 2 List the values of P, r and T. Check that r and T are in the same time terms. Need to convert 18 months into years. WRITE PrT I = ---------100 P = $210 r = 8% p.a. T = 18 months = 1 1--2- years 3 Substitute into the formula. 4 Use a calculator to evaluate. Add the interest to the principal (total amount received). Write your answer. 5 6 210 × 8 × 1 1--2I = -----------------------------100 I = $25.20 A=P+I A = $210 + $25.20 Total amount received at the end of the investment is $235.20. Ch 13 FM YR 12 Page 611 Monday, November 13, 2000 3:28 PM Chapter 13 Simple interest 611 remember remember 1. Simple interest is the percentage of an amount borrowed or invested multiplied by the number of time periods, (usually years). The interest is added to the principal as payment for the use of the money or as return on the money invested. 2. A = P + I where A = Total amount ($) P = Principal, or amount borrowed or invested ($) I = Simple interest charged or earned ($) PrT 3. I = ---------100 I = Simple interest charged or earned ($) P = Principal (money invested or loaned) ($) r = Rate of interest earned per period (% per period) T = Time, the number of periods over which the agreement operates 4. Interest rate, r, and time, T, must be stated and calculated in the same time terms. 13A 13.1 SkillS HEET Mat 1 1 Find the interest charged on the following amounts borrowed for the d reads hca L Sp he periods and at the rates given. a $680 for 4 years at 5% p.a. b $210 for 3 years at 9% p.a. Simple interest c $415 for 5 years at 7% p.a. d $460 at 12% p.a. for 2 years 3 1 GC pro e $1020 at 12 --2- % p.a. for 2 years f $713 at 6 --4- % p.a. for 7 years 1 g $821 at 7 --4- % p.a. for 3 years h 11.25% p.a. on $65 for 6 years Simple i 6.15% p.a. on $21.25 for 9 years j 9.21% p.a. on $623.46 for 4 years interest k 13 3--4- % p.a. on $791.35 for 5 years. et Example EXCE WORKED Simple interest gram 13.2 SkillS HEET 2 Find the interest charged or earned on the following loans and investments: a $690 loaned at 12% p.a. simple interest for 15 months b $7500 invested for 3 years at 1% per month simple interest c $25 000 borrowed for 13 weeks at 0.1% per week simple interest d $250 invested at 1 3--4- % per month for 2 1--2- years. WORKED Example 2 3 Find the amount to which each investment has grown after the investment periods shown in the following examples: a $300 invested at 10% p.a. simple interest for 24 months b $750 invested for 3 years at 1% per month simple interest c $20 000 invested for 3 years and 6 months at 11% p.a. simple interest d $15 invested at 6 3--4- % p.a. for 2 years and 8 months e $10.20 invested at 8 1--2- % p.a. for 208 weeks. Ch 13 FM YR 12 Page 612 Monday, November 13, 2000 3:28 PM 612 Further Mathematics 4 multiple choice If John had $63 in his bank account and earned 9% p.a. over 3 years, the simple interest earned would be: A $5.67 B $1701 C $17.01 D $22.68 E $27.00 5 multiple choice If $720 was invested in a fixed deposit account earning 6 1--2- % p.a. for 5 years, the interest earned at the end of 5 years would be: A $234.00 B $23 400.00 C $23.40 D $216.00 E $350.00 6 multiple choice A 4-year bond paid 7.6% p.a. simple interest. If Sonja bought a bond worth $550, the interest she earned would be: A $16.72 B $167.20 C $717.20 D $1672 E $30.40 7 multiple choice Bodgee Bank advertised a special offer. If a person invests $150 for 2 years, the bank will pay 12% p.a. simple interest on the money. At the expiry date, the investor would have earned: A $300 B $36 C $186 D $48 E $24 8 multiple choice Maclay invested $160 in a bank for 6 years earning 8% simple interest each year. At the end of the 6 years, he will receive in total: A $928 B $236.80 C $76.80 D $768 E $208 9 multiple choice Simple interest was calculated on a term deposit of 4 years at 3 1--2- % per year. When Ashleigh calculated the total return on her investment of $63.50, it was: A $72.39 B $7.75 C $71.24 D $8.89 E $75.50 10 multiple choice Joanne asked Sally for a loan of $125 to buy new shoes. Sally agreed on the condition that Joanne paid it back in two years at 3% p.a. simple interest. The amount Joanne paid Sally at the end of the two years was: A $200 B $7.50 C $130.50 D $132.50 E $128.75 Ch 13 FM YR 12 Page 613 Monday, November 13, 2000 3:28 PM Chapter 13 Simple interest 613 11 multiple choice Betty invests $550 in an investment account earning 4% p.a. simple interest over 6 years. Ron puts his $550 in a similar investment earning 5% p.a. simple interest for 5 years. The difference in their earnings at the end of the investment period is: A $55 B $5.50 C $7.50 D $0 E $595 12 multiple choice Two banks pay simple interest on short-term deposits. Hales Bank pays 8% p.a. over 5 years and Countrybank pays 10% p.a. for 4 years. The difference between the two banks’ final payout figure if $2000 was invested in each account is: A $0 B $800 C $2800 D $150 E $400 13 Robyn wishes to purchase a new dress worth $350 to wear to the school formal. If she borrows the total amount from the bank and pays it off over 3 years at 11% p.a. simple interest, what is the total amount Robyn must pay back to the bank? 14 The Sharks Building Society offers loans at 8 1--2- % p.a. simple interest for a period of 18 months. Andrew borrows $200 from Sharks to buy Monique an engagement ring. Calculate the amount of interest Andrew is to pay over the 18 months. 15 Silvio invested the $1500 he won in Lotto with an insurance company bond that pays 12 1--4- % p.a. simple interest provided he keeps the bond for 5 years. What is Silvio’s total return from the bond at the end of the 5 years? 16 The insurance company that Silvio used in the previous question allows people to withdraw part or all the money early. If this happens the insurance company will only pay 6 3--4- % p.a. simple interest on the amount which is withdrawn over the period it was invested in the bond. The part which is left in the bond receives the original agreed interest. Silvio needed $700 for repairs to his car 2 years after he had invested the money but left the rest in for the full 5 years. How much interest did he earn from the bond in total? 17 Jill and John decide to borrow money to improve their boat, but cannot agree which loan is the better value. They would like to borrow $2550. Jill goes to the Big-4 Bank and finds that they will lend her the money at 11 1--3- % p.a. simple interest for 3 years. John finds that the Friendly Building Society will lend the $2550 to them at 1% per month simple interest for the 3 years. a Which institution offers the best rates over the 3 years? b Explain why. Ch 13 FM YR 12 Page 614 Monday, November 13, 2000 3:28 PM 614 Further Mathematics Finding P, r and T In many cases we may wish to find the principal, interest rate or period of a loan. In these situations it is necessary to rearrange or transpose the simple interest formula after (or before) substitution, as the following example illustrates. WORKED Example 3 A bank offers 9% p.a. simple interest on an investment. At the end of 4 years the interest earned was $215. How much was invested? THINK WRITE 1 Write the simple interest formula. 2 List the values of I, r and T. Check that r and T are in the same time terms. 3 Substitute into the formula. 4 Make P the subject by multiplying both sides by 100 and dividing both sides by (9 × 4). Use a calculator to evaluate. Write your answer. 5 6 PrT I = ---------100 I = $215 r = 9% p.a. T = 4 years P×r×T I = ---------------------100 P×9×4 215 = ---------------------100 215 × 100 P = -----------------------9×4 P = 597.22 The amount invested was $597.22. Transposed simple interest formula It may be easier to use the transposed formula when finding P, r or T. Simple interest formula transposes: to find the principal to find the interest rate to find the period of the loan or investment 100 × I P = ----------------r×T 100 × I r = ----------------P×T 100 × I T = ----------------P×r Ch 13 FM YR 12 Page 615 Monday, November 13, 2000 3:28 PM Chapter 13 Simple interest 615 WORKED Example 4 When $720 is invested for 36 months it earns $205.20 simple interest. Find the yearly interest rate. THINK WRITE/DISPLAY 1 Write the simple interest formula. 2 List the values of P, I and T. T must be expressed in years so that r can be evaluated in % per year. 3 Substitute into the formula. 4 Evaluate on a calculator. Remember to bracket (720 × 3). 5 Write your answer. 100 × I r = ----------------P×T P = $720 I = $205.20 T = 36 months = 3 years 100 × 205.20 r = ------------------------------720 × 3 The interest rate offered was 9.5% per annum. WORKED Example 5 An amount of $255 was invested at 8.5% p.a. How long will it take, to the nearest year, to earn $86.70 in interest? THINK WRITE/DISPLAY 1 Write the simple interest formula. 2 Substitute the values of P, I and r. The rate, r is expressed per annum so time, T, will be evaluated in the same time terms, namely years. 3 Substitute into the formula. 4 Evaluate on a calculator. Remember to bracket (255 × 8.5). 5 Write your answer. 100 × I T = ----------------P×r P = $255 I = $86.70 r = 8.5% p.a. 100 × 86.70 T = ---------------------------255 × 8.5 The period of the investment was 4 years. Ch 13 FM YR 12 Page 616 Monday, November 13, 2000 3:28 PM 616 Further Mathematics remember remember When finding P, r or T: 1. substitute the given values into the formula and then rearrange to isolate the pronumeral, or 2. transpose the simple interest formula 100 × I (a) to find the principal P = ----------------r×T 100 × I (b) to find the interest rate r = ----------------P×T 100 × I (c) to find the period of the loan or investment T = ----------------P×r and substitute the given values into the transposed formula. 13B 13.3 SkillS HEET WORKED Example 3 Mat d hca Finding P, r and T WORKED Example 4 WORKED Example 5 Finding P, r and T 1 For each of the following, find the principal invested. a Simple interest of 5% p.a., earning $307 interest over 2 years b Simple interest of 7% p.a., earning $1232 interest over 4 years c Simple interest of 8% p.a., earning $651 interest over 18 months d Simple interest of 5 1--2- % p.a., earning $78 interest over 6 years e Simple interest of 6.25% p.a., earning $625 interest over 4 years 2 For each of the following, find the interest rate offered. Express rates in % per annum. a Loan of $10 000, with a $2000 interest charge, for 2 years b Investment of $5000, earning $1250 interest for 4 years c Loan of $150, with a $20 interest charge, for 2 months d Investment of $1400 earning $178.50 interest for 6 years e Investment of $6250 earning $525 interest for 2 1--2- years 3 For each of the following, find the period of time (to the nearest month) for which the principal was invested or borrowed. a Investment of $1000 at simple interest of 5% p.a. earning $50 b Loan of $6000 at simple interest of 7% p.a. with an interest charge of $630 c Loan of $100 at simple interest of 24% p.a. with an interest charge of $6 d Investment of $23 000 at simple interest of 6 1--2- % p.a. earning $10 465 e Loan of $1 500 000 at simple interest of 0.125% per month earning $1875 4 Lennie Cavan earned $576 in interest when she invested in a fund paying 9.5% simple interest for 4 years. How much did Lennie invest originally? 5 Lennie’s sister Lisa also earned $576 interest at 9% simple interest, but she only had to invest it for 3 years. What was Lisa’s initial investment? 6 Jack Kahn put some money away for 5 years in a bank account which paid 3 3--4- % interest. He found from his bank statement that he had earned $66. How much did Jack invest? Ch 13 FM YR 12 Page 617 Monday, November 13, 2000 3:28 PM Chapter 13 Simple interest 617 7 James needed to earn $225 in one year. He invested $2500 in an account earning simple interest at a rate of 4.5% p.a. paid monthly. How many months will it take James to achieve his aim? 8 Carol has $3000 to invest. Her aim is to earn $450 in interest at a rate of 5% p.a. Over what term would she invest? 9 multiple choice Peter borrowed $5000 and intended to pay it back in 3 years. The terms of the loan indicated Peter was to pay 9 3--4- % p.a. interest. The interest Peter paid on the loan was: A $146 250 B $446.25 C $1462.50 D $121.88 E $1211.88 10 multiple choice Joanne’s accountant found that for the past 2 years she had earned a total of $420 interest in an account paying 6% simple interest. When she calculated how much she invested the amount was: A $350 B $3500 C $50.40 D $7000 E $70.00 11 multiple choice A loan of $1000 is taken over 5 years. The simple interest is calculated monthly. The total amount repaid for this loan is $1800. The simple interest rate per year on this loan is closest to: A 8.9% B 16% C 36% D 5% E 11.1% 12 multiple choice Jarrod decides to buy a motorbike at no deposit and no repayments for 3 years. He takes out a loan of $12 800 and is charged at 7.5% p.a. simple interest over the 3 years. The lump sum Jarrod has to pay in 3 years time is: A $960 B $13 760 C $2880 D $12 800 E $15 680 13 multiple choice Chris and Jane each take out loans of $4500 and are offered 6 1--4- % p.a. simple interest over a 3-year period. Chris’s interest is paid monthly whereas Jane’s is paid yearly. The difference in the total amount of interest each person pays after the 3 years is: A none B $877.50 C $10 530 D $9652.50 E $1000 14 Alisha has $8900 that she is able to invest. She has a goal of earning at least $1100 in 2 years or less. Do any of the following investments satisfy Alisha’s goal? a 10% p.a. for 15 months b 4 1--4- % p.a. earning $1200 c After 100 weeks a final payout of $10 500 d After 2 years at 0.6% per month Ch 13 FM YR 12 Page 618 Monday, November 13, 2000 3:28 PM 618 Further Mathematics Bonds, debentures and term deposits Debentures If a company needs money, one option is for it to offer a debenture (a legal document detailing an investment agreement) for sale to the public. An investor will pay an amount of money (principal) to the company, and in return the company agrees to pay the investor interest at regular intervals (monthly, quarterly or yearly). At the end of the agreed term the principal is returned to the investor. The advantage of the debenture is two-fold: first, the company has the use of the money during the agreed period to make more money for the company and second, the investor knows what their return will be for each period and is guaranteed the return of the principal. Term deposits TOP INVESTOR RATES 1 to 5 yr effective rates are shown in brackets. Source: CANNEX (Polifax 019 725 660). BEST BANK TERM DEPOSITS Rate Period Bank 4.70 30 days HSBC 5.00 90 days Arab Bank 5.12 180 days HSBC 5.25 270 days Arab Bank (5.58) Suncorp Metway 5.00 1 year 5.66/5.70 (5.78) 2 years HSBC/PIBA 6.20 (6.20) 3 years HSBC 6.16 (6.30) 4 years HSBC 6.33 (6.48) 5 years HSBC Period 30 days 90 days 180 days 270 days 1 year 2 years 3 years 4 years 5 years BEST OTHERS Rate Institution 3.95 GIO 4.75 GIO 4.95 GIO Greater BS/HC CU 5.00 5.25 GIO 5.49 GIO 5.70 AGC 6.00 Police CU 6.20 AGC (5.35) (5.60) (5.82) (6.09) (6.34) Term deposits allow an investor to lend money to a bank or building society for a particular length of time. The money cannot be withdrawn during the agreed period but earns a better interest rate than in a normal savings account. At the end of the term the interest plus the principal is paid back to the investor. The advantage of the term deposit is that the money is secure and the interest rate is better than that on a savings account. The disadvantage, of course, is that if the money is needed during the period it cannot be withdrawn (except under special circumstances agreed to by the bank). Investment bonds Investment bonds are another form of investment which is offered to the investor by a bank or the government, and interest is paid on the investment monthly, quarterly, six monthly or annually. The one advantage is that the bond can be sold to someone else during the period before the maturation date. This allows the investor some flexibility if the money is needed during the period of investment. All the above investment types offer advantages to the investor and to the institution. The institution has the use of the money over a fixed period and the investor receives higher than normal interest. All of these investments carry some risk and individuals must decide on which type to use based on personal circumstances. Bonds, debentures and term deposits are simple interest accounts. Ch 13 FM YR 12 Page 619 Monday, November 13, 2000 3:28 PM Chapter 13 Simple interest 619 WORKED Example 6 Jaclyn buys $50 000 worth of debentures in a company. She earns 9.5% p.a. simple interest, paid to her quarterly (that is, every 3 months). If the agreed period of the debenture was 18 months: a calculate the amount of interest Jaclyn will earn for each quarter b calculate the total amount collected at the end of the term. THINK WRITE a b 1 Write the simple interest formula. 2 List the values of P, r and T. Convert the interest rate period to quarters. 3 Substitute into the formula and evaluate. 4 Write your answer. There are 6 quarters in 18 months. Alternatively, use the simple interest formula with the new data. 1 2 Write your answer. PrT a I = ---------100 P = $50 000 r = 9.5% per year = 9.5% ÷ 4 per quarter = 2.375% per quarter T = 1 quarter 50 000 × 2.375 × 1 I = --------------------------------------------100 = 1187.50 Jaclyn will earn $1187.50 for each quarter. b Total interest = $1187.50 × 6 = $7125 or 50 000 × 2.375 × 6 I = --------------------------------------------100 = 7125 The total interest earned is $7125. WORKED Example 7 Townbank offers a term deposit account paying investors 12.5% p.a. simple interest on investments over $100 000 for 2 years or more. Peta decides to invest $150 000 in this account for 2 years. How much interest will Peta earn at the end of the investment? THINK WRITE 1 Write the simple interest formula. PrT I = ---------100 2 List the values of P, r and T. Check that r and T are described in the same time terms. P = $150 000 r = 12.5% p.a. T = 2 years 3 Substitute into the formula and evaluate. 4 Write your answer. 150 000 × 12.5 × 2 I = --------------------------------------------100 = $37 500 Peta’s $150 000 invested for 2 years will earn $37 500. Ch 13 FM YR 12 Page 620 Monday, November 13, 2000 3:28 PM 620 Further Mathematics WORKED Example 8 An investment bond is offered to the public at 9% p.a. Louise buys a bond worth $2000 that will mature in 2 years. How much in total will Louise receive at the end of the 2 years? THINK WRITE 1 Write the simple interest formula. 2 List the values of P, r and T. 3 Substitute into the formula. 4 5 Use a calculator to evaluate. Add interest to principal. 6 Write your answer. PrT I = ---------100 P = $2000 r = 9% p.a. T = 2 years 2000 × 9 × 2 I = -----------------------------100 I = $360 A=P+I A = 2000 + 360 = 2360 The $2000 investment bond will mature at the end of 2 years to a total of $2360 at simple interest of 9% p.a. remember remember 1. Simple interest accounts include bonds, debentures and term deposits. 2. Read the question carefully: does it ask for the interest or the final total amount? 13C d hca WORKED Mat Example Spreadshe et EXCEL Simple interest 6 WORKED Example 7 Simple interest WORKED GC p am rogr Simple interest Example 8 Bonds, debentures and term deposits 1 Spice Clothing company offers debentures paying 8% p.a. interest paid quarterly for a period of 2 years. When $20 000 worth of Spice debentures are purchased, calculate the total return on the investment. 2 Harry decided to invest $2000 in a term deposit for 18 months. The bank offered 10.5% p.a. interest paid each half-year. Calculate the interest Harry would earn on the investment. 3 An investment bond is advertised as paying 10 1--2- % p.a. interest on a 3-year investment. Elise purchased a bond for $3000, but needed to sell it after 18 months. How much will Elise receive at the end of her 18-month investment? 4 Rabbit debentures, worth $10 000, were purchased for a period of 15 months. The debenture paid 12% p.a., payable each 3 months. What was the investment worth at the end of the 15 months? Ch 13 FM YR 12 Page 621 Monday, November 13, 2000 3:28 PM Chapter 13 Simple interest 621 5 JNK Bank offers term deposits on amounts above $5000 at 12% p.a. simple interest payable each quarter for periods longer than 2 years. Mr Smith invests $6000 in this term deposit for 2 1--2- years. What is Mr Smith’s final return on his money? 6 Mark purchases a $2500 investment bond earning 12 1--4- % p.a. interest paid yearly. The bond matures after 2 years. What interest will Mark earn? 7 multiple choice Debentures in TRADEX are issued at 9% p.a. simple interest. The interest gained on an investment of $7000 over 3 years would be: A $630 B $1890 C $18 900 D $7630 E $21 000 8 multiple choice The rate of interest on a term deposit for 3 months is 4.25% per year. If $10 000 is invested in the term deposit, the amount of interest earned over the 3 months is: A $106.25 B $425 C $141.67 D $1062.50 E $1275 9 multiple choice State government bonds pay interest of 7 1--4- % p.a. simple interest. Philippa invested $2500 in the bonds which mature in 5 years. Philippa’s income each quarter would be: A $181.25 B $2718.77 C $45.31 D $725 E $72.50 10 multiple choice ElCorp offers company debentures earning 8 1--2- % p.a. interest for an investment of $5000 for 2 years. The interest on the investment is: A $170 B $212.50 C $825 D $850 E $85 11 multiple choice A term deposit is advertised stating that if $2500 is invested for 2 years the interest earned is $285. The rate of interest per annum is: A 10% B 17.5% C 5.7% D 11.4% E 10.5% Ch 13 FM YR 12 Page 622 Monday, November 13, 2000 3:28 PM 622 Further Mathematics 12 multiple choice An investment bond of $7500 pays interest of $1125 at 3.75% p.a. interest. The time the bond is taken for is: A 3 years B 4 1--2- years C 3 1--2- years D 4 years E 5 years 13 multiple choice A principal amount is invested in a bond that will accumulate to a total of $64 365 after 4 months at 6 1--2- % p.a. The principal is: A $60 000 B $63 000 C $6300 D $50 000 E $5000 14 The following term deposit rates were advertised in a magazine Toni Ford had $5500 to invest. Calculate her return if she invested the money in a term deposit with this bank for: a 35 days b 120 days c 1 year. Hint: Express days as a fraction of a year. Term Rate 30–59 days 4.2% p.a. 60–149 days 4.7% p.a. 150–269 days 5.0% p.a. 270–365 days 5.4% p.a. 15 Dennis and Delia have $7500 to invest. They know that they will need the money in 18 months but are not sure how to invest it. While reading a magazine, they see the following three advertisements: i investment bonds offered at 12 1--2- % p.a. interest paid each 6 months ii debentures in a company paying 12% p.a. with interest paid each quarter Work ET SHE 13.1 iii a term deposit paying 11 3--4- % p.a. interest paid each 3 months. a Calculate their total return on each investment. b What did you notice about the time in which the interest was calculated? Ch 13 FM YR 12 Page 623 Monday, November 13, 2000 3:28 PM Chapter 13 Simple interest 623 Bank savings accounts Most banks offer their customers savings accounts with interest that is usually paid on 1. the minimum monthly balance, or 2. the daily balance. The interest is added at a specified time — say once or twice a year — as nominated by the bank, for example, on the first day of June and December of each year. The more frequently the interest is added, the better for the customers. Savings accounts — minimum monthly balances To calculate interest on a minimum monthly balance saving account, the bank looks at the balances of the account for each month and calculates the interest on the smallest balance that appears in each month. WORKED Example 9 At the beginning of March, Ryan had $621 in his savings bank account. On 10 March he deposited $60. If the bank pays 8% p.a. interest paid monthly and calculated on the minimum monthly balance, calculate the interest Ryan earns in March. THINK WRITE The smallest balance for March is Minimum monthly balance for March is $621. 1 $621, as the only other transaction in that month increased the balance. PrT I = ---------2 Write the simple interest formula. 100 P = $621 3 List the values of P, with r and T in 8 months. - % per month r = ----12 T = 1 month 4 Substitute into the formula and evaluate. 5 Write your answer. 8 -×1 621 × ----12 I = ----------------------------100 = 4.14 The interest earned for the month of March was $4.14. Ch 13 FM YR 12 Page 624 Monday, November 13, 2000 3:28 PM 624 Further Mathematics The minimum monthly balance method is used in the next worked example. WORKED Example 10 Minimum monthly balance method Date Deposit Withdrawal 3/7 $100 $337.50 7/7 $500 $837.50 21/7 28/7 $678 $ 50 Balance $159.50 $209.50 The above passbook page shows the transactions for July. Find the interest that will be earned in July if the bank pays 7% p.a. simple interest on the minimum monthly balance. THINK WRITE Minimum monthly balance for July is $159.50. 1 To find the smallest balance for July, look at all the running balances. Also check balances at the start and end of the month. Notice that the balance on 1 and 2 July, if shown, would have been $237.50. 2 Write the simple interest formula. PrT I = ---------100 3 List the values of P, r and T in months. P = $159.50 r= 7 ------ % 12 per month T = 1 month 4 Substitute into the formula and evaluate. 7 -×1 159.50 × ----12 I = -----------------------------------100 = 0.93 5 Write your answer. The interest earned for July was $0.93. Savings accounts — daily balances To calculate the interest on a daily balance saving account, the bank looks at the balances of the account. The number of days each balance is maintained is used to calculate the interest. When doing these calculations for yourself, you need to set out your workings carefully, for example using tables. Let’s investigate worked example 10 again, using the daily balance method. Ch 13 FM YR 12 Page 625 Monday, November 13, 2000 3:28 PM Chapter 13 Simple interest 625 WORKED Example 11 Daily balance method Use the daily balance method to find the interest that will be earned in July, if the bank pays 7% p.a. simple interest on the daily balance. THINK 1 2 3 4 5 WRITE Set up a table showing each new balance and the number of days the balance applies. Look at all running balances including those for 1 and 31 July. Calculate the interest for each balance. As the interest rate is in % per annum, express the number of days as a fraction of a year; for 2 - of a year. example, 2 days = -------365 Sum the interest. The calculations were to hundredths of a cent for accuracy. Round off to the nearest cent. Write your answer. Balance $ Number of days the balance applies $237.50 Simple interest calculations $ Interest earned $ 2 2 237.50 × 7 × -------365 ------------------------------------100 $0.0911 $337.50 4 4 337.50 × 7 × -------365 -------------------------------------100 $0.2589 $837.50 14 14 837.50 × 7 × -------365 ------------------------------------100 $2.2486 $159.50 7 7 159.50 × 7 × -------365 ------------------------------------100 $0.2141 $209.50 4 4 209.50 × 7 × -------365 -------------------------------------100 $0.1607 Interest for month = $2.9734 $2.9734 ≈ $2.97 The interest earned for July was $2.97. The daily balance method offers more interest than the minimum monthly balance method, as it credits the customer for all monies in the account, including the $600 deposited for 14 days. remember remember 1. Two methods used by banks for calculating interest on savings accounts are: (a) minimum monthly balances (b) daily balances. 2. Daily balances offer the best interest rate for investors. 3. Look at the balances on the first and last day of the month when establishing the minimum monthly balance or daily balances. 1 - of a year. 4. Express days as a fraction of a year; for example, 1 day = -------365 Ch 13 FM YR 12 Page 626 Monday, November 13, 2000 3:28 PM 626 Further Mathematics 13D Bank savings accounts GC p am rogr Simple WORKED Example interest 9 d hca Mat EXCE et reads L Sp he Simple interest SkillS HEET 13.4 WORKED Example 10 WORKED Example 11 1 A bank savings passbook showed that the opening balance for the month was $2150. That month Paul paid the following bills out of the account: Electricity $21.60 Telephone $10.30 Rent $52.00 Paul also deposited his wage of $620 for the month into the account. a What was Paul’s minimum monthly balance? b If the bank pays 5.5% p.a. paid monthly on the minimum monthly balance, how much interest did Paul earn in the month? Date Deposit Withdrawal Balance 2 1/5 $27.50 3/5 $12 $39.50 7/5 $16 $23.50 19/5 $ 8 $15.50 27/5 $10 $25.50 Roberta’s passbook shows the above transactions for May. Find the interest Roberta will earn in May if the bank pays 6% p.a. simple interest: a on the minimum monthly balance b on the daily balance. 3 For the month of July, Rhonda received $3.20 in interest on her savings account. Rhonda’s minimum balance in July was $426.20. What was the per annum simple interest rate offered by the bank? 4 Kristen receives the following statement from her bank. Due to a computer error the interest and balances were not calculated. Kristen rang the bank and was told that she received interest at a rate of 6 3--4- % p.a. paid monthly on her minimum monthly balance. Copy out Kristen’s statement and fill in the balances and interest payments. 1998 Transaction Debit Credit Balance 1 May Balance B/F 2132.20 3 May Cheq 4217 460.27 7 May Deposit 230.16 17 May Cheq 4218 891.20 26 May Wages 1740.60 31 May Interest _______ 2 June Deposit 415.10 8 June Cheq 4220 2217.00 19 June Cheq 4219 428.50 21 June Cheq 4222 16.80 23 June Wages 1740.60 30 June Interest _______ 1 July Deposit 22.80 4 July Cheq 4221 36.72 18 July Cheq 4223 280.96 26 July Wages 1740.60 31 July Interest _______ Ch 13 FM YR 12 Page 627 Monday, November 13, 2000 3:28 PM Chapter 13 Simple interest 627 5 Using the bank statement from question 4, another bank offers to show Kristen that daily balance interest credited each quarter is more rewarding. The interest is still 6.75% p.a. but is only credited at the end of the quarter, that is, on 31 July. Calculate: a the interest for the quarter ending July b the increase in interest earned using the daily balance method. Hint: This could be done using a spreadsheet. See the section on Spreadsheet Applications later in this chapter. 6 Clark Kent has the following income and expenses for August and September. Income: $1410.20 salary each fortnight beginning 4 August $461.27 income tax refund on 5 September $68.20 cheque from health fund on 10 August Expenses: $620.80 rent on 20 August and 20 September $180.64 telephone account on 2 September $150.26 electricity account on 15 August $180.00 Visa account on 30 August $327.60 health fund on 5 August and 5 September Draw up a statement (as for question 4) for Clark, remembering that he receives 7 1--2- % interest paid on the last day of each month on the minimum monthly balance in the account. 7 If the savings interest rate is 2 1--2- % p.a., calculate the interest credited at the end of each quarter for the following accounts using: i the minimum monthly balance ii the daily balance. Also calculate: iii the increase in interest earned using the daily rather than the minimum monthly balance method. a 3rd quarter statement for July, August and September Date 3/7 7/8 21/8 28/8 20/9 Deposit $ $ Withdrawal $100 500 670 $420 $10 000 Balance $ 750.00 $ 1250.00 $ 1920.00 $ 1500.00 $11 500.00 b 1st quarter statement for January, February and March in 2000 c Date 31/12/1999 1/2/2000 1/3/2000 28/3/2000 Deposit $100 $600 Date 3/10 17/12 21/12 22/12 28/12 Deposit $2100 $3500 Withdrawal $100 $ 50 Withdrawal $1900 $400 $650 Balance $400.00 ? ? ? Balance $2450.00 $5950.00 $4050.00 $3650.00 $3000.00 Ch 13 FM YR 12 Page 628 Monday, November 13, 2000 3:28 PM 628 Further Mathematics Hire-purchase People buy on hire-purchase when they cannot afford to buy the goods for cash. A deposit is usually paid and the balance is paid over a fixed period of time. The retailer arranges a contract with a financial institution and the purchaser pays regular instalments including interest at a flat rate to the financial institution. A flat rate is the same as simple interest rate. The interest charged is added onto the balance owing and then divided into the equal instalments. Advantages of this form of buying are: 1. the purchaser has the use of the goods while paying them off 2. the cost of the goods is spread over a long term in small amounts. The disadvantages are more complex: 1. the purchaser pays more for the goods in the long run 2. the goods are legally owned by the finance company until they are fully paid off 3. any forfeit on making the regular payments entitles the finance company to repossess the goods as well as retain all past payments made. The main stages of hire-purchase interest and total price calculations are: Step 1. Check the price of the goods. Step 2. Pay any deposit. Step 3. Set up the balance as a loan. Loan amount = price of goods − deposit Step 4. Calculate the interest on the loan using the simple interest formula. Step 5. The total amount to be repaid is the sum of the balance and the interest. Step 6. Establish regular payments/instalments. total amount Instalment amount = ----------------------------------------------------number of instalments Step 7. Total cost of goods = deposit + loan amount + interest or = deposit + instalment amount × number of instalments Ch 13 FM YR 12 Page 629 Monday, November 13, 2000 3:28 PM Chapter 13 Simple interest 629 WORKED Example 12 A sapphire ring with a marked price of $1800 is offered to the purchaser on the following terms: $200 deposit and the balance to be paid over 24 equal monthly instalments with interest charged at 11.5% p.a. flat rate. Find: a the total interest paid b the monthly repayments. THINK a 1 Write the cash price. 2 Determine the deposit. 3 Calculate the amount of the loan required. b 4 List P, r and T. 5 Write the simple interest formula, substitute into it and evaluate. 6 1 Write your answer. Add the interest to the principal. 2 Calculate the monthly repayments. WRITE a Cash price = $1800 Deposit = $200 Balance or loan amount = cash price − deposit = $1800 − $200 = $1600 P = $1600 r = 11.5% p.a. T = 2 years PrT I = ---------100 1600 × 11.5 × 2 I = -------------------------------------100 I = $368 Total interest to be paid is $368. b Total repayment amount = $1600 + $368 = $1968 total amount Regular payment = -----------------------------------------------------number of repayments $1968 = --------------24 = $82 3 Write your answer. The regular monthly repayments are $82. Ch 13 FM YR 12 Page 630 Monday, November 13, 2000 3:28 PM 630 Further Mathematics WORKED Example 13 A car is purchased on hire-purchase. The cash price is $21 000 and the terms are a deposit of 10% of the price, then the balance to be paid off over 60 equal monthly instalments. Interest is charged at 12% p.a. a What is the monthly instalment? b What is the total cost of the car? THINK WRITE a 1 Write the cash price. a Cash price = $21 000 Deposit = 10% × $21 000 2 Calculate the deposit, that is, 10% of $21 000. = $2100 Calculate the amount of the loan Loan amount = $21 000 − $2100 3 required. = $18 900 P = $18 900 4 List P, r, and T. Check that r and T r = 12% p.a. are in the same time terms. Convert the time period into years to match T = 60 months the % rate per annum. = 5 years PrT I = ---------5 Write the simple interest formula, 100 substitute into it and evaluate. 18 900 × 12 × 5 I = ------------------------------------100 I = 11 340 Total amount = 18 900 + 11 340 6 Add the interest to the principal to find the total amount of the loan to be repaid. = $30 240 total amount Regular payment = -----------------------------------------------------7 Calculate the monthly instalment. number of repayments $30 240 = ------------------60 = $504 The monthly instalment is $504. 8 Write your answer. b 1 Calculate the cost of the car. b Total cost = deposit + instalment amount × number of instalments = 2100 + 504 × 60 = $32 340 or Total cost = deposit + loan + interest = 2100 + 18 900 + 11 340 = $32 340 The total cost of the car is $32 340. 2 Write your answer. Weekly instalment advertising Many retailers use the option of hire-purchase to attract new sales. They also choose to advertise the instalment amount as it can seem to be very manageable. Buyers should investigate the entire arrangement offered and find answers to questions such as: 1. What is the interest rate? 2. How does it compare to bank rates? 3. What is the total cost of the item? 4. How much interest is charged? Ch 13 FM YR 12 Page 631 Monday, November 13, 2000 3:28 PM Chapter 13 Simple interest 631 WORKED Example 14 The following advertisement for a computer was found in a newspaper. Computer for sale Cash price $3695 or pay only a third deposit and 104 weekly instalments of only $25.97. If there is a total of 104 weekly instalments and a third deposit, find: a the interest charged b the interest rate c the total cost of the computer. THINK a 1 2 3 4 5 b 1 2 c 1 2 WRITE a Cash price = $3695 Deposit = 1--3- of $3695 = $1231.67 Calculate the amount of the loan. Loan amount = $3695.00 − $1231.67 = $2463.33 Calculate the total cost of the loan, that Total cost of loan = $25.97 × 104 is, the total of the loan and the interest = $2700.88 charged paid by weekly instalments. Calculate the interest charged and Interest charged = total amount − loan write your answer. I=A−P = 2700.88 − 2463.33 = 237.55 Interest on the $2463.33 loan is $237.55 Use the transposed simple interest b P = $2463.33 formula to find r, the interest rate on I = $237.55 the loan. Check that T is expressed T = 104 weeks in years to evaluate the interest rate = 2 years in % per annum. 100 × I r = ----------------P×T 100 × 237.55 r = ------------------------------2463.33 × 2 = 4.82 . . . Write your answer. The interest rate for this hire-purchase is 4.8% p.a. Calculate the total cost of the c Total cost = deposit + loan + interest computer. = 1231.67 + 2463.33 + 237.55 = $3932.55 Write your answer. The total cost for the computer including interest on the loan is $3932.55. Write the cash price. Calculate the deposit. Ch 13 FM YR 12 Page 632 Monday, November 13, 2000 3:28 PM 632 Further Mathematics remember remember The main stages of interest and total price calculations are: 1. Loan amount = price of goods − deposit 2. Flat rate interest on the loan is calculated using the simple interest formula. total amount 3. Instalment amount = ----------------------------------------------------number of instalments 4. Total cost of goods = deposit + loan amount + interest or = deposit + instalment amount × number of instalments 13E WORKED Example 12 WORKED Example 13 Hire-purchase 1 Debbie and Peter purchased a lounge suite on hire-purchase. The cash price was $2500. Peter and Debbie paid $250 deposit and signed an agreement to pay the balance in 36 equal monthly instalments. If the hire-purchase company charges 14% p.a. simple interest, find: a the total interest paid b the monthly repayments. 2 When buying new appliances for a recently renovated kitchen, Cheryl bought, from the same supplier, a refrigerator worth $490, a stove worth $350 and a dishwasher worth $890. If she paid $450 deposit and paid the balance over 48 months in equal monthly instalments at 12% p.a. simple interest, find: a Cheryl’s monthly instalments b the total amount Cheryl paid for the goods. 3 While on holidays in Noosa, Jan saw a bracelet she could not live without. The marked price was $2000. The jewellery shop owner offered her a discount of 15% if she paid a deposit of $250. Jan paid the deposit and signed a hire-purchase agreement that she would pay the balance of the bracelet’s cost at 15% p.a. flat rate with 24 equal monthly instalments. a What was the price of the bracelet after the 15% discount? b Calculate the balance Jan was to pay back. c Calculate the interest Jan paid. d Calculate Jan’s monthly instalment. e How much did Jan pay altogether for the bracelet? WORKED Example 14 4 The cash price of a suit is $1800. If a customer pays a deposit of $300 and pays equal monthly instalments of $60 over 2 1--2- years, calculate: a the amount of interest charged b the flat rate of interest c the total paid for the suit. 5 A car has a marked price of $7500. The dealer gives two choices of payment: i no deposit, with the $7500 paid in equal monthly instalments of $250 for 3 years ii $1500 deposit, paying interest of 12% p.a. and making equal monthly repayments for 3 years. a Calculate the interest rate in choice i. b Which deal is best for the purchaser? Why? Ch 13 FM YR 12 Page 633 Monday, November 13, 2000 3:28 PM Chapter 13 Simple interest 633 6 multiple choice An electric guitar is bought on hire-purchase for a $250 deposit and monthly instalments of $78.50 for 3 years. The cash price for this guitar is $2500. The interest rate is closest to: A 9.5% B 7% C 8.5% D 8% E 7.5% 7 multiple choice ‘Carpeting the home is not cheap’, Rob stated. ‘Hire-purchase is the answer’, replied Tom. The cost of the carpet for the house is $9500. Rob and Tom place a deposit of $1500 and plan to pay it back weekly over 4 years at 13% interest per year. The weekly instalment is: A $253.37 B $62.20 C $46.20 D $58.46 E $462.00 8 multiple choice A salesman told a couple that if they bought a television at $890 today, he would allow a deposit of $100 plus $8.65 weekly for 2 years. The interest rate charged is: A 10% B 7% C 6.5% D 9 1--2- % E 7.5% 9 For the video camera in the advertisement, find: i the total paid ii the interest rate for both Option a and Option b. THIS VIDEO CAMERA CAN BE YOURS. EASY TERMS CASH PRICE $780 or Option a NO DEPOSIT AND $37.70 MONTHLY PAYMENTS FOR 2 YEARS or Option b $100 DEPOSIT AND $26.72 PAID MONTHLY FOR 30 MONTHS 10 A company advertised a dining room suite for $2500. You could pay: a cash and receive a 10% discount, or b $200 deposit and 5% p.a. interest on the remainder for 3 years, or c $300 deposit and 4.5% p.a. on the remainder for 3 1--2- years, or d $400 deposit and 4% p.a. interest on the remainder for 4 years. What is the total paid on each deal? 11 Carefully read the advertisement at right for the cash price and regular instalments for the colour television. The term of the repayments is for 3 years with 20% deposit. Calculate: a the flat interest rate b the total cost of the TV under the hirepurchase plan c the increase in cost over a cash sale. $1095 or $15.40 fortnightly Ch 13 FM YR 12 Page 634 Monday, November 13, 2000 3:28 PM 634 Further Mathematics Effective rate of interest $10,000 PERSONAL LOAN from $149 per fortnight Based on a 3 year term at a fixed rate of 9.95%* p.a. When purchasing goods on hire-purchase or through a personal loan, the finance company lending the money hopes to make the deal look as attractive as possible. Some details, therefore, are not prominently stated to the customer. One such detail is the effective rate of interest. The amount borrowed reduces over the term of the loan, but the customer is still paying interest on the total initial loan amount. The effective interest rate is the equivalent reducing balance interest rate taken over the contract period. There are two ways of converting flat rate to effective rate. 1. Estimation Effective interest rate is a little less than 2 × flat interest rate. 2. Calculation 2n Effective interest rate = ------------ × flat rate where n is the number of payments. n+1 That is, on a loan of $100 at 10% interest over 4 years with yearly repayments, the interest charged is: I = 100 × 0.10 × 4 = $40. 2×4 The effective interest rate is ------------ × 10% = 16% (assuming yearly repayment). 4+1 This means that, even though the person is paying $40 interest, the effective interest rate over the period is actually 16%, not 10%. The longer the period of the loan, the higher the effective interest rate. This is shown clearly in the following table. Year Principal owing Repayment of principal Flat rate of interest paid Effective rate of interest paid 1 100 25 10% of 100 = 10 16% of 100 = 16 2 75 25 10% of 100 = 10 16% of 75 = 12 3 50 25 10% of 100 = 10 16% of 50 = 8 4 25 25 10% of 100 = 10 16% of 25 = 4 $100 $40 $40 Total interest $40 Flat rate 10% Effective rate 16% Ch 13 FM YR 12 Page 635 Monday, November 13, 2000 3:28 PM Chapter 13 Simple interest 635 WORKED Example 15 Jason decides to borrow money for a holiday. If a personal loan is taken over 4 years with equal quarterly repayments at 12% p.a. flat rate (simple interest), calculate the effective rate of interest. THINK WRITE 1 Write the flat rate and number of instalments. Flat rate = 12% n=4×4 = 16 2 Write the formula for effective rate of interest. 2n Effective rate = ------------ × flat rate n+1 Substitute n = 16 and r = 12. 2 × 16 Effective rate = --------------- × 12 16 + 1 = 22.588 The effective interest rate is 22.6% p.a. for a flat rate loan of 12% with sixteen instalments. 3 4 Write your answer. Check the answer by estimating the rate which is less than 2 × 12% (or 24%) p.a. remember remember 1. The effective interest rate is a true indication of the interest rate on a loan that is calculated using a flat interest rate when the loan is progressively being reduced, such as in hire-purchases. 2. Estimation of effective interest: Effective interest rate is a little less than 2 × flat interst rate. Calculation of effective interest: 2n Effective interest rate = ------------ × flat rate where n is the number of payments. n+1 3. The fewer the payments, the closer the flat rate is to being a true indication of the rate charged. For example, 12% flat rate with 1 payment only: 2×1 Effective rate = ------------ × 12% = 12% 1+1 Ch 13 FM YR 12 Page 636 Monday, November 13, 2000 3:28 PM 636 Further Mathematics 13F WORKED Example 15 d hca Mat EXCE et reads L Sp he Effective rate of interest 1 William is to purchase a new video recorder. If William pays $125 monthly instalments over 3 years at an interest rate of 11.5% p.a. simple interest, what effective interest rate is he paying? 2 Item Cash price $ Deposit $ Monthly instalment $ Interest rate Term of loan Effective rate of interest a Television $875 $150 8% p.a. 2 years b New car $23 990 $2000 10% p.a. 5 years c Clothing $550 $100 7.5% p.a. 1 year $1020 $50 6 3--4- % p.a. 18 months $250 $75 9% p.a. 15 months d Refrigerator e Tools For each of the items in the above table, calculate: i the total amount of interest charged on each item ii the total amount paid over the period given for each item iii the monthly instalment on each item iv the effective interest rate. 3 The cash price for a car is $4600. If the car is purchased on time payments the cost will be $5200. A deposit of $100 is required and the agreement is that the car will be fully paid for in 3 years, paid in equal monthly instalments. Find: a the monthly instalment b the simple (flat) interest rate per year c the effective interest rate. 4 A camera valued at $1200 is purchased using a hire-purchase agreement. A deposit of $200 is required and equal monthly instalments of $75 are paid over the 18-month agreed period. Calculate: a the flat (simple) interest rate per annum b the effective interest rate. 5 The bank approves a personal loan of $5000. A flat interest rate of 12.5% p.a. is charged, with repayments to be made over a 9-month period in equal weekly instalments. Calculate: a the weekly instalment b the effective interest rate. 6 Calculate the effective interest rate on a loan of $1000 if the monthly repayments are $60 and the loan is to be repaid over 2 years. (Hint: First calculate the simple interest rate.) Ch 13 FM YR 12 Page 637 Monday, November 13, 2000 3:28 PM Chapter 13 Simple interest 637 7 Carefully read the advertisement (including the small print) for the purchase of the refrigerator below and calculate: a the flat interest rate b the effective interest rate c the total cost under the hire-purchase plan d the increase in cost over a cash sale. $599 or $4.21 weekly (one third deposit over two years) 8 Copy and complete the following table. Deposit Instalment (monthly) Period Simple interest rate $2500 $500 2 years 10% p.a. $150 $50 6 months 9.5% p.a. $685 $75 9 months 6 3--4- % p.a. $128 $ nil $11.20 1 year $6500 $500 $325 2 years $10 000 $1500 5 years Effective interest rate ET SHE Work Cash price 10% p.a. 13.2 Ch 13 FM YR 12 Page 638 Monday, November 13, 2000 3:28 PM 638 Further Mathematics Spreadsheet applications Spreadshe et EXCEL Accountants, financial planners, banks and other financial institutions use spreadsheets to record and perform calculations. Many calculation tasks are similar in nature and tedious; therefore, once a spreadsheet is set up, some of these tasks can be done more quickly and easily. Another advantage of the use of a spreadsheet is solving the ‘what if’ question. This function allows the numbers entered on the spreadsheet to be changed and an answer to be calculated to predict what would happen in a particular scenario. This is particularly useful when looking at factors such as how much a person can borrow and pay back, changes in terms, and changes in interest rates. Your Maths Quest CD contains the Excel files ‘Simple interest’ and ‘Effective rate of interest’. These may be used to investigate various scenarios by typing new values in the yellow cells or by modifying the spreadsheet in some way. A screen shot of the file ‘Simple Interest’ is shown below. Spreadshe et EXCEL Simple interest Effective rate of interest Which is the best deal? 1 Find three advertisements that offer products on hire purchase. 2 Investigate each offer by calculating: a the total amount of interest to be paid b the total amount to be paid over the term of the loan c the monthly repayment d the effective interest rate. 3 Compare this with other methods of financing the purchase of the product. 4 Write a brief report on the advantages and disadvantages of each method. Ch 13 FM YR 12 Page 639 Monday, November 13, 2000 3:28 PM Chapter 13 Simple interest 639 summary Simple interest formula • A = P + I where A = Total amount ($) P = Principal or amount borrowed or invested ($) I = Simple interest charged or earned ($) PrT • I = ---------I = Simple interest charged or earned ($) 100 P = Principal (money invested or loaned) ($) r = Rate of interest earned per period (% per period) T = Time, the number of periods over which the agreement operates • Interest rate, r, and time period, T, must be stated and calculated in the same time terms. Finding P, r and T • To find the principal • To find the interest rate • To find the period of the loan or investment Bonds, debentures and term deposits • • • • 100 × I P = ----------------r×T 100 × I r = ----------------P×T 100 × I T = ----------------P×r Term investments with governments are called bonds. Term investments with companies are called debentures. Term investments with banks are called term deposits. All three are investments for a fixed period of time offering a simple interest rate. Savings banks — minimum monthly and daily balances • Two methods used by banks for calculating interest on savings accounts are: 1. minimum monthly balances 2. daily balances. • Daily balances offer the best interest rate for investors. • Look at the balances on the first and last day of the month when establishing the minimum monthly balance or daily balances. 1 - of a year. • Express days as a fraction of a year; for example, 1 day = -------365 Hire-purchase • Hire-purchase is a loan for goods with interest calculated using flat rate (simple) interest and regular payments. • The main stages of calculations are: 1. Loan amount = price of goods − deposit 2. Flat rate interest on the loan is calculated using the simple interest formula. total amount Instalment amount = ----------------------------------------------------number of instalments Total cost of goods = deposit + loan amount + interest or = deposit + instalment amount × number of instalments Effective rates of interest • The effective interest rate is a true indication of the interest rate on a loan. It is calculated using a flat interest rate when the loan is progressively being reduced, such as in hire-purchases. 1. Estimation: Effective interest rate is a little less than 2 × flat interest rate 2. Calculation: 2n Effective interest rate = ------------ × flat rate where n is the number of payments. n+1 Ch 13 FM YR 12 Page 640 Monday, November 13, 2000 3:28 PM 640 Further Mathematics CHAPTER review Multiple choice 13A 1 Two banks pay simple interest on short-term deposits. Bank A pays 6% p.a. over 4 years and Bank B pays 6.5% p.a. for 3 1--2- years. The difference between the two banks’ final payout figure if $5000 was invested in each account is: A $0 B $1200 C $1137.50 D $150 E $62.50 13A 2 Clayton invested $360 in a bank for 3 years at 8% simple interest each year. At the end of the 3 years, the total amount he will receive is: A $86.40 B $236.80 C $28.80 D $388.80 E $446.40 13A 3 Philip borrowed $7000 and intended to pay it back in 4 years. The terms of the loan indicated Philip was to pay 9% p.a. interest. The interest Philip paid on the loan was: A $25 200 B $630 C $7630 D $9520 E $2520 13B 4 A loan of $5000 is taken over 5 years. The simple interest is calculated monthly. The interest bill on this loan is $1125. The simple interest rate per year on this loan is: A 3% B 4 1--2- % C 3.75% D 5% E 3.5% 13B, C 5 The principal invested in an investment bond that will accumulate $2015 after 6 months invested at 6 1--2- % p.a. is: A $60 000 B $62 000 C $6200 D $50 000 E $5000 13B 6 A loan of $10 000 is taken over 10 years. The total interest bill on this loan is $2000. The simple interest rate per year on this loan is: A 3% B 4 1--2- % C 2% D 5% E 2.5% 13C 7 A 6-year bond pays 8 1--2- % p.a. simple interest. If Rhonda buys a bond worth $500, the interest she would earn would be: A $250 B $255 C $2550 D $233.75 E $230 13C 8 Simple interest was calculated on a term deposit of 5 years at 3 3--4- % p.a. When Leigh 13C 9 State government bonds pay interest of 7 3--4- % p.a. simple interest. Jess invested $3500 in the calculated her total return on her investment principal of $350, her return was: A $415.63 B $400 C $65.63 D $131.25 E $481.25 bonds which mature in 5 years. Jess’s income each quarter would be: A $113.00 B $1356.25 C $3567.81 D $67.81 E $82.50 Ch 13 FM YR 12 Page 641 Monday, November 13, 2000 3:28 PM Chapter 13 Simple interest 641 10 In the bank statement shown below the minimum balance for the month is: Date 5/4 7/4 9/4 23/4 Transaction Transfer from CBR Salary Cheque — 23456 ATM — Rowville A $456.50 B $1956.50 Deposit Withdrawal Balance $456.50 $1956.50 $576.50 $451.50 $100 $1500 $1380 $125 C $576.50 D $451.50 13D E $356.50 11 A pearl necklace is purchased on hire-purchase for $225 deposit with equal monthly payments of $80 for 2 years. The cash price is $2000. The interest rate is: A 3.5% B 6% C 4% D 8% E 7.5% 13E 12 A hire-purchase contract specifies that there are to be monthly payments for 2 years. The flat rate of interest is 6.3% p.a. The effective interest rate for this contract is closest to: A 12.1% B 11.6% C 8.4% D 6.3% E 12.6% 13F Short answer 1 Cynthia invested $270 with a building society in a fixed deposit account that paid 8% p.a. simple interest for 4 years. How much did Cynthia receive at the end of the 4 years? 13A 2 A bank offers 8.5% p.a. simple interest on an investment. At the end of 3 years the interest earned was $765. How much was invested? 13B 3 If $725 is invested for 3 years and earns $206.65 interest, calculate the yearly interest rate. 13B 13B 4 Jack put some money away for 4 1--2- years in a bank account which is paying 3 3--4- % p.a. interest. He found on his bank statement he had earned $67.50. How much did Jack invest? 5 Jacob needed to earn $225 in one year. He invested $2000 in an account earning simple interest at a rate of 4.5% p.a. paid monthly. How many months will it take Jacob to achieve his aim? 13B Ch 13 FM YR 12 Page 642 Monday, November 13, 2000 3:28 PM 642 Further Mathematics 13C 6 Steve invested the $1800 he won at the races in an insurance company bond that pays 12 1--2- % p.a. provided he keeps the bond for 4 years. What is Steve’s total return from the bond at the end of the 4 years? 13C 7 Jocelyn buys $3500 worth of debentures in a company. She earns 8.5% p.a. simple interest paid to her quarterly. If the agreed period of the debenture was 28 months, calculate the amount of interest Jocelyn will earn. 13C 8 The bank offers a term deposit account paying investors 10.5% p.a. on investments over $10 000 for 2 years. Paul decides to invest $12 000 in this account. How much interest will he earn at the end of the investment? 13C 9 An investment bond is offered to the public at 10% per year. Louis buys a bond worth $4000 that will mature in 2 1--2- years. How much in total will Louis receive at the end of the 2 1--2- years? 13D 10 At the beginning of July, Ross had $580 in his savings bank account. On 15 July he withdrew $80. If the bank pays 8% p.a. interest paid monthly, calculate the interest Ross earns in July: a if calculated on the minimum monthly balance b if calculated on the daily balance. 13D 11 Date 1/5 3/5 7/5 19/5 27/5 Deposit Withdrawal Balance $28.80 $302.20 $273.40 $12 $6 $10 Deborah’s passbook shows the above transactions for May. Calculate the interest Deborah will earn in May if the bank pays 4 3--4- % p.a. simple interest monthly: a on the minimum monthly balance b on the daily balance. Ch 13 FM YR 12 Page 643 Monday, November 13, 2000 3:28 PM Chapter 13 Simple interest 643 12 The cash price of a car is $18 000. If a customer pays a deposit of $3000 and pays equal monthly instalments of $300 over 5 years, calculate: a the amount of interest charged b the flat rate of interest c the total paid for the car d the effective interest rate. 13 The cash price for a bicycle is $460. If the bike is purchased on time payments the total cost will be $550. A deposit of $50 is required and the agreement is that the bike will be fully paid for in 2 years, in equal monthly instalments. Find: a the monthly instalment (round up to the nearest cent) b the simple interest rate per year (to 1 decimal place) c the effective interest rate (to 1 decimal place). 13E, F 13E, F Ch 13 FM YR 12 Page 644 Monday, November 13, 2000 3:28 PM 644 Further Mathematics Analysis 1 Date 4 August 8 August 19 August 27 August 28 August Description ATM Deposit EFTPOS Salary ATM Debit Credit 100.00 Balance 325.60 975.60 119.50 1527.40 2383.50 1983.50 a Complete the missing credits, debits and balances in the shaded areas of the above account. b The bank is offering 2.4% p.a. on the minimum monthly balance. What is the interest rate per month? c Calculate the interest that was earned for the month of August. 2 Geoff wants to buy a windsurfer. Its retail price is $3995. Geoff’s first option for financing the purchase is using hire-purchase. The terms offered by Your Money Finance Company is 10% deposit with fortnightly instalments over 2 years at an interest rate of 7.8% per annum. a How much will Geoff need to withdraw from his savings account to pay the deposit? b Calculate the fortnightly repayments and total interest charge. c What is the total cost of the windsurfer? d A personal loan is advertised at 13.5% per annum. For Geoff to compare the interest rate he needs to convert the hire-purchase flat rate of interest to the effective interest rate. Calculate the effective interest rate. CHAPTER test yyourself ourself 13 3 Another option is for Geoff to save up until he has the cash to pay for the windsurfer. He can place the balance of his savings account (shown above in question 1) into a term deposit offering 5.6% per annum for a 2-year term. a Calculate the total value of his investment at the end of 2 years. b Geoff uses the term deposit investment towards the purchase of the windsurfer. What extra fortnightly savings will be needed over the next 2 years to make up the balance of $3995? c What is the main attraction of the hire-purchase option over the options in 3a and b?

© Copyright 2019