1. business and industrial

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?