Financial Mathematics
Outcomes: MAO-WM-01, MA5-FIN-C-01, MA5-FIN-C-02
Lesson
Concept
1
Computation with Percentages and Money
2
3
4
5
6
7
Converting Salaries
Calculating Wages
Overtime and Wage Increase
Holiday Loading and Holiday Pay
Deductions and Medicare
Calculating Tax
8
Buying on Terms
9
10
Simple Interest
Compound Interest Formula
11
More Compound Interest
Classwork
Cambridge Year 9 Advance Textbook
4 - 6, 8 - 13
Oxford Year 11 Textbook
1A: 4 – 8, 10
1A: 11, 12, 15, 17, 18, 20, 26
1B: 2, 4, 6, 7, 10, 12 - 15
1D: 2, 5 - 13
10A: 2 - 9
10B: 2, 3, 7, 9, 10, 12
Oxford Year 9 Textbook
5L: 3-10
Cambridge Year 9 Advance Textbook
1K: 3, 5, 6, 8 - 15
1M: 5 -8, 12, 15
Oxford Year 10 Textbook
9C: 2, 4, 7, 9, 13, 15, 18, 21, 24
Ext.
20
16 - 18
16
13, 18
26 - 29
Name:_______________________
Checked
Lesson 1: Computation with Percentages and Money
Lesson Goal: Review of Computation with percentages and money
To find an original amount, use the unitary method or use division.
Exercise 1F Cambridge Year 9
Examples:
1) Express 0.45 as a percentage.
2) Express 25% as a decimal.
1
3) Express 3 % as a fraction.
4
4) Write 50c out of $2.50 as a percentage.
5) Find 15% of $35.
6) Determine the original amount if 5% of the amount is $45 .
Lesson 2: Converting Salaries
Lesson Goal: Lean how to convert between salaries
‘Per annum’ means ‘for every year’
Converting from yearly salaries to monthly, fortnightly and weekly amounts:
1 year = 365 days
= 12 months
= 52 weeks
= 26 fortnights
Exercise 1A Oxford Year 11
Examples:
1) Ken earns an annual salary of $59 735 and works a 38 hour week. Find out how much
he earns:
a) Monthly
b) Fortnightly
c) Weekly
d) Kens wife Brooke works part time in retail and earns $21.80 per hour. Who has
the higher hourly rate?
e) If Brooke works on average 18 hours per week, what is her yearly income?
2) Michiko works part-time and earns $283 per week.
a) How much does Michiko earn per year?
b) How much does Michiko earn per month?
3) Find the difference between these salaries if Amel earns $3560 a month and Ahmed
earns $827 a week. (Hint: Make sure to convert to yearly)
Lesson 3: Calculating Wages
Lesson Goal: Learn how to calculate wages.
Exercise 1A Oxford Year 11
Examples:
1) Musa is paid $21.65 per hour for casual work. How much does he earn if he works:
a) 6 hrs in 1 day?
b) 6 hrs per day for 5 days
2) Angus works a 35-hour week and is paid $17.80 per hour. How much does he earn in a
year?
3) Convert an annual salary of $47 424 for a 35-hour week to:
a) A weekly salary
b) An hourly salary
4) Pete works as a casual at KFC. He is paid $13.45 an hour from Monday to Friday and
$18.85 an hour on weekends. During a week of his school holidays he worked from 2:20
pm until 7:20 pm Monday to Friday and 9:30 am until 1:30 pm on Saturday and Sunday.
a) Find the total hours he worked on weekdays?
b) Find the total hours did he worked on weekends?
c) Calculate his income for the week.
Lesson 4: Overtime and Wage Increase
Lesson Goal: Learn how to calculate overtime and wage increase.
Most employees work under an award that states the maximum number of normal work
hours per day or per week. Hours worked in excess of these are overtime and are paid at a
higher hourly rate than normal time. The two main overtime rates are time-and-a-half (× 1.5)
and double-time (× 2).
Exercise 1B Oxford Year 11
Examples:
1) Convert an hourly rate of $16.20 to an overtime rate of:
a) Time and a half
b) double time
2) Janice works 10 hours on a Sunday. The Sunday rate of pay is $12.80 per hour for the
first 4 hours, then ‘time-and-a-half’ for additional hours worked. Calculate Janice’s
wage for Sunday.
3) The following wage table shows the hours worked by employees at Active Sports.
Calculate the weekly wage for each if:
- the normal hourly rate is $19.40
- overtime of time-and-a-half is paid for hours worked beyond 4 hours from Monday
to Friday
- time-and-a-half is the hourly rate on Saturdays.
4) Vicki earns $15.45 per hour as a cleaner. If she cleans the toilets, she receives an
allowance of $24.00 per day. Find her weekly pay if she works for 4 hours each
weekday and cleans the toilets on Wednesday and Friday.
Lesson 5: Bonuses, Holiday Loading and Holiday Pay
Lesson Goal: Learn how to calculate bonuses holiday loading and holiday pay.
Employees sometimes get extra payments in addition to their normal wages.
A bonus is a gift or incentive for employees who have worked hard over a period of time.
Employers use a bonus to encourage the employees to work even harder!
Annual leave loading is an increase in an employee’s pay while on holidays. The holiday
1
loading is a fixed percentage of the employee’s normal pay. It is usually at a rate of 17 %.
2
Holiday loading – If asked to find this you only need to find the extra pay gained from the
1
loading �17 %�.
2
1
Holiday pay – If asked to find this you need to find the holiday pay plus the loading �117 %�.
Exercise 1D Oxford Year 11
2
Examples:
1) Renee worked extremely hard throughout the year. Her reward was a bonus of 6% of
her yearly income. If her normal yearly income was $58 000, calculate the value of her
bonus.
2) Lee is a permanent part-time employee and is entitled to 4 weeks annual leave and a
1
leave loading of 17 %. If his weekly pay is $368, find his holiday pay.
2
3) Paul earns $28.60 an hour and works a 38-hour week.
1
a) Calculate his holiday loading if he is given 17 % of 4 weeks wages.
2
b) Calculate his holiday pay.
Lesson 6: Deductions and Medicare
Lesson Goal: Learn how to calculate taxable income and Medicare levy.
Taxable income is calculated using total income and allowable deductions.
Total income includes income from all sources throughout the year such as;
-
wages and salaries, bonuses, interest earned, commissions and allowances.
The total income may be subject to deductions that reduce the amount used to calculate the
tax payable.
Tax deductions may include;
-
tools used for work, safety equipment, self-education expenses, union fees, donations
over $2, car travel expenses, uniforms and their cleaning, and tax agent fees.
Taxable income = total income − allowable deductions
An extra tax that may be payable is the Medicare levy. Medicare is the public hospital medical
system available without charge for Australians. The Medicare levy is currently calculated at
1.5% of taxable income.
Exercise 10A Oxford Year 11
Examples:
1) Sara uses the following information to calculate her taxable income: wages $35 980,
interest $569 (joint account), bonus $200, cost of uniforms $140, tax agent fee $80, use
of car $298. Calculate her taxable income.
2) Annie uses the following information to calculate her taxable income: wages $45 320,
interest $665 (joint account), bonus $800, cost of uniforms $340, tax agent fee $60, use
of car $45. Complete the following to calculate Annie’s taxable income.
3) Calculate the Medicare levy payable on a taxable income of $35 908.
Lesson 7: Calculating Tax
Lesson Goal: Learn how to calculate tax payable.
The calculation of income tax is done using a tax table. The following table shows the tax
rates for 2012–13.
Exercise 10B Oxford Year 11
Examples:
1) Use the tax table to calculate the tax payable on these incomes.
a) $53 251
b) $26 784
c) $105 631
2) Julia works as a secretary and receives a yearly salary of $34 479 plus an income of
$1950 per year from baby-sitting. Her total tax deductions are $1570. During the year
she paid tax instalments amounting to $3265.
a) Find her total income.
b) Find her taxable income.
c) Find the tax payable on her taxable income.
d) Find her tax refund or balance payable.
Lesson 8: Buying on Terms
Lesson Goal: Learn how to buy on terms and compare with buying items outright.
When you buy on terms, you pay an initial deposit upfront, and the item is received
immediately. The balance of the price or the amount remaining is borrowed and repaid with
simple interest included in equal repayments over a fixed time period. Buying outright will be
the cheaper option. However, buying on terms is also a good option if you do not have the
total amount upfront and require the item immediately.
Exercise 5L Oxford Year 9
Examples:
1) A camping set costing $998 can be bought on terms for $99 deposit and 24 monthly
instalments of $55.
a) Calculate the total cost of buying the camping set on terms.
b) How much would you save by paying cash?
2) A computer costing $1498 can be bought on terms for $300 deposit and 36 monthly
repayments of $46.50.
a) Calculate the total cost of buying the computer on terms.
b) Find the total amount of interest charged.
c) Calculate the amount of interest paid annually.
d) What was the amount of money borrowed?
e) Calculate the annual rate of interest charged.
3) A large screen TV set can be bought for $2998 cash or on the following terms: deposit
$299, with the balance to be repaid over 2 years in 24 equal monthly repayments.
Simple interest is charged on the balance at 12% p.a. If the TV is bought on terms:
a) Calculate the balance owing after the deposit is paid.
b) Calculate the interest charged on the balance owing.
c) What is the monthly repayment?
EXAMPLE 2
Isabella lay-bys a dress costing $224 and pays a deposit of $50. She wants to collect the dress in about
3 months to wear to a wedding. If she pays off the balance by making 3 equal monthly instalments, calculate:
a the balance to be paid
b the amount of each monthly instalment.
Solve
Think
Apply
Always subtract
the deposit before
calculating payments.
a
Balance = $224 − $50
= $174
Subtract the $50 deposit from the
balance to be paid.
b
$174
Monthly instalment = _____
3
= $58
She owed $174, which is divided
by the 3 monthly instalments.
3 Martin lay-bys a suit costing $485. He pays a deposit of $80 and pays off the balance by making 4 equal
monthly payments. Calculate:
a the balance to be paid
b the amount of each monthly payment.
4 Yvonne lay-bys a gas barbecue costing $778. She is required to pay a 10% deposit and to pay the balance in
12 equal weekly instalments. Calculate:
a the deposit
c the amount of each weekly instalment.
b the balance to be paid
5 Josh lay-bys a Blu-ray player costing $456. He is required to pay a 15% deposit and to pay the balance in
6 equal fortnightly instalments. Calculate:
a the deposit
c the amount of each fortnightly instalment.
b the balance to be paid
6 List some advantages and disadvantages of using the lay-by method to purchase goods.
L Buying on terms
Buying on terms is sometimes
called hire-purchase.
When an item is bought on terms, a deposit is paid and the item is received immediately. The balance of the
price is borrowed and this balance plus simple interest is repaid in equal instalments over a fixed period.
EXAMPLE 1
NUMBER & ALGEBRA
A camping set costing $998 can be bought on terms for $99 deposit and 24 monthly instalments of $55.
a Calculate the total cost of buying the camping set on terms.
b How much would you save by paying cash?
Solve
124
Think
Apply
Make sure that the
deposit is added to the
amount paid.
a
Total cost = $99 + 24 × $55
= $1419
Calculate all amounts paid: the
deposit plus all monthly payments.
b
Amount saved by paying cash
= $1419 − $998 = $421
Deduct the cost of purchase for cash
from the cost of paying on terms.
Insight Mathematics 9 stages 5.2/5.3 Australian Curriculum
Exercise 5L
1 A surfboard costing $699 can be bought on terms for $79 deposit and 24 monthly instalments of $33.
a Calculate the total cost of buying the surfboard on terms.
b How much would you save by paying cash?
2 A laptop computer costing $2298 can be bought on terms for $229 deposit and 18 monthly repayments
of $135.60.
a Calculate the total cost of buying the computer on terms.
b How much would you save by paying cash?
3 A home theatre system costing $1598 can be bought on the following terms: 10% deposit and 48 weekly
instalments of $37.15.
a Calculate the total cost of buying the system on terms.
b How much would you save by paying cash?
4 A sound system costing $879 can be bought on the following terms: 15% deposit and 26 fortnightly
repayments of $39.98.
a Calculate the total cost of buying the sound system on terms.
b How much would you save by paying cash?
EXAMPLE 2
A computer costing $1498 can be bought on terms for $300 deposit and 36 monthly repayments of $46.50.
a Calculate the total cost of buying the computer on terms.
b Find the total amount of interest charged.
c Calculate the amount of interest paid annually.
d What was the amount of money borrowed?
e Calculate the annual rate of interest charged.
Think
a
Total cost = $300 + 36 × $46.50
= $1974
Add the deposit to all the
payments.
b
Total interest = $1974 − $1498
= $476
The amount paid in excess of the
purchase price is the interest.
c
total interest
Annual interest = ______________
number of years
$476.00
= _______ = $158.67
3
Convert the total interest to
amount of interest per year by
dividing by the number of years.
d
Amount borrowed
= balance owing after paying the deposit
= $1498 − $300 = $1198
The $300 deposit is subtracted
from the balance owing to find
the amount borrowed.
e
Annual interest rate
annual interest
= _______________ × 100%
amount borrowed
$158.67
= _______ × 100% = 13.2%
$1198
Divide $158.67 by the amount
borrowed and multiply by 100 to
convert the rate to a percentage.
Apply
Pay particular
attention to the
deposit. It is the
amount of money
paid initially. It is
always added to
amounts you paid and
subtracted from the
amount owing.
NUMBER & ALGEBRA
Solve
Chapter 5 Financial mathematics
125
5 A $1499 digital SLR camera can be bought on terms for $200 deposit and 24 monthly repayments of $74.69.
a Calculate the total cost of buying the camera on terms.
b Find the total amount of interest charged.
c Calculate the amount of interest paid annually.
d What was the amount of money borrowed?
e Calculate the annual rate of interest charged.
6 A trailer costing $1890 can be bought on terms for $100 deposit and 36 monthly repayments of $66.50.
a Calculate the total cost of buying the trailer on terms.
b Find the total amount of interest charged.
c Calculate the amount of interest paid annually.
d What was the amount of money borrowed?
e Calculate the annual rate of interest charged.
7 A lounge suite was advertised for $5990 or $500 deposit and 48 monthly repayments of $187.58.
a Calculate the total cost of buying the lounge suite on terms.
b Find the total amount of interest charged.
c Calculate the amount of interest paid annually.
d What was the amount of money borrowed?
e Calculate the annual rate of interest charged.
EXAMPLE 3
A large screen TV set can be bought for $2998 cash or
on the following terms: deposit $299, with the balance to
be repaid over 2 years in 24 equal monthly repayments.
Simple interest is charged on the balance at 12% p.a. If
the TV is bought on terms:
a Calculate the balance owing after the deposit is paid.
b Calculate the interest charged on the balance owing.
c What is the monthly repayment?
Solve
Think
a
Balance owing = $2998 − $299
= $2699
Subtract the deposit.
b
Interest = $2699 × 0.12 × 2
= $647.76
Calculate the total
interest.
c
Balance owing + interest = $2699 + $647.76
= $3346.76
$3346.76
Monthly repayment = ________
24
= $139.45 to nearest cent
Add the balance owing
and the interest. Divide
by the number of months
to find the repayment.
Apply
Always subtract
the deposit before
calculating the interest.
Add the interest and
the principal. To check,
multiply the payments by
the time period and make
sure that they exceed the
balance owing.
8 Peter buys a second-hand car advertised for $9600 on the following terms: deposit $2000, the balance to be
NUMBER & ALGEBRA
repaid over 2 years in equal monthly repayments. Simple interest is charged at 12% p.a.
a Calculate the balance owing.
b Calculate the interest charged on the balance owing.
c What is the monthly repayment?
9 Angela buys a motorbike advertised for $12 900 on the following terms: deposit $3000, the balance to be
repaid over 3 years in equal monthly repayments. Simple interest is charged at 9% p.a.
a Calculate the balance owing.
b Calculate the interest charged on the balance owing.
c What is the monthly repayment?
126
Insight Mathematics 9 stages 5.2/5.3 Australian Curriculum
10 Adrienne buys a washing machine advertised for $499 on the following terms: deposit 10% and the balance
repaid over 2 years in equal monthly repayments. Simple interest is charged at 15% p.a.
b Calculate the balance owing.
d What is the monthly repayment?
a Calculate the deposit.
c Find the interest charged on the balance owing.
11 Robin buys a new car advertised for $19 900 on the following terms: deposit 15%, the balance to be repaid
over 4 years in equal monthly repayments. Simple interest is charged at 11.9% p.a.
a Calculate the deposit.
b Calculate the balance owing.
c Find the interest charged on the balance owing.
d What is the monthly repayment?
12 List some advantages and disadvantages of purchasing goods on terms.
Investigation 3 Monthly repayments
The spreadsheet below calculates the monthly repayments for items bought on terms.
A
B
C
D
E
F
G
H
Item
Cash
price
($)
Deposit
($)
Interest rate
(% p.a.)
Repayment
period
(years)
Balance
owing
($)
Interest on
balance
($)
Monthly
repayment
($)
1
2
Gaming
computer
2998
298
12.0
2
3
LED TV
1899
189
15.0
2
4
Furniture
4672
250
11.6
3
5
Ducted air
conditioning
7659
1000
14.2
4
6
Refrigerator
and freezer
3628
628
9.9
3
1 Copy the spreadsheet.
2 In cell F2 type the formula: =B2−C2. This is the balance owing after the deposit is paid.
3 In cell G2 type the formula: =F2*D2/100*E2. This is the amount of interest charged on the balance.
4 In cell H2 type the formula: =(F2+G2)/(E2*12). This is the monthly repayment.
5 To find these values for the other items:
• Highlight cells F2 to H6.
• Go to Edit.
• Select Fill Down.
NUMBER & ALGEBRA
6 The gaming computer is to be paid off over 3 years instead of 2.
a What would be the monthly repayment?
b How much more interest would be paid?
(Hint: Change the repayment period to 3 and use the arrow key to move right.)
7 Try changing the repayment period and/or the interest rate for the other items to investigate the effect on the
monthly repayment and the amount of interest paid.
8 Check the advertised monthly repayments for several items advertised in newspapers and magazines. If
payments for the same items differ, investigate for hidden charges.
Chapter 5 Financial mathematics
127
Lesson 9: Simple Interest
Lesson Goal: Learn the simple interest formula and how to use it.
When you invest money in a financial institution, such as a bank, the bank pays for the use of
the money. This payment by the bank is called interest and is calculated as a percentage of
the amount invested. Similarly, when you borrow money a charge is made for the use of the
bank’s money. This charge also is called interest and is calculated as a percentage of the
amount borrowed. If the interest is calculated as a fixed percentage of the original amount
invested (or borrowed), it is called simple interest.
Simple Interest Formula:
Where:
𝐼𝐼 = 𝑃𝑃𝑅𝑅𝑇𝑇
𝐼𝐼 = amount of interest
𝑃𝑃 = principal (the initial amount borrowed or invested)
𝑅𝑅 = interest rate p.a. expressed as a decimal (𝑅𝑅 =
𝑇𝑇 = time in years
𝑟𝑟
100
)
Exercise 1K Cambridge Year 9
Examples:
1) Calculate the simple interest received when $8000 is invested for 3 years at 4.5% p.a.
2) Use the simple interest formula to calculate the simple interest earned on an
investment of $10 800 at 3.9% p.a. for 5 years.
3) Calculate the amount to which $7000 will grow in 3 years if invested at 6.5% p.a. simple
interest.
4) Calculate the simple interest earned on $6000 at 8% p.a. for 16 months.
5) Rene invested $4700 at 6% p.a. simple interest. How long did it take to earn $1128 in
interest?
Lesson 10: Compound Interest and Depreciation Formula
Lesson Goal: Learn how to use the Compound Interest Formula.
Compound Interest Formula:
Depreciation Formula:
Where:
𝐴𝐴 = 𝑃𝑃 (1 + 𝑅𝑅)𝑛𝑛
𝐴𝐴 = 𝑃𝑃 (1 − 𝑅𝑅)𝑛𝑛
𝐴𝐴 = final amount (includes principal and interest).
𝑃𝑃 = principal (the initial amount borrowed or invested)
𝑅𝑅 = interest rate p.a. expressed as a decimal (𝑅𝑅 =
𝑛𝑛 = number of compounding periods
𝑟𝑟
100
)
Exercise 1M Cambridge Year 9
Examples:
1) Complete the following using the compound interest formula.
a) Find the amount to which $6000 grows if it is invested for 5 years at 3% p.a.
compound interest.
b) The total amount of interest earned over this period
2) Calculate the amount to which $10 000 grows if it is invested for 3 years at 12% p.a.
interest, compounded:
a) Monthly:
b) Quarterly:
c) Six-monthly:
3) Naomi bought a computer three years ago for $5680. She assumed the value of the
computer would depreciate at a rate of 25% p.a. What is the value of the computer at
the end of the three years (assuming her assumption was correct)?
Lesson 11: More Compound Interest
Lesson Goal: Put the compound interest formula into practise more.
At times the final amount of an investment is given, and calculations are used to find the
principal, the time period or the interest rate.
Exercise 9C Oxford Year 10
Examples:
1) Calculate how much money (principal) must be invested at 4% p.a. interest
compounding annually to have $10 000 at the end of 3 years.
2) Calculate how much money must be invested at 6% p.a. interest compounding
quarterly to have $5000 after 2 years.
3) What annual compound interest rate is required to increase $2000 to $2500 over 5
years?
4) What interest rate compounding monthly is required to increase $500 to $700 over 3
years?
5) The amount of $7300 is obtained when $6000 is invested at 5% interest, compounding
annually. Use guess, check and refi ne to calculate the value of n.
10 Calculate the interest charged on these credit card cash advances.
a $600 for 19 days when the annual interest rate is 16%
b $350 for 22 days when the annual interest rate is 16%
c $200 for 25 days when the annual interest rate is 19%
d $500 for 15 days when the annual interest rate is 21%
e $400 for 13 days when the annual interest rate is 14%
C Compound interest calculations
At times the final amount of an investment is given and calculations are used to find the principal, the time
period or the interest rate.
EXAMPLE 1
Calculate how much money (principal) must be invested at 4% p.a. interest compounding annually to have
$10 000 at the end of 3 years.
Solve
10 000 = P(1 + 0.04)3
= P(1.04)3
10 000
______
=P
(1.04)3
P = $8889.96
Think
Apply
A = P(1 + R)
Substitute the values into the formula.
As the interest compounds annually,
R = 0.04 and n = 3.
Carefully check the
compounding period.
Substitute the answer into
the formula to check.
n
Exercise 9C
1 Complete the following to find the amount that must be invested at 6% p.a. interest compounding annually
to have $4000 at the end of 5 years.
□ □
___ = P 1 +____
100
= P(1.06)□
4000
_____
=P
□
P = ___
(
)
2 Calculate the amount that must be invested at 7% p.a. interest compounding annually to have $6000 at the
end of 4 years.
NUMBER & ALGEBRA
3 Calculate the amount that must be invested at 2.5% p.a. interest compounding annually to have $1000 at
the end of 7 years.
4 Calculate the amount that must be invested at 3.6% p.a. interest compounding annually to have $800 at the
end of 6 years.
5 Calculate the amount that must be invested at 1.8% p.a. interest compounding annually to have $400 at
the end of 10 years.
CChapter 9 Financial mathematics
Ch
219
EXAMPLE 2
Calculate how much money must be invested at 6% p.a. interest compounding quarterly to have $5000
after 2 years.
Solve
Think
Apply
5000 = P(1 + 0.015)8
= P(1.015)8
5000
_______
=P
(1.015)8
P = $4438.56
A = P(1 + R)n
The investment compounds quarterly.
Divide 0.06 by 4 to obtain R.
Multiply 2 by 4 to obtain n.
Calculate the values of R and n.
In general, R is divided by the
number of compounding periods
per annum and n is multiplied.
6 Complete the following to find the amount of money that must be invested at 8% p.a. interest compounding
quarterly to have $3000 at the end of 3 years.
R = ___ ÷ 4 = ___ and n = 3 × ___ = ___
A = P(1 + R)n
___ = P(1 + 0.02)□
___ = P(1.___)□
3000
P = _____ = ___
□
7 Calculate the amount that must be invested at 3.2% p.a. interest compounding quarterly to have $7000 at
the end of 8 years.
8 Calculate the amount that must be invested at 6% p.a. interest compounding monthly to have $500 at the
end of 6 years.
9 Calculate the amount that must be invested at 4.8% p.a. interest compounding monthly to have $1200 at the
end of 5 years.
10 Calculate the amount that must be invested at 2.4% p.a. interest compounding monthly to have $10 000 at
the end of 8 years.
11 Calculate the amount that must be invested at 5.1% p.a. interest compounding monthly to have $450 at the
end of 3 years.
EXAMPLE 3
What annual compound interest rate is required to increase $2000 to $2500 over 5 years?
Solve
NUMBER & ALGEBRA
2500 = 2000(1 + R)5
2500
_____
= (1 + R)5
2000
1.25 = (1 + R)5
____
5
√1.25 = (1 + R)
1.0456… = 1 + R
∴ R = 0.0456… ≈ 0.046
The interest rate required is 4.6% p.a.
220
Insight Mathematics 10 stages
e 5.1/5.2 Australian
es
stralian Curriculum
Curriculu
Think
n
A = P(1 + R)
Substitute the values. As the
compounding period is annual,
n = 5.
Apply
Substitute the values
and solve the resulting
equation. The value of R
is for that compounding
time period.
r
R = ____ so r = R × 100
100
Licensed to Lisa Campbell, from Casula High School until 2020-01-01.
12 Complete to find the annual compound interest rate
required to increase $1000 to $1200 over 4 years.
A = ___, P = ___ and ___ = 4
A = P(1 + R)n
1200 = ___ (1 + R)□
1200
_____
= (1 + R)□
□
___ = (1 + R)□
4
___
√1.2 = 1 + R
R = ___ − 1 ≈ 0.0466
The interest rate required is ___% p.a.
13 What annual compound interest rate is required to increase $500 to $800 over 6 years?
14 What annual compound interest rate is required to increase $3500 to $4000 over 4 years?
15 What annual compound interest rate is required to increase $200 to $300 over 8 years?
16 What annual compound interest rate is required to increase $450 to $500 over 3 years?
EXAMPLE 4
What interest rate compounding monthly is required to increase $500 to $700 over 3 years?
Solve
Think
Apply
700 = 500(1 + R)36
700
____
= (1 + R)36
500
1.4 = (1 + R)36
___
36
√1.4 = 1 + R
1.009 39 = 1 + R
R = 0.009 39
r = 0.00939 × 12 × 100
= 11.27% p.a.
The interest rate required is 11.27% p.a.
A = P(1 + R)n
A = 700, P = 500
3 years is 36 months so n = 36.
Substitute the values and
calculate. The number of
time periods depends on the
frequency of compounding
the interest. The value
of R is in terms of the
compounding period.
Subtract 1 from both sides.
This is R for 1 month.
Multiply R by 12 to make the
rate annual. Multiply by 100 to
make it a percentage.
17 Complete to find the interest rate compounding monthly required to increase $900 to $1200 over 6 years.
72
NUMBER & ALGEBRA
A = ___, ___ = 900 and n = ___ × 12 = ___
A = P(1 + R)n
1200 = ___(1 + R)□
1200
_____
= (1 + R)□
□
___ = (1 + R)72
_____
√1.333 = ___
R = ___ − 1 = ___
r = ___ × 12 × ___ = ___
18 What interest rate compounding monthly is required to increase $8000 to $10 000 over 5 years?
19 What interest rate compounding monthly is required to increase $100 to $125 over 4 years?
CChapter 9 Financial mathematics
Ch
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221
20 What interest rate compounding quarterly is required to increase $400 to $450 over 3 years?
21 What interest rate compounding quarterly is required to increase $900 to $1200 over 6 years?
22 What interest rate compounding half yearly is required to increase $1000 to $1300 over 8 years?
EXAMPLE 5
The amount of $7300 is obtained when $6000 is invested at 5% interest, compounding annually. Use
guess, check and refine to calculate the value of n.
Solve
A = P(1 + R)n
7300 = 6000(1.05)n
6000(1.05)3 = $6945.75
6000(1.05)6 = $8040.57
6000(1.05)4 = $7293.04
The time is approximately 4 years.
Think
Apply
Substitute the values into the
compound interest formula.
Try n = 3
too small
Try n = 6
too big
Try n = 4
very close
Try different values for n.
If the value is less, then use
a bigger value of n.
23 The amount of $2020 is obtained when $1500 is invested at 5.1% p.a. interest, compounding annually.
Use guess, check and refine to calculate the value of n.
A = P(1 + R)n
2020 = ___(1 + 0.051)n = 1500(1.051)n
Try n = 4: 1500(1.051)4 = ___
too ___
too ___
Try n = 8: 1500(1.051)8 = ___
6
very close
Try n = 6: 1500(1.051) = ___
The time is approximately 6 years.
24 The amount of $2700 is obtained when $2000 is invested at 3.4% p.a. interest, compounding annually.
Use guess, check and refine to calculate the value of n.
25 The amount of $800 is obtained when $600 is invested at 4.2% p.a. interest, compounding annually.
Use guess, check and refine to calculate the value of n.
Extension
EXAMPLE 6
How long would it take to obtain $1000 if $600 is invested at 4.8% p.a. interest, compounding monthly?
Use the guess, check and refine method. Give the answer to the nearest month.
Solve
n
NUMBER & ALGEBRA
A = P(1 + R)
1000 = 600(1 + 0.004)n
600(1.004)100 = $894
600(1.004)150 = $1091.96
600(1.004)130 = $1008.17
600(1.004)128 = $1000.15
The time is 10 years 8 months.
222
Think
Apply
Substitute using
R = 0.048 ÷ 12 = 0.004
Try n = 100 too small
Try n = 150 too big
Try n = 130 very close
Try n = 128 correct
Thus n = 128 months or 10 years 8 months.
The value of n will
be approximate
and may take many
attempts to obtain
the correct value.
Insight Mathematics 10 stages
e 5.1/5.2 Australian
es
stralian Curriculum
Curriculu
Licensed to Lisa Campbell, from Casula High School until 2020-01-01.
26 The amount of $600 is obtained when $550 is invested at 3.2% p.a. interest, compounding quarterly.
Use guess, check and refine to calculate the value of n.
27 The amount of $1400 is obtained when $1300 is invested at 5% p.a. interest, compounding quarterly.
Use guess, check and refine to calculate the value of n.
28 The amount of $820 is obtained when $750 is invested at 3.6% p.a. interest, compounding monthly.
Use guess, check and refine to calculate the value of n.
29 The amount of $5100 is obtained when $5000 is invested at 2.4% p.a. interest, compounding monthly.
Use guess, check and refine to calculate the value of n.
Investigation 2 Using a spreadsheet
The following spreadsheets can be used for compound interest calculations. Enter the formulas as shown to
perform the calculations. Use the spreadsheets to confirm the answers to the questions in Exercise 9C.
1 This spreadsheet can be used to calculate the final amount of an investment when the principal, interest rate
and time period are known.
A
B
1
C
D
Compound interest calculator
2
Principal
Interest rate as a
percentage
Number
of years
Number of compounding
periods per year
3
8889.96
6
3
1
4
5
Final amount
6
=A3*(1+B3/(100*D3))^(C3*D3)
7
8
9
2 This spreadsheet can be used to calculate the interest rate required to achieve a final amount of an investment
when the principal and time period are known.
A
1
B
C
D
Compound interest calculator
2
Principal
Final amount
Number
of years
Number of compounding
periods per year
3
2000
2500
5
1
5
Interest rate
Annual rate as a percentage
6
=(B3/A3)^(1/C3)-1
=D3*A6*100
NUMBER & ALGEBRA
4
7
8
9
CChapter 9 Financial mathematics
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