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CREDIT AND LOANS

79

33

Credit and loans

FINANCIAL MATHEMATICS

At some stage in your life you will probably need to borrow money. This may be for a home, a car, a computer or a holiday. When taking out a loan you must consider how much money you need, your ability to repay the loan, how much interest you will pay as well as the terms and conditions of the loan. There are many institutions willing to lend money, but the borrower must be certain that they can pay off the loan. It is not always best to take up offers such as ‘Buy now and pay nothing for 6 months’ or ‘We will arrange finance for approved customers’ or ‘Buy now and pay only $20 per week’. Credit does not mean free.

You will compare different loan and credit options in this chapter.

In this chapter you will learn how to:

n

calculate the principal, interest and repayments for a flat rate loan

n

consider borrowing money ‘on terms’ and deferred payment plans

n

construct and calculate values in a table of loan repayments

n

use published tables to determine monthly repayments on a reducing balance loan

n

use technology to compare loans

n

calculate credit card payments involving interest-free periods, interest rates, fees and charges.

!NNC Yr12 maths ch 03 Page 79 Wednesday, October 4, 2000 1:42 AM

80

NEW CENTURY MATHS GENERAL: HSC

FLAT RATE LOANS

A

flat rate loan

is one where flat (or simple) interest is charged on the amount borrowed (or principal) for the term of the loan. The term of the loan, usually from 1 to 5 years, is decided by the lender and is based on the borrower’s capacity to meet the repayments. There may be a small charge to establish the loan. This is to cover administrative costs.

This type of loan is used for purchasing goods such as a car, computer or stereo. Fees and charges often apply and can be included in the amount borrowed or paid up front.

Some examples of lending institutions offering flat rate loans are:

n

banks

n

building societies (e.g. IMB)

n

credit unions

n

insurance companies (e.g. AMP, GIO, NRMA)

n

finance companies (e.g. AVCO).

Example 1

Mark borrowed $2500 at a flat interest rate of 9% p.a. to buy a new computer. What is his monthly repayment (to the nearest cent) if he pays off the loan in 5 years?

Solution

P

=

$2500,

r

=

0.09,

n

=

5Interest

=

Prn

=

$2500

×

0.09

×

5

=

$1125Amount to repay

=

principal

+

interest

=

$2500

+

$1125

=

$3625

Monthly repayment

=

5 years

=

5

×

12 months

=

60 months

=

$60.4166 …Mark’s monthly repayment is $60.42.

Example 2

Donna wanted to buy a car stereo for $1500. Her bank offered her a loan at a flat interest rate of 12% p.a. to be repaid in fortnightly instalments over 2 years. The bank charges were stamp duty of $7.50, loan insurance of $8.50 and an establishment fee of $12. If Donna included the bank charges in the amount she borrowed, find:(a) the total amount to repay(b) the fortnightly repayment

Flat rate interest or simple interest is calculated using the formulaI = Prn

where P = principalr = rate per period expressed as a decimal n = number of periods.

$362560

---------------


CREDIT AND LOANS

81

Solution

(a) Total amount borrowed

=

$1500

+

$7.50

+

$8.50

+

$12

=

$1528

P

=

$1528,

r

=

0.12,

n

=

2Interest

=

Prn

=

$1528

×

0.12

×

2

=

$366.72Total amount to repay

=

amount borrowed

+

interest

=

$1528

+

$366.72

=

$1894.72

(b) Fortnightly repayment

=

2 years

=

52 fortnights

=

$36.4369 …Each fortnightly repayment is $36.44.

1.

Guy borrowed $9000 to buy a second-hand car on a flat rate loan at 10% p.a. interest over 5 years.(a) What must he repay altogether? (b) What is his monthly repayment?

2.

Heidi obtained a holiday loan of $3500 at 15% p.a. flat rate interest to be paid back in fortnightly instalments over 3 years.(a) How much will her holiday cost altogether?(b) How much is each instalment?

3.

Brooke borrowed $9500 to help furnish her new home. She took a flat rate loan at 10.2% p.a. interest and paid off the loan in weekly instalments over 4 years. How much was each repayment?

4.

Elton obtained a $1200 loan at a flat interest rate of 11.2% p.a. to help with the purchase of a new piano. He paid off the loan in monthly instalments over 3 years.(a) How much interest did he pay? (b) How much was each repayment?

5.

Mabel bought a new washing machine and dryer and borrowed $1200 to help with the purchase. If the simple interest rate was 14.6% p.a. and the term of the loan was 2 years, how much was each fortnightly instalment?

6.

Gaylene bought a new bed from Bob’s Beds for $1850. She paid off the bed in weekly instalments over the next 18 months at a flat interest rate of 18.55% p.a. If she included $9 stamp duty, a dealer’s commission of $18.50 and loan protection insurance of $45 in the amount borrowed, find:(a) the total amount borrowed (b) the amount of interest she paid(c) the weekly repayment(d) how much she would have saved if she had paid the charges up front

7.

Katie borrowed $8500 over 3 years at a flat interest rate of 13.5% p.a. to help towards her wedding. She included the $7.50 stamp duty and $15 per year loan insurance in the amount borrowed and repaid the loan in monthly instalments.(a) How much did she borrow altogether?(b) How much interest did she pay?(c) What was her monthly repayment?

Interest is paid on the whole amount borrowed, including bank charges.

$1894.7252

----------------------

Exercise 3-01: Flat rate loans


82


8.

The Education Credit Union published this table for flat rate loans. Giselle borrowed$8000 over 4 years.(a) How much does she repay per month?(b) What is the total amount to repay the loan?(c) What is the interest charged?(d) Calculate the flat interest rate per annum.

9.

Ryan borrowed $2200 from the Education Credit Union to purchase new ski equipment. He agreed to repay the loan over 2 years. Use the table in question 8 to find:(a) how much he repays per month(b) the total amount to be repaid(c) the amount of interest charged(d) the simple interest rate per annum

10.

Cameron borrowed $6300 from the Education Credit Union to consolidate his debts. He arranged to repay the loan over 5 years. Use the table in question 8 to find:(a) the monthly repayment(b) the total amount of interest

11.

A bank advertised the following fixed rates per annum for secured personal loans.

Three people had their loans approved:Alexandra (A) borrowed $22 500 over 5 years for a new car.Basil (B) borrowed $9000 over 36 months for a second-hand motorbike.

Cecilia (C) borrowed $24 000 over 4 years for a new boat.

For each person find:(a) the interest rate per annum charged by the bank(b) the amount of interest charged(c) the monthly repayment

12.

The following fixed interest rates per annum, determined by the amount borrowed and the term of the loan, were published by a bank.

Loan securityAmount of loan

Greater than $20 000 Less than $20 000

New motor vehicle 9.4% 9.4%

Used motor vehicle 9.9% 10.75%

New or used motorbike, caravan or boat 10.49% 10.75%

Loan amountTerm

1–2 years 3–5 years 6–7 years

Greater than $10 000 10.49% 10.99% 11.99%

Less than or equal to $10 000 10.99% 11.49% 12.49%

Note:

30 months (2 years) loans will receive the 3–5 years interest rate; 66 months (5 years) will receive the 6–7 years interest rate.

Years to repay loan

Monthly repayment (per $1000)

1 $91.25

2 $49.58

3 $35.69

4 $28.75

5 $24.58

12---

12---

12---


CREDIT AND LOANS

83

Consider these cases:Adrian (A) gets a loan of $7500 for a second-hand car to be paid back over 2 years.Beryl (B) obtains a loan of $10 200 over 4 years for a new bathroom.Christian (C) takes out a $9800 loan for an overseas holiday, to be repaid in 30 months.

For each person find:(a) the interest rate on their loan(b) the amount of interest they will pay(c) their monthly repayment

13.

The OzExpress Credit Union advertised the following personal loans.

Calculate the repayment for the following OzExpress Credit Union personal loans.(a) $2000 for a new computer to be repaid monthly over 3 years(b) $32 000 for a new car to be repaid fortnightly over 8 years(c) $4500 for a Fiji holiday to be repaid weekly over 3 years(d) $15 400 for a used car to be repaid monthly over 6 years

1.

Collect current interest rates for flat rate loans (often called

personal loans

) from as many different lending institutions as you can. These can be found by approaching a bank (or other lending institution) and asking for pamphlets, by consulting the money section of a newspaper or by accessing Internet websites.

Here are seven websites that may be of help:www.westpac.com.auwww.national.com.auwww.commbank.com.auwww.anz.com.auwww.colonial.com.auwww.teacherscreditunion.com.auwww.stgeorge.com.au

2.

Investigate the different types of flat rate loans and their current interest rates.

3.

What terms and conditions for flat rate loans does each lending institution impose on its borrowers?

Loan type Terms and conditions Interest rate (p.a.)

New Car Loan Loan availability—up to $40 000Maximum term—10 years

9.25%

Car Loan Loan availability—up to $40 000Maximum term—8 years

9.90%

One-Stop Travel Loan Loan availability—up to $40 000Minimum loan—$2000Maximum term—normally 5 years

9.50%

New Computer Loan Loan availability—up to $40 000Minimum loan—$2000Maximum term—normally 4 years

9.50%

Investigation: Current interest rates


84


1.

Create a spreadsheet to calculate monthly, fortnightly or weekly repayments for a flat rate loan by entering amounts in cells B1, B2, B3 and B4 and appropriate formulas in cells B6, B7 and B8.

2.

Show that the fortnightly repayment required to pay off a loan of $10 000 in 8 years at 9% p.a. interest is $82.69.

3.

What weekly repayment is needed to pay off a loan of $7600 at 8% p.a. interest in 4 years?

BUYING ON TERMSMany shops and businesses sell goods to customers on terms. Another name for this is time payment as the customer signs an agreement to pay for the goods over a certain period of time. It is also called hire purchase because the customer actually hires (or borrows) the goods until they are paid off.

If you buy on terms, you do not actually own the goods until they are fully paid for, and goods can be repossessed (taken back) if you fail to meet the repayments.

Example 3A refrigerator is advertised for $1200 cash or $200 deposit and $57 per month for 2 years.(a) How much more will the refrigerator cost if bought on terms?(b) What is the flat rate of interest charged?

A B C D

1 Principal (P) $10 000.00

2 Interest rate (% p.a.) 9

3 Term (years) 8

4 Periods per year 12

5

6 Total interest $7200.00

7 Total repaid $17 200.00

8 Repayment per period $179.17

9

10

Spreadsheet activity: Calculating flat rate loan repayments

Just for the record

AUSTRALIA ’S FIRST BANK

The first bank in Australia, the Bank of New South Wales, was opened in Macquarie Place, Sydney on 8 April 1817 in what was the British colony of NSW. In the 1970s it became known as ‘The Wales Bank’. In 1982 it expanded to the Western Pacific region and changed its name as it was no longer restricted to New South Wales. What is its new name?


CREDIT AND LOANS 85

Solution(a) Deposit= $200

Total repayments= $57 × 24= $1368

Total cost on terms= $1368 + $200= $1568

Difference between cash price and terms price= $1568 – $1200= $368

By buying on terms the refrigerator will cost $368 more than the cash price.

(b) The buyer borrows $1000 since $200 of the $1200 are paid in cash.P = $1000, I = $368, n = 2

I = Prn368= 1000 × r × 2

= 2000 × r

r =

= 0.184The flat rate of interest charged is 18.4% p.a.

Deferred payment plan

Some stores advertise deferred payment schemes to entice customers to purchase goods, in the expectation that they are receiving something for nothing.

If the customer pays for the goods before the end of the ‘interest-free period’, this type of plan may be beneficial. A customer who pays for the goods after the interest-free period will be charged interest from the date of purchase.

Example 4Michelle wants to purchase a home gym for $900. She cannot afford to pay cash and must choose one of the following options:

The Loan Arranger: Pay 10% deposit and 12 monthly repayments of $80.

Flashy Finance: No deposit, nothing to pay for 6 months, then 12 monthly repayments of $100.

(a) For each option find:(i) the total cost (ii) the amount of interest

(iii) the flat interest rate per annum (correct to 1 decimal place)(b) Which of the two loan options should Michelle choose? Why?

Solution(a) The Loan Arranger

(i) Deposit = 10% × $900 Total repayments= $80 × 12

= $90 = $960Total cost= deposit + total repayments

= $90 + $960= $1050

The deposit is sometimes called a down payment.

The difference is the interest on the loan.

3682000------------

No depositNothing to pay until June

Buy now and pay nothing for 6 months!


86 NEW CENTURY MATHS GENERAL: HSC

(ii) Interest = $1050 − $900= $150

(iii) I = $150, P = $810, n = 1I = Prn

150 = 810 × r × 1

r =

= 0.1851 …The flat interest rate is 18.5% p.a.

Flashy Finance(i) Total cost= $100 × 12

= $1200(ii) Interest = $1200 − $900

= $300

(iii) I = $300, P = $900, n = 1.5I = Prn

300 = 900 × 1.5 × r= 1350 × r

r =

= 0.2222 …The flat interest rate is 22.2% p.a.

(b) Michelle should choose The Loan Arranger. By paying a deposit and then paying the rest over 12 months she pays less interest.

1. Darby and Joan retired and bought a second-hand motorhome for $19 500 to travel around Australia. They paid $1500 deposit and 36 monthly instalments of $685.(a) How much did they pay altogether for the motorhome?(b) How much interest did they pay?(c) What was the flat rate of interest charged?

2. Budget Bob sells used cars on terms. For sale he has the following:

(a) How much would you pay for the Kombi Van if you took Bob’s terms?(b) How much would you save on the Celica by paying cash rather than terms?(c) What is the total amount a buyer would pay for the Ute on terms?(d) What is the flat rate of interest per annum, correct to 1 decimal place, charged on

each vehicle?(e) Can you give a reason why Bob charges different rates of interest for different cars?

Amount borrowed= $900 − $90= $810

150810---------

With this option, interest is charged for 18 months; that is, n = 1.5 years.

3001350------------

Exercise 3-02: Buying on terms

BB UU DD GG EE TT BB OO BB ’’ SS BB EE AA UU TT DD EE AA LL SS

☞ Toyota Celica $16 000 cash or $500 deposit and $95 per week over 5 years

☞ Holden Ute $12 500 cash or $300 deposit and $78 per week over 4 years

☞ Kombi Van $6200 cash or $180 deposit and $70 per week over 2 years

Price includes registration, stamp duty and delivery for approved customers

!NNC Yr12 maths ch 03 6 Wednesday, October 4, 2000 1:42 AM

CREDIT AND LOANS 87

3. Del and Barry bought an old cottage for $11 000 and had it transported to their farm in the country. They paid the vendor 10% deposit and agreed to pay him the rest at $85 per week over 5 years.(a) How much did they actually pay for the house?(b) What was the vendor’s flat interest rate, correct to 3 significant figures?

4. The Ripoff Finance company had the following hire purchase agreement with Mr B. Spender:

Cash price of yacht $17 500Deposit $450Trade-in allowance on old yacht $1200Stamp duty $245.30Registration $362.50

36 monthly repayments of $680(a) What was the total amount paid for the yacht if the deposit, stamp duty and

registration were paid by Mr Spender at the time of purchase?(b) What was the flat interest rate charged p.a. by the finance company? Answer correct

to 1 decimal place. (Note: Principal borrowed = cash price − deposit − trade-in)

5. Jayden bought a new bedroom suite from Dad’s Discounts. The cash price was $2850 but Jayden chose to take Dad’s terms and paid 10% deposit and $42.50 per week for 18 months.(a) How much did she actually pay for the bedroom suite?(b) How much would she have saved by paying cash?(c) What was the flat rate of interest charged?

6. Billy Joe bought a new stereo TV, cash price $999, and chose the deferred payment plan offered at Main Man discounts. He paid $20 deposit and his first payment of $60 was 3 months later. If he made 24 monthly repayments, find:(a) the total cost of the TV(b) the flat rate of interest charged (correct to 1 decimal place)

7. Steven purchased an engagement ring and paid $100 down, no payments for 6 months and then $60 a fortnight for 3 years.(a) How much more than the advertised price of $3000 did he pay?(b) What was the flat rate of interest charged?

8. Brian wants to buy a new computer desk that retails for $1150. He chooses the deferred payment plan of no deposit, nothing to pay for 6 months and then 12 monthly instalments of $130.(a) Find the total cost of the desk using the deferred payment plan.(b) How much extra does the plan cost?(c) What is the flat interest rate charged?

9. Write down one advantage and one disadvantage of:(a) paying cash for goods(b) buying on terms(c) using a deferred payment plan



REDUCING BALANCE LOANSLoans taken over long periods, such as home loans, are usually calculated using reducible interest. This means that interest is calculated on the balance still owing, not on the total principal borrowed as with flat rate interest. The balance of the loan reduces after each repayment and continues in this way until the loan is fully repaid—that is, until the balance owing is zero.

Example 5Shelley borrowed $15 000 to buy a new car at 8% p.a. reducible interest. She made monthly payments of $180.(a) Draw up a table showing the progress of the loan in the first 6 months.(b) How much had Shelley paid off the principal after her 6th payment?(c) How much interest did she pay in the first 6 months?(d) How much interest did she save compared with a flat interest rate loan at 8% p.a.?

Solution(a) For the first month:

Amount owing (principal) at the start of the month $15 000

Interest on this principal for 1 month $15 000 × = $100

Amount owing before repayment $15 000 + $100 = $15 100Balance owing after repayment $15 100 − $180 = $14 920∴ Principal for the second month $14 920Continue in this way until you have calculated the values for the first 6 months.

(b) At the end of the first 6 months:Amount owing= $14 511.94

Amount paid off principal= $15 000 − $14 511.94= $488.06

(c) Interest paid in first 6 months= $591.94 Find the sum of the interest column.

Reducing balance loan table

Loan for Shelley’s new car

Amount borrowed $15 000

Interest rate p.a. 8%

Monthly repayment (R) $180

No. of months(n)

Principal ($P)

Interest($I )

Amount owing before repayment $(P + I )

Balance$(P + I − R)

1 15 000 100 15 100 14 920

2 14 920 99.47 15 019.47 14 839.47

3 14 839.47 98.93 14 938.40 14 758.40

4 14 758.40 98.39 14 856.79 14 676.79

5 14 676.79 97.85 14 774.64 14 594.64

6 14 594.64 97.30 14 691.94 14 511.94

591.94

0.0812

----------


CREDIT AND LOANS 89

(d) For a flat rate of interest for 6 months:P = $15 000, r = 0.08, n = 0.5 n = 6 months = 0.5 years

I = Prn= 15 000 × 0.08 × 0.5= $600

Interest saved for first 6 months= $600 − $591.94= $8.06

1. (a) Copy the table in Example 5 and add rows to show the progress of the loan for the next 6 months.

(b) How much had Shelley paid off the principal after 12 months?(c) How much interest did she pay in the first year?(d) Calculate her saving in interest for the first year compared to the same loan at a flat

rate interest of 8% p.a.

2. Isabelle borrowed $100 000 to purchase her first studio apartment. The bank offered to lend her the money at 10% p.a. reducible interest and fortnightly repayments of $540.

(a) Copy and complete the table above showing the progress of her loan for the first6 fortnights.

(b) How much had she paid off the principal after 6 repayments?(c) How much interest did she pay:

(i) in the first 3 fortnights? (ii) in the next 3 fortnights?(d) What is happening to the amount of interest as the number of repayments increases?

3. Greg borrowed $2000 to buy new golf clubs. The interest rate is 10% p.a. and he makes quarterly repayments of $500.(a) Draw up a loan repayment table and calculate values until the balance is zero. The

last payment may be less than $500.(b) How much will his last payment be?(c) How much interest will he pay?(d) How many years will it take to pay off the loan?(e) How much more interest would he pay if the loan had a flat rate interest of 10% p.a.?

4. Gillian borrowed $2500 at 7% p.a. interest for a South Pacific Cruise. She repaid the loan in regular fortnightly repayments of $300.(a) How many weeks did it take to pay off the loan completely?(b) What was the amount of the last payment?(c) How much interest was charged on this loan?(d) How much interest would she have paid if this loan had had a flat interest rate of

7% p.a.?

No. offortnights (n)

Principal($P)

Interest($I )

Amount owing$(P + I )

Balance$(P + I − R)

1 100 000 384.62 100 384.62 99 844.622 99 844.62 384.02 100 228.64 99 688.643456

Exercise 3-03: Reducing balance loans



USING PUBLISHED LOAN REPAYMENT TABLESBecause the calculations for large loans are complicated, financial institutions publish tables related to loans. For example, the table below gives the monthly repayments for a $1000 reducing balance loan for various terms and interest rates.

Monthly repayments—reducing balance loan of $1000

Example 6(a) Mr and Mrs Pitt obtain a premium home loan of $370 000 at 7% p.a. reducible interest

for a term of 20 years. Find:(i) the monthly repayment (ii) the total amount repaid(iii) the total interest paid

(b) How much more interest would they pay by choosing a 25-year term rather than a 20-year term?

Solution(a) Term = 20 years, interest rate = 7% p.a.

From the table above:

(i) Monthly repayment for $1000= $7.75Monthly repayment for $370 000= 370 × $7.75

= $2867.50

(ii) Total amount repaid= $2867.50 × 20 × 12= $688 200

(iii) Interest paid= total amount repaid − amount borrowed= $688 200 − $370 000= $318 200

(b) Term = 25 years, interest rate = 7% p.a.From the table above:

Monthly repayment for $1000= $7.07Monthly repayment for $370 000= 370 × $7.07

= $2615.90Total amount repaid= $2615.90 × 25 × 12

= $784 770Interest paid= $784 770 – $370 000

= $414 770Difference in interest paid= $414 770 − $318 200

= $96 570They would pay $96 570 more interest by choosing a 25-year term.

Interest rate (%)Term (years)

5 10 15 20 25 30

56789

101112

18.8719.3319.8020.2820.7621.2521.7422.24

10.6111.1011.6112.1312.6713.2213.7714.35

7.918.448.999.56

10.1410.7511.3712.00

6.607.167.758.369.009.65

10.3211.01

5.856.447.077.728.399.099.80

10.53

5.376.006.657.348.058.789.52

10.29


CREDIT AND LOANS 91

1. Use the table of loan repayments on page 90 to determine monthly repayments for the following loans.(a) $80 000 at 12% over 15 years (b) $125 000 at 9% over 25 years(c) $230 000 at 6% over 20 years (d) $97 000 at 11% over 10 years

2. Use the table of loan repayments on page 90 to determine, for each loan below:(i) the monthly repayment(ii) the total amount repaid(iii) the total interest paid

(a) $380 000 at 7% over 30 years (b) $67 000 at 8% over 25 years(c) $420 000 at 10% over 30 years (d) $365 000 at 9% over 20 years

3. This table gives the monthly repayments required to pay off a loan in 10 years.

Monthly loan repayments—10-year term

For each of the following 10-year loans find:(i) the total repayments(ii) the amount of interest paid over the term of the loan

(a) $20 000 at 8% p.a. (b) $100 000 at 6% p.a.(c) $5000 at 7% p.a. (d) $50 000 at 10% p.a.(e) $100 000 at 8% p.a. (f) $25 000 at 7% p.a.

Principal borrowedInterest rate

6% p.a. 7% p.a. 8% p.a. 10% p.a.

$5000$20 000$50 000

$100 000

$55.52$222.05$555.11

$1110.21

$58.06$232.22$580.55

$1161.09

$60.67$242.66$606.64

$1213.28

$66.08$262.31$660.76

$1321.51

FINDING THE KEY

When you study a novel, your aim is to find the underlying theme of the book. The same is true when you study a mathematics topic. Look for ‘the key’, the main idea that threads through the topic: the reason for studying it. Once you understand and appreciate the point of a topic, a lot of its concepts and ideas will fall into place.

For example, what are the main themes of topics in this chapter? To help you find the key:n Analyse the content of the chapter for meaning and relevance. Why did you learn the

topics? How and where will you use the mathematics?n Use your own words to write your understanding of the topics in the chapter so that you

take ‘ownership’ of your mathematics.n Use the chapter introduction and chapter review to help you.n Ask yourself and your teacher questions about the topic. Have an opinion. Sound

informed.

Finding the key will unlock the door to mathematical understanding!

Study tips

Exercise 3-04: Using published loan repayment tables



4. The table below shows the monthly repayment required to pay off a reducing balance loan ranging from $50 000 to $100 000 at an interest rate of 8.50% p.a. over terms from 15 to 30 years. For each loan below find:

(i) the monthly repayment(ii) the amount of interest paid over the term of the loan

(a) $60 000 for 19 years (b) $75 000 for 25 years(c) $100 000 for 30 years (d) $80 000 for 17 years(e) $95 000 for 18 years (f) $50 000 for 15 years(g) $175 000 for 25 years (h) $290 000 for 15 years

Monthly loan repayments (interest 8.50% p.a.)

5. Use the table of monthly loan repayments on page 90 to:(a) find the interest rate per annum on a:

(i) 5-year loan if the monthly repayment is $21.74 per $1000(ii) 20-year loan if the monthly repayment is $8.36 per $1000

(b) find the term of the loan at:(i) 7% p.a. interest if the monthly repayment is $7.75(ii) 11% p.a. interest if the monthly repayment is $9.80

If you pay half the monthly repayment each fortnight you actually pay 13 monthly repayments per year instead of 12. This means the balance of the loan reduces more rapidly, you are charged less interest and your loan is paid off in less time.

The graph shows the progress of a $200 000 loan at 6.7% p.a. reducible interest over 30 years where repayments are made monthly and fortnightly. By making fortnightly repayments, the loan is paid off6 years earlier. Write two or three sentences describing this graph.

Principal($)

Term (years)

15 16 17 18 19 20 25 30

50 000 492.37 477.25 464.15 452.73 442.73 433.92 402.62 384.46

55 00060 00065 00070 00075 000

541.61590.85640.09689.32738.56

524.98572.70620.42668.15715.87

510.57556.98603.39649.81696.22

498.01543.28588.55633.83679.10

487.00531.27575.54619.82664.09

477.31520.70564.09607.48650.87

442.88483.14523.40563.66603.93

422.91461.35499.80536.24576.69

80 00085 00090 00095 000

100 000

787.80837.03886.27935.51984.74

763.60811.32859.05906.77954.50

742.64789.05835.47881.88928.30

724.37769.64814.92860.19905.46

708.36752.63796.91841.18885.45

694.26737.65781.05824.44867.83

644.19684.45724.71764.97804.23

615.14653.58692.03730.47768.92

Think: Fortnightly vs monthly repayments

Reducing balance loan (interest 6.7% p.a.)

200 000

180 000

160 000

140 000

120 000

100 000

80 000

60 000

40 000

20 000

05 10 15 20 25 30

Years

Bal

ance

($)

Monthly repayments

Fortnightly repayments


CREDIT AND LOANS 93

USING TECHNOLOGY TO COMPARE HOME LOANSFor this section you will need access to the Internet, a graphics calculator and a spreadsheet package.

Using the InternetSome good sources of information on current loan rates, terms, conditions, fees and charges are the websites of banks and credit unions (page 83). Here are three more websites to investigate:

www.nrma.com.au (has a loan repayment calculator with graph)www.banksmart.com.au (general information on all lenders)www.yourmortgage.com.au (calculators and glossary of mortgage terms)

Use a loan calculator to:n estimate what repayments might be at a higher or lower interest raten estimate the impact of changing from monthly to fortnightly repaymentsn change the term of the loan and see the effect on repayments.

The financial mode of a graphics calculatorThe financial mode (or TVM—time, value, money) of a graphics calculator can be used to simulate a reducing balance loan by using the compound interest formula.

On a Casio graphics calculator, access TVM mode then Compound Interest.

By entering five out of these six quantities, you can find the unknown quantity:n = number of periods (and hence payments) in the term of the loan

I% = interest rate per annumPV = present value or loan amount borrowed

PMT = payment for each periodFV = future value or unpaid balance (this is zero when the loan is paid off)P/Y= number of periods per year

Example 7Use a graphics calculator to find the number of payments needed to pay off a loan of $200 000 at 8% p.a. if monthly repayments of $1500 are made.

Solutionn = Leave n blank.

I% = 8PV = 200 000

PMT = −1500FV = 0P/Y= 12

The number of payments (n) is 330.68, which would be 330 payments of $1500 and one payment of $1020 to finish the loan.

The payment is entered as a negative because this is being subtracted from the loan balance.



Constructing a spreadsheetThe spreadsheet below shows the progress of a home loan of $200 000 with an interest rate of 8% p.a. and a monthly repayment of $1500. The screen has been split to show the beginning and end of the loan. Construct your spreadsheet as follows.

1. Enter the information in rows 1–4 and row 7 (cells B1 to B4 may need to be formatted).

2. Enter the number 1 in cell A8 then the formulas as shown in this table into cells B8 to F8 and A9 and B9. Copy the formulas down in each column.

3. Your spreadsheet will look like this.

Note: Including a payment (R) column means you can increase, decrease or miss a repayment at different times during the loan and see the effect on the term of the loan.

4. Use the Chart option to draw a line graph showing the progress of the loan.

5. When is the loan balance $100 000?

6. What is the balance of the loan at the end of 10 years?

A B C D E F

1 Amount borrowed (principal P) $200,000

2 Interest rate per annum 8.00%

3 Repayment per period ($) 1500

4 No. periods per year 12

7 Period no. Principal (P) Interest (I) Payment (R) P +I P+I-R

8 1 =B1 =B8*$B$2/$B$4 =$B$3 =B8+C8 =E8-D8

9 =A8+1 =F8 =B9*$B$2/$B$4 =$B$3 =B9+C9 =E9-D9

10 =A9+1 =F9 =B10*$B$2/$B$4 =$B$3 =B10+C10 =E10-D10

250000

200000

150000

100000

50000

0

Am

ou

nt

ow

ing

Number of payments

1 34

67

10

0

13

3

16

6

19

9

23

2

26

5

29

8

33

1


CREDIT AND LOANS 95

Example 8Use your spreadsheet to find:(a) the term of a $200 000 loan with an interest rate of 8% p.a. and a monthly repayment of

$1500(b) the final repayment required to reduce the balance to zero(c) how long it would take to repay the loan if the monthly repayment was doubled to $3000

per month(d) the monthly repayment needed to repay the loan in 10 years(e) the final repayment required to reduce the balance to zero in 10 years

Solution(a) 331 monthly repayments are made, so the term of the loan is 331 months or 27 years and

7 months.(b) The final repayment required to reduce the balance to zero is $1022.71.(c) By changing cell B3 to $3000, it can be seen that the loan would be repaid in 89 months

or 7 years and 5 months.(d) Split the screen so you can view the rows for periods 1 to

12 and 118 to 120. Keep trying different values in cell B3 until the period 120 is the last period for the loan.The monthly repayment required to repay the loan in 10 years is $2430.

(e) The final repayment required to reduce the balance to zero in 10 years is $1799.18.

Equipment: A graphics calculator and your home loan spreadsheet.

Use a graphics calculator for questions 1 to 4.

1. Find the monthly repayment for each of the following loans.(a) a $300 000 home loan over 30 years at 6.4% p.a.(b) a $15 800 reducing balance loan over 9 years at 9.3% p.a.

2. Find the fortnightly repayment for each of these loans.(a) a $26 000 reducing balance loan with an interest rate of 7.35% p.a. over 6 years(b) a $356 000 home loan over 30 years at 8.35% p.a. interest

3. Find the principal that can be borrowed for:(a) 20 years at 7% p.a. if a monthly repayment of $5000 is made(b) 10 years at 6.5% p.a. if a weekly repayment of $260 is made

4. Find the interest rate per annum on a reducing balance loan of:(a) $115 000 repaid at $1270 per month over 13 years(b) $175 000 if fortnightly repayments of $740 are made over 15 years

Use your home loan spreadsheet for questions 5 to 6.

5. Find the terms of these loans and the final repayments.(a) $200 000 with an interest rate of 8% p.a. and a monthly repayment of $2400(b) $156 000 with an interest rate of 7.3% p.a. and a monthly repayment of $1680

6. Find the monthly repayment and final repayment needed to pay off each of these loans.(a) $300 000 with an interest rate of 8% p.a. in 10 years(b) $260 000 with an interest rate of 8.2% p.a. in 15 years

10 years = 120 months

Exercise 3-05: Using technology to compare home loans



Use the most appropriate technology for questions 7 to 8.

7. (a) What is the monthly repayment on a loan of $240 000 at 7.5% p.a. interest over a term of 20 years?

(b) What effect does missing one repayment have on the term of the loan?(c) What happens if the monthly repayment is reduced to:

(i) $1500? (ii) $1000?(d) What is the least amount (to the nearest dollar) that must be repaid per month to have

a reducing balance?

8. Consider a loan of $20 000 over 5 years at 6% p.a. interest.(a) What is the monthly repayment?(b) What is the effect on the monthly repayment if:

(i) the amount borrowed is halved?(ii) the amount borrowed is doubled?(iii) the term of the loan is halved?(iv) the term of the loan is doubled?

(c) What is the effect on the term of the loan if:(i) the monthly repayment is increased by $50?(ii) the monthly repayment is doubled?(iii) fortnightly repayments are made?(iv) the 10th payment is missed?

DEVELOPING AN EXAM TECHNIQUE

Getting used to completing exams and knowing what to expect in them minimises exam anxiety. Getting into an exam ‘routine’ saves time and worry and allows you to stay focused on answering questions and using your time wisely rather than panicking.

n Use the reading time to plan your exam. How much can you do here without actually writing? Browse through the exam paper to see the work that is ahead of you.

n Calculate the average amount of time you should spend on each question/section. Take note of the marks allocated per question/section. Try to stick to your schedule.

n Easier questions are usually the first ones. Do an easy question first to boost your confidence. It will also save time.

n Allocate more time to harder questions. Leave them if you get stuck and come back to them later.

n Attempt every question. It is better to do a bit of every question and get some marks than to leave questions entirely blank and receive no marks for them.

n Show all working. Even if you get the wrong answer, you will be awarded some marks for correct working. Draw diagrams if necessary.

Study tips


CREDIT AND LOANS 97

CREDIT CARD PAYMENTSMany people prefer to shop without cash by using credit cards, sometimes referred to as ‘plastic money’. Credit cards act as short term loans and are a good way to purchase goods if managed properly. Unfortunately, credit cards encourage some people to spend beyond their means.

Monthly statements are issued to consumers listing the purchases for the previous month. Fees, charges and interest are added.

There are two main types of credit cards:n no interest-free period and no annual feen an interest-free period and an annual fee.

A period is only classed as ‘interest-free’ if the account is paid in full before this period ends; otherwise, interest is charged from the date of purchase. Cash advances do not have an interest-free period and are charged interest from the date of the advance. The due date on a statement is the day when the interest-free period ends.

Example 9Manuel has a credit card with no interest-free period and an interest rate of 14% p.a. He makes the following purchases for the period 1 August to 31 August:

2 August Dinner set $65.5016 August Pair trousers $85.0023 August Haircut $24.0026 August Dinner $36.8029 August White shirt $32.00

(a) What is the total amount of his purchases?(b) Manuel pays his account in full on 3 September. How much does he pay?

Solution(a) Total purchases= $65.50 + $85.00 + $24.00 + $36.80 + $32.00

= $243.30(b) Interest is charged on each purchase from the date of purchase until the date payment is

received. For example, the dinner set is bought on 2 August and paid for on 3 September.Number of days= 29 + 3 = 32

Interest rate per annum= 14% = 0.14

Interest rate per day= ≈ 0.000 384

Total interest= $1.6672 … ≈ $1.67Manuel’s total payment= $243.30 + $1.67

= $244.97

Purchase amount No. of days interest Interest to 3 September ($)

$65.50$85.00$24.00$36.80$32.00

32181185

65.50 × 0.000 384 × 32 = 0.8048 …85.00 × 0.000 384 × 18 = 0.5875 …24.00 × 0.000 384 × 11 = 0.1013 …36.80 × 0.000 384 × 8 = 0.1130 …32.00 × 0.000 384 × 5 = 0.0614 …

0.14365----------

Don’t round off interest until the end.



Example 10Consider this statement for a credit card account with an interest-free period of 55 days.

(a) What is Mr Spender’s credit limit?(b) For this period, what was the total amount of:

(i) purchases?(ii) withdrawals (cash advances)?(iii) interest and other charges?

(c) How much was paid by the cardholder and on what date?(d) On what dates does the interest-free period start and end?(e) What interest rate is charged (i) per annum (ii) per day if the account is not paid in full

by the due date?(f) What interest will be charged if this account is paid in full on 6 February. Why?

Your Account Summary

Mr B Spender Balance from previous statement $816.97

25 Fast Lane Payment and other credits $500.00 CR

Decimal Point 2178 Purchases, cash advances $1250.50

Interest and other charges $24.65

Closing balance $1592.12

Minimum payment required $80.00Due date 7 February 2004

Statement period: 14 December 2003 to 13 January 2004Account no: 2483 0222 1223 6517Credit limit: $2000Available credit: $404.88

Your Transaction Record

Date Reference Details Amount15 Dec 2003 3146532 Payment – thank you $500.00 CR

15 Dec 2003 Interest $3.6518 Dec 2003 4367549 Elio’s Restaurant $85.6021 Dec 2003 3020542 ATM withdrawal $100.0021 Dec 2003 3020542 Cash advance fee ATM $1.0031 Dec 2003 5746380 Sydney Opera House $750.002 Jan 2004 Annual fee $20.007 Jan 2004 2345698 Drew’s Menswear $256.5012 Jan 2004 6348921 Shelley’s Scissor Magic $58.40

Annual percentage rate Daily percentage rate15.95% 0.0437%


CREDIT AND LOANS 99

Solution(a) $2000(b) (i) $85.60 + $750.00 + $256.50 + $58.40 = $1150.50

(ii) $100.00 (ATM withdrawal)(iii) $3.65 + $1.00 + $20.00 = $24.65 (or from account summary)

(c) $500 on 15 December 2003(d) Interest-free period starts on 14 December 2003 and ends on 7 February 2004.(e) (i) 15.95% (ii) 0.0437%(f) Unlike the other items, cash advances (ATM withdrawals) do not have an interest-free

period, so interest will be charged for the 47 days from 21 December to 6 February.Interest on cash advance= $100.00 × 0.0437% × 47

= $2.0539≈ $2.05

1. Ian has a credit card with no interest-free period and an interest rate of 14% p.a. He makes the following purchases for the period 1 October to 31 October:

4 October Swim trunks $38.5018 October Fins and mask $120.0019 October Swimming lessons $65.0025 October Wetsuit $135.5028 October Towel $32.00

(a) What is the total amount of his purchases?(b) Ian pays his account in full on 5 November. How much does he pay?

2. Mrs Hoggett has a credit card with no interest-free period and an interest rate of 15% p.a. She makes the following purchases for the period 1 May to 31 May:

2 May Dress $76.5010 May Shoes $85.0013 May Hat $25.0021 May Pig $38.5028 May Hay $126.0028 May Mower $399.00

(a) What is the total amount of her purchases?(b) Mrs Hoggett pays her account on 10 June. How much does she pay in interest for

her May purchases?

Just for the record

FIRST CREDIT CARD

The first international credit card was the Diners Club card in 1950. The American inventor of the Diners Club card, Frank McNamara, came up with the idea when he was entertaining friends at a restaurant and was short of cash. However, it was not until 1974 that Australia had its first credit card: Bankcard.

Exercise 3-06: Credit card payments



3. Andre has a credit card with an interest-free period of 40 days and an interest rate of 18% p.a. He makes the following purchases for the period 1 November to 30 November:

4 November Tennis racquet $256.509 November Shoes $255.00

18 November Clothes $480.5020 November Tennis lessons $138.0025 November Haircut $26.00

(a) What is the total amount of his purchases?(b) Andre pays his account in full on 24 December. How much interest does he pay?

4. Tiger has a credit card with an interest-free period of 55 days and an interest rate of 17% p.a. He makes the following purchases for the period 1 February to 28 February:

2 February Golf club $234.0011 February Golf shoes $165.0015 February Green fees $48.5022 February Golf lessons $84.5026 February Tees $6.00

(a) What is the total amount of his purchases?(b) Tiger does not pay the due amount until 1 May. How much does he pay altogether?

5. Consider the monthly statement for Mr Spender in Example 10 (page 98).(a) Draw up his next monthly account for the period 14 January to 13 February, given

that:n he pays his last account in full on the due daten he makes the following purchases:

18 January Elio Restaurant $95.601 February Big M Groceries $138.50

10 February Robert’s Roses $55.00n he withdraws $100 from an ATM on 28 January (incurring a fee of $1).

(b) How much interest is due on 13 February?(c) The minimum payment (5% of account or $5, whichever is greater) is due 55 days

from the first day of the account period. How much is the minimum payment (to the nearest dollar) and when is it due?

1. Collect monthly statements for as many credit, charge, store and debit cards as you can and use these to determine the advantages and disadvantages of each type. Look at interest rates, terms and conditions as well as fees and charges. Also consider any rewards or benefits to the consumer.

2. Collect newspaper articles on the use and misuse of credit in the community.

Investigation: Plastic credit


CREDIT AND LOANS 101

Chapter review

Credit and loans1. Flat rate loans2. Buying on terms3. Reducing balance loans4. Using published loan repayment tables5. Using technology to compare home loans6. Credit card payments

In the Preliminary Course, you learnt about saving and investing money. This chapter, Credit and loans, examined the financial mathematics of using credit and borrowing money. You should be competent in making interest calculations involving flat rate and reducing balance loans as well as credit cards. You have constructed spreadsheets, used the financial function on a graphics calculator and investigated loans, credit cards and associated fees and charges. You should be able to make an informed decision on different loan and credit options.

Make a summary of this topic. Use the chapter outline above as a guide. An incomplete mind map has also been started below. Use your own words, symbols, diagrams, boxes and reminders. Use the questions in Your say below to think about your understanding of the topic. Gain a ‘whole picture’ view of the topic and identify any weak areas.

n Have you satisfied the outcomes listed at the front of this chapter?n What was the most important thing that you learned?n How did you feel about the topic? Did you enjoy it?n What was new?n What are your weaknesses? What will you need to study more?n How will you revise and summarise this topic?

Topic summary

Credit and loans

Loans• reducing balance• flat rate

Loan repayment tables

Who has the best loan?

Spreadsheets

Credit cards Buying on terms

Your say: Reflecting about the topic ● ● ● ●



1. Jack borrows $3000 to buy a second-hand car on a flat rate loan at 12% p.a. interest over 4 years.(a) How much will he repay altogether?(b) What is his monthly repayment?

2. Huynh bought a motorcycle for $15 000. The flat rate loan was at 15% p.a. interest over 4 years, to be repaid in equal monthly instalments. If there were additional charges of $100 delivery, $35 stamp duty, $250 registration and third-party insurance, and $80 per year loan insurance, find:(a) the total amount borrowed(b) the amount of interest charged(c) the monthly instalment

3. Juliet bought a new bedroom suite from Romeo’s Discounts. The cash price was $999 but Juliet chose to buy on terms and paid 10% deposit and $20 per week for 15 months.(a) How much did she actually pay for the bedroom suite?(b) What was the flat rate of interest charged per annum?

4. Liz and Phil retired and wanted to go on a QE2 cruise. The price of the cruise was $18 000 each. They paid a $6000 deposit each and paid off the remainder in 24 monthly instalments of $1120.(a) How much did they pay altogether for the cruise?(b) How much interest did they pay?(c) What was the flat rate of interest charged per annum?

5. Xi borrowed $10 000 for a new bathroom at 10% p.a. reducible interest. He made monthly payments of $120.(a) Draw up a table showing the progress of the loan in the first 5 months.(b) How much had Xi paid off the principal after his 5th payment?(c) How much interest did he pay in the first 5 months?(d) How much interest did he save in the first 5 months compared with a flat interest

rate of 10% p.a.?

6. Tina has a credit card with an interest-free period of 45 days and an interest rate of 16% p.a. She makes the following purchases for the period 1 November to 30 November:

7 November Clothes $325.509 November Shoes $295.00

17 November Clothes $570.5024 November Singing lessons $146.0025 November Makeup $54.0026 November Beauty treatment $137.00

(a) What is the total amount of her purchases?(b) Tina pays her account in full on 15 December. How much interest does she pay?(c) How much interest would she pay if she settled her account on 31 December?

Chapter assignment


CREDIT AND LOANS 103

7. Monthly loan repayments—25-year term

Use the loan repayment table over a 25-year term to find the following.(a) the monthly repayment for a loan of $90 000 at 7% p.a.(b) the amount borrowed at 8.5% p.a. if the monthly repayment is $725(c) the interest rate per annum for a loan of $80 000 if the monthly repayment is $671(d) the monthly repayment for a loan of $100 000 if the interest rate is 7% p.a.(e) the amount borrowed at 7.5% p.a. interest if the monthly repayment is $665(f) the interest rate per annum for a principal of $70 000 if the monthly repayment

is $564

8. Monthly loan repayments (interest 14% p.a.)

The table above gives the monthly repayments on a reducing balance loan with an interest rate of 14% p.a. Calculate for each loan below:

(i) the monthly repayment(ii) the total amount repaid(iii) the total interest paid(iv) the interest charged if the loan was at a flat interest rate of 14% p.a.

(a) $15 000 over 36 months (b) $20 000 over 18 months(c) $40 000 over 5 years (d) $35 000 over 2 years

Principal ($)

Interest rate (% p.a.)

6.00 6.25 6.50 6.75 7.00 7.50 8.00 8.50 9.00

10 00050 00060 00070 00080 00090 000

100 000

64322387451515580644

66330396462528594660

68338405473540608675

69345415484553622691

71353424495565636707

74370443517591665739

77386463540617695772

81403483564644725805

84420504587671755839

Amount of loan ($)Period of loan (months)

18 24 36 48 60

10 000 619 480 342 273 233

11 00012 00013 00014 00015 000

681743805867929

528576624672720

376410444478513

301328355383410

256279302326349

16 00017 00018 00019 00020 000

9911053111411761238

768816864912960

547581615649684

437465492519547

372396419442465



9. Edita has a gross income of $52 000 p.a. and wants to purchase a studio apartment. The bank will allow her to repay up to 20% of her gross income per annum. The current loan rate is 6.84% p.a. and fees and charges amount to $4000. She wants to borrow the maximum possible and make monthly repayments.(a) How much can she repay:

(i) per annum? (ii) per month?(b) Would the bank lend her $150 000 over 10 years? Justify your answer.(c) If no, what amount will the bank lend Edita over 10 years?

10.

The graph shows the progress of a reducing balance loan of $400 000 with monthly repayments of $3000. From the graph, estimate:(a) the number of monthly repayments to pay off the loan(b) the balance owing on the loan after 10 years(c) the time taken to reduce the balance to $200 000

11. The graph compares the progress of a $100 000 loan when repayments are made monthly and fortnightly.(a) What are two benefits of paying

fortnightly instead of monthly?(b) Estimate the amount owing on the

loan after 10 years if repayments are made:(i) monthly(ii) fortnightly

(c) Estimate the number of years it takes to reduce the balance to $50 000 if repayments are made:(i) monthly(ii) fortnightly

Reducing balance loan—

9624 72 120 14448 168 192 216 240 264

No. of repayments

400 000

350 000

300 000

250 000

200 000

150 000

100 000

50 000

0

Am

ount

ow

ing

($)

monthly repayments

Reducing balance loan

164 12 20 248 28

Loan term (years)

100 000

75 000

50 000

25 000

0

Loan

am

ount

($)

Monthly

Fortnightlyrepayments

repayments


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