credit & borrowing - cambridge.pdf

1C h a p t e r

1Credit and borrowing

Syllabus topic FM4 Credit and borrowingIn this chapter you will learn to:

Calculate the principal, interest and repayments for flat-rate loans

Calculate the values using a table of home loan repayments

Calculate future value and present value

Compare different options for borrowing money

Calculate credit card payments, interest charges and balances

1.1 Flat-rate loansInterest is paid for borrowing money. There are different ways of calculating interest. Flat-rate loans use simple interest. Simple interest (or flat interest) is a fixed percentage of the amount borrowed and is calculated on the original amount. For example, if we borrow $10 000 from a bank at a simple interest rate of 6% per annum (per year) we would be required to pay $600 each year. That is,

= =Interest $10000 0.06 (or6

100) $600

Flat-rate loans are calculated on the initial amount borrowed or the principal. The amount owed on the loan is calculated by adding the interest to the principal.

Flat-rate loans

I = Prn A = P + I

I Interest (simple or flat) to be paid for borrowing the money

P Principal is the initial amount of money borrowed

r Rate of simple interest per period, expressed as a decimal

n Number of time periods

A Amount owed or total to be paid

1.11.21.5

ISBN: 9781107654594 Photocopying is restricted under law and this material must not be transferred to another party

The Powers Family Trust 2013 Cambridge University Press

2 HSC Mathematics General 2

Example 1 Calculating the interest on a flat-rate loan

Abbey applied for a flat-rate loan of $40 000 at 9% per annum simple interest. She plans to repay the loan after two years and six months.a How much interest will be paid?b What is the total owing at the end of two years and six months?

Solution1 Write the simple interest formula.2 Substitute P = 40 000, r = 0.09

and n = 2.5 into the formula.3 Evaluate.4 Write the answer in words.5 Write the amount owed formula.6 Substitute P = 40 000 and

I = 9000 into the formula.7 Evaluate.8 Write the answer in words.

a I = Prn = 40 000 0.09 2.5 = $9000

Simple interest owed is $9000.b A = P + I = 40 000 + 9000 = $49 000

Amount owed is $49 000.

Example 2 Finding the principal for a flat-rate loan

Noah applied for a flat-rate car loan with an interest rate of 9% p.a. He was told the total simple interest would be

$6300 for 31

2 years. What was the principal?

Solution1 Write the simple interest formula.2 Substitute I = 6300, r = 0.09 and

n = 3.5 into the formula.3 Make P the subject of the formula by

dividing both sides by (0.09 3.5).4 Evaluate.5 Write the answer in words.

I = Prn 6300 = P 0.09 3.5

P =

6300

0 09 3 5( . . ) = $20 000

Principal is $20 000.



3Chapter 1 Credit and borrowing

Example 3 Using a graphics calculator for a flat-rate loan

Mary and Lucas plan to borrow $300 000 at

81

2% p.a. simple interest for 3 years. Answer

the following questions by using a graphics calculator.a How much simple interest will they pay over the

3 years?b What is the total amount owed after 3 years?

Solution1 Select the TVM (Time, Value, Money) menu.2 Select Simple Interest (F1).

3 Enter the time period n = 3 365 = 1095 (simple interest period is calculated in days).

4 Enter the interest rate I% = 8.5.5 Enter the principal or present value

PV = 300 000. In the TVM mode, all money we pay out is negative and money we receive is positive. In this example $300 000 is received.

6 To calculate the simple interest, select SI.7 Write the answer in words.

a

They will pay $76 500.8 To calculate the total amount owed, select

SFV (Simple Final Value).

9 Write the answer in words.

b

Total owed $376 500.




Loan repaymentsA loan repayment is the amount of money to be paid at regular intervals over the time period. The interval is often fortnightly or monthly.

Loan repayments

Loan repayment = Total to be paid Number of repayments

Example 4 Calculating a loan repayment

Jessica wishes to buy a lounge suite priced at $2750. She chooses to buy it on terms by paying a 10% deposit and borrowing the balance. Interest is charged at 11.5% p.a. on the amount borrowed. Jessica makes fortnightly repayments over 3 years. Calculate her fortnightly repayments.

Solution 1 Calculate the deposit by multiplying

10% or 0.10 by $2750. 2 Calculate the balance by subtracting

the deposit ($275) from the cost price ($2750).

3 Write the simple interest formula. 4 Substitute P = 2475, r = 0.115 and

n = 3 into the formula. 5 Evaluate. 6 Write the loan repayment formula. 7 Calculate the total to be paid by

adding the balance ($2475) and the interest ($853.875).

8 Calculate the number of repayments by multiplying the fortnights in a year (26) by the number of years (3).

9 Evaluate correct to two decimal places.


Deposit = 10% of $2750 = 0.10 2750 = $275

Balance = 2750 275 = $2475

I = Prn = 2475 0.115 3 = $853.875

RepaymentTotal to be paid

Number of repayme=

nnts

Pr ncipal Interest

Number of repayments=

+i

=

=

+

=

=

(2475 853.875)

(342.67788462$42.68

26)

Fortnightly repayments are $42.68.




Exercise 1A 1 Calculate the amount of simple interest for each of the following loans:

a Principal = $25 000, Interest rate = 11% p.a., Time period = 4 years.

b Principal = $400 000, Interest rate = 81

4% p.a., Time period = 5 years.

c Principal = $560 000, Interest rate = 6.75% p.a., Time period = 15 years.d Principal = $7400, Interest rate = 7% p.a., Time period = 18 months.e Principal = $80 000, Interest rate = 9.25% p.a., Time period = 30 months.

2 Calculate the amount owed for each of the following loans:a Principal = $800, Simple interest rate = 6% p.a., Time period = 3 years.

b Principal = $5200, Simple interest rate = 16% p.a., Time period = 71

2 years.

c Principal = $12 500, Simple interest rate = 11.4% p.a., Time period = 4.5 years.

d Principal = $6000, Simple interest rate = 41

2% p.a., Time period = 6 months.

e Principal = $40 000, Simple interest rate = 7.75% p.a., Time period = 42 months.

3 A sum of $170 000 was borrowed for 3 years.a Find the simple interest owed if the rate of interest is 6.5% per annum.b What is the amount owed at the end of 3 years?

4 Hayley intends to borrow $2700 to build a driveway for her new house. She is offered a flat-rate loan with a simple interest rate of 14.5% per annum. How much interest will be owed after 3 months? Answer correct to the nearest cent.

5 Ethan borrowed $1800 at 6% per annum. What is the simple interest owed between 30 June and 1 September?

6 Ruby borrows $36 000 for 31

2 years. What is the rate of simple interest if she will owe

$8820 in interest?

7 Chloe has paid $49 500 interest on a $220 000 loan at a flat interest rate of 10%. What was the term of the loan?




8 Create the spreadsheet below.

a Cell D5 has a formula that calculates the simple interest. Enter this formula.b Fill down the contents of D6 to D8 using the formula for D5.c Cell E5 has a formula that calculates the amount owed. Enter this formula.d Fill down the contents of E6 to E8 using the formula for E5.e Change the interest rate from 8% to 10%.f Change the time period from 20 years to 15 years.

9 Bailey buys a television for $1800. He pays it off monthly over 2 years at a flat interest rate of 12.5% per annum.a How many months will it take Bailey to pay for the television?b What is the interest charged for the 2 years?c How much per month will he pay? Give your answer to the nearest cent.

10 Mitchell approached a bank for a business loan of $22 000. The interest rate is 10.5% p.a. flat. He decides to repay the loan over a period of 4 years.a What is the principal?b What is the rate of interest?c What will be the amount of

interest charged over that period?d What will be the monthly

repayment? Give the answer correct to the nearest cent.

11 Jordan decides to buy a car for $23 000. He has saved $9000 for the deposit and takes out a flat-rate loan over 2 years for the balance. The interest charged is 13% per annum.a What is the balance?b What is the total amount of interest to be paid?c What will be his monthly repayment? Answer correct to the nearest cent.

1A




Development

12 Mia borrowed $400 000 at a flat rate of interest of 8.5% per annum. This rate was fixed for 2 years on the principal. She pays back the interest only over this period.a How much interest is to be paid over the 2 years?b After paying the fixed rate of interest for the first year, Mia finds the bank will

decrease the flat interest rate to 7.5% if she pays a charge of $2000. How much will she save by changing to the lower interest rate for the last year?

13 Cooper plans to borrow money to purchase a car and considers the following fortnightly repayment guide. He decides to borrow $19 000 and pay back the loan in fortnightly instalments over 2 years. What is the flat rate of interest per annum on this loan, correct to two decimal places?

14 A truck is advertised at $36 000. It can be bought on terms for a 20% deposit and repayments of $276 per week for 3 years. Assume there are 52 weeks in the year.a What is the deposit?b Calculate the total cost of the truck if bought on these terms.c What is the total interest paid?d What is the flat interest rate for the loan, correct to one decimal place?

15 Determine the flat rate of interest charged on a painting that has a cash price of $7500. The painting was purchased on terms with a 20% deposit and the balance to be paid at $370 per month for 2 years.

16 Grace takes a loan of $30 000 over 60 months for a swimming pool. The repayment rate is $677.50 per month.a How much will Grace pay back

altogether?b What is the flat interest rate per annum for the loan, correct to one decimal place?c Grace would like to increase the loan to $40 000 to landscape the pool. What would

be her monthly repayment assuming the same time period and flat interest rate? Answer correct to the nearest cent.

Amount borrowed

Fortnightly repayments

1 year 2 years 3 years

$18 000 $755 $427 $305

$18 500 $783 $429 $307

$19 000 $804 $431 $309




1.2 Table of loan repaymentsA home loan or mortgage is a loan given to buy a house or a unit. The interest on a home loan is often calculated per month on the amount of money owing and repayments are made monthly. The amount owing after each month becomes the new principal for the next month. Each calculation results in a smaller amount of interest and is called reducible interest. These calculations are often displayed in a table.

Table of loan repayments

Amount owed and the interest paid reduce after each loan repayment.

Example 5 Calculating the values in a table of loan repayments

Riley has taken out a home loan of $400 000. The flat rate of interest is 9% p.a. and the monthly repayment (R) is $3120. Complete the table below for one month to answer these questions.a What interest is owed after one month?b Determine the value of P + I.c Determine the value of P + I R.

Months (n) Principal (P ) Interest (I ) P + I P + I + R

1 $400 000.00 a b c

Solution

1 Write the simple interest formula.2 Substitute P = 400 000, r = 0.09

and n =1

12 into the formula and

evaluate.3 Write the answer in the table.

a =

=

=

I Prn

400000 0.091

12$3000

Interest owed is $3000.

4 Add the principal ($400 000) and the interest ($3000).

5 Write the answer in the table.

b P + I = 400 000 + 3000= $403 000

6 Subtract the monthly repayment ($3120) from the amount owing (P + I or $403 000).

7 Write the answer in the table.

c P + I R = 400 000 + 3000 3120= $399 880

Months (n) Principal (P ) Interest (I ) P + I P + I R

1 $400 000.00 $3000 $403 000 $399 880

1.5




Example 6 Calculating the values in a table of loan repayments

What are the missing values in the table of home loan repayments shown below?

Amount borrowed $150 000 This table assumes the same number of days in each

month. I = Prn or I P

r==

12

Annual interest rate (r) 7%

Monthly repayment (R) $1200

Month (n) Principal (P) Interest (I) P + I P + I R

1 $150 000.00

2

Solution1 Write the simple interest formula.2 Substitute P = 150 000, r = 0.07 and

n = 1

12 into the formula.

3 Evaluate.4 Add the principal ($150 000) and the

interest ($875).5 Subtract the monthly repayment

($1200) from the amount owing (P + I or $150 875).

6 The answer for P + I R is the principal for the next month ($149 675). It is the amount owing after one month. Write it in the table for the second month.

7 Repeat the above steps for the second row to determine the amount owing after 2 months.

8 Notice the amount of interest in the second month ($873.10) is less than the amount of interest in the first month ($875).

First monthI Prn=

=

=

150 000 0 071

12875

.

$

P + I = 150 000 + 875= $150 875

P + I R = 150 875 1200= $149 675

New principal is $149 675.

Second monthI = Prn

= 149 675 0.07 1

12= $873.10

P + I = 149 675 + 873.10= $150 548.10

P + I R = 150 548.10 1200= $149 348.10


1 $150 000.00 $875.00 $150 875.00 $149 675.00

2 $149 675.00 $873.10 $150 548.10 $149 348.10




Exercise 1B 1 Kayla borrows $170 000 for a home at an interest rate of 6% p.a. with a monthly

repayment of $1000.

Months (n) Principal (P) Interest (I ) P + I P + I R

1 $170 000.00 a b c

2 c d e f

3 f g h i

Answer correct to the nearest cent. Use this approximation in subsequent questions.a Determine the interest, I, charged for the first month.b Determine the value of P + I for the first month.c Determine the value of P + I R for the first month.d Determine the interest, I, charged for the second month.e Determine the value of P + I for the second month.f Determine the value of P + I R for the second month.g Determine the interest, I, charged for the third month.h Determine the value of P + I for the third month.i Determine the value of P + I R for the third month.

2 Chris borrowed $250 000 at 7.2% p.a. for a unit. The interest is charged monthly and the monthly repayment is $1650. Complete the following table.


1 $250 000.00 $1500.00 $251 500.00 $249 850.00

2 $249 850.00 $1499.10 $251 349.10 $249 699.10

3

4

5

Answer correct to the nearest cent. Use this approximation in subsequent questions.a What is the principal at the beginning of the third month?b Calculate the interest charged for the third month.c How much is owed at the end of the third month?d What is the principal at the beginning of the fourth month?e Calculate the interest charged for the fourth month.f How much is owed at the end of the fourth month?g What is the principal at the beginning of the fifth month?h Calculate the interest charged for the fifth month.i How much is owed at the end of the fifth month?




Development

3 Complete the table of home loan repayments shown below. Use your calculator answer to complete each cell of the table, not an approximation. Answer correct to the nearest cent.

Amount borrowed $300 000This table assumes the same

number of days in each month. I = Prn or I P

r==

12




1 $300 000.00 $1750.00 $301 750.00 $299 750.00

2 $299 750.00 $1748.54 $301 498.54 $299 498.54

3

4

5

6

7

8


Amount borrowed $520 000This table assumes there

are 26 fortnights in a year.

I = Prn or I Pr

= 26


Fortnightly repayment (R) $1800

Fortnight (n) Principal (P) Interest (I) P + I P + I R

1 $520 000.00 $1600.00 $521 600.00 $519 800.00

2 $519 800.00 $1599.38 $521 399.38 $519 599.38

3

4

5

6

7

8




5 Create the spreadsheet below.

a Cell B9 has a formula that refers to cell B4 or the amount. Enter this formula.b Cell C9 has a formula that calculates the simple interest. Enter this formula.c Cell D9 has a formula that calculates the amount owing. Enter this formula.d Cell E9 has a formula that calculates the amount owing after a repayment has been

made. Enter this formula.e Fill down the contents of B9:E9 to B13:E13.

6 Dylan borrowed $240 000 for an investment property. The interest rate is 10% p.a. and he makes monthly repayments of $2300. Construct a table of home loan repayments for the first two months to answer the following questions.a How much interest was paid in the first month?b What is the balance owing after one month?c How much has the principal been reduced during the first month?d How much interest was paid in the first two months?e What is the balance owing after two months?f How much has the principal been reduced during the first two months?

7 Charlotte borrowed $480 000 for an inner city apartment. The interest rate is 8% p.a. and she makes fortnightly repayments of $1600. Construct a table of home loan repayments for the first three fortnights.a What is the balance owing after the first fortnight?b How much interest was paid in the first fortnight?c How much has the principal been reduced during the first fortnight?d What is the balance owing after three fortnights?e How much interest was paid in three fortnights?f How much has the principal been reduced during the three fortnights?

1B




1.3 Future value formulaCompound interest is calculated on the initial amount borrowed or invested plus any interest that has been charged or earned. It calculates interest on the interest. In the preliminary course we used the formula A = P(1 + r)n to calculate the compound interest. In the financial world, the compound interest formula is known as the future value formula and is expressed as FV = PV(1 + r)n. The amount (A) is the future value (FV) and the principal (P) is the present value (PV).

Future value formula

FV = PV(1 + r)n or =+

PVFV

r(1 )n or I = FV PV

FV Future value of the loan or amount (final balance)

PV Present value of the loan or principal (initial quantity of money)

r Rate of interest per compounding time period expressed as a decimal

n Number of compounding time periods

I Interest (compounded) earned

Example 7 Calculating future value and present value

a Blake invests $7000 over 5 years at a compound interest rate of 4.5% p.a. Calculate the future value after 5 years. Answer correct to the nearest cent.

b Calculate the present value of an annuity that has a future value of $500 000 over 8 years with an interest rate of 8.5% per annum compounded monthly.

Solution

1 Write the future value formula. a FV = PV(1 + r)n

2 Substitute PV = 7000, r = 0.045 and n = 5 into the formula.

= 7000(1 + 0.045)5

= $8723.27

3 Evaluate to the nearest cent. Future value is $8723.27.


5 Write the future value formula. b

6 The investment is compounding per month hence the rate (r) and time period (n) are expressed in months.

7 Substitute FV = 500 000, r =0 085

12

. and n = 8 12 = 96.

8 Evaluate to the nearest cent.

9 Write the answer in words. Present value is $253 916.41.

+

=

+

=

=

FV

r(1 )

500000

(10.085

12)

$253916.41

PV n

96




Exercise 1C 1 Calculate the future value, to the nearest cent, for each of the following:

a Present value = $400, Compound interest rate = 3% p.a., Time period = 2 years.

b Present value = $3000, Compound interest rate = 51


c Present value = $18 000, Compound interest rate = 10% p.a., Time period = 21

2

years.

d Present value = $65 000, Compound interest rate = 5.9% p.a., Time period = 31

4

years.

2 Use the formula FV = PV(1 + r)n to calculate the value of an investment of $16 000, over a period of 2 years with an interest rate of 5% compounding annually.

3 Sophia and Isaac invested $27 000 for 6 years at 9% p.a. interest compounding annually. What is the amount of interest earned in the first year?

4 Calculate the present value, to the nearest cent, for each of the following:a Future value = $34 000, Interest rate = 4% p.a., Time period = 4 years.

b Future value = $200 000, Interest rate = 121


c Future value = $4600, Interest rate = 15% p.a., Time period = 21

2 years.

d Future value = $60 000, Interest rate = 6.25% p.a., Time period = 11

4 years.

5 Calculate the present value of an investment that has a future value of $5000 after 4 years and earns 9% p.a. compound interest, paid annually.




Development

6 Calculate the future value, to the nearest cent, for each of the following:a Present value of $680 invested for 4 years at 5% p.a. compounded biannually.b Present value of $5000 invested for 6 years at 6% p.a. compounded quarterly.c Present value of $1400 invested for 3 years at 4.2% p.a. compounded monthly.d Present value of $780 invested for 5 years at 9.8% p.a. compounded weekly.e Present value of $290 invested for 7 years at 10% p.a. compounded fortnightly.

7 Calculate the present value, to the nearest dollar, for each of the following:a Future value of $1243, interest rate at 6% p.a. compounded biannually for 5 years.b Future value of $8200, interest rate at 4% p.a. compounded quarterly for 8 years.c Future value of $1580, interest rate at 4.8% p.a. compounded monthly for 4 years.d Future value of $19 600, interest rate at 8% p.a. compounded weekly for 3 years.e Future value of $3800, interest rate at 5% p.a. compounded fortnightly for 7 years.

8 Find the future value of a bank account after 3 years if the present value of $4000 earns 4.6% p.a. interest compounding quarterly.

9 Alexander invested $16 400 over 6 years at 7.4% p.a. interest compounding monthly.a Calculate the value of the investment after 4 years.b Calculate the compound interest earned.

10 What sum of money would Max need to invest to accumulate a total of $100 000 at the end of 7 years at 8% p.a. interest compounding biannually? Answer to the nearest cent.

11 What sum of money needs to be invested to accumulate a total of $40 000 after 10 years at 9.25% p.a. interest compounding monthly? Answer to the nearest cent.

12 How much more interest is earned on $60 000 if interest of 8% p.a. is compounded quarterly over 6 years than if simple interest of 8% is earned over the same time?

13 Mikayla invests $200 000 for 10 years at 6% p.a. interest compounded quarterly. Abby also invests $200 000 for 10 years, but her interest rate is 6% p.a. compounded monthly.a Calculate the value of Mikaylas investment at maturity.b Show that the compounded value of Abbys investment is greater than the value of

Mikaylas investment.c Explain why Abbys investment is worth more than Mikaylas investment.




1.4 Comparing loansComparing loans and making the best choice is not simply about choosing a loan with the lowest interest rate. Borrowers also need to consider the following factors: Flexibility ability to redraw money and make extra repayments. This allows the loan to

meet changing needs without incurring extra costs, for example, if you get a higher paying job and want to increase the amount of your repayments.

Comparison rate interest rate on the loan that includes the interest and any fees or charges. It takes into account the amount of the loan, the term of the loan and the number of repayments. Comparison rates calculators are available on the internet to compare loans. The calculator shown below (from www.nmb.com.au) compares different loan amounts, standard and introductory interest rates, terms and fees.

Input Loan Details

Loan Amount :

Terms (years) :

Loan 1

$150.000 $150.000

25 25

Introductory Term(months) :

0 0

Introductory InterestRate :

0.00 0.00

Standard Interest Rate : 6.00 6.00

Establishment Fees : $0 $0

Monthly Fees : $10 $0

Annual Fees : $0 $0

Discharge Fees :

Calculate Clear InputsScroll down to view results.

$0 $0

Loan 2

Comparison Rate ResultsTotal Term of Loan(months) :Introductory Payment(monthly) :

Loan 1

300 300

$0.00 $0.00

- no. of introductorypayments :

0 0

Standard Payment(monthly) :

$966.45 $966.45

- no. of standardpayments :

300 300

Comparison Rate : 6.11% 6.00%

Loan 2

The above table shows the comparison of two loans that are identical except for a $10 monthly fee. The comparison rates are 6% and 6.11%.




Effective interest rateThe effective interest rate is another method of comparing interest rates with different time periods. It is an equivalent interest rate if compounding happened annually. The effective interest rate is a single rate that takes into account different rates and time periods.

Effective interest rate

E = (1+ r)n - 1

E Effective rate of interest per annum as a decimal

r Rate of interest per compounding period expressed as a decimal

n Number of compounding time periods per annum

Example 8 Calculating the effective interest rate

Calculate the effective annual interest rate of a home loan with an interest rate of 7.25% p.a. compounded monthly. Give your answer as a percentage correct to two decimal places.

Solution1 Write the effective interest formula.2 Substitute r = (0.0725 12) and n = 12 into

the formula.3 Evaluate.4 Express the answer as a percentage correct

to two decimal places.5 Write the answer in words.

E r(1 ) 1

10.0725

121

0.074958297427.50%

n

12

= +

= + =

=

Effective interest rate is 7.50%.

Consider a bank loan with an annual interest rate of 12% p.a. The table below shows the effective annual interest rate if the compounding period is annual, biannual, quarterly or monthly. When the interest is compounding monthly, quarterly or biannually, the amount of interest paid is more than if the interest is compounding annually.

Compounding period Interest rate Time periods Effective interest rate

Annual 12% 1 12.00%

Biannual 6% 2 12.36%

Quarterly 3% 4 12.55%

Monthly 1% 12 12.68%

1.31.4




Exercise 1D 1 The table below compares loans offered by the same bank.

a Which loan type has the lowest interest rate?b Which loan type has the highest application fee?c What is the service fee for the basic loan?d What is the interest rate for a variable loan?e What is the legal fee for the intro loan?f What is the difference in the interest rates between the variable and basic loans?

2 The table below compares the features offered by different loans.

a Which loan has the lower comparison rate?b Which loan has the greater wealth package saving?c What is the interest rate for the fixed loan?d What is the interest rate for the standard variable loan?e What is the monthly loan service fee for the fixed loan?f What is the upfront establishment fee for both loans?g What is the difference between the interest rate and the comparison rate for the

standard variable loan?h What is the difference between the interest rate and the comparison rate for

the fixed loan?




Development

3 Use the formula E = (1 + r)n - 1 to calculate the effective annual interest rate. Give your answer as a percentage correct to two decimal places.a Interest rate of 6% p.a. compounding biannually.b Interest rate of 7% p.a. compounding biannually.c Interest rate of 8% p.a. compounding quarterly.d Interest rate of 6.4% p.a. compounding quarterly.e Interest rate of 10% p.a. compounding monthly.f Interest rate of 14% p.a. compounding monthly.g Interest rate of 7.6% p.a. compounding half-yearly.h Interest rate of 12.36% p.a. compounding half-yearly.

4 Use the formula E = (1 + r)n - 1 to calculate the effective annual interest rate. Give your answer as a percentage correct to two decimal places.a Interest rate of 7.2% p.a. compounding fortnightly.

b Interest rate of 71

2% p.a. compounding fortnightly.

c Interest rate of 4.8% p.a. compounding weekly.d Interest rate of 9.6% p.a. compounding weekly.

e Interest rate of 141

2% p.a. compounding daily.

f Interest rate of 10.8% p.a. compounding daily.

5 A finance company advertises home loans with an interest rate of 9% p.a. compounded monthly. What is the effective interest rate over a 12-month period? Answer correct to two decimal places.

6 Lily and Jacob would like to borrow $200 000 for a home to be repaid in equal monthly instalments of $1800 over 30 years.a How much is paid on the loan for one

year?b Determine the total amount to be repaid

on the loan.c Calculate the total interest payment.d What is the equivalent annual flat rate of interest? (Answer correct to one decimal

place.)e What is the effective interest rate if the annual interest is compounded monthly?

Answer correct to one decimal place.




1.5 Credit cardsCredit cards are used to buy goods and services and pay for them later. The time when interest is not charged on your purchases is called the interest-free period. If payment is not received when the statement is due then interest is charged from the date of purchase. Interest on credit cards is usually calculated daily on the outstanding balance using compound interest.

Credit cards

=Daily interest rateAnnual interest rate

365

A = P(1 + r)n I = A P

A Amount owing on the credit card

P Principal is the purchases made on the credit card plus the outstanding balance

r Rate of interest per compounding time period expressed as a decimal

n Number of compounding time periods

I Interest (compound) charged for the use of their credit card

Example 9 Calculating the cost of using a credit card

Samantha has a credit card with a compound interest rate of 18% p.a. and no interest-free period. Samantha used her credit card to pay for clothing costing $280. She paid the credit card account 14 days later. What is the total amount she paid for the clothing, including the interest charged?

Solution1 Write the formula for compound interest.2 Substitute P = 280, r = (0.18 365) and

n = 14 into the formula.3 Evaluate.4 Express the answer correct to two decimal

places.5 Answer the question in words.

A P r n= +

= +

=

=

( )

.

.

1

280 10 18

365

281 9393596

14

$$ .281 94

Clothing costs $281.94




Credit card statementsCredit card statements are issued each month and contain information such as account number, opening balance, new charges, payments, refunds, reward points, payment due data, minimum payment and closing balance.

Example 10 Reading a credit card statement

CapitalBank

MR JoHN CITIzEN

123 SAMPLE STREET

SUBURBIA NSW 2000

Mastercard 0000 1801 0002 1010

opening balance $207.72 Payment due date

30 November

New charges $460.14 Minimum payment

$25.00

Payments/refunds $207.72 Closing balance

$460.14

Capital Awards 0000123456

opening points balance 50,500 Total points balance

Total points earned 460 34,910

Points redeemed 15,600

Answer the following questions using the above credit card statement.a What is the credit card account number?b What is the opening balance?c What is the payment due date?d What is the minimum payment?e What is the closing balance?

Solution

1 Read number after MasterCard. a 0000 1801 0002 10102 Read opening balance. b opening balance is $207.72.3 Read the box Payment due date. c Payment due date is 30 November.4 Read the box Minimum payment. d Minimum payment is $25.00.5 Read the box Closing balance. e Closing balance is $460.14.




Exercise 1E 1 Use the credit card statement opposite to

answer these questions.a What is the due date?b What is the cost of the purchases?c What is the closing account balance?d What is the minimum amount due?e What payment was made last month?f How much interest was charged?g What was the opening balance?h What is the cardholders credit

balance?

2 A credit card has a daily interest rate of 0.05% per day. Find the interest charged on these outstanding balances. Answer correct to the nearest cent.a $840 for 12 daysb $742.40 for 20 daysc $5680 for 30 daysd $128 for 18 dayse $240 for 6 daysf $1450 for 15 days

3 Calculate the amount owed, to the nearest cent, for each of the following credit card transactions. The credit card has no interest-free period.a Transactions = $540, Compound interest rate = 14% p.a., Time period = 15 days.b Transactions = $270, Compound interest rate = 11% p.a., Time period = 9 days.c Transactions = $1400, Compound interest rate = 18% p.a., Time period = 22 days.d Transactions = $480, Compound interest rate = 16% p.a., Time period = 18 days.e Transactions = $680, Compound interest rate = 10% p.a., Time period = 9 days.

4 Calculate the interest charged for each of the following credit card transactions. The credit card has no interest-free period. Answer correct to the nearest cent.a Transactions = $680, Compound interest rate = 15% p.a., Time period = 20 days.b Transactions = $740, Compound interest rate = 12% p.a., Time period = 13 days.c Transactions = $1960, Compound interest rate = 17% p.a., Time period = 30 days.d Transactions = $820, Compound interest rate = 21% p.a., Time period = 35 days.e Transactions = $1700, Compound interest rate =19% p.a., Time period = 32 days.

Account Summary

Payment Summary

Opening Balance

Closing Account Balance

Payments and Other Credits

Purchases

Cash Advances

Interest and Other Charges

Cardholder Credit Balances

Card Balances Renewal

Monthly Payment

Due Date

Minimum Amount Due

$743.42

$172.91

$743.42

$172.91

$0.00

$0.00

4,511.88

$4,684.79

$10.00

21/04/2012

$10.00




5 Luke has a credit card with a compound interest rate of 19.99% per annum.a What is the daily percentage interest rate, correct to two decimal places?b Luke has an outstanding balance of $4890 for a period of 30 days. How much

interest, to the nearest cent, will he be charged?

6 Andrews credit card charges 0.054% compound interest per day on any outstanding balances. How much interest is Andrew charged on an amount of $450, which is outstanding on his credit card for 35 days? Answer correct to the nearest cent.

7 Joel has a credit card with an interest rate of 0.04% compounding per day and no interest-free period. He uses his credit card to pay for a mobile phone costing $980. Calculate the total amount paid for the mobile phone if Joel paid the credit card account in the following time period. Answer correct to the nearest cent.a 10 days later b 20 days laterc 30 days later d 40 days latere 50 days later f 60 days later

8 Olivia received a new credit card with no interest-free period and a daily compound interest rate of 0.05%. She used her credit card to purchase food for $320 and petrol for $50 on 18 July. This amount stayed on the credit card for 24 days. What is the total interest charged? Answer correct to the nearest cent.

9 Alyssa uses a credit card with a no interest-free period and a compound interest rate of 15.5% p.a. from the purchase date. During April she makes the following transactions.

Transaction details

04 April IGA Supermarket $85.00

09 April KMart $115.00

12 April David Jones $340.00

27 April General Pants $80.00

28 April JB-HIFI $30.00

a What is the daily compound interest rate, correct to three decimal places?b Alyssas account is due on 30 April. What is the total amount due?c How much interest has Alyssa paid on the IGA transaction during the month?

Answer correct to the nearest cent.d How much interest has Alyssa paid on the KMart transaction during the month?

Answer correct to the nearest cent.




Development

10 Charlies credit card has up to 55 days interest free and is due on the 22nd of each month. The interest rate is 19.4% p.a. compounding daily. Charlie buys furniture costing $5160 on 25 October. How much interest is he charged if he pays the balance on 22 December?Interest is charged from the date of purchase if the total amount owing has not been paid by the due date.

11 Harry buys two Blu-ray DVDs for $29.90 each and a shirt for $84.95 on his credit card. This amount stays on his credit card for 75 days. There is a 45-day interest-free period and a daily interest of 0.05% compound on his credit card.a How much did Harry spend on his credit card?b Calculate the amount Harry owes on the credit

card.c What was the interest charged on these

purchases?d What would be the interest charged if Harrys

credit card did not have a 45-day interest-free period?

12 Sarah and Joshua each use their credit cards to buy holiday packages to Adelaide. The cost of the package is $1700 for each person.a The charge on Sarahs credit card is 0.9% compound interest per month on the

unpaid balance. It has no interest-free period. Sarah pays $800 after one month and another $500 the next month. How much does she still owe on her credit card?

b The charge on Joshuas credit card is interest-free in the first month, and 1.4% compound interest per month on any unpaid balance. Joshua pays $800 after one month and another $500 the next month. How much does he still owe on his credit card?

13 Emilys August credit card statement shows an opening balance of $1850, a purchase of $2450 on 5 August, and another purchase of $55 on 14 August. The minimum payment is 3% of the closing balance. The initial credit charge is 1.6% compounding per month of any amount outstanding.a What is the closing balance on this credit card for August?b Calculate the amount of interest charged for the month of August.c What is the minimum payment, to the nearest cent, required for August?d What is the opening balance for October if Emily paid the minimum payment in

September for interest charged in August and made no purchases in September?1.5



Review25Chapter 1 Credit and borrowing

Chapter summary Credit and borrowing Study guide 1

Flat-rate loans I = Prn A = P + I I Interest (simple or flat) earned for the use of moneyP Principal is the initial amount of money borrowed r Rate of simple interest per period expressed as a decimal n Number of time periods A Amount of final balance

Table of loan repayments The interest on a home loan is often calculated per month on the amount of money owing, and repayments are made per month. The amount owing after each month becomes the new principal for the next month. Each calculation results in a smaller amount of interest and is called reducible interest. These calculations are often displayed in a table.

Future value formulaFV = PV(1 + r)n or =

+PV

FV

r(1 )n or I = FV PV

FV Future value of the loan or amount (final balance)PV Present value of the loan or principal (initial quantity of

money)r Rate of interest per compounding time period as a decimaln Number of compounding time periodsI Interest (compounded) earned

Comparing loans Comparing loans and making the best choice is not simply about choosing a loan with the lowest interest rate. Borrowers need to consider flexibility, fees and the comparison rate.

Credit cards =Daily interest rateAnnual interest rate

365

A = P(1 + r)n I = A P A Amount owing on the credit cardP Principal is the purchases plus the outstanding balance r Rate of interest per compounding time period as a decimal n Number of compounding time periods I Interest charged for the use of their credit card



Revi

ewHSC Mathematics General 226

1 What is the interest earned on $1400 at 7% p.a. simple interest for 3 years?A $98 B $294 C $498 D $1694

2 David wants to earn $9000 a year in interest. How much must he invest if the simple interest rate is 15% p.a.? Answer to the nearest dollar.A $1350 B $10 350 C $60 000 D $600 000

Use the table below to answer questions 3 and 4. The table uses an interest rate of 11% p.a. with a monthly repayment of $1250.


1 $120 000 $1100

3 What is the value of P + I ?A $118 900 B $120 011 C $121 100 D $121 250

4 What is the value of P + I R?A $119 850 B $119 900 C $120 000 D $122 200

5 Holly invests $8000 at 10% p.a. interest compounding annually. What is the future value after 3 years? Answer to the nearest dollar.A $242 B $2648 C $8242 D $10 648

6 Nathan borrows $3000 at 10% p.a. interest compounding annually. What is the amount owed after 2 years? Answer to the nearest dollar.A $3030 B $3060 C $3600 D $3630

7 What is the future value after 3 years of $6000 invested at 7% p.a. interest compounding monthly? Answer to the nearest dollar.A $1350 B $1397 C $7350 D $7398

8 Calculate the present value of an amount invested for 4 years at an interest rate of 4.5% p.a. compounded quarterly, if it has a future value of $20 000?A $16 722 B $16 771 C $19 125 D $23 920

9 A credit card has a daily interest rate of 0.05% per day and no interest-free period. Find the interest charged on $1530 for 14 days. Answer correct to the nearest cent.A $0.77 B $10.74C $76.50 D $1540.71

Sample HSC Objective-response questions



Review27Chapter 1 Credit and borrowing

Sample HSC Short-answer questions

1 William takes out a flat-rate loan of $60 000 for a period of 5 years, at a simple interest rate of 12% per annum. Find the amount owing at the end of 5 years.

2 Amelia would like to purchase a $2000 television from electronics shop. However, to buy the television she has applied for a flat-rate loan over 2 years at 15% p.a. How much does Amelia pay altogether for the TV?

3 Paige takes out a loan of $21 000 over 36 months. The repayment rate is $753.42 per month.a How much will Paige pay back

altogether?b What is the equivalent flat interest rate per annum for the loan, correct to one decimal

place?

4 James borrows $280 000 and repays the loan in equal fortnightly repayments of $1250 over 20 years. What is the flat rate of interest per annum on Jamess loan, correct to two decimal places?


Amount borrowed $450 000This table assumes the same

number of days in each month.

I = Prn or = I Pr

12

Annual interest rate (r) 6.25%



1 $450 000.00 $2343.75

2

3

4



Revi

ewHSC Mathematics General 228

6 Julia has been given a home loan of $400 000 at 8% p.a. compounded monthly. The loan is to be repaid in 300 equal monthly instalments of $3087.26.a Determine the amount to be repaid on this loan.b How much interest is paid on this loan?c Using the formula E = (1 + r)n - 1, find the effective interest rate of the loan per annum.

Give your answer as a percentage correct to two decimal places.

7 Calculate the future value, to the nearest cent, for each of the following:a Present value = $920, invested for 4 years at 5% p.a. compounded monthly.b Present value of $2100, invested for 3 years at 6.1% p.a. compounded monthly.

8 Calculate the present value, to the nearest cent, for each of the following:a Future value = $26 000, Interest rate = 4.9% p.a., Time period = 3 years.b Future value of $10 400, Interest rate at 9% p.a. compounded quarterly for 5 years.

9 What sum of money would Emma need to invest to accumulate a total of $200 000 at the end of 10 years at 12% p.a. interest compounding biannually? Answer to the nearest cent.

10 Madison has a credit card with an interest rate of 17% p.a. compounding daily and no interest-free period. Madison used her credit card to pay for shoes costing $170. She paid the credit card account 26 days later. What is the total amount she paid for the shoes, including the interest charged?

11 Benjamin uses a credit card that has no interest-free period and a compound interest rate of 18.5% p.a. from the purchase date. During February he makes the following transactions.

Transaction details

06 February Coles $278.00

07 February Myer $87.00

18 February Big W $259.00

18 February Jag $120.00

20 February Bunnings $460.00

a What is the daily compound interest rate, correct to three decimal places?b Benjamins account is due on 28 February. What is the interest charged for the

transaction at Bunnings? Answer correct to the nearest cent.

Challenge questions 1



credit & borrowing - cambridge.pdf

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