3 Interest

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Review of percentage

The simplest way to change a fraction or decimal to a percentage is to multiply by 100%. Multiplying by 100% does not change the value of the fraction or percentage because

100% = 100 × = 1.

A percentage is a special way of writing a ratio. The percentage sign, %, means ‘per hundred’ or To change a fraction to a percentage, the denominator is changed to 100 and replaced by the percentage sign.

1 100 ---.

!

1 100

---Change the following to percentages.

a b 0.624 c 5 out of 8 d 1.37 e

Solution

a Method 1

Write the denominator as 100. =

means 75 per 100 or 75%. = = 75%

Method 2

Multiply by 100%. = × 100%

= × = 75%

b Multiply by 100%. 0.624= 0.624 × 100% = 62.4%

c Write as a fraction. 5 out of 8=

Multiply by 100%. ₌_×_100%

Simplify. = 62.5%

d Multiply by 100%. 1.37= 1.37 × 100% = 137%

e Multiply by 100%. = × 100%

Evaluate using your calculator and round off. ≈ 14.29%

3 4

--- 1

7

---3 4

--- 3×25

4×25

---75 100

--- 75

100

---3 4

--- 3

4

---3 4

---1

100 1

--- %

25

5 8

---5 8

---1 7

--- 1

7

---Example

1

(3)

When we say that one quantity is expressed as a percentage of another, we mean that a fraction has been changed to a percentage. The ﬁrst quantity is the top number (numerator) and the second is the bottom number (denominator) of the fraction. The fraction is changed to a percentage by multiplying by 100% as shown in Example 1.

In business calculations, percentages are nearly always expressed as a percentage of costs, and a proﬁt or loss is normally expressed as a percentage of the buying price.

An amount is often stated as a percentage of a total or original amount. To calculate the amount it is easiest to change the percentage to a decimal and multiply. To change a percentage to a decimal, you just divide by 100%.

Gianna bought a wetsuit in December for $350 and sold it in March for $150. What was her loss as a percentage of:

a the buying price? b the selling price?

Solution

Calculate the loss. Loss= $350 − $150 = $200

a Write loss as fraction of buying price. % loss= × %

Substitute. = × 100%

Evaluate and round off. ≈ 57.1%

b Write loss as fraction of selling price. % loss= × %

Substitute. = × 100%

Evaluate and round off. ≈ 133.3%

loss buying price

--- 100

1

---200 350

---loss selling price

--- 100

1

---200 150

---Example

2

A clothing retailer uses a markup of 85%. A particular brand of jeans is bought for $76. What will be the selling price of the jeans?

Solution

This can be worked out in two ways. You can work out the markup and add it on, or work out the selling price percentage ﬁrst. The second method is sometimes called the

percentage increase method. Method 1

Calculate the markup. Markup= 85% of $76

Change 85% to a decimal (85% = = 0.85). = 0.85 × $76

Evaluate. = $64.60

Calculate selling price (SP). SP= $76 + $64.60 = $140.60

Method 2

Write the selling price (SP) as a % of the buying price (BP).

SP= BP + 85% (of BP)

= (100% + 85%) of BP

= 185% of BP

Change 185% to a decimal. = 1.85 × $76

Evaluate. = $140.60

85 100

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In some cases you will know a final amount and a percentage and want to find the original amount. In this case you need to work backwards to find the required figure. You should write down the percentages in the following form to solve the problem.

NEW amount = percentage of OLD amount

!

Carl sold a portable CD player for $120, making a loss of 20%. What did Carl pay for the CD player?

Solution

Here, the new amount (the amount Carl received from the sale) is less than the old amount (the price he originally paid).

Use the rule for new and old amounts. NEW amount= percentage of OLD amount

Use the information from the problem. Selling price= 80% of buying price

Change 80% to a decimal and substitute relevant values.

$120= 0.80 × buying price

Divide both sides by 0.80 and rearrange. Buying price= $120 ÷ 0.80

Evaluate. = $150

Write the answer. Carl paid $150 for the CD player.

Example

4

1 Change the following to percentages.

a b 0.576 c 2

d 14 out of 60 e 3 : 5 f 0.075

g 1.46 h 17 out of 20 i four in every ﬁve

2 Connie bought a canoe for $950 and sold it for $800. What is her loss as a percentage of: a the buying price? b the selling price?

3 Hans bought an old personal computer for $700. He spent $480 upgrading the machine and

installing extra RAM. Hans then sold it for $1850. Work out:

a his total outlay

b the proﬁt as a percentage of his outlay

c the selling price as a percentage of the buying price.

4 Viktor got a $40 discount when he bought a jacket marked at $220. a What was the discount as a percentage of the marked price? b What was the actual price as a percentage of the marked price? c What was the discount as a percentage of the actual price?

5 Work out:

a 15% of $230 b 26.8% of $385 c 17% of $7890

d 6 % of $480 e 4.25% of $299.95 f 12.4% of $2300

g 1.35% of $6390 h 3 % of $4820 i 21.7% of $98.40

j 9.8% of $512.95 15

16

--- 3

4

---1 2

---1 4

---Exercise 3.1

Review of percentage

Additional exercise

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3.2

Interest

When you use credit, borrow money, lend money or invest money, payment is made for the use of the money. The amount can be calculated in different ways, and the method of calculation makes a difference to the total amount paid.

6 Kym bought a wide screen TV and home entertainment system for $4500 and sold it at a

proﬁt of 28%. How much did she sell it for?

7 Jimbo sees a shirt marked at $125 with a discount of 16%. What will he pay if he buys it?

8 The price of butter increased by 15% between 1999 and 2003. If it was $1.95 in 1999, what

was it in 2003?

9 Trish is a traveller for a distributor of hardware. She works on a commission of 23% plus a

retainer of $350 a fortnight. What would she earn if she sold $2574 worth of bathroom ﬁttings in a fortnight?

10 Running costs of a car include petrol, servicing, registration and insurance. The cost of

petrol is about 40% of total running costs. Bradley bought a second-hand car and found he spent $35 a week on petrol. What would he expect to spend on the car altogether in a year?

Modelling and problem solving

11 Cassie paid $904.80 for a new TV. The retailer used a markup of 45% and charged handling

and processing fees of 30%. The wholesaler used a markup of 20%. These were applied one after the other. What was the cost to the wholesaler?

12 Robbie is a second-hand dealer who bought an old table for resale. He spent $250 on

repairs and sold it for $1079.50. If he made 27% proﬁt, how much did he pay for the table?

The payment of money for a loan is called interest, and the amount of money borrowed or lent is called the principal.

The interest rate is worked out as a percentage of the principal for 1 year. When the interest is not actually for 1 year, a nominal interest rate is calculated. In this case a pro-rata amount is used to make 1 year’s worth of interest at the same rate. The phrase ‘interest rate’ nearly always means the nominal interest rate.

!

Anna received payments of $38 a quarter from $2500 worth of Australian Savings Bonds.

a What was the amount of interest for 1 year? b What was the interest rate?

Solution

a Calculate yearly interest (4 quarters in a year). Interest= 4 × $38 = $152

b The principal is $2500 and the pro-rata amount of interest for 1 year is $152.

Find interest as a % of principal and evaluate. % interest= × 100% = 6.08%

Write the answer. The interest rate was 6.08%.

152 2500

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Samantha and Julie own a boutique and needed a loan to purchase stock. They obtained a 90-day loan of $12 000 but had to pay back $12 532.

a How much interest did they pay? b What was the interest rate?

Solution

a Calculate the interest ($) for 90 days. Interest= $12 532 − $12 000 = $532

b The interest rate is calculated on the amount that would be paid over a whole year.

Write interest for 90 days. Interest for 90 days= $532

Calculate interest for 1 day. Interest for 1 day= $532 ÷ 90

Calculate interest for 1 year. Interest for 1 year= $532 ÷ 90 × 365

≈ $2157.56

Calculate interest as a percentage. Interest rate≈ × 100%

≈ 17.98%

Write the answer. The interest rate was 17.98%. 2157.56 12 000

---Example

6

1 Jose has $7000 in an investment account. He gets $26.25 a month interest. What is the

interest rate?

2 Sondra borrows $8000 on a 30-day promissory note. She has to repay $8131.51 at the end

of the 30 days. What is the interest rate?

3 Lien bought some bonds for $3000. She is paid interest of $40 a quarter. What is the

interest rate?

4 Colin has an interest-only loan of $59 000. He has to pay interest of $295 each fortnight.

What is the interest rate?

5 Trinh has a ﬁxed-deposit income-earning investment of $35 000. He gets $30.50 a week

from his investment. What is the interest rate?

6 Robyn borrowed $25 000 on a 90-day promissory note. She had to repay $26 171 at the end

of the 90 days. Unfortunately, she had to reﬁnance the loan and reborrowed the principal for a further 60 days, repaying $25 904 at the end of the next 60 days. What was the interest rate in each case?

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3.3

Simple interest

If you invest or borrow money, the time for which money is invested or lent is called the

term of the investment or loan. In the case of simple interest, the total interest is

calculated by multiplying the interest for 1 year by the term. If the term is less than 1 year, it is a fraction.

To calculate simple interest:

1 multiply the principal by the interest rate expressed as a decimal; 2 then multiply by the term.

You can use a formula for simple interest. The principal is multiplied by the interest as a decimal, and then multiplied by the number of years (term). Replacing principal by P, interest rate as a decimal by i and number of years by n, we get the following:

If the interest rate is expressed as the percentage r%, we must divide r by 100 to make it the decimal i shown in the formula. The ﬁnal amount A shows the total of the interest and principal.

Simple interest formulas

I = Pin = and A = P + I

where: P= principal = amount of money borrowed or invested i= interest rate per year as a decimal

r%= interest rate per year as a percentage n= time in years

A= ﬁnal amount = principal + interest. Prn

100

---!

Use the formula to calculate the amount of interest that Julia would pay if she borrowed $7500 at 8.3% simple interest for 5 years.

Solution

Write known information. P= $7500, r = 8.3, n = 5

Write the formula for simple interest. I=

Substitute and evaluate. = = $3112.50

State the result. Julia would pay $3112.50 interest.

Prn

100

---7500××××8.3××××5 100

---Example

7

Find the quarterly interest paid when Hassan invests $16 000 in bonds paying 7.5% interest.

Solution

Write the known information. P= $16 000, r = 7.5

Quarterly interest, so n = = Substitute. = 16 000 × ×

Prn

100

---3

12 --- 1

4

--- . 7.5

100

--- 1

4

(8)

You can use the simple interest formula in reverse to calculate interest rates. The principal required to get a particular amount of interest can also be worked out this way.

On a calculator, you can use the bracket function or memory for the denominator. Alternatively, you can just divide by each number on the bottom.

Either:

16 000 7.5 1 100 4 →

Or:

100 4 →

16 000 7.5 1 →

Or:

16 000 7.5 100 4 →

State the result. Hassan gets $300 interest each quarter.

✕ ✕

÷

( ✕ )

=

₃₃₃_{3 0}_{0 0}₀₀₀₀₀

✕

=

Min M ₄₄₄_{4 0}₀_{0 0}₀₀₀₀

✕ ✕

÷

MR

=

₃₃₃_{3 0}_{0 0}₀₀₀₀₀

✕

÷

=

₃₃_{3 0}₃₀₀_{0 0}₀₀₀

David has some bonds worth $6000 paying interest of $82.50 every quarter. What interest rate are the bonds earning?

Solution

Write the known information. P= $6000, I = $82.50 per quarter

Quarterly interest, so n = Substitute. 82.50= 6000 × × Multiply by the numbers in the denominator and

divide by those in the numerator and rearrange. r=

Evaluate. = 5.5

Write the result. David’s bonds earn 5.5% interest.

Prn

100

---1

4

--- . r

100

--- 1

4

---82.50××××100××××4 6000

---Example

9

A ﬁxed-deposit income-producing account is advertised as having 7.2% interest and paying interest every fortnight. What amount would you have to invest to get $150 a fortnight?

Solution

Write the known information. I = $150 per fortnight, r = 7.2

Fortnightly interest, so n = = Substitute. 150=

Work backwards. = P

Rearrange and evaluate. P≈ $54 166.67

Write the result. You would have to invest $54 166.67.

Prn

100

---2 52 --- 1

26

--- . P××××7.2××××1

100××××26

---150××××100××××26 7.2

(9)

Simple interest calculations

Let’s see how you can set up a spreadsheet to help carry out simple interest calculations. Open a blank spreadsheet.

1 Enter the headings ‘Simple interest’ into cell B1, ‘Interest rate (i)’ into B2, ‘Principal

(P)’ into B3, ‘End of year’ into A5, ‘Principal’ into B5, ‘Yearly interest’ into C5, ‘Amount’ into D5 and ‘Total interest’ into E5.

2 Next enter the years for which interest will be calculated by placing ‘1’ into cell A6,

‘=A6+1’ into A7, ‘=A7+1’ into A8, ‘=A8+1’ into A9 and so on down to A15.

3 Enter the principal for each year by entering ‘=C3’ into each cell from B6 to B15. (Remember, the principal doesn’t change with simple interest.)

4 Enter the yearly interest calculation by placing ‘=C3*C2’ into each cell from C6 to C15.

5 Enter the calculation of the amount at the end of each year by placing ‘=C3+C6’ into cell D6, ‘=C3+C7’ into D7, ‘=C3+C8’ into D8 and so on down to D15.

6 Finally, enter the calculation of the total interest by placing ‘=C6’ into cell E6, ‘=E6+C7’ into E7, ‘=E7+C8’ into E8, ‘=E8+C9’ into E9 and so on down to E15.

Next format cell C3 and the region B6:E15 as currency with 2 decimal places and the ‘$’ symbol, using the Format command in the toolbar.

• Use the spreadsheet to carry out some simple interest calculations by entering an annual interest rate as a decimal into cell C2 and a principal into C3. Try different values for interest and principal and note the effect.

Technology

Spreadsheet

(10)

Exercise 3.3

Simple interest

1 Calculate the simple interest for the following loans or investments. a $7000 at 15% for 5 months b $1500 at 8% for 4 years c $6900 at 4% for 18 months d $55 000 at 13% for 3 weeks e $4600 at 17% for 3 years

2 Work out the simple interest for the following.

a $6000 at 12% for 5 years b $7500 at 6.7% for 3 years c $8400 at 5.8% for 5 months d $900 at 4.6% for 15 weeks e $30 000 at 14.1% for 90 days

3 Greg had to borrow $45 000 for 60 days for his business. He has to pay 24% simple

interest. What interest is due at the end of the term, and how much must he pay back?

4 Elaine got $500 credit for 5 weeks at 15% simple interest. How much was due at the end of

the 5 weeks?

5 Gabriella invested $75 000 in 30-day bonds at 6.4% simple interest. How much did she

get back?

6 Andrew borrowed $12 000 for 4 years at 19.6% simple interest. How much interest would

he pay?

7 Junko has a small business importing Japanese art. She takes a business loan of $40 000

for 90 days at 16.5% simple interest. How much interest does she pay, and what amount is due at the end of 90 days?

8 Peter invests $12 000 in bonds paying 5.7% interest with monthly payments. How much

does he get each month?

9 Rajendra bought some $100 Australian Savings Bonds paying 6.2% interest. He bought

180 bonds and receives interest each quarter. How much does he get each quarter?

10 Elena borrowed $8000 at 13.8% simple interest over 3 years. How much interest does she

pay, and what total amount does she have to pay back?

Modelling and problem solving

11 Kim has $8500 invested in bonds that pay $46.05 a month. What is the interest rate?

12 Andrew wants to get $175 a week by investing in an account that pays 4.6% interest. How

much must he invest?

13 Kirsten has $7300 to invest. Her bank offers $120 a quarter for such an investment.

Another bank says that it pays 6.6% on investments in multiples of $1000.

a What interest rate does her bank offer?

b How much a quarter would she get from the other bank?

14 John bought 750 shares for $3450. He got half-year dividends of 11 cents a share. What is

the equivalent interest rate?

15 How much would you have to put in an investment account to get $300 a month interest

at 5.6%?

16 A company borrowed $200 000 on the short-term money market for 60 days. The company

had to repay $207 000. What interest rate did the company pay? 1

2

---Additional exercise

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3.4

Compound interest

When you have money invested at simple interest, the interest is paid to you periodically. It could be every month, every quarter or some other interest period. The amount you have invested (the principal) stays the same.

In the case of compound interest, the interest is not paid at the end of the interest period. Instead it is added to the investment. This means that the interest increases during the term of the investment because the principal increases. The interest period is called a rest and can be yearly, quarterly, monthly or even daily. During each rest you get interest on a compound of the principal and the previous interest. A rest of 1 month is sometimes stated as compounding monthly.

17 An executive on a salary of $2400 a week wants to make sure that she gets at least 60%

of her salary on retirement. She can get 6% interest. How much must she have invested to reach her goal?

18 Katrina has $9500 invested in an account paying 4.5% interest ‘on call’. She gets the

interest each quarter. She could also put money into an account paying 5.3%, but it would then be a 3-month ﬁxed deposit.

a How much does she get each quarter?

b How much would she have to put into the second account to get the same payment?

Suzanne invested $5400 at 6% compound interest with monthly rests. How much did she have at the end of 4 months, and how much interest did she earn?

Solution

There are two methods of calculation.

Method 1

Use the simple interest formula for each rest and add the interest to the amount invested to calculate the next principal.

Monthly rests, so n =

Write the known information. P = $5400, r = 6

Calculate interest for 1st month. I1= = $27.00 Calculate the amount after 1 month. P1=$5400+ $27.00 =$5427.00

Calculate interest for 2nd month. I2= ≈ $27.14 Calculate the amountafter 2 months. P2=$5427.00+ $27.14 =$5454.14

Calculate interest for 3rd month. I3= ≈ $27.27

Calculate the amountafter 3 months. P3=$5454.14+ $27.27 =$5481.41

Calculate interest for 4th month. I4= ≈ $27.41 Calculate the amount after 4 months. P4=$5481.41+ $27.41 =$5508.82

1 12 --- .

Prn

100

---5400××××6××××1 100××××12

---5427.00××××6××××1 100××××12

---5454.14××××6××××1 100××××12

---5481.41××××6××××1 100××××12

(12)

Method 2

Use the percentage increase method to get the new principal after each rest.

Each month, the amount increases by 6% ÷ 12 = 0.5%, so each month the principal is 100.5% of the previous principal.

Both methods produce the same result.

Calculate the amount after 1 month. A1= 100.5% of $5400 Change the percentage into a decimal. = 1.005 ×$5400

Evaluate. =$5427.00

Calculate the amount after 2 months. A2= 100.5% of $5427.00

Change the percentage into a decimal. = 1.005 ×$5427.00

Evaluate. ≈$5454.14

Calculate the amount after 3 months. A3= 100.5% of $5454.14

Change the percentage into a decimal. = 1.005 × $5454.14

Evaluate. ≈ $5481.41

Calculate the amount after 4 months. A4= 100.5% of $5481.41 Change the percentage into a decimal. = 1.005 ×$5481.41

Evaluate. ≈$5508.82

State the result. Suzanne had $5508.82 after 4 months. The interest was $5508.82 − $5400 = $108.82.

Constant multiplier function

The constant multiplier function on a calculator can be used for a shortcut using the second method shown in Example 11. Since the previous amount is always multiplied by the same number, you put in that number as the constant multiplier and press the equals key the required number of times.

On many calculators the constant multiplier is obtained by keying in the desired

multiplier and then pressing the multiply key twice. You need to work out how to do this for your calculator. Let’s see how this would be done for the situation described in Example 11.

Enter the constant multiplier. 1.005 →

Now key in the starting amount. 5400

Press to ﬁnd the amount after 1 year. →

Press again for the amount after 2 years. →

And again for the amount after 3 years. →

And once more for the amount after 4 years. →

You can see that this answer is not exactly the same as the one obtained in Example 11. This is because the amounts were not rounded to the nearest cent at each step.

✕ ✕ K _1111....0_{0 0}₀₀₀₀_{0 5}₅₅₅

=

K 55 455444 2222 7777

=

K 555 454 5445 455444 ....11113333 5555

=

K55 455444 8888 1111....4444 00 500555 6666 7777 5555

=

K 555 555 055000 88 ....888 88 111128 2 777227 0000 3333

(13)

3.5

Compound interest formula

Even if you use a calculator shortcut, the calculation of compound interest can be very tedious. A formula can be used instead to save some steps. We can make the formula by replacing the quantities in the second method shown in Example 11 by symbols.

At each step of the calculation, the previous amount is multiplied by where i is

the interest rate per year as a decimal and k is the number of rests per year.

In Example 11, = = 0.005. We again use P for the principal and A for the amount of

principal plus interest. Let’s see how this works.

Use either method shown in Example 11 on pages 83–84 to ﬁnd the ﬁnal amount and interest earned for each of the following investments at compound interest.

a $6000 at 9% with quarterly rests for 1 years b $8300 at 6% with monthly rests for 5 months c $12 000 at 5.4% with 6-monthly rests for 2 years

d $7500 at 15% with monthly rests for 2 years e $9400 at 7.2% with quarterly rests for 3 years

f $1500 at 7.8% with fortnightly rests for a year

g $8150 at 6.3% with monthly rests for 2 years h $5600 at 7.1% with weekly rests for 6 months

1 2

---Exercise 3.4

Compound interest

Did you know?

There are many examples of consumers being exploited by unscrupulous individuals.

Payday lending is one such example. Payday lending is the practice of advancing consumers a small amount of money until the next payday in exchange for a fee.

In December 2000 the 7.30 Report on ABC TV conducted a segment on payday lending in which someone who had used payday lending said:

‘Basically, for each $100 you borrow, you pay back an extra $22 (for each week the loan is unpaid) … That seemed alright because I wasn’t borrowing high amounts.’

The problems of payday lending are not just associated with high interest rates. There are also excessive charges. A person can easily be caught in a ‘debt trap’ when he/she rolls over a payday loan instead of paying it back. The combination of interest and fees can be financially crippling for some consumers.

In another example of this practice, a Queensland loan of $120 in 1999 was repeatedly rolled over to the next payday period when the borrower could not pay. Within 3 months the borrower owed $680 plus fees of $272—a total of $952.

For each of the examples mentioned here, calculate the nominal interest rate and discuss this with others in the class.

1 i k --+

 

 _,

i k -- 0.06

(14)

---Calculate the amount after 1 rest. A1= P ×

Calculate the amount after 2 rests. A2= A1×

Substitute for A1. = P × ×

Simplify. = P ×

Calculate the amount after 3 rests. A3= A2 ×

Substitute for A2. = P × ×

Simplify. = P ×

We can follow this pattern to form a general formula for compound interest.

Unless it is otherwise stated, assume that rests are annual. 1 i k --+     1 i k --+     1 i k --+  

  ₁ i

k --+     1 i k --+  

 2

1 i k --+     1 i k --+  

 2 ₁ i

k --+     1 i k --+  

 3

Compound interest formulas

A = P = P and I = A − P

where: P = principal, I = interest and A = ﬁnal amount i = interest rate per year as a decimal

k = number of rests per year t = number of years

n = kt = number of rests.

If rests are annual, k = 1, n = t and we can write the formula as A = P(1 + i)n_. 1 i

k --+

 

 k t ₁ i

k --+

 

 n

!

Use the compound interest formula to ﬁnd the amount and interest earned when $6200 is invested at 7.5% interest with monthly rests for 4 years.

Solution

Write the formula for compound interest. A= P

List known values. P= $6200, r = 7.5, k = 12, t = 4

Express the interest rate as a decimal. i= = 0.075 per year

Calculate the monthly interest rate. = 0.075 ÷ 12 = 0.006 25 per month

Calculate the number of rests. n= 12 × 4 = 48

Substitute into the formula. A= 6200(1 + 0.006 25)48

Simplify. = 6200 × 1.3485 …

Evaluate and round off. ≈ $8361.31

State the result. The amount is $8361.31 and the interest is $8361.31 − $6200 = $2161.31.

1 i

k

---+

 

 n

(15)

Power calculations

The calculator key is used for power calculations. Let’s see how to use this key to complete the calculation 6200(1 + 0.006 25)48_{from Example 12.}

7.5 100 12 1 →

48 →

6200 →

You may have to use the second function (or inverse) key on your calculator to work out the power. You can also use the bracket keys to work out the answer in one long

calculation, but this is prone to error.

x y

÷

+

=

1111....0000 00 600666 2222 5555

x y

=

_1111....33_{3 4}₃₄₄_{4 8}₈₈_{8 5}_{5 9}₅₅_{9 9}₉₉₉₉_{9 11115}_{5 1111}₅₅

✕

=

₈₈88 3333 66 1111....33366 3 11114444 7777 3333 8888

Technology

Compound interest using a graphics calculator

You can use a graphics calculator to ﬁnd A, i, n or P in the compound interest formula

A = P

When using a graphics calculator for financial calculations, it is very important to remember a basic finance/accounting convention that the calculators use. Cash outflows (money that you pay) are considered negative, while cash inflows (money paid to you) are considered positive.

All calculators use the standard ﬁnancial symbols:

N = Total number of rests (interest/payment periods)

I% = Nominal interest rate expressed as a percentage per annum PV = Present value—for an investment it is the amount invested

PMT= Payment in each interest period—you do not pay anything for an investment FV = Future value—amount at the end of the investment period

P/Y = Payments/year—the number of rests each year for an investment C/Y = Compounding periods/year—the same as P/Y for most situations.

For an investment at compound interest, PV is negative because you pay that out at the beginning. FV is positive because it is paid to you at the end.

For all the calculators, the same method of calculation is used. The known values are entered into the ﬁnancial solving function, and the desired quantity is selected and calculated.

Let’s see how to use the graphics calculator to calculate the amount and interest earned when $3200 is invested for 2 years with monthly rests at 15.9% compound interest. For this problem, the ﬁnance variable values to be entered into the calculator are

N = 24, I% = 15.9, PV =−3200, PMT = 0, P/Y = 12

Note that all calculators will automatically set C/Y the same as the entered value of P/Y.

Remember to use the key to enter negative amounts. 1 i

k --+

 

 n

(–)

Technology

Graphics

(16)

If the amount of interest is known, the compound interest formula can be used in reverse to ﬁnd an unknown principal or interest rate.

Casio CFX-9850GB PLUS

Press the key and select the TVM submenu by

pressing the key.

Press (Compound interest).

Enter the known values—it doesn’t matter what is in FV as that’s the amount we want to ﬁnd.

Press (FV) and press to return to the screen shown here.

Texas Instruments TI-83

You select the ﬁnance functions differently for the TI-83 and the TI-83 PLUS.

Press (FINANCE) or

(1:Finance) on the TI-83 PLUS. Select 1:TVM Solver.

Enter the known values—it doesn’t matter what is in FV as that’s the amount we want to ﬁnd.

Move the cursor up to FV.

Press (SOLVE).

All calculators

You should get the value FV = 4388.8291 …, so the investment grows to about $4388.83 and the interest earned is about $4388.83 − $3200 = $1188.83.

Sharp EL-9650

The Sharp instructions are given on the CD-ROM. MENU

EXE

F2

F5 EXIT

2nd x-1 APPS ENTER

ALPHA ENTER

Calculator

instructions

Areti has the opportunity to invest some money at 9% compound interest with monthly rests. How much should she invest if she wants to have $10 000 available in 5 years?

Solution

List known values. A= $10 000, r = 9, k = 12, t = 5

Calculate other values. = 0.09 ÷ 12 = 0.0075 per month

n= 12 × 5 = 60

Substitute for A, and n. 10 000= P × (1 + 0.0075)60

Simplify. 10 000= P × 1.5656 …

Rearrange. P= 10 000 ÷ 1.5656 …

Evaluate and round off. ≈ $6387.00

Write the result. Areti should invest $6387.00. 1 i

k

---+

 

 n

i k

---i k

(17)

Use your graphics calculator to ﬁnd what principal must be invested to accumulate to $15 000 after 6 years at 17% compound interest.

Solution

For this problem, the ﬁnance variable values to be entered into the calculator are N = 6, I% = 17, PMT = 0, FV = 15 000, P/Y = 1

Casio CFX-9850GB PLUS

Use the TVM . Press (Compound interest). Enter the known values—it doesn’t matter what is in PV.

Press (PV) .

Press (FINANCE) or

(1:Finance) on the TI-83 PLUS. Select the TVM Solver.

Enter the known values—it doesn’t matter what is in PV.

Move the cursor up to PV. Press (SOLVE).

All calculators

You should get the value PV =−5847.5788 …, so $5847.58 must be invested.

Sharp EL-9650

The Sharp instructions are given on the CD-ROM.

MENU F2

F3 EXIT

2nd x-1 APPS ENTER

ALPHA ENTER

Example

14

Calculator

instructions Graphics

calculator

Two years ago, Tanh placed $8300 in an investment paying compound interest with quarterly rests. If his money has grown to $9500, what interest has he been earning?

Solution

List known values. A= $9500, P = $8300, k = 4, t = 2

Calculate other values. n= 4 × 2 = 8

Substitute into the formula. 9500= 8300 ×

Rearrange and simplify. = = 1.1445 …

Use the or calculator key to

find the 8th root. = ≈ 1.0170 …

Isolate ≈ 1.0170 … − 1

Evaluate. = 0.0170 … per quarter

Calculate the nominal interest and round off. i= 4 × 0.0170 … = 0.0681 per year

Evaluate r. r≈ 6.81

State the result. Tanh has been getting about 6.81% interest. 1 i

k

---+

 

 n

1 i

k

---+

 

 8

1 i

k

---+

 

 8 9500

8300

---x

y

x1/y

1 i

k

---+ 8 1.1445 …

i k

-- i

k

(18)

It is more difﬁcult to ﬁnd how long an investment will take to reach a particular amount. After setting up the formula, a certain amount of trial and error is needed.

Use your graphics calculator to ﬁnd what interest rate is needed for $7600 to double in 5 years if it is compounding quarterly.

Solution

For this problem, the ﬁnance variable values to be entered into the calculator are N = 20, PV =−7600, PMT = 0, FV = 15 200, P/Y = 4

Casio CFX-9850GB PLUS

Use the TVM .

Enter the known values—it doesn’t matter what is in I%.

Press (I%) .

Press (FINANCE) or

(1:Finance) on the TI-83 PLUS.

Select the TVM Solver.

Enter the known values—it doesn’t matter what is in I%. Move the cursor up to I%.

Press (SOLVE).

All calculators

You should get the value I% = 14.1059 …, so the interest rate needed is about 14.1%.

Sharp EL-9650

F2

F2 EXIT

2nd x-1 APPS ENTER

ALPHA ENTER

Example

16

Graphics

calculator

Calculator

instructions

Steve has $5000 invested at 8.4% compound interest with monthly rests. How long will it take to accumulate to $8000?

Solution

List known values. A= $8000, P = $5000, r = 8.4, k = 12

Calculate other values. = 0.084 ÷ 12 = 0.007 per month

Substitute into the formula. 8000= 5000 × (1 + 0.007)n Rearrange and simplify. 1.007n₌ ₌_1.6

1 i

k

---+

 

 n

i k

---8000 5000

(19)

Use trial and error starting with n = 20. 1.00720_≈_1.1497 Not enough; try n = 50. 1.00750_≈_1.4173 Still not enough; try n = 80. 1.00780_≈_1.7473 Too much; try n = 70. 1.00770_≈_1.6295 Still too much; try n = 65. 1.00765_≈_1.5737 Again not enough; try n = 67. 1.00767_≈_1.5958 Still not enough; try n = 68. 1.00768_≈_1.6070

Close enough. State the result. After 67 rests the amount will be just under $8000, and after 68 rests it will be just over $8000. So it will take 68 months (5 years and 8 months) to reach $8000.

Use your graphics calculator to ﬁnd how long it will take $4000 to accumulate to at least $10 000 at 14.5% compound interest with monthly rests.

Solution

For this problem, the ﬁnance variable values to be entered into the calculator are I% = 14.5, PV =−4000, PMT = 0, FV = 10 000, P/Y = 12

Casio CFX-9850GB PLUS

Use the TVM .

Enter the known values—it doesn’t matter what is in n.

Press (n) .

Press (FINANCE) or

Select the TVM Solver.

Enter the known values—it doesn’t matter what is in N. Move the cursor up to N.

Press (SOLVE).

All calculators

You should get the value N = 76.2881 …, so the time is about 77 ÷ 12 = 6 years 5 months.

Sharp EL-9650

F2

F1 EXIT

2nd x-1 APPS ENTER

ALPHA ENTER

Example

18

Graphics

calculator

Calculator

(20)

Compound interest calculations

Let’s see how you can set up a spreadsheet to help carry out compound interest calculations. Open a blank spreadsheet.

1 Enter the headings ‘Compound interest’ into cell B1, ‘Interest rate (i)’ into B2,

‘Principal (P)’ into B3, ‘Rests per year (k)’ into B4, ‘End of year’ into A6, ‘Principal’ into B6, ‘Yearly interest’ into C6 and ‘Total interest’ into D6.

2 Next enter the years for which interest will be calculated by placing ‘1’ into cell A7,

‘=A7+1’ into A8, ‘=A8+1’ into A9, ‘=A9+1’ into A10 and so on down to A16.

3 The principal at the end of each year will be different because it will have interest earned

for the year added in. Begin by entering the formula ‘=C$3*(1+C$2/C$4)^(A7*C$4)’ into cell B7. Then enter ‘=C$3*(1+C$2/C$4)^(A8*C$4)’ into cell B8,

‘=C$3*(1+C$2/C$4)^(A9*C$4)’ into B9 and so on down to B16.

4 Enter the yearly interest calculation by placing ‘=B7−C3’ into cell C7, ‘=B8−B7’ into C8, ‘=B9−B8’ into C9 and so on down to C16.

5 Finally, enter the calculation of the total interest by placing ‘=B7−C3’ into cell D7, ‘=B8−C3’ into D8, ‘=B9−C3’ into D9 and so on down to D16.

Next format cell C3 and the region B7:D16 as currency with 2 decimal places and the ‘$’ symbol, using the Format command in the toolbar.

• Use the spreadsheet to carry out some compound interest calculations by entering an interest rate per year as a decimal into cell C2, a principal into C3 and the number of rests per year into C4. Try different values for interest and principal and note the effect. • A copy of the spreadsheet is on the CD-ROM. Compare yours with the one provided.

• You can change the years over which the calculations are made by entering the year of your choice into cell A7. Try beginning at a year other than year 1.

Challenge

• All calculations except the ﬁrst yearly interest calculation in C7 are correct. Devise a formula for C7 that would result in the correct value regardless of the value in A7.

A more sophisticated version of this spreadsheet is also provided on the CD-ROM. Use it to check your answers to the following exercise.

1

Technology

Spreadsheet

Compound interest 1

Spreadsheet

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1 Use the formula to work out the ﬁnal amount and interest earned for each investment. a $15 000 at 9.6% compounding monthly for 2 years

b $8950 at 6.9% for 3 years with monthly rests c $4500 at 4.5% compounding quarterly for 3 years

d $540 at 11% for 20 years with annual rests e $7540 at 7.5% for 10 years with semi-annual rests

f $2500 at 7.8% for 2 years compounding weekly

g $5800 at 9.1% compounding fortnightly for 5 years h $6000 at 10.5% compounding daily for 2 years i $4250 at 5.6% compounding daily for 6 months

j $25 000 at 9.4% compounding daily for 10 years

Modelling and problem solving

2 How much should Felix invest to get back $5000 in 3 years’ time at 8.1% interest,

calculated monthly?

3 Phan needs to have $3500 available in 2 years’ time. How much should she put in an

account paying 6.5% with weekly rests to make sure she has it?

4 Anthoula got $2000 interest over 5 years when she left $3000 in an account paying

compound interest with monthly rests. What interest rate was paid?

5 Antonio left 250 000 lire in an account when he emigrated to Australia. After 20 years with

compound interest calculated annually it had grown to 1 200 000 lire. What interest rate was paid?

6 How long would it take $4000 to grow to $5000 at 8% interest, calculated quarterly?

7 Tania’s credit union pays 5% interest (compounded daily). How long would it take $10 000

to increase to $10 500 in this account?

8 How long will it take to double your money at 8% compound interest (calculated

annually)? You can work this out using an amount like $10 000.

9 Nikita had $5000 in an investment and after 4 years with monthly rests it had grown to

$6300. What interest rate was Nikita getting?

10 How much should Renee invest in an account paying 6% interest calculated semi-annually

to have $12 500 in 7 years’ time?

11 How long would it take Jarrod to triple his investment of $15 000 at 7.5% interest,

calculated fortnightly?

Exercise 3.5

Compound interest formula

Additional

exercise

3.5

How long does it take to double, triple, or quadruple your money at compound interest?

The formula n = can be used to work out approximately how many years it takes to

double your money at a nominal interest rate of r% compounded annually. Work in groups of three or four to decide on the accuracy of the formula.

70 r

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3.6

Nominal and effective rates of interest

The amount of interest paid in a compound interest calculation depends primarily on the nominal interest rate and the term. However, it is also affected by the frequency of rests.

In order to compare compound interest rates, we must take account of the number of rests in a year. The effective interest rate is worked out to take the number of rests into account. As a decimal, it is the interest that would be paid on $1 invested for 1 year. As a percentage, it is the interest that would be paid on $100 invested for 1 year. The effective interest rate is sometimes called the actual interest rate. You can use the effective interest rate to compare rates of interest for which the rests are different.

• Each person should work out the times it takes to double $10 000 to $20 000 with different interest rates. One person does the rates from 4% to 7%, another 8% to 11%, and so on. • Work out the actual times using the method shown in Example 17 and compare them with

the results obtained by using the formula n =

• Try changing the number of rests in a year to see how that affects the accuracy. • Discuss your ﬁndings as a group and with the class as a whole. How accurate is the

formula?

Extension

The formula is derived from something called the Maclaurin Series. It is possible to work out corresponding formulas for the times to triple, quadruple, etc. In general, the

approximation is that the time taken to increase the money by a factor f is given by

n =

where ln f is the natural logarithm of the factor f. This function (ln) is available on your calculator.

Work in your groups to test the accuracy of the formula for a factor other than 2. 70

r --- .

100× lnf r

---Work in groups of four for this investigation.

1 Each person in the group is to work out the amount of interest he/she would get by

investing $100 000 at 12% compound interest over 2 years. In each case, add the interest into the account when it is paid. Each person should use a different rest period: annual, monthly, weekly or daily.

2 When you have completed the calculations, compare the total interest that each person gets.

Draw a graph showing the number of rests in 1 year on the horizontal axis and the amount of interest on the vertical axis.

3 Discuss the effect of more frequent payment of compound interest at the same nominal rate. 4 Which would be better: 15% compounded daily or 16% compounded annually?

5 What do you think would happen to the amount of interest if it were paid each hour or each

second? Perhaps the whole class could investigate this by extending the graph.

(23)

It is possible to develop a formula to work out the effective interest rate. A simple formula for effective interest is:

It is generally easier to use the method shown in Example 19 than to use the formula.

All graphics calculators have in-built formulas for effective interest rates.

What is the effective rate of interest for an investment earning 14% compounded monthly?

Solution

To work out the effective rate as a percentage, we calculate the interest on $100 for 1 year.

List known values. P= $100, r = 14, k = 12, t = 1

Calculate other values. = 0.14 ÷ 12 = 0.0116 … per month

n= 12 × 1 = 12

Substitute into the formula. A= 100 × (1 + 0.0116 …)12 Evaluate and round off. ≈ $114.93

Calculate interest. I≈ $114.93 − $100

= $14.93

State the result. The interest over the year is about $14.93, so the effective rate of interest is about 14.93%.

1 i

k

---+

 

 n

i k

---Example

19

Effective interest rate

e =

where: e= effective yearly interest rate as a decimal i= nominal yearly interest rate as a decimal k= number of rests in a year.

1 i k --+

 

 k_–₁

!

What is the effective rate of interest for an investment earning 7% compounded monthly?

Solution

Write the formula for effective interest. e= − 1

List known values. i= 0.07, k = 12

Substitute into the formula. e= − 1

Evaluate and round off. ≈ 0.0723

(= 7.23%)

State the result. The effective interest rate is about 7.23%. 1 i

k

---+

 

 k

1 0.07 12

---+

 

 12

(24)

Use your graphics calculator to calculate the effective interest rate for an investment at 4.6% compounded monthly.

Solution

Casio CFX-9850GB PLUS

Use the TVM .

Press (Conversion).

Press 12 to store the number of rests (n) and

4.6 to store the interest rate (I%).

Press (EFF) to calculate the effective interest rate. The result is 4.6982 …, so the effective interest rate is about 4.70%.

Press (FINANCE) or

Select C: EFF(. Press 4.6 12 to store the interest rate and number of rests.

The result is 4.6982 …, so the effective interest rate is about 4.70%.

Sharp EL-9650

F5

EXE

F1

2nd x-1 _APPS _ENTER

,

) ENTER

Example

21

Calculator

instructions Graphics

calculator

1 Alison can invest money at 7.5% interest compounded monthly. What is the effective

interest rate?

2 Peter invested his money at 6.7% compounded daily. What is the effective rate of interest?

3 Georgio has the choice of three investments at his bank. He can invest at 7.6% compounded

quarterly, 7.5% compounded monthly or 7.4% compounded daily. Calculate the effective rate for each option and decide which is best.

4 Kimiko’s credit union has a different choice: she can invest at 8.5% compounded quarterly,

8.3% compounded monthly or 8.1% compounded daily. Calculate the effective rate for each option and decide which is best.

5 Use your graphics calculator to ﬁnd the effective interest rate for each of the following. a 6.25% compounded monthly b 5.05% compounded quarterly

c 7.02% compounded daily d 8.31% compounded semi-annually (half-yearly)

6 Use your graphics calculator to determine which investment would provide the better

return: 8.04% compounding monthly or 7.95% compounding daily.

Exercise 3.6

Nominal and effective rates of interest

Graphics

calculator

Chapter

summary

(25)

Chapter

Review

Communication and justification

1 How do you calculate a percentage proﬁt?

2 Explain the difference between interest and interest rate.

3 What is meant by a nominal interest rate?

4 Explain the difference between simple and compound interest.

5 What is a rest in the calculation of compound interest?

Knowledge and procedures

6 Write the following as percentages.

a b 1.34 c 2 d 0.036

7 Felicity bought some second-hand painters’ trestles for $250. After she had ﬁnished

painting the house she sold them for $290. Work out:

a her proﬁt as a percentage of the buying price b the selling price as a percentage of the buying price.

8 Work out:

a 23% of $245 b 152% of $5300 c 2.3% of $180.

9 Vlad bought a car for $4300 and sold it at a proﬁt of 18%. How much did he sell it for?

10 Simone bought a set of spanners marked at $85 for a discount of 15% at a sale. What

did she pay?

11 At the same sale she paid only $420 for a block and tackle with a 30% discount. What

was the original price?

12 Tom works on a commission of 5 %. In a fortnight when he earned $1100 gross,

what was the value of goods he sold?

13 Gerhardt owns a small business and took out a 90-day promissory note for $25 000.

He had to pay back $26 110. What was the interest rate?

14 Robbie has an investment of $28 000 that pays $37.50 a week interest. What is the

interest rate?

15 Write the formula for simple interest.

16 Find the simple interest on $6500 at 15% over 4 years.

17 Calculate the simple interest for each of the following. a $5000 at 8% for 3 years

b $18 000 at 16.8% for 3 months c $6200 at 5.5% for 4 years

18 David borrowed $45 000 for 60 days at 17.5% simple interest. How much did he have

to pay back?

19 Carmen bought 340 bonds paying 6.7% interest with fortnightly payments. The bonds

were $200 each. How much interest did she get each fortnight?

Ex 3.1 Ex 3.2 Ex 3.2 Ex 3.3-4 Ex 3.4 Ex 3.1

14 25

--- 1

2

---Ex 3.1 Ex 3.1 Ex 3.1 Ex 3.1 Ex 3.1 Ex 3.1

1 2

---Ex 3.2 Ex 3.2 Ex 3.3 Ex 3.3 Ex 3.3

1 2

(26)

20 John was getting $45 a month from his investment of $15 000. What was the simple

interest rate?

21 Write the formula for compound interest.

22 Find the ﬁnal amount and interest when $7800 is invested at 11.5% compounding

monthly over 4 years.

23 Calculate the ﬁnal amount and interest earned for each of the following investments. a $5000 at 7% compounding annually for 4 years

b $12 000 at 4.8% compounding monthly for 3 years c $7500 at 9.2% compounding weekly for 5 years

24 What is the effective interest rate for: a 7.5% compounding monthly? b 13.4% compounding daily? c 8.4% compounding quarterly?

Modelling and problem solving

25 Charles invested $8000 for 4 years with interest at monthly rests. He ended up with

$9500. What interest rate was paid?

26 Julie inherited a bank account that had been getting 5% interest compounded each

half-year for 15 years. The amount in the account was $7340 when she inherited it. How much was in the account originally?

27 How long would it take $8000 to grow to $12 000 at 15% interest, compounding

weekly?

28 If $5000 amounts to $7000 over 5 years with interest compounding quarterly, what is

the interest rate?

29 What sum should be invested at age 20 to get $1 000 000 at age 65, if an average

compound interest rate of 8% may be assumed?

3 Interest

Contents

Review of percentage

a b 0.624 c 5 out of 8 d 1.37 e

Solution

a Method 1

---Example

a the buying price? b the selling price?

Solution

a Write loss as fraction of buying price. % loss= × %

---Example

percentage increase method. Method 1

Solution

$120= 0.80 × buying price

Example

1 Change the following to percentages.

a his total outlay

---Exercise 3.1

6 Kym bought a wide screen TV and home entertainment system for $4500 and sold it at a

10 Running costs of a car include petrol, servicing, registration and insurance. The cost of

Modelling and problem solving

11 Cassie paid $904.80 for a new TV. The retailer used a markup of 45% and charged handling

a What was the amount of interest for 1 year? b What was the interest rate?

Solution

a Calculate yearly interest (4 quarters in a year). Interest= 4 × $38 = $152

a How much interest did they pay? b What was the interest rate?

Solution

a Calculate the interest ($) for 90 days. Interest= $12 532 − $12 000 = $532

---Example

1 Jose has $7000 in an investment account. He gets $26.25 a month interest. What is the

5 Trinh has a ﬁxed-deposit income-earning investment of $35 000. He gets $30.50 a week

Simple interest

term of the investment or loan. In the case of simple interest, the total interest is

Simple interest formulas

Solution

Solution

Solution

---Example

Simple interest calculations

1 Enter the headings ‘Simple interest’ into cell B1, ‘Interest rate (i)’ into B2, ‘Principal

Technology

2 Work out the simple interest for the following.

5 Gabriella invested $75 000 in 30-day bonds at 6.4% simple interest. How much did she

8 Peter invests $12 000 in bonds paying 5.7% interest with monthly payments. How much

Modelling and problem solving

11 Kim has $8500 invested in bonds that pay $46.05 a month. What is the interest rate?

b How much a quarter would she get from the other bank?

Compound interest

17 An executive on a salary of $2400 a week wants to make sure that she gets at least 60%

b How much would she have to put into the second account to get the same payment?

Solution

Method 1

Method 2

Constant multiplier function

Compound interest formula

g $8150 at 6.3% with monthly rests for 2 years h $5600 at 7.1% with weekly rests for 6 months

---Exercise 3.4

Compound interest formulas

Solution

Power calculations

Technology

N = 24, I% = 15.9, PV =−3200, PMT = 0, P/Y = 12

Technology

Casio CFX-9850GB PLUS

All calculators

F5 EXIT

2nd x-1 APPS ENTER

ALPHA ENTER

Solution

Casio CFX-9850GB PLUS

Sharp EL-9650

MENU F2

2nd x-1 APPS ENTER

ALPHA ENTER

Solution

Casio CFX-9850GB PLUS

All calculators

F2 EXIT

2nd x-1 APPS ENTER

ALPHA ENTER