Chapter 5

Section 5.1

Simple Interest

The Mathematics of Finance

In these sections we will learn about of the more practical uses of mathematics that is, the mathematics of finance. This is concerned with how mathematics is used to calculate interest and finance charges.

Terminology

The principal (P) represents the amount of money initially put into an investment such as a savings account or a loan. (This is sometimes referred to as the present value (PV) of the investment.

The amount of money earned by the investment is called the interest (I). This represents the profit for the person investing their money.

The rate (r) is the percentage of the principal earned on the investment over a certain time period. The rate is usually given on a yearly basis unless it is otherwise stated.

The term (t) is the length of time the investment is to last. It is important the time units for the term match the time units for the rate. In other words if the rate is on yearly basis the term must be measured in years.

The future value (FV) of an investment is the total value of all money in the investment including both principal and interest.

Simple Interest

When the interest earned in an investment is based on a straight percentage per-year money is invested we call that simple interest (or add on interest). For simple interest there are formulas that relate several of the previous ideas we mentioned.

Interest

When the rate (r) and the term (t) are both expressed in terms of years the interest (I) earned by an investment of principal (P) simple interest is given by:

PrtI Future Value

The future value (FV) of a simple interest investment of principle (P) for a term of t years at a yearly rate of r is given by the following formulas: )1( rtPFV

PrtPFV

IPFV

Example

A three-year CD pays 4.5% yearly rate and is a simple interest investment. Find the interest and future value of a $10,000 investment.

In this example we have P = 10,000 and r = .045 and t = 3.

350,13)045(.000,10 PrtI 350,11)3)045(.1(000,10)1(

350,11350,1000,10

rtPFV

IPFV

Changing Time Units and Dimensional Analysis

Because simple interest is based on a yearly rate and not all investment are for a full number of years we will need to be able to change certain lengths of time into years. This is done by a process called dimensional analysis.

Unit ConversionTo convert from one unit to another we use an idea called dimensional analysis or cancellation of units. You start with the units you have and on the other side of the equation you put the units you want to change to. You write down fraction or fractions with the correct units so they cancel and leave you with what is on the other side. For example if we wish to change 40 hours into weeks.

40 hour

Convert 40 hours into weeks.

1 day

24 hour

1 week

7 day

= 40

(24)(7)

week = .238095 week

cancel hours cancel days

A shoe company borrowed $12,000 on May 1st that they agreed to pay back at 8.4% simple interest on July 15th. Find the amount they must pay on July 15th.

The amount the shoe company pays on July 15th is the future value (FV). The principal P = 12,000 and the interest rate r = .084. The difficulty here is to find the term t.

There are 31 days in May, 30 days in June and 15 days in July the loan runs. This is a total of 76 days.

76 days

365 days

1 year=

365

76year 90.209,12

365

76)084(.1000,12

FV

A person buys a two year old used car for $9,000. They put $1,000 down and finance the rest at 7.2% for 48 months of add on interest. How much are their monthly payments. (Note: The interest here is add on! Later we will talk about compound.)

304,104)072(.1000,8 FVThe principal P = 9,000-1,000=8,000 is the amount they finance. The rate r = .072 the time is 4 years (i.e. 48 months = 4 years).

The future value (FV) is being divided into 48 monthly payments.

67.21448

304,10PaymentMonthly

Credit Card Financing and Average Daily Balance

The amount of money owed on a credit card changes during a month depending on charges and payments made during the month. The balance (i.e. the amount owed on the card) will go up and down. In order to keep things fair to both the customer and the credit card company the interest charged on the card for the month is for the average daily balance. In other words the average daily balance serves as the principal for the loan. The average daily balance is a weighted average of the current balances for the month.

To find the average daily balance we need to know what the beginning balance was and any payments and charges made during the month. We then fill in a table to find the number of days the account had a certain balance.

From the table we multiply each balance by the number of days at that balance add them up and divide by number of days in the period to get the average daily balance.

Period Begins Period Ends Number days Balance

Date 1 Date 2 d1 b1

Date 3 Date 4 d2 b2

n

nn

ddd

bdbdbd

21

2211balancedailyaverage

A credit card customer started on May 1st with a balance of 253.60 on their credit card. The transaction to the right were made throughout the month.

a) Find the average daily balance for the month of May.

b) If the customer is charge 18% simple interest find the finance charge for the month of May.

Transactions for May

May 4 Gas (charge) 45.00

May 8 Payment 120.00

May 15 Food (charge) 27.30

May 27 Books (charge) 98.41

Period Begins Period Ends Number of Days Balance

May 1 May 3 3 253.60

May 4 May 7 4 298.60

May 8 May 14 7 178.60

May 15 May 26 12 205.90

May 27 May 31 5 304.31

19.23231

75.7197

512743

31.304590.2051260.178760.298460.2533balancedailyaverage

a)

b) To find the simple interest (finance charge) P = 232.19, r = .18 and t = 365

31

55.3365

31)18(.19.232 I