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Section 8.3 – Simple Interest
Objective #1: Calculating simple interest.
When a person gets a loan from the bank, that person has to pay back the
amount of the loan plus interest. The interest is the fee that the person has
to pay for using the money from the bank. In the same way, if a person
opens a savings account, the bank pays that person interest since the bank
gets to use that person’s money while it is in the account.
The formula for simple interest is:
I = Prt
The Interest, I, is the fee paid for borrowing money. The amount of the loan
or the amount in a savings account is called the principal, P. The interest
rate, r, is the rate that interest is charged on the amount borrowed
converted to a decimal or fraction. The interest rate is usually given as a
percentage and is assumed to be a yearly percentage unless it is states
otherwise. The amount of time, t, is the length of time of the loan in years.
In instances where the interest rate is given as monthly interest rate or a
daily interest rate, then correspondingly, the time will have to be in months
or in days. The amount, A, that is paid back is the principal plus interest
(A = P + I).
Many financial institutions do not like the fact that the number of days in a
month is inconsistent. Some months have 31 days, some have 30, some
have 28 and occasionally, there is a month that has 29 days. To avoid
these inconsistencies, banks will assume that there are 30 days in a month.
As a consequence, they will then assume that there are
12•30 = 360 days in a year. This is known as “Banker’s Rule”:
Banker’s Rule:
1 month = 30 days
1 year = 12 months = 360 days
As you can imagine, there is potential for great confusion between financial
institutions and the consumer since each has a different definition of what a
month is. To avoid confusion, many times these institutions will state the
length of time in days rather than in months. For example, many stores will
offer “90 days same as cash” instead of “three months same as cash” since
three months to financial institution is not the same as three months for a
consumer. By offering it in days, everyone understands the exact length
being discussed. Now, let’s try some examples:
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Solve the following:
Ex. 1
Max borrowed $4,500 at 6% simple interest for 2 years. What
was the interest he had to pay and the total he had to pay back?
Solution:
In this problem, P = $4,500, r = 6% = 0.06, and t = 2 years. Plugging
into I = Prt, we get:
I = ($4500)(0.06)(2) = $540
The amount to be paid back, A, is principal plus interest which is
A = $4500 + $540 = $5040.
So, the interest is $540 and the amount he had to pay back was
$5040.
Ex. 2
Alisa had a loan of $600 at 7.8% simple interest for 120 days.
How much did she have to pay back?
Solution:
In this problem, P = $600, r = 7.8% = 0.078, and t = 120 days.
Before we plug into the formula, we will need to convert the days
into years. Using Banker’s Rule,
t = 120 days = (120 days) / (360 days) =
Plugging into I = Prt, we get:
I = ($600)(0.078)(
1
)
3
1
3
years.
= $15.60
So, the interest is $15.60. The amount to be paid back, A, is
principal plus interest which is A = $600 + $15.60 = $615.60.
Thus, she had to pay back $615.60.
€
Objective #2: Using the future value formula.
The amount a person has to payback A = P + I. Since I = Prt, we can
rewrite this as A = P + I = P + Prt or P(1 + rt). The amount to be paid back,
A, is called the future value and the principal, P, is called the present value.
Future Value for Simple Interest
The future value, A, of p dollars at a simple interest rate r (as a decimal)
for t years is given by A = P(1 + rt).
Solve the following:
Ex. 3
A loan of $3500 is made at a simple interest rate of 4.5% for
three months. Find the loan's future value.
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Solution:
In this problem, P = $3500, r = 4.5% = 0.045, and t = 3/12 = 0.25
years. Plugging into the formula A = P(1 + rt), we get:
A = 3500(1 + (0.045)(0.25)) = 3500(1.01125) ≈ $3539.38
The future value is $3539.38.
Ex. 4
Alejandro borrowed $3600 for nine months. If he paid back
$3863.25, what was the simple interest rate?
Solution:
Since P = $3600, t = 9/12 = 0.75 years, and A = $3863.25, we can
plug into A = P(1 + rt) and solve for r:
3863.25 = 3600(1 + r(0.75)) (distribute)
3863.25 = 3600 + 2700r
(subtract 3600 from both sides)
263.25 = 2700r
(divide both sides by 2700)
0.0975 = r or r = 0.0975 = 9.75%
The interest rate is 9.75%.
Ex. 5
Juanita has to repay $6,809.60 for the principal she borrowed
at 7.2% for 3 years. How much was the principal?
Solution:
Since A = $6809.60, r = 7.2% = 0.072, and t = 3 years, we can
plug into A = P(1 + rt) and solve for P:
6809.60 = P(1 + 0.072(3))
6809.60 = 1.216P
(divide both sides by 1.216)
5600 = P or P = $5600
The principal was $5600.
Ex. 6
Mike received a payday loan of $260. If, at the end of four
weeks, he paid off his balance of $328.26, what was the annual
simple interest rate of the loan?
Solution:
Since P = $260, t = 4/52 =
1
13
years, and A = $328.26, we can
plug into A = P(1 + rt) and solve for r:
1
328.26 = 260(1 + r( ))
(distribute)
13
328.26 = 260
(subtract 260 from both sides)
€ + 20r
68.26 = 20r
(divide both sides by 20)
3.413 = r or r = 3.413 = 341.3%
The interest rate€is 341.3%.