Section 4.1 – Simple Interest and Compound Interest: Future
Value and Present Value
Simple Interest
People can earn interest on investments, or may owe interest on loans. Interest is the amount of
money, in addition to the initial investment or loan amount, which is either paid to an investor or
charged by a lender.
The principal is the original amount of the investment or loan. Simple interest is the most
straightforward type of interest, since the interest on the investment or loan is based only on the
principal, and not based on any additional interest that accrues over time. Simple interest is
usually used for short-term investments.
We will first show an example of simple interest using a chart, and then give a formula for
simple interest which can be used for future problems.
Example 1: Norma invests $500 in an account which earns 8% simple interest per year.
A.
How much interest will she earn by the end of the third year?
B.
How much will she have in her account after three years?
Solution: The table below shows the interest earned each year, as well as the account balance at
the end of each year. Since the principal earns 8% per year, we multiply the principal by 0.08 to
find the annual interest.
Year Interest Earned Year-End Account Balance
1
500 ( 0.08 ) = 40
500 + 40 = 540
2
500 ( 0.08 ) = 40
540 + 40 = 580
3
500 ( 0.08 ) = 40
580 + 40 = 620
A.
Norma earns $40 per year for 3 years, so she will earn $120 in interest by the end of the
third year.
B.
Norma will have $620 in her account after three years.
***
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Section 4.1
We will now develop the formula for simple interest. Notice in Example 1 that Norma earns the
same amount each year. We could bypass the chart and conclude that the interest earned in three
years is 500 ( 0.08 )( 3) , or $120. The formula for simple interest is shown below.
Simple Interest Formula
I = Prt
where
I = interest
P = principal* (the original amount of the investment or loan)
r = interest rate (written as a decimal, rather than a percent)
t = time in years
*The principal will later be referred to as present value.
Example 2: How much simple interest will be earned on a $500 deposit in an account that earns
3.25% per year over a period of 7 months? (Round the answer to the nearest cent.)
Solution: P = 500 and r = 0.0325 . The time, 7 months, is
7
7
of a year, so t = .
12
12
7
I = Prt = ( 500 )( 0.0325 )   = 9.48
 12 
The simple interest earned is $9.48.
***
Example 3: Terry deposited $800 in a simple interest bearing account that earns 5.25% per year.
Today, the account’s balance is $821. When did she deposit the original amount?
Solution: P = 800 and r = 0.0525 , and I = 821 − 800 = 21 . We substitute these values into the
simple interest formula and then solve for t, as shown below:
I = Prt
21 = 800 ( 0.0525 )( t )
21 = 42t
t=
21 1
=
42 2
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Section 4.1
The time, t, is
1
1 6
of a year. Since = , Terry deposited the money 6 months ago.
2
2 12
***
Future Value with Simple Interest
In some real-world cases we may want to know the accumulated amount of an investment. The
accumulated amount is called the future value. The future value is the principal plus any
interest. We have already shown examples in this section with future value, though this
terminology was not used. In Example 1, the future value was $620.
Let F represent the future value. Then
F = P+I
= P + Prt
= P (1 + rt )
Future Value Formula, with Simple Interest
F = P (1 + rt )
where
F = future value
P = principal* (the original amount of the investment or loan)
r = interest rate (written as a decimal, rather than a percent)
t = time in years
*The principal will later be referred to as present value.
The term future value refers to the account value at some point in time after the initial
investment, regardless of whether that time is actually in the future. Recall the wording in
Example 3:
Terry deposited $800 in a simple interest bearing account that earns 5.25% per year.
Today, the account’s balance is $821. When did she deposit the original amount?
In Example 3, the account balance of $821 represents the future value (the principal plus the
interest), even though $821 refers to the value of the account today.
Example 4: Drew borrowed $750 at 10% simple interest per year. How much is due when the
loan matures in 6 months?
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Section 4.1
Solution: P = 750 and r = 0.10 . We want to find the future value in 6 months. The time, 6
1
months, is of a year, so t = 0.5 . Using the future value formula,
2
F = P (1 + rt ) = 750 1 + 0.10 ( 0.5 )  = 787.50
The amount due is $787.50.
Notice that the future value formula saves time, but is based on the simple interest formula. The
above example could also have been solved by first computing the interest, and then adding it to
the original loan amount, as follows: I = Prt = 750 ( 0.10 )( 0.5 ) = 37.50 . The total amount due
when the loan matures is 750 + 37.50 = 787.50 .
***
Example 5: Nancy deposits $1000 in an account earning 4.2% simple interest per year. How
much will she have in this account after 2 years?
Solution: P = 1000 , r = 0.042 , and t = 2 . Using the future value formula,
F = P (1 + rt ) = 1000 1 + 0.042 ( 2 )  = 1084
The future value is $1084.
***
Present Value with Simple Interest
In other cases we may be interested in finding out how much we need to invest now in order for
an account to have a certain value in the future. In this case, we are finding the principal which
can also be referred to as the present value.
Before giving the formula for present value, we will first show that it is simply a rearrangement
of the future value formula.
Example 6: Caroline is earning a rate of 6% simple interest per year at her bank. How much
does she need to deposit now in order to have $2000 in the account in 3 years?
Solution: We want to know the present value, P, which will yield a future value of $2000 in 3
years. We can substitute the known values into the future value formula, and then solve for P.
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Section 4.1
F = P (1 + rt )
2000 = P 1 + 0.06 ( 3) 
2000 = 1.18 P
2000
= 1694.92
P=
1.18
Caroline should invest $1694.92 now in order to have $2000 in 3 years.
***
It can save time if we have a formula which is already solved for P:
F = P (1 + rt ) is the formula for future value with simple interest.
Solving for P,
F
−1
P=
, or equivalently, P = F (1 + rt ) .
1 + rt
This formula is called the present value formula for simple interest.
Present Value Formula, with Simple Interest
P = F (1 + rt )
−1
where
F = future value
P = principal, or present value
r = interest rate (written as a decimal)
t = time in years
Example 7: Solve Example 6 by using the present value formula.
−1
−1
Solution: P = F (1 + rt ) = 2000 1 + 0.06 ( 3)  = 1694.92 . Caroline should invest $1694.92.
***
Earlier in this section, it was noted that the future value of an account does not always
correspond with a date in the future; it instead refers to the account value at some point in time
after the initial investment is made. In a similar way, present value does not always refer to
today’s value; it can instead refer to the value of the account at whatever time the investment was
made (i.e., the principal). This use of the present value formula is shown below in Example 8.
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Section 4.1
Example 8: Your simple interest bearing account has a current balance of $1830.60. It has
earned 2.55% simple interest per year for 8 months. How much did you originally invest?
Solution: We want to find P, the principal. We can also consider P to be the present value at the
time when the investment was made. The current account balance of $1830.60 corresponds with
the variable F for future value, since it represents the account value 8 months after the
8
8 2
investment was made. Since 8 months is
of a year, t =
= .
12
12 3
Using the present value formula with simple interest,
−1
P = F (1 + rt )
−1

 2 
= 1830.60 1 + 0.0255    = 1800
 3 

We can conclude that $1800 was originally invested in the account.
***
Compound Interest
Recall that for simple interest, the interest was based only on the original amount, the principal.
Compound interest, on the other hand, is interest charged or earned on the original principal
and also on any previously charged or earned interest.
Before learning the formula for compound interest, we will illustrate the concept with two
examples:
Example 9: Norma invests $500 at 8% per year compounded annually. This means that interest
is added to the account balance at the end of each year. How much will she have in her account
after three years? Solve this problem using a table which shows the interest earned during each
compounding period. (A formula will be shown later in this section, so that a table will not be
necessary.)
Solution: The table below shows the interest which compounds each year:
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Section 4.1
Year Interest Earned
Year-End Account Balance
1
500 ( 0.08 ) = 40
500 + 40 = 540
2
540 ( 0.08 ) = 43.20
540 + 43.20 = 583.20
3
583.20 ( 0.08 ) = 46.66
583.20 + 46.66 = 629.86
Norma will owe $629.86 after three years.
***
In this textbook, the number of compounding periods per year will be any one of: annually,
semiannually, quarterly, or monthly. An example of semiannual compounding is shown below.
Example 10: Norma invests $500 at 8% per year compounded semiannually. This means that
interest is added to the account balance at the end of each 6-month period. How much will she
have in her account after three years? Solve this problem using a table which shows the interest
earned during each compounding period. (A formula will be shown later in this section, so that a
table will not be necessary.)
Solution: The interest is compounded every 6 months, which is every 0.5 years. The interest will
be compounded six times during the 3 years (36 months) which it is invested. Since the interest
0.08
rate is 8% per year, the interest during each 6 month period is
, which is 0.04 .
2
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Section 4.1
Time Elapsed Interest Earned
Account Balance at the End
of Each Compounding Period
500 ( 0.04 ) = 20
500 + 20 = 520
520 ( 0.04 ) = 20.80
520 + 20.80 = 540.80
1.5 years
540.80 ( 0.04 ) = 21.63
540.80 + 21.63 = 562.43
2 years
562.43 ( 0.04 ) = 22.50
562.43 + 22.50 = 584.93
2.5 years
584.93 ( 0.04 ) = 23.40
584.93 + 23.40 = 608.33
3 years
608.33 ( 0.04 ) = 24.33
608.33 + 24.33 = 632.66
0.5 years
1 year
Norma will owe $632.66 after three years.
***
Future Value with Compound Interest
In the example above, the time period was only three years. If there were more compounding
periods (such as monthly compounding for 20 years), the problem could become tedious. We
have a formula that will quickly calculate the desired amount.
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Section 4.1
Future Value Formula, with Compound Interest
If interest is compounded m times per year, then
n
F = P (1 + i ) , where i =
r
and n = mt
m
F = future value
P = present value (principal)
i = interest rate per compounding period
r = interest rate (written as a decimal, rather than a percent)
m = number of compounding periods per year
n = the total number of compounding periods at time t
t = time in years
Example 11: Solve Example 10 using the future value formula for compound interest.
Solution: Recall: Norma invests $500 at 8% per year compounded semiannually. We want to
know how much she will have in her account after three years.
r
, n = mt , and m represents the number of
m
compounding periods per year. The formula is not as complicated as it seems if we think through
the meaning of the variables.
n
We want to use the formula F = P (1 + i ) , where i =
The principal, P, is $500.
We are told that Norma invests the money at an 8% yearly rate, compounded semiannually.
Semiannual compounding means that the interest will be compounded twice per year, so m = 2 .
i represents the interest rate per compounding period. This means that every 6 months, the
r 0.08
= 0.04 .
account earns 4% interest. This corresponds to the formula i = =
m
2
n represents the total number of compounding periods. The interest is compounded twice a year
( m = 2 ) for three years ( t = 3 ), so there are 6 total compounding periods, giving us the value
n = 6 . This corresponds to the formula n = mt = 2 ( 3) = 6 . We can see the 6 compounding
periods in the chart that we created for Example 10.
n
6
F = P (1 + i ) = 500 (1 + 0.04 ) = $632.66
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Section 4.1
Norma will have $632.66 in her account after three years. This is the same amount we obtained
using the chart in Example 10.
***
Note: In Examples 10 and 11, the answers were the same, to the nearest cent, when using the
chart or the formula to compute the answer. It is possible, however, to have very small
differences between answers using these two methods, due to the fact that we round all interest
and balances to the nearest cent as we progress from one compounding period to the other with
the chart method.
In Examples 1, 9, and 10, Norma invested $500 for 3 years at an 8% interest rate. However, each
of the three accounts used a different method for computing interest. Let us do a quick analysis
of the results for these three examples to see which method earns the most money.
Principal
Rate
Time
Interest Type
Future Value
Example 1
$500
8%
3 years
Simple
Interest
$620
Example 9
$500
8%
3 years
Annual
Compounding
$629.86
Example 10
$500
8%
3 years
Semiannual
Compounding
$632.66
Notice that compounded interest (Examples 9 and 10) earned more than simple interest
(Example 1), since Norma was earning interest on her interest with compounding. Furthermore,
notice that the semiannual compounding earned more than the annual compounding. Given the
same principal, interest rate, and time, more money will be earned when there are more
compounding periods per year, since the ‘interest on the interest’ is compounded more often.
Example 12: Sally deposited $7000 in an Individual Retirement Account (IRA) that pays 3.25%
per year compounded monthly. How much will she have in this account at the end of 25 years
when she retires?
Solution: The interest is compounded monthly, which is 12 times per year. Therefore, m = 12 .
i is the interest rate for each of the monthly compounding periods, so i =
r 0.0325
=
.
m
12
n represents the total number of compounding periods. The interest is compounded 12 times per
year for 25 years, so n = mt = 12 ( 25) = 300 .
Using the future value formula for compound interest,
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Section 4.1
n
 0.0325 
F = P (1 + i ) = 7000 1 +

12 

300
= 15, 757.43
Sally’s IRA balance in 25 years will be $15,757.43.
***
Example 13: Ted found his dream job and must move across the country. He realizes he needs
to borrow money for moving expenses. His local credit union loans him $5000 at 6.3% per year
compounded quarterly. How much will he owe the credit union when his loan matures in 2
years? Assume that he makes no payments until the total balance comes due.
Solution: The interest is compounded quarterly, which is 4 times per year. Therefore, m = 4 .
i is the interest rate for each of the quarterly compounding periods, so i =
r 0.063
=
.
m
4
n represents the total number of compounding periods. The interest is compounded 4 times per
year for 2 years, so n = mt = 4 ( 2 ) = 8 .
Using the future value formula for compound interest,
8
F = P (1 + i )
n
 0.063 
= 5000 1 +
 = 5665.84
4 

Ted will owe $5665.84.
***
Present Value with Compound Interest
When working with compound interest, there are times when we are interested in finding the
present value, or principal. It can save time if we have a formula which is already solved for P:
n
F = P (1 + i ) is the formula for future value with simple interest.
Solving for P,
F
−n
P=
, or equivalently, P = F (1 + i ) .
n
(1 + i )
This formula is called the present value formula for compound interest.
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Section 4.1
Present Value Formula, with Compound Interest
If interest is compounded m times per year, then
−n
P = F (1 + i ) , where i =
r
and n = mt
m
P = present value (principal)
F = future value
i = interest rate per compounding period
r = interest rate (written as a decimal, rather than a percent)
m = number of compounding periods per year
n = the total number of compounding periods at time t
t = time in years
Example 14: Heather would like to have $40,000 in 18 years to give to her niece to help pay for
her college education. Her local bank has an account that pays 2.55% per year compounded
semiannually. How much must Heather deposit today in this account to have the desired funds in
18 years?
Solution: Heather wants the account to have a future value of $40,000, so F = 40, 000 .
The interest is compounded semiannually, which is 2 times per year. Therefore, m = 2 .
i is the interest rate for each of the semiannual compounding periods, so i =
r 0.0255
=
.
m
2
n represents the total number of compounding periods. The interest is compounded 2 times per
year for 18 years, so n = mt = 2 (18) = 36 .
Using the present value formula for compound interest,
P = F (1 + i )
−n
 0.0255 
= 40, 000 1 +

2 

−36
= 25, 350.06
Heather should deposit $25,350.06.
***
Example 15: Craig recently bought a lake house and now wishes to buy a sailboat. He wishes to
have a down payment of $3000 for the sailboat in 3 years. How much does he need to deposit
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Section 4.1
now in an account that pays 4% per year compounded monthly to have the desired funds in 3
years?
Solution:
Craig wants the account to have a future value of $3000, so F = 3000 .
The interest is compounded monthly, which is 12 times per year. Therefore, m = 12 .
i is the interest rate for each of the monthly compounding periods, so i =
r 0.04
=
.
m 12
n represents the total number of compounding periods. The interest is compounded 12 times per
year for 3 years, so n = mt = 12 ( 3) = 36 .
Using the present value formula for compound interest,
P = F (1 + i )
−n
 0.04 
= 3000 1 +

12 

−36
= 2661.29
Craig should deposit $2661.29.
***
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Section 4.1