ASIAN DEVELOPMENT FOUNDATION COLLEGE
P. BURGOS STREET.
TACLOBAN CITY
S.Y. 2020-2021
MODULE 7
MATHEMATICS IN
THE
MODERN WORLD
Distance Learning for an Individualized Learning Instructions
FREDERICK NORADA LIPON
ASIAN DEVELOPMENT FOUNDATION COLLEGE
P. BURGOS STREET
TACLOBAN CITY
S.Y. 2020-2021
Program Title:
Subject: MATHEMATICS IN THE MODERN WORLD
Topic/s: SIMPLE AND COMPOUND INTEREST
Module: 7
LEARNING TARGETS:
a. Define the concept of interest and show how it relates to the time value of money.
b. Distinguish between simple and compound interest and demonstrate how to calculate
each.
c. Solve the following problems in Simple and Compound interest,
SIMPLE AND COMPOUND INTEREST
Interest on loans of a year or less id frequently calculated as SIMPLE INTEREST, is a
type of interest that is charged (or paid) only on the amount borrowed (or invested) and not on
the past Interest. The amount borrowed is called the principal. The rate of interest is given as a
percentage per year, expressed as a decimal. For example, 6%= 0.06 and 11 ½ % = 0.115. The
time the money is earning interest is calculated in years. One year’s interest is calculated by
multiplying the principal times the interest rate, or Pr. If the time that the money earns interest is
other than one year, we multiply the interest for one year by the number of years, or Prt.
SIMPLE INTEREST
I= Prt
WHERE;
P = is the principal;
r = is the annual interest rate (expressed as a decimal);
t = is the time in years
EXAMPLE 1: SIMPLE INTEREST
To buy furniture for a new apartment, Pamela Shipley borrowed $ 5000 at 8% simple interest for
11 months. How much interest will she pay?
SOLUTION:
Since 8% is the yearly interest rate, we need to know the time of the loan in years. We can convert
11 months into years by dividing 11 months by 12 (the number of months per year). Use the
formula I=Prt, with P= $ 5000, r= 0.08, t= 11/12 (in years). The total interest she will pay is
I= 5000(0.08) (11/12)
=366.67 or $366.67
A deposit of P dollars today at a rate of interest r for t years produces interest of I=Prt. The interest,
added to the original principal P, gives
P + Prt = P(1+rt)
This amount is called the future value of P dollars at an interest rate r for the time t in years.
When loans are involved, the future value is often called the maturity value of the loan. This idea
is summarized as follows.
Future or Maturity Value for Simple Interest
The future or maturity value A of P at a simple interest rate r for t years is
A= P(1+rt)
EXAMPLE 2: Maturity Values
Find the maturity value for each loan at simple interest.
(a). A loan of $2500 to be repaid in 8 months with an interest of 4.3%
SOLUTION:
The loan is for 8 months, 0r 8/12= 2/3 of a year. The maturity value is
A = P(1+rt) P= $2500, r= 0.043, t= 2/3
= 2500[ 1+0.043(2/3)]
=2500(1+0.028667)
= 2571. 67
Or $2571.67. (The answer is rounded to the nearest cent, as is customary in financial problems).
Of this maturity value,
I=A-P
$2571.67-$2500 = $71.67 represents the interest.
(b). A loan of $11, 280 for 85 days at 7% interest
SOLUTION:
It is common to assume 360 days in a year when working with simple interest. We shall usually
make such as assumption. Using P= 11, 280, r= 0.07, and t= 85/360, the maturity value in this
example is
A= 11, 280[ 1+0.07(85/360)] = 11, 466.43 or $11,466.43
CAUTION When using the formula for future value, as well as the other formulas in this
chapter, we often neglect the fact that in real life, money amounts are rounded to
the nearest penny. As some consequences, when the amounts are rounded, their
values may differ by a few cents from the amounts given by these formulas. For instance, in
Example 2(a), the interest in each monthly payment would be $2500(0.043/12) = $8.96, rounded
to the nearest penny. After 8 months, the total is 8($8.96) =$71.68, which is 1¢ more than we
computed in the example.
In part (b) of example 2 we assumed 360 days in a year. Historically, to simplify calculations, it
was often assumed that each year had twelve 30-day months., making a year 360 days long.
Treasury bills sold by the U.S. government assume a 360 day in calculating interest. Interest
found using a 360 day is called ordinary interest., and interest found using a 365-day year is called
exact interest.
The formula for future value has four variables, P, r, t, and A. we can use the formula to find any
of the quantities that these variables represent, as illustrated in the next example.
EXAMPLE 3: Alicia Rinkie wants to borrow $8000 from Robyn Martin. She is willing to pay bak
$8180 in 6 months. What interest rate will she pay?
SOLUTION:
Use the formula for the future value, with A= 8180, t= 6/12=0.5 and solve for r=?
A= P(1+rt)
8180= 8000(1+0.5r)
8180=800+4000r distributive property
180=4000r subtract 8000
R= 0.045 divide by 4000.
Thus, the interest rate is 4.5% (written as a percent)
When you deposit money in the bank and earn interest, it is as if the bank borrowed the
money from you. Reversing the scenario in Example 3, if you put $8000 in a bank account that
pays simple interest at a rate of 4.5% annually, you will have accumulated $8180 after 6 months.
COMPOUND INTEREST
As mentioned earlier, simple interest is normally used for loans or investments of a year or less.
For longer periods compound interest is used. With compound interest, interest is charged (or
paid) on interest as well as on principal. For example: if $1000 is deposited at 5% interest for 1
year, at the end of the year the interest is $1000(0.05) (1) = $50. The balance in the account is
$1000+$50= $1050. If this amount is left at 5% interest for another year, the interest is calculated
on $1050 instead of the original $1000, so the amount in the account at the end of the second
year is $1050+1050(0.05) (1) = $1102. 50. Note that simple interest would produce a total amount
of only
$1000[1+(0.05) (2)] = $1100.
The additional $2.50 is the interest on 50 at 5%for one year.
To find the formula for compound interest, first suppose that P dollars is deposited at a rate of
interest r per year. The amount on deposit at the end of the first year is found by the simple
interest formula, with t= 1
A= P(1+r.1) = P(1+r)
If the deposit earns compound interest, the interest earned during the second year is paid on
the total amount on the deposit at the end of the first year. Using the formula, A= P(1+rt) again,
with P replaced by P (1+r) and t= 1, gives the total amount on deposit at the end of the second
year.
A= [P(1+r)] (1+r.1) = P(1+r)2
In the same way, the total amount on deposit at the end of the year is
P(1+r)3
Generalizing, if P is the initial deposit, in t years the total amount on deposit is
A= P(1+r) t
Called the compound interest
Note: Compare this formula for compound interest with the formula for simple interest.
Compound Interest A= P(1+r) t
Simple Interest A= P(1+rt)
The important distinction between the two formulas is that in the compound interest formula, the
number of years, t, is an exponent, so that money grows much rapidly when interest is
compounded.
COMPOUND AMOUNT
A= P(1+i) n
Where i= r/m and n=mt
A is the future (maturity) value,
P is the principal
r is the annual interest rate
m is the number of compounding periods per year
t is the number of years
n is the number of compounding periods
i is the interest rate per period.
EXAMPLE 4: Compound Interest
Suppose $1000 is deposited for 6 years in an account paying 4. 25% per year compounded
annually.
(a). Find the compound amount.
SOLUTION
Since interest is compounded annually, the number of compounding periods per year is m = 1.
The interest rate per period is i=r/m 0.045/1 = 0.0425 and the number of compounding periods is
n=mt=1(6) = 6. (Notice that when interest is compounded annually, i= r and n=t). Using the formula
for the compound amount with P=1000, i=0.0425, and n=6 gives
A= P(1+i) n
= 1000 (1+0.0425)6
= 1000(1.0425) 6
= 1283.68 or $1283.68
(b). Find the amount of Interest earned.
SOLUTION:
Subtract the initial deposit from the compound amount.
I= A – P = $1283.68-$1000= 283.68
EXAMPLE 5
If 60000 amounts to 68694 in 2 years, the find the rate of the interest.
SOLUTION:
Given: A= 68694
P= 60000
N= 2 years
R=?
A= P(1+r/100) ^n
68694 = 60000(1+r/100) ^2
68694/60000 = (1+r/100) ^2
11449/10000 =(1+r/100) ^2
1+r/100= square root (11449/10000) = square root of 1.1449
1+r/100=1.07
r/100=1.07-1= 0.07
r= 0.07x100= 7%
WORKSHEET SIMPLE INTEREST
Use simple interest to find the ending balance (Show your solution)
1. $34,100 at 4% for 3 years
2. $210 at 8% for 7 years
3. $4,000 at 3% for 4 years
4. $20,600 at 8% for 2 years
5. $14,000 at 6% for 9 years
6. Mitch took out a school loan for $24, 000. The interest rate on the loan was 8%. Mitch paid
off the loan in just 5 years. How much did she pay together?
7. If the sum of money produces $3900 as interest in 3 years and 3 months at 16% per year
simple interest, find the principal.
8. In how many years will $400 yield an interest of $112 at 14% simple interest.
9. In how many years will $1200 yield an interest of $13230 at 10% simple interest.
10. Trevor deposits $1500 in a simple interest account that pays 4.3% interest annually. After
15 years, how much total money would be in the account?
WORKSHEET COMPOUND INTEREST
1. $500 Invested at 4% compounded annually for 10 years
2. $600 invested at 6% compounded annually for 6 years
3. $750 invested at 3% compounded annually for 8 years
4. If you deposit $6500 into an account paying 8% annual interest compounded monthly,
how much money will be in the account after 7 years?
5. How much money would you need to deposit today at 9% annual interest compounded
monthly to have $ 1200 in the account after 6 years.?
FORMULAS
SIMPLE INTEREST I= Prt
T= I/(PxR)
P=I(RxT)
R=I/(PXT)
COMPOUND INTEREST A= P(1+i) n
NOTE: WRITE YOUR ANSWER IN A PAPER OR BOND PAPER. DON’T FORGET TO SHOW
YOUR SOLUTIONS.
#KEEPSAFE
#God Bless
