Basic Mathematics for College Students

6.5 Interest 563









That is, during the second year, we earn interest on the interestas well as on the original
$2,000 principal. Using , we can find the interest earned in the second year.

This is the simple interest formula.

Substitute for P, r , and t.

Do the multiplication.

In the second year, $105 of interest is earned. The account now contains that interest
plus the $2,100 principal, for a total of $2,205.
As the figure below shows, we calculated the simple interest two times to find the
compound interest.

After 1 year, After another year,
calculate the simple calculate the simple
interest: interest:
$100 earned $105 earned
$2,000 $2,100 $2,205
Original principal New principal New principal
If we compute only the simple intereston $2,000, at 5% for 2 years, the interest
earned is. Thus, the account balance would be $2,200.
Comparing the balances, we find that the account earning compound interest will
contain $5 more than the account earning simple interest.
In the previous example, the interest was calculated at the end of each year, or
annually.When compounding, we can compute the interest in other time spans, such
as semiannually(twice a year),quarterly(four times a year), or even daily.

I$2,0000.05 2 $200

I$105

I$2,1000.05 1

IPrt

IPrt

EXAMPLE (^5) Compound Interest As a special gift for her newborn
granddaughter, a grandmother opens a $1,000 savings account in the baby’s name.
The interest rate is 4.2%, compounded quarterly. Find the amount of money the
child will have in the bank on her first birthday.
StrategyWe will use the simple interest formula four times in a series of
steps to find the amount of money in the account after 1 year. Each time, the time
tis.
WHYThe interest is compounded quarterly.
SolutionIf the interest is compounded quarterly, the interest will be computed
four times in one year. To find the amount of interest $1,000 will earn in the first
quarter of the year, we use the simple interest formula, where tis of a year.
This is the simple interest formula.
Substitute for P, r , and t.
Multiply: $1,000 0.042 $42.
Do the multiplication.
Do the division.
The interest earned in the first quarter is $10.50. This now becomes part of the
principal for the second quarter.
Add the original principal and the
interest that it earned to find the
second-quarter principal.
P2nd Qtr$1,000$10.50$1,010.50

I$10.50

I

$42

4

I$42

1

4

I$1,0000.042

1

4

IPrt

P1st Qtr$1,000 r4.2%0.042 t

1

4

Interest earned in the first quarter:

14

14

IPrt

Self Check 5

COMPOUND INTERESTSuppose

$8,000 is deposited in an account

that earns 2.3% compounded

quarterly. Find the amount of

money in an account at the end

of the first year.

Now TryProblem 33

10.5

(^4)  42.0
 4
02
 0
20
 20
0