Intermediate Algebra (11th edition)

Solving a Compound Interest Problem for t

Suppose inflation is averaging 3% per year. Approximate the time it will take for

prices to double. Round to the nearest hundredth.

We want the number of years tfor Pdollars to grow to 2Pdollars at a rate of 3%

per year. In the compound interest formula, we substitute 2Pfor A, and let

and

Substitute in the compound

interest formula.

Divide by P. Simplify.

Property 3

Power rule

Interchange sides. Divide by log 1.03.

Use a calculator.

Prices will double in about 23.45 yr. (This is called the doubling timeof the money.)

To check, verify that 1.0323.45L2.

tL23.45

t=

log 2

log 1.03

log 2=t log 1 1.03 2

log 2=log 1 1.03 2 t

2 = 1 1.03 2 t

2 P=P a 1 +

0.03

1

b

1 t

n= 1.

r= 0.03

EXAMPLE 7

SECTION 10.6 Exponential and Logarithmic Equations; Further Applications 617

NOW TRY

Interest can be compounded annually, semiannually, quarterly, daily, and so on. The

number of compounding periods can get larger and larger. If the value of nis allowed to

approach infinity, we have an example of continuous compounding.The formula for

continuous compounding is derived in advanced courses, and is an example of expo-

nential growth involving the number e.

Solving a Continuous Compound Interest Problem

In Example 6we found that $1000 invested for 5 yr at 3% interest compounded quar-

terly would grow to $1161.18.

(a)How much would this same investment grow to if interest were compounded

continuously?

Continuous compounding formula

Let and

Multiply in the exponent.

Use a calculator. Round to the nearest cent.

Continuous compounding would cause the investment to grow to $1161.83. This is

$0.65 more than the amount the investment grew to in Example 6,when interest was

compounded quarterly.

A= 1161.83

A= 1000 e0.15

A= 1000 e0.03^152 P=1000, r=0.03, t=5.

A= Pert

EXAMPLE 8

Continuous Compound Interest Formula

If a principal of Pdollars is deposited at an annual rate of interest rcompounded

continuously for tyears, the final amount Aon deposit is given by

APert.

NOW TRY
EXERCISE 7
Approximate the time it
would take for money
deposited in an account paying
4% interest compounded
quarterly to double. Round to
the nearest hundredth.

NOW TRY ANSWER
7.17.42 yr