# Precalculus : Compound Interest Problems

## Example Questions

### Example Question #1 : Compound Interest Problems

If you invest  into a savings account which earns an interest rate  per year, how much would it take for your deposit to double?

Explanation:

The equation of the value for this problem is

.

We can divide by R to get

.

We want to solve for n in this case, which is the amount of years. If we use the natural log on both sides and properties of logarithms, we get

.

If we solve for n, we get

### Example Question #7 : Exponential Equations And Inequalities

If you deposit  into a savings account which earns a  yearly interest rate, how much is in your account after two years?

Explanation:

Since we are investing for two years with a yearly rate of 5%, we will use the formula to calculate compound interest.

where

is the amount of money after time.

is the principal amount (initial amount).

is the interest rate.

is time.

Our amount after two years is:

### Example Question #1 : Compound Interest Problems

If you deposit  into a savings account which compounds interest every month, what is the expression for the amount of money in your account after  years if you earn a nominal interest rate of  compounded monthly?

Explanation:

Since  is the nominal interest rate compounded monthly we write the interest term as  as it is the effective monthly rate.

We compound for  years which is  months. Since our interest rate is compounded monthly our time needs to be in the same units thus, months will be the units of time.

Plugging this into the equation for compound interest gives us the expression:

### Example Question #1 : Exponential Equations And Inequalities

John opens a savings account and deposits  into it. This savings account gains  interest per year. After  years, John withdraws all the money, and deposits it into another savings account with  interest per year.  years later, John withdraws the money.

How much money does John have after this  year period? (Assume compound interest in both accounts)

Explanation:

Plugging our numbers into the formula for compound interest, we have:

.

So John has about  after the first three years.

After placing his money into the other savings account, he has

after  more years.

So John has accumulated about .

### Example Question #1 : Compound Interest Problems

Suppose you took out a loan  years ago that gains  interest. Suppose that you haven't made any payments on it yet, and right now you owe  on the loan. How much was the loan worth when you took it out?

Explanation:

The formula for the compund interest is as follows:

By substuting known values into the compound interest formula, we have:

.

From here, substitute known values.

Divide by

### Example Question #1 : Compound Interest Problems

How many years does it take for  to grow into  when the is deposited into a savings account that gains  annual compound interest?

years

years

years

years

Explanation:

To find the number of years required, we solve the compound interest formula for .

The formula is as follows:

Substitute known values.

Divide by .

Take the natural log of both sides.

Use the log power rule.

Divide by .

use a calculator to simplify.