Lesson 5 Money Matters Develop Understanding

Learning Focus

Model the growth of a savings account earning compound interest.

Solve exponential equations using logarithms.

What does the structure of an equation tell us about the context that it models and how to solve it?

Open Up the Math: Launch, Explore, Discuss

Part I: As an enterprising young mathematician, you know that your superior knowledge of mathematics will help you make better decisions about all kinds of things in your life. One important area is money $$$. So, you’ve been contemplating the world and wondering how you could maximize the money that you make in your savings account.

You’re young and you haven’t saved much money yet. As a matter of fact, you only have $$ 100$ , but you really want to make the best of it. You like the idea of compound interest, meaning that the bank pays you interest on all the money in your savings account, including whatever interest that they had previously paid you. This sounds like a very good deal. Let’s look a little closer at how this turns out.

1.

If your saving account pays a generous $5 %$ per year and is compounded only once each year, how much money would be in the account at the end of one year?

2.

Model the amount of money in the account after $t$ years. Include tables, graphs, and equations in your model.

3.

How much money will be in the account after $20 years$ ?

4.

When will the amount of money in the account be $$ 188.57$ ? Explain how you found your solution.

5.

Use a second method to find a solution to problem 4. Explain your second method.

Pause and Reflect:

A formula that is often used for calculating the amount of money in an account that is compounded annually is:

$A = P {(1 + r)}^{t}$ Where:

$A = amount of money in the account after t years$

$P = principal, the original amount of the investment$

$r = the annual interest rate$

$t = the time in years$

6.

Apply this formula using $$ 100$ as the initial investment and a $5 %$ interest rate, and compare the result to the model that you created.

7.

Based upon the work that you did in creating your model, explain the $(1 + r)$ part of the formula.

In our earlier work in the unit (You’re a Lizard Wizard) we thought about what happens if we look at exponential growth with smaller intervals. In the world of finance, banks often compound savings accounts more often than once a year.

8.

Predict how the formula for compound interest would change if the account were compounded monthly.

9.

The compound interest formula for any number of compounds per year is:

$A = P {(1 + \frac{r}{n})}^{n t}$

Where:

$A = amount of money in the account after t years$

$P = principal, the original amount of the investment$

$r = the annual interest rate$

$n = the number of compounding periods each year$

$t = the time in years$

a.

Explain why the interest rate in the base is divided by $n$ .

b.

Explain why the number of years in the exponent is multiplied by $n$ .

10.

If you start with $$ 100$ , and the bank pays $5 %$ compounded monthly, how much money will you have in your account after $20 years$ ?

11.

How does the amount of money in the account compounded monthly compare with the amount after $20 years$ when the account was compounded annually?

Just to make our lives a little easier, let’s say we could find an investment that pays $12 %$ a year, compounded monthly. (When you start looking at investments, you’ll find that it is nearly impossible to get this amount from a bank, but we’ll go ahead and dream big!)

12.

Use an algebraic method to solve each problem, assuming we are still starting with $$ 100$ :

a.

How much money will be in the account after $10 years$ ?

b.

When will the account have $$ 400$ in it?

c.

When will the account have $$ 1,000$ in it?

d.

How long will it take to double the money in the account?

Ready for More?

Use all your equation-solving skills to find the rate of interest on an account that is compounded quarterly, starts with $$ 500$ , and has $$ 805.16$ after $8 years$ .

Takeaways

Solving an exponential equation with a variable in the exponent:

Vocabulary

natural logarithm
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we modeled the growth of a savings account earning compound interest with an exponential function. We interpreted the compound interest formula to understand how it changes when the number of compounding periods each year is changed. We learned to use logarithms to solve exponential equations of any base when the variable is in the exponent.

Retrieval

Identify the type of function represented as linear, exponential, or quadratic.

1.

$f (x) = x^{2} + 3 x + 2$

2.

$n$	$g (n)$
$1$	$10$
$2$	$20$
$3$	$40$
$4$	$80$

3.

$x$	$h (x)$
$1$	$48$
$2$	$32$
$3$	$16$
$4$	$0$

Solve the following equations.

4.

$5^{x} = 625$

5.

$3^{x} = 729$