Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 19: Exponents and Logarithms

Lesson 19.2: Show Me the Money

In this Grade 4 AoPS Introduction to Algebra lesson, students learn the difference between simple interest and compound interest, including how to apply the compound interest formula when interest is compounded annually or multiple times per year. Using real-world loan and investment scenarios, students practice calculating total amounts owed using the expressions (1 + r/100)^n · k and (1 + r/100m)^(nm) · k. The lesson builds algebraic fluency with exponents in a financial context as part of Chapter 19's focus on Exponents and Logarithms.

Section 1

Simple Interest

Property

If an amount of money, $P$ , called the principal, is invested for a period of $t$ years at an annual interest rate $r$ , the amount of interest, $I$ , earned is

I = Prt

where

$I$ = interest
$P$ = principal
$r$ = rate
$t$ = time

Interest earned according to this formula is called simple interest.

Examples

How much interest will be earned on a principal of 8,000 dollars invested at an interest rate of 3% for 5 years?

Using the formula $I = Prt$ , we substitute the values: $I = (8000)(0.03)(5) = 1200$ . The interest earned is 1,200 dollars.

A loan of 4,000 dollars was repaid with 640 dollars in interest after 2 years. What was the interest rate?

Using $I = Prt$ , we have $640 = (4000)(r)(2)$ . This simplifies to $640 = 8000r$ . Solving for $r$ gives $r = 0.08$ , so the interest rate was 8%.

Section 2

Compound Interest

Property

If a principal of $P$ dollars is invested at an annual interest rate $r$ (expressed as a decimal) compounded $n$ times yearly, the amount $A$ after $t$ years is given by

A = P\left(1 + \frac{r}{n}\right)^{nt}

Examples

To find the amount after 3 years when 1000 dollars is invested at 5% compounded annually, use $A = 1000(1 + \frac{0.05}{1})^{1 \cdot 3} = 1000(1.05)^3 \approx 1157.63$ dollars.
If you invest 500 dollars at 6% interest compounded monthly for 10 years, the formula is $A = 500(1 + \frac{0.06}{12})^{12 \cdot 10} = 500(1.005)^{120}$ .
For a 2000 dollars investment at 4% compounded quarterly for 5 years, the amount would be $A = 2000(1 + \frac{0.04}{4})^{4 \cdot 5} = 2000(1.01)^{20}$ .

Explanation

This formula calculates how your money grows when interest is added multiple times per year. The more frequently it's compounded, the faster your investment grows because you start earning interest on your previously earned interest.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Simple Interest

Property

If an amount of money, $P$ , called the principal, is invested for a period of $t$ years at an annual interest rate $r$ , the amount of interest, $I$ , earned is

I = Prt

where

$I$ = interest
$P$ = principal
$r$ = rate
$t$ = time

Interest earned according to this formula is called simple interest.

Examples

How much interest will be earned on a principal of 8,000 dollars invested at an interest rate of 3% for 5 years?

Using the formula $I = Prt$ , we substitute the values: $I = (8000)(0.03)(5) = 1200$ . The interest earned is 1,200 dollars.

A loan of 4,000 dollars was repaid with 640 dollars in interest after 2 years. What was the interest rate?

Using $I = Prt$ , we have $640 = (4000)(r)(2)$ . This simplifies to $640 = 8000r$ . Solving for $r$ gives $r = 0.08$ , so the interest rate was 8%.

Section 2

Compound Interest

Property

If a principal of $P$ dollars is invested at an annual interest rate $r$ (expressed as a decimal) compounded $n$ times yearly, the amount $A$ after $t$ years is given by

A = P\left(1 + \frac{r}{n}\right)^{nt}

Examples

To find the amount after 3 years when 1000 dollars is invested at 5% compounded annually, use $A = 1000(1 + \frac{0.05}{1})^{1 \cdot 3} = 1000(1.05)^3 \approx 1157.63$ dollars.
If you invest 500 dollars at 6% interest compounded monthly for 10 years, the formula is $A = 500(1 + \frac{0.06}{12})^{12 \cdot 10} = 500(1.005)^{120}$ .
For a 2000 dollars investment at 4% compounded quarterly for 5 years, the amount would be $A = 2000(1 + \frac{0.04}{4})^{4 \cdot 5} = 2000(1.01)^{20}$ .

Explanation

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Simple Interest

Property

If an amount of money, $P$ , called the principal, is invested for a period of $t$ years at an annual interest rate $r$ , the amount of interest, $I$ , earned is

I = Prt

where

$I$ = interest
$P$ = principal
$r$ = rate
$t$ = time

Interest earned according to this formula is called simple interest.

Examples

How much interest will be earned on a principal of 8,000 dollars invested at an interest rate of 3% for 5 years?

Using the formula $I = Prt$ , we substitute the values: $I = (8000)(0.03)(5) = 1200$ . The interest earned is 1,200 dollars.

A loan of 4,000 dollars was repaid with 640 dollars in interest after 2 years. What was the interest rate?

Using $I = Prt$ , we have $640 = (4000)(r)(2)$ . This simplifies to $640 = 8000r$ . Solving for $r$ gives $r = 0.08$ , so the interest rate was 8%.

Section 2

Compound Interest

Property

If a principal of $P$ dollars is invested at an annual interest rate $r$ (expressed as a decimal) compounded $n$ times yearly, the amount $A$ after $t$ years is given by

A = P\left(1 + \frac{r}{n}\right)^{nt}

Examples

To find the amount after 3 years when 1000 dollars is invested at 5% compounded annually, use $A = 1000(1 + \frac{0.05}{1})^{1 \cdot 3} = 1000(1.05)^3 \approx 1157.63$ dollars.
If you invest 500 dollars at 6% interest compounded monthly for 10 years, the formula is $A = 500(1 + \frac{0.06}{12})^{12 \cdot 10} = 500(1.005)^{120}$ .
For a 2000 dollars investment at 4% compounded quarterly for 5 years, the amount would be $A = 2000(1 + \frac{0.04}{4})^{4 \cdot 5} = 2000(1.01)^{20}$ .