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

Lesson 19.3: Interest-ing Problems

In this Grade 4 AoPS Introduction to Algebra lesson, students apply compound interest formulas to solve real-world problems involving future value, present value, and unknown interest rates. Using the compound interest formula and its inverse, learners practice isolating variables with exponents to find quantities such as the present value of a future sum or the annual percentage rate on a loan. This lesson is part of Chapter 19's coverage of exponents and their practical applications within the AMC 8 and AMC 10 competition math context.

Section 1

Solving Complex Interest Rate Equations

Property

To find unknown interest rates in complex scenarios, set up equations using the compound interest formula $A = P(1 + r)^n$ and solve by isolating the interest rate term: $(1 + r)^n = \frac{A}{P}$ , then $r = \left(\frac{A}{P}\right)^{\frac{1}{n}} - 1$

Examples

Section 2

Compound Interest with Compounding Periods

Property

The future value $A$ of a principal amount $P$ (present value) after $t$ years with an annual interest rate $r$ compounded $n$ times per year is given by the formula:

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

To find the present value $P$ needed to achieve a future amount $A$ , the formula is rearranged:

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

Examples

To find the future value of a 1,000 dollar investment after 5 years at a 6% annual rate compounded semi-annually ( $n=2$ ), you calculate:

A = 1000\left(1 + \frac{0.06}{2}\right)^{2 \times 5} = 1000(1.03)^{10} \approx 1343.92 \text{ dollars}

To find the present value you must invest to have 5,000 dollars in 3 years at a 4% annual rate compounded quarterly ( $n=4$ ), you calculate:

P = \frac{5000}{\left(1 + \frac{0.04}{4}\right)^{4 \times 3}} = \frac{5000}{(1.01)^{12}} \approx 4437.04 \text{ dollars}

Explanation

This formula is an extension of simple compound interest, accounting for interest being calculated more than once a year. The annual rate $r$ is divided by the number of compounding periods $n$ to get the rate per period. The exponent $nt$ represents the total number of times interest is compounded over the investment''s lifetime. This skill allows for more precise calculations of future and present value in real-world financial scenarios where interest is often compounded semi-annually, quarterly, or even monthly.

Section 3

Present Value Calculations

Property

Present value is the current worth of a future amount of money, calculated using:

PV = \frac{FV}{(1 + r)^n}

where

PV

is present value,

FV

is future value,

r

is the interest rate (as a decimal), and

n

is the number of compounding periods.

Examples

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 19: Exponents and Logarithms

Back to AoPS: Introduction to Algebra (AMC 8 & 10)

Lesson 1
Lesson 19.1: Exponential Functions
Learn Practice
Lesson 2
Lesson 19.2: Show Me the Money
Learn Practice
Lesson 3Current
Lesson 19.3: Interest-ing Problems
Learn Practice
Lesson 4
Lesson 19.4: What is a Logarithm?
Learn Practice

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Solving Complex Interest Rate Equations

Property

Examples

Section 2

Compound Interest with Compounding Periods

Property

The future value $A$ of a principal amount $P$ (present value) after $t$ years with an annual interest rate $r$ compounded $n$ times per year is given by the formula:

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

To find the present value $P$ needed to achieve a future amount $A$ , the formula is rearranged:

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

Examples

To find the future value of a 1,000 dollar investment after 5 years at a 6% annual rate compounded semi-annually ( $n=2$ ), you calculate:

A = 1000\left(1 + \frac{0.06}{2}\right)^{2 \times 5} = 1000(1.03)^{10} \approx 1343.92 \text{ dollars}

To find the present value you must invest to have 5,000 dollars in 3 years at a 4% annual rate compounded quarterly ( $n=4$ ), you calculate:

P = \frac{5000}{\left(1 + \frac{0.04}{4}\right)^{4 \times 3}} = \frac{5000}{(1.01)^{12}} \approx 4437.04 \text{ dollars}

Explanation

Section 3

Present Value Calculations

Property

Present value is the current worth of a future amount of money, calculated using:

PV = \frac{FV}{(1 + r)^n}

where

PV

is present value,

FV

is future value,

r

is the interest rate (as a decimal), and

n

is the number of compounding periods.

Examples

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 19: Exponents and Logarithms

Back to AoPS: Introduction to Algebra (AMC 8 & 10)

Lesson 1
Lesson 19.1: Exponential Functions
Learn Practice
Lesson 2
Lesson 19.2: Show Me the Money
Learn Practice
Lesson 3Current
Lesson 19.3: Interest-ing Problems
Learn Practice
Lesson 4
Lesson 19.4: What is a Logarithm?
Learn Practice

Overview

Lesson overview

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

Overview

Section 1

Solving Complex Interest Rate Equations

Property

Examples

Section 2

Compound Interest with Compounding Periods

Property

The future value $A$ of a principal amount $P$ (present value) after $t$ years with an annual interest rate $r$ compounded $n$ times per year is given by the formula:

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

To find the present value $P$ needed to achieve a future amount $A$ , the formula is rearranged:

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

Examples

To find the future value of a 1,000 dollar investment after 5 years at a 6% annual rate compounded semi-annually ( $n=2$ ), you calculate:

A = 1000\left(1 + \frac{0.06}{2}\right)^{2 \times 5} = 1000(1.03)^{10} \approx 1343.92 \text{ dollars}

To find the present value you must invest to have 5,000 dollars in 3 years at a 4% annual rate compounded quarterly ( $n=4$ ), you calculate:

P = \frac{5000}{\left(1 + \frac{0.04}{4}\right)^{4 \times 3}} = \frac{5000}{(1.01)^{12}} \approx 4437.04 \text{ dollars}

Explanation

Section 3

Present Value Calculations

Property

Present value is the current worth of a future amount of money, calculated using:

PV = \frac{FV}{(1 + r)^n}

where

PV

is present value,

FV

is future value,

r

is the interest rate (as a decimal), and

n

is the number of compounding periods.

Examples

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 19: Exponents and Logarithms

Back to AoPS: Introduction to Algebra (AMC 8 & 10)

Lesson 1
Lesson 19.1: Exponential Functions
Learn Practice
Lesson 2
Lesson 19.2: Show Me the Money
Learn Practice
Lesson 3Current
Lesson 19.3: Interest-ing Problems
Learn Practice
Lesson 4
Lesson 19.4: What is a Logarithm?
Learn Practice

Lesson 19.3: Interest-ing Problems

Property

Examples

Property

Examples

Explanation

Property

Examples

Chapter 19: Exponents and Logarithms

Lesson 19.1: Exponential Functions

Lesson 19.2: Show Me the Money

Lesson 19.3: Interest-ing Problems

Lesson 19.4: What is a Logarithm?

Lesson overview

Property

Examples

Property

Examples

Explanation

Property

Examples

Chapter 19: Exponents and Logarithms

Lesson 19.1: Exponential Functions

Lesson 19.2: Show Me the Money

Lesson 19.3: Interest-ing Problems

Lesson 19.4: What is a Logarithm?

Lesson 19.1: Exponential Functions

Lesson 19.2: Show Me the Money

Lesson 19.3: Interest-ing Problems

Lesson 19.4: What is a Logarithm?