Compound Interest

Q: How can I check my work?

Balance divided by principal equals the accumulated compound factor. A numerical check can detect an error; when claiming an identity for all inputs, also explain the algebraic property that makes it true.

Q: Which mistakes should I watch for?

Using the percent rate itself as the growth factor. Adding the same amount each period in an exponential model. Including zero in the range of an unshifted nonzero exponential.

Q: Can I use another valid method?

Yes. Explain your assumptions, show enough intermediate work to justify the result, and verify it in the original problem. Equivalent expressions and valid alternative methods receive credit.

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By Cambridge Veritas · Algebra I learning resources

Introduction

Compound Interest helps you connect a mathematical rule to a result you can explain and check. Use principal, annual decimal rate, number of compounding periods, and years consistently in A=P(1+r/n)^nt. This lesson belongs to exponential functions and builds toward independent Algebra I problem solving.

A.9C · Readiness

Alignment uses 19 TAC §111.39(c). Review lessons sample their listed standards. Readiness and supporting labels describe STAAR emphasis, not whether a topic is required.

Learning objectives

Identify the quantities and representations used in compound interest.
Apply the lesson's rule to solve a numerical or contextual problem.
Explain each mathematical step using the correct vocabulary.
Verify an answer and correct a likely error.

Prerequisite knowledge

Evaluate powers and convert percentages to decimals.

Previous lesson: Initial Value

Key mathematical vocabulary

constant: A value fixed within a problem.
domain: The set of permitted inputs.
range: The set of outputs produced.
exponent: A number specifying a power.
factor: An expression multiplied by another to form a product.

Formulas and rules

f(x)=ab^x, b>0, b≠1
Growth factor=1+r; decay factor=1−r
A=P(1+r/n)^nt

Exponential growth and decay

Use principal, annual decimal rate, number of compounding periods, and years consistently in A=P(1+r/n)^nt.

An exponential model has the form f(t)=ab^t, where b>0 and b≠1 for a nonconstant exponential. The initial value is a=f(0). Each increase of 1 in t multiplies the output by b rather than adding a constant.

For a positive initial amount, b>1 models growth and 0<b<1 models decay. A rate r written as a decimal gives growth factor 1+r or decay factor 1−r. Compound interest is A=P(1+r/n)^nt, with compatible units. For a≠0, the mathematical domain is all real numbers; the sign of a determines whether the range is y>0 or y<0. Context may restrict time to t≥0.

A method you can explain

Read the question and identify the known information, unknown quantities, and any domain restrictions. Select a suitable representation and tool.
Apply the relationship explained above. Use A=P(1+r/n)^nt with P=1200, r=0.2, n=1, t=2.
Write each calculation or logical inference on a separate line. Keep units and signs visible. When two methods are possible, compare which one uses the given information most directly.
Check the result in the ORIGINAL conditions, not merely the last simplified line. Balance divided by principal equals the accumulated compound factor.
Interpret what the answer means. Decide whether its size, sign, and precision make sense; identify a restriction or assumption that would change the conclusion.

Five worked examples

Example 1 · Beginner

Invest $1200 at 20% annual interest compounded annually. Find the balance after 2 years, rounded to cents.

Use A=P(1+r/n)^nt with P=1200, r=0.2, n=1, t=2.
The periodic factor is 1+0.2/1=1.2; the number of periods is 2.
A=1200(1.2)²=1728.00 dollars. Keep intermediate powers unrounded; this assumes a fixed rate with no deposits or fees.

Answer: $1728.00