Solve problems involving the simple interest formula $I = Prt$ and compound interest (TEKS A.12E).

What this topic is asking

TEKS A.12E places simple and compound interest in the Number and Algebraic Methods category, the Texas Algebra I take on personal financial literacy. STAAR expects you to compute interest and final balances and to interpret the results. The two formulas are not on the reference sheet, so they must be memorized. Compound interest is also your first concrete exponential growth model, which links this dot point to the Exponential Functions category.

Simple interest: linear growth

Simple interest is charged or earned only on the original principal, so the same amount is added each year.

For $P = 600$, $r = 0.03$, $t = 4$: $I = 600 \times 0.03 \times 4 = 72$, and the balance is $600 + 72 = 672$. Because the increase is a fixed amount each year, simple interest is a linear model: the balance forms an arithmetic sequence.

Compound interest: exponential growth

Compound interest earns interest on the accumulated balance, so the growth speeds up over time. Compounded annually:

where $A$ is the final amount, $P$ the principal, $r$ the annual rate as a decimal, and $t$ the number of years. The factor $(1 + r)$ is applied once per year, which makes the balance a geometric sequence and an exponential model.

Reading the question: interest versus balance

The most common error is answering the wrong quantity. "How much interest" wants $I$ (simple) or $A - P$ (compound); "what is the balance" or "final amount" wants $A$. A multiple-choice item almost always lists both as options, so decide which the prompt asks for before you pick.

How STAAR examines interest

• Multiple choice. Compute simple interest or a compound balance, with the other quantity offered as a distractor.
• Number entry. Enter a final balance to the nearest dollar; round only at the end.
• Connection to exponentials. Compound interest is the bridge to $f(x) = ab^x$, where $a = P$ and $b = 1 + r$.

A clarifying idea is that the decimal conversion of the rate is where many points are lost: $5\%$ is $0.05$, not $5$, and $r$ in the compound formula is the decimal too, so $1 + r = 1.05$.

Why compounding pulls ahead

Over a single year, simple and compound interest at the same rate give the same result, because there has been no balance growth to compound yet. The gap opens from year two onward: simple interest keeps charging the rate on the original principal, while compound interest charges it on a balance that already includes last year's interest. The longer the time and the higher the rate, the wider the gap, which is the practical point of the financial-literacy standard. On STAAR you may be asked to compare the two over a few years and state which earns more, or by how much; the answer is always that compounding earns at least as much, and strictly more once $t > 1$. Recognizing compound interest as the exponential factor $(1 + r)$ applied $t$ times is what makes that comparison immediate rather than something you have to recompute term by term.

Try this

Q1. Find the simple interest on $1,500 at 4% for 2 years. [1 point]

• Cue. $I = 1500(0.04)(2) = 120$.

Q2. Find the balance on $500 compounded annually at 8% for 2 years, to the nearest dollar. [2 points]

• Cue. $A = 500(1.08)^2 = 500(1.1664) = 583.2 \approx 583$.