SAT Compound Interest and Exponential Growth: Mastering Repeated Multiplication

Linear growth adds the same amount each period (arithmetic progression); exponential growth multiplies by the same factor each period (geometric progression). A savings account earning 5% interest annually grows exponentially: the amount multiplies by 1.05 each year, not adds a fixed dollar amount. This distinction is critical for SAT word problems testing whether you understand growth types. Linear growth: A(t)=initial+rate*time; Exponential growth: A(t)=initial*(1+rate)^time.

Real-world contexts for exponential growth include: bank interest (money multiplies by a growth factor), population growth (population multiplies each generation), radioactive decay (amount remaining multiplies by a decay factor), and disease spread (infections multiply each transmission round). SAT questions often disguise exponential growth in unfamiliar contexts, testing whether you recognize the structure beneath the story.

The compound interest formula is A=P(1+r/n)^(nt), where P is principal (starting amount), r is annual interest rate (as a decimal), n is compounds per year, and t is years. For annual compounding (n=1), this simplifies to A=P(1+r)^t. Practice identifying each variable in a word problem: locate the starting amount, the interest rate and compounding frequency, and the time period to solve, then substitute directly into the formula.

Common errors include: forgetting to convert percentage to decimal (5% becomes 0.05, not 5), misidentifying which variable to solve for, or confusing simple interest (A=P(1+rt)) with compound interest. Always verify your answer by checking reasonableness: Does the final amount exceed the starting amount? Does the growth match the given rate?

Some SAT questions require solving for time or rate in an exponential equation, not just calculating final amount. If you know the initial amount, final amount, and rate, you must solve for time: A=P(1+r)^t becomes t=log(A/P)/log(1+r). Understand that logarithms are the inverse of exponents: if 2^3=8, then log_2(8)=3, so use logarithms to "undo" exponents and isolate the variable you are solving for.

On the SAT Math section with calculator, you can compute logarithms directly. On the no-calculator section, exponential equations are usually structured to have integer exponents you can solve by inspection or trial. Master both approaches to handle either section confidently.

Create five real-world scenarios: (1) an investment growing at 7% annually for 10 years, (2) a population doubling every 20 years, (3) a radioactive substance with a 5,730-year half-life, (4) bacteria tripling every 2 hours, and (5) currency inflation at 3% per year. For each, set up and solve for final amount, time to reach a target, or required rate to meet a goal. Solve all five scenarios, verify answers are reasonable, and time yourself: this routine should take 15 minutes once you are fluent with the formula.

After drilling, tackle official SAT practice problems involving compound interest and exponential growth. You will notice these problems are actually quite predictable once you recognize the exponential structure. Confidence in this skill unlocks consistent correct answers on a frequently tested topic.