ACT Math: Compound Interest and Exponential Growth Formulas

Compound interest is interest earned on interest. Formula: A=P(1+r/n)^(nt). Components: A = final amount. P = principal (initial investment). r = annual interest rate (as decimal). n = compounding frequency per year (1=annual, 2=semiannual, 4=quarterly, 12=monthly, 365=daily). t = time in years. Example: Invest $1000 at 5% annual interest, compounded quarterly, for 2 years. A=1000(1+0.05/4)^(4×2)=1000(1.0125)^8≈1104. Compare to simple interest (I=Prt): 1000+1000(0.05)(2)=1100. Compound interest yields slightly more due to re-compounding. On the ACT, you'll: (1) Calculate final amount given P, r, n, t, (2) Compare compounding frequencies, (3) Work backward to find P, r, or t. Note: More frequent compounding (daily vs. annual) yields slightly more money but with diminishing returns. Compound interest shows exponential growth; the formula is essentially y=a(b)^t where b=(1+r/n).

Why it matters: Understanding compound interest is essential for saving, investing, and loans. Long-term investing relies on compounding; short-term differences are small.

Mistake 1: Using annual rate without converting for compounding frequency. If compounding quarterly, r/4 must appear in the formula, not r. Mistake 2: Forgetting that n is frequency per year, not total times. If t=2 years and n=4 (quarterly), total compounding periods = n×t = 8. Mistake 3: Confusing simple and compound interest. Simple: I=Prt (linear). Compound: A=P(1+r/n)^(nt) (exponential). Mistake 4: Misreading problem for compounding frequency. "Annual" = n=1, "quarterly" = n=4, "monthly" = n=12. Always verify: Is this compound (re-earning interest on interest) or simple (flat interest)? What's the compounding frequency?

Checklist: (1) Identify P, r, n, t from the problem. (2) Convert r to decimal and r/n for the frequency. (3) Plug into formula. (4) Calculate step-by-step. (5) Verify the answer is reasonable (more than principal, less than doubling if rates are moderate).

Problem 1: $500 at 6% annual, compounded annually, 3 years. A=500(1+0.06/1)^(1×3)=500(1.06)^3≈595.51. Problem 2: $2000 at 4% annual, compounded quarterly, 5 years. A=2000(1+0.04/4)^(4×5)=2000(1.01)^20≈2440.84. Problem 3: $1000 at 5% annual, compounded monthly, 2 years. A=1000(1+0.05/12)^(12×2)=1000(1.00417)^24≈1104.89. Problem 4: Compare $1000 at 3% compounded annually vs. quarterly for 10 years. Annual: 1000(1.03)^10≈1344.. Quarterly: 1000(1.0075)^40≈1349. Quarterly yields about $5 more. Problem 5: How long to double $1000 at 7% compounded annually? 2000=1000(1.07)^t. 2=(1.07)^t. t=log(2)/log(1.07)≈10.2 years. For each, show formula, substitution, and calculation steps.

Daily drill: Solve one compound interest problem daily. Alternate between calculating final amount, comparing frequencies, and finding time. Practice using logarithms for solving exponential equations.

About 1 compound interest or exponential growth question per ACT Math section combines formulas, exponents, and real-world context. If you master the formula, you answer these quickly. Compound interest questions are high-value: they're practical (students care about money), they combine multiple skills (algebra, exponents, formulas), and many students either avoid or misapply the formula, making correct answers stand out.

Spend 2 days on compound interest. Memorize the formula, practice calculations, and learn the impact of compounding frequency. By test day, you'll solve these confidently and gain reliable points.