1. The problem is to understand how money grows using different financial methods: simple interest, compound interest, and annuities. 2. **Simple Interest:** Money grows linearly over time. The formula is: $$A = P(1 + rt)$$ where $A$ is the amount after time $t$, $P$ is the principal (initial investment), $r$ is the annual interest rate (as a decimal), and $t$ is time in years. This means interest is earned only on the initial principal. 3. **Compound Interest:** Interest is earned on the principal plus the interest accumulated so far. The formula is: $$A = P\left(1 + \frac{r}{n}\right)^{nt}$$ where $n$ is the number of compounding periods per year. This leads to exponential growth. 4. **Annuities:** A series of equal payments made at regular intervals. For an ordinary annuity, the future value is: $$A = P \times \frac{(1 + r)^t - 1}{r}$$ where $P$ is the payment each period, $r$ is the interest rate per period, and $t$ is the number of payments. 5. To help the young professional: - Use simple interest if the investment is short term. - Use compound interest for long-term savings to maximize growth. - Use annuities for regular savings or investments over time, like monthly deposits. 6. Example: If $P=1000$, $r=0.05$, $t=3$ years, and compounding annually ($n=1$): - Simple interest grows to: $$A = 1000(1+0.05 \times 3) = 1000(1.15) = 1150$$ - Compound interest grows to: $$A = 1000\left(1 + \frac{0.05}{1}\right)^{1 \times 3} = 1000(1.05)^3 \approx 1157.63$$ Final answer: Different methods have different growth patterns, so choosing based on goals and time horizon is crucial.