Interest Blueprint NotesWealth Blueprint Name: Date: Compound Interest Discovery The Foundation Interest is the amount of money ___________ or ___________ for the use of money. Unlike simple interest, compound interest is calculated on the __________________ amount AND the __________________ interest from previous periods. It is essentially "interest on interest." Discrete (Periodic) Compounding \[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \] A = _______________________ Final Balance r = _______________________ Annual Interest Rate (Decimal) P = _______________________ Principal (Initial Amount) n = _______________________ Compoundings per Year t = ____________________________________________________________________ Time (in Years) Decoding Frequency (n) PeriodValue of nPeriodValue of nAnnually________Monthly________Semi-Annually________Weekly________Quarterly________Daily________ Continuous Compounding When interest is compounded instantly every moment, we use the natural base e. This is often called the "PERT" formula. \[ A = Pe^{rt} \] What is e? Irrational number \(\approx\) ________ Keyword "____________________" Blueprint Checkpoint Identify the formula needed for each scenario: 1. A savings account compounds interest monthly. 2. A high-yield investment compounds continuously. 3. A student loan requires quarterly compounding.
Interest Blueprint KeyWealth Blueprint [KEY] Compound Interest Discovery The Foundation Interest is the amount of money earned or paid for the use of money. Unlike simple interest, compound interest is calculated on the principal (initial) amount AND the accumulated interest from previous periods. It is essentially "interest on interest." Discrete (Periodic) Compounding \[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \] A = Final Account Balance Final Balance r = Annual Interest Rate Annual Interest Rate (Decimal) P = Principal (Starting Amt) Principal (Initial Amount) n = Compounding Periods Compoundings per Year t = Number of Years Elapsed Time (in Years) Decoding Frequency (n) PeriodValue of nPeriodValue of nAnnually1Monthly12Semi-Annually2Weekly52Quarterly4Daily365 Continuous Compounding When interest is compounded instantly every moment, we use the natural base e. This is often called the "PERT" formula. \[ A = Pe^{rt} \] What is e? Irrational number \(\approx\) 2.718... Keyword "Continuously" Blueprint Checkpoint Identify the formula needed for each scenario: Periodic [n = 12] 1. A savings account compounds interest monthly. Continuous 2. A high-yield investment compounds continuously. Periodic [n = 4] 3. A student loan requires quarterly compounding.
Balance Builder WorksheetBalance Builder Name: Date: Interest Application Practice Discrete Compounding \[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \] Continuous Compounding \[ A = P \cdot e^{rt} \] Phase 1: Periodic Growth 1. $2,500 at 4.5% annual interest, 6 years, quarterly. P: r: n: t: 2. Bond of $10,000, 7.2%, 10 years, monthly. P: r: n: t: 3. Interest earned on $1,200 at 3%, 2 years, daily. P: r: n: t: 4. $4,000 at 2.8%, 5 years, compounded semi-annually. P: r: n: t: 5. $500 at 1.5% interest for 1 year, weekly. P: r: n: t: 6. $15,000 at 5% interest for 4 years, quarterly. P: r: n: t: 7. A loan of $800 at 12% interest for 0.5 years (6 months), compounded monthly. P: r: n: t: Phase 2: Instantaneous Growth 8. $5,000 at 6% interest for 5 years, continuous. P: r: t: 9. Compare $1,000 at 5% for 20 years: continuous vs. annual. P: r: t: 10. $2,200 at 3.5% interest, 8 years, continuous. P: r: t: 11. $7,500 earns 4.25% interest for 12 years, continuous. P: r: t: 12. Find value of $3,000 at 9%, 3 years, continuous. P: r:
Balance Builder KeyBalance Builder [KEY] Interest Application Practice Phase 1: Periodic Growth 1. $2,500, 4.5%, 6y, quarterly: \( A = 2500(1 + \frac{0.045}{4})^{24} \approx \$3,271.61 \) 2. $10,000, 7.2%, 10y, monthly: \( A = 10000(1 + \frac{0.072}{12})^{120} \approx \$20,495.35 \) 3. Interest on $1,200, 3%, 2y, daily: \( A \approx \$1,274.20 \rightarrow Interest = \$74.20 \) 4. $4,000, 2.8%, 5y, semi-ann: \( A = 4000(1 + \frac{0.028}{2})^{10} \approx \$4,596.53 \) 5. $500, 1.5%, 1y, weekly: \( A = 500(1 + \frac{0.015}{52})^{52} \approx \$507.55 \) 6. $15,000, 5%, 4y, quarterly: \( A = 15000(1.0125)^{16} \approx \$18,298.32 \) 7. $800, 12%, 0.5y, monthly: \( A = 800(1.01)^{6} \approx \$849.22 \) Phase 2: Continuous Growth 8. $5,000, 6%, 5y, continuous: \( A = 5000 \cdot e^{0.06 \cdot 5} \approx \$6,749.29 \) 9. $1,000, 5%, 20y: \( Cont: \$2,718.28 \ | \ Ann: \$2,653.30 \) 10. $2,200, 3.5%, 8y, continuous: \( A = 2200 \cdot e^{0.035 \cdot 8} \approx \$2,910.89 \) 11. $7,500, 4.25%, 12y, continuous: \( A = 7500 \cdot e^{0.0425 \cdot 12} \approx \$12,489.92 \) 12. $3,000, 9%, 3y, continuous: \( A = 3000 \cdot e^{0.09 \cdot 3} \approx \$3,929.89 \) 13. $50,000, 2.5%, 15y, continuous: \( A = 50000 \cdot e^{0.025 \cdot 15} \approx \$72,749.49 \) Phase 3: Decision Key 14. Credit A: \( \$1,195.62 \) | B: \( \$1,194.83 \) Key: Card B (Lower Cost). 15. Bank A: \( \$3,615.68 \) | B: \( \$3,644.24 \) Key: Bank B (Grows faster).
