Compound Interest Calculation

Compound Interest Problem

Given Data

• Principal (P): $3,000
• Time (t): 7 years
• Interest Rate (r): 7% per annum
• Compounding Frequency: Monthly
• Withdrawals: None assumed

Formulas

1. Compound Interest Formula:
   • The formula to calculate the amount A accumulated after a certain time when interest is compounded is given by:
     $A = P \left(1 + \frac{r}{n}\right)^{nt}$
   • Where:
     • A = the amount of money accumulated after n years, including interest.
     • P = principal amount (the initial amount of money).
     • r = annual interest rate (decimal).
     • n = number of times that interest is compounded per year.
     • t = the number of years the money is invested for.

Variable Substitution

• Convert the interest rate from percentage to decimal:
  • $r = 7\% = 0.07$
• Since interest is compounded monthly,
  • $n = 12 ext{ (months)}$
• Substitute the values into the formula:

Calculation

1. Substitute values into the formula:
   $A = 3000 \left(1 + \frac{0.07}{12}\right)^{12 \times 7}$
2. Calculate the monthly interest rate:
   $\frac{0.07}{12} = 0.00583333$ (approx.)
3. Calculate the exponent term:
   $nt = 12 \times 7 = 84$
4. Rewrite the formula with calculated values:
   $A = 3000 \left(1 + 0.00583333\right)^{84}$
5. Simplify further:
   • Calculate:
     $A = 3000 \left(1.00583333\right)^{84}$
6. Find the value of $\left(1.00583333\right)^{84}$:
   • This value is approximately 1.747422.
7. Final computation for A: $A = 3000 \times 1.747422$
   • Thus,
     $A \approx 5242.27$ (rounding to the nearest cent).

Conclusion

• The amount accumulated after investing $3,000 for 7 years at an interest rate of 7% compounded monthly is approximately $5,242.27.