To calculate how long it will take for Mr. Reyes' account to reach $51,000 by investing $290 monthly at an 8.3% interest rate, we can use the future value of an annuity formula:
1. Identify the variables:
- Present value (PV) = $0 (initial investment)
- Monthly investment (PMT) = $290
- Interest rate (r) = 8.3% or 0.083 (decimal form)
- Future value (FV) = $51,000
2. Use the future value of an annuity formula:
FV = PMT * [(1 + r)^n - 1] / r
3. Plug in the values and solve for the number of months (n):
$51,000 = $290 * [(1 + 0.083)^n - 1] / 0.083
4. Simplify the equation and solve for n:
$51,000 = $290 * [(1.083)^n - 1] / 0.083
$51,000 * 0.083 = $290 * (1.083)^n - $290
$4,233 = $290 * (1.083)^n - $290
$4,523 = $290 * (1.083)^n
(1.083)^n = $4,523 / $290
(1.083)^n = 15.6
5. Take the natural logarithm of both sides to solve for n:
n * ln(1.083) = ln(15.6)
n = ln(15.6) / ln(1.083)
n ≈ 44.82
Therefore, it will take approximately 44.82 months for Mr. Reyes' account to reach $51,000 by investing $290 monthly at an 8.3% interest rate. Rounded to 2 decimal places, it will take 44.82 months or approximately 3 years and 8.82 months.
