Answer: So, it will take approximately 13 years and 4.8 months to save $1708.00 by making deposits of $204.00 at the end of every six months into an account earning interest at 5% compounded semi-annually. Rounding to the nearest month, it will take 13 years and 5 months.
Explanation:
To determine how long it will take to save $1708.00 by making deposits of $204.00 at the end of every six months into an account earning interest at 5% compounded semi-annually, we can use the formula for the future value of a series of deposits:
A = P [(1 + r/n)^(nt) - 1] / (r/n)
Where:
A = the future value (in this case, $1708.00)
P = the periodic deposit amount ($204.00)
r = the annual interest rate (5% or 0.05 as a decimal)
n = the number of times the interest is compounded per year (semi-annually, so n = 2)
t = the number of years (which we want to find)
We need to solve for t:
$1708.00 = $204 [(1 + 0.05/2)^(2t) - 1] / (0.05/2)
First, simplify the equation inside the brackets:
$1708.00 = $204 [(1 + 0.025)^(2t) - 1] / 0.025
Now, isolate the term with the exponent:
[(1 + 0.025)^(2t) - 1] = ($1708.00 * 0.025) / $204
[(1.025)^(2t) - 1] = 0.13210784314
Next, add 1 to both sides of the equation:
(1.025)^(2t) = 1 + 0.13210784314
(1.025)^(2t) = 1.13210784314
Now, take the natural logarithm (ln) of both sides to solve for 2t:
ln[(1.025)^(2t)] = ln(1.13210784314)
2t * ln(1.025) = ln(1.13210784314)
Now, divide both sides by 2 * ln(1.025):
t = ln(1.13210784314) / (2 * ln(1.025))
t ≈ 13.40
Now, convert the decimal part into months. Since there are 12 months in a year, multiply the decimal part by 12:
0.40 * 12 ≈ 4.8 months
So, it will take approximately 13 years and 4.8 months to save $1708.00 by making deposits of $204.00 at the end of every six months into an account earning interest at 5% compounded semi-annually. Rounding to the nearest month, it will take 13 years and 5 months.