Question

In the following ordinary annuity, the interest is compounded with each payment, and the payment is made at the end of the compounding period.

Find the amount of time needed for the sinking fund to reach the given accumulated amount. (Round your answer to two decimal places.)

$5500 yearly at 8% to accumulate $100,000.

Answer 1

Solution :-

Given data :

Amount per year is $ 5500

Accumulated amount is $ 100,000

Interest rate = 8% = 0.08

Here we need to calculate the amount of time needed for the sinking fund to reach the given accumulated amount.

$100,000=\frac{5500(1 + \frac{0.08}{12})^{12t}-1}{\frac{0.08}{12}}$

$100,000=\frac{5500(1 +0.0066)^{12t}-1}{0.0066}$

$100,000=\frac{5500(1.0066)^{12t}-1}{0.0066}$

$100,000*0.0066=5500(1.0066)^{12t}-1$

$6600=5500(1.0066)^{12t}-1$

$6600+1=5500(1.0066)^{12t}$

$6601=5500(1.0066)^{12t}$

$(1.0066)^{12t}=\frac{6601}{5500}$

$(1.0066)^{12t}=1.200$

Now, Apply natural logarithm on both sides, we get

$ln{(1.0066)^{12t}=ln{(1.200)}$

$12t* {ln{(1.0066)}=ln{(1.200)}$

$12t* 0.0065=0.1823$

$12t=\frac{0.1823}{ 0.0065}$

$12t=28.0461$

$t=\frac{28.0461}{12}$

$t=2.34$

$\therefore$ The amount of time needed for the sinking fund to reach the given accumulated amount is t = 2.34 yr