# An amount say $800 is invested at a rate of interest of 5% for 30 years. What is the amount at the end of the period if it is? ## (a) compounded every year (b) compounded monthly (c) compounded every week (d) compounded continuously. ##### 1 Answer Apr 12, 2017 (a) $3457.55 (b) $3574.20 (c) $3582.77 (d) $3585.35 #### Explanation: When we invest an amount say $P$at a rate of interest of $r$for $t$years, the amount at the end of the period (at simple interest) becomes $P \left(1 + \frac{r t}{100}\right)$. However, when compounded, it depends on the fixed period after which it is compounded or rather how many times in a year interest is compounded. If it is compounded after every $\frac{1}{n}$year, the amount becomes $P {\left(1 + \frac{r}{n \times 100}\right)}^{n t}$Here we have P=$800, r=5% and $t = 30$ years

(a) If compounded yearly i.e. once a year, as $n = 1$, it becomes

800(1+5/100)^30=800xx1.05^30=800xx4.321942=$3457.55 (b) If compounded monthly i.e. $12$times a year, as $n = 12$, it becomes 800(1+5/1200)^(30xx12)=800xx1.004166^360=800xx4.467744=$3574.20

(c) If compounded weekly i.e. $52$ times a year, as $n = 52$, it becomes

800(1+5/5200)^(30xx52)=800xx1.00096154^1560=800xx4.47846=$3582.77 (d) If compounded continuously, it approximates $P {e}^{\frac{r t}{100}}$i.e. 800e^((5xx30)/100)=800xxe^1.5=800xx4.481689=$3585.35