Question

A person invests 5500 dollars in a bank. The bank pays 4.5% interest compounded semi-annually. To the nearest tenth of a year, how long must the person leave the money in the bank until it reaches 6900 dollars?

Answer 1

Answer:

$\large \boxed{\text{5.1 yr}}$

Step-by-step explanation:

The formula for interest compounded periodically is  

$A = P \left(1 + \dfrac{r}{n} \right)^{nt}$

where

A = Accrued Amount

P = Principal Amount

r = annual interest rate as a decimal

n = number of payments per year

t = time in years

(a) Data

A = $6900

P = $5500

r = 0.045

n = 2

(b) Calculation

$\begin{array}{rcl}A & = & P \left(1 + \dfrac{r}{n} \right)^{nt}\\\\6900 & = & 5500 \left (1 + \dfrac{0.045}{2} \right)^{2t}\\\\6900 & = & 5500 (1 + 0.0225)^{2t}\\\dfrac{6900}{5500} & = & (1.0225)^{2t}\\\\\ln \left (\dfrac{6900}{5500}\right ) & = &2t \ln1.0225\\\\0.227& = &2t\times 0.0223\\& = &0.0445t\\t & = & \dfrac{0.227}{0.0445}\\\\& = & \textbf{5.1 yr}\\\end{array}\\\text{The must leave their money in the bank for \large \boxed{\textbf{5.1 yr}} }$