Question

Answer this question using two different strategies.You have decided to make an investment of $2000 in order to save for the purchase of a newcar. You expect to need about $22000 for the car of your choice. Predict when yourinvestment will be enough to purchase the car if you are earning 8.2% compounded quarterly.This is the compound interest formula to determine the final amount invested:A=P(1+i)n where:

Answer 1

The Solution:

Given:

`\begin{gathered} Formula:\text{ }A=P(1+i)^n \\ \text{ In this case,} \\ A=\text{ \$}22000 \\ P=\text{ \$}2000 \\ n=4* t=4t\text{ \lparen t=number of years\rparen} \\ i=(8.2)/(100*4)=(8.2)/(400)=0.0205 \end{gathered}`

Substitute these values in the formula.

`\begin{gathered} 22000=2000(1+0.0205)^(4t) \\ \\ Dividing\text{ both sides by 2000, we get} \\ \\ (1.0205)^(4t)=(22000)/(2000) \\ \\ (1.0205)^(4t)=11 \end{gathered}`

Taking the ln of both sides, we get:

`\begin{gathered} \ln(1.0205)^(4t)=\ln(11) \\ \\ 4t\ln(1.0205)=\ln(11) \\ \\ 4t=(\ln(11))/(\ln(1.0205)) \end{gathered}`
`t=(\ln(11))/(4\ln(1.0205))=29.54134\approx30\text{ years}`

Therefore, the correct answer is 30 years.