I. How Fast Does Your Money Grow?

Barry heard in his Personal Finance class that he should start investing as soon as

possible. He had always thought that it would be smart to start investing after he

finishes college and when his salary is high enough to pay the bills and to have money

left over. He projects that will be $5-10$ years from now. Barry wants to compare the

difference between investing now and investing later. A financial planner who spoke to

the class suggested that a Roth IRA $($Individual Retirement Account$)$ would be a more

profitable investment over the long term than a regular IRA, so Barry wants to seriously

consider the Roth IRA.

$($a$)$ If Barry purchases a $$2,000$ Roth IRA when he is $25$ years old and expects to

earn an average of $6\%$ per year compounded annually over $35$ years $($until he is

$60,$ how much will accumulate in the investment?

$($b$)$ If Barry doesn't put the money in the IRA until he is $35$ years old, how much

money will accumulate in the account by the time he is $60$ years old? How much

less will he earn because he invested $10$ years later?

$($c$)$

Interest rate is critical to the speed at which your investment grows. If $$1$ is

invested at $2\%$ compounded annually, it takes approximately $34\mathrm{.}9$ years to

double. If $$1$ is invested at $5\%$ compounded annually, it takes approximately

$14\mathrm{.}2$ years to double. Determine how many years it takes $$1$ to double if

invested at $10\%$ compounded annually; at $12\%$ compounded annually.$|$

$($d$)$ At what interest rate would you need to invest to have your money double in $10$

years if it is compounded annually?