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The following formula describes the future value of periodic monthly deposits in an investment such as a Roth IRA. $P$ represents the amount of each periodic deposit, $r$ represents the interest rate (APR) expressed in decimal form, $t$ represents the number of years for which monthly deposits are made, and B represents the account balance.

$$B=\frac { P{ (1+\frac { r }{ 12 } ) }^{ 12t }-P }{ \frac { r }{ 12 } }$$

Violet is currently 30 years old and plans to start a Roth IRA that has a typical rate of return of 4.6% APR. If she wants to have an account balance of a half-million dollars by the time she reaches age 70, how much money per month will she need to start depositing now?

Abby is currently 25 years old and plans to start periodic deposits of $250 per month into a Roth IRA that has a typical rate of return of 4.3% APR. How much additional interest would Abby earn by waiting until age 70 to retire and stop making periodic deposits instead of waiting until age 65 to retire and stop making periodic deposits?

Violet’s monthly deposit

Abby’s additional interest

$ 80

$ 305

$ 363

$ 434

$ 77,983

$ 92,983

$ 318,655

$ 411,638

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