Question

a lady invests $500 each semiannual period into an IRA paying 10% compounded semiannually. Given that she is 30 yrs old, finds the amount she will have accumulated at age 54

Answer 1

No. of instalments = (54 - 30) x 2 = 48 instalments.

Annual rate = 10%

There are two interpretations on how to take semi-annual rate. Some just take 5%. A more acceptable approach is:
(1+10%)^6/12, that is 1.0488, which gives the rate as 1.0488-1 = 4.88%. I will work out using both the interpretations.

If the rate is taken as 4.88%

Future value of annuity for 48 years at 4.88% is 181.27. It can be obtained from excel using the formula =FV(4.88%,48,-1). Or it can be found using the formula (1.0488⁴⁸ - 1) / 4.88% in normal calculator.
So, the amount will be $90,635, being $500 x 181.27.

If we take the other interpretation and take the rate as 5%

The amount will be $500 x (1.05⁴⁸ - 1) / 5% = $94,012.70

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