Answer to Question #161168 in Financial Math for Ndlp

Question

At the end of every 6 months, Marcia deposits $100 in a savings account

T that pays 4%/a compounded semi-annually. She made the first deposit

when her son was 6 months old, and she made the last deposit on her son’s 21st birthday. The money remained in the account until her son turned 25, when Marcia gave it to him. How much did he receive?

Answer 1

The formula for compound interest, including principle sum is:

$A=\frac{P\times[(1+\frac{r}{k})^{Nk}-1]}{\frac{r}{k}}$

where,

A = the balance in the account after N years.

P = the regular deposit (the amount you deposit each year, each month, etc.)

r = the annual interest rate in decimal form = $\frac{4}{100}=0.04$

k = the number of compounding period in one year = 2

N = the number of years that interest is compounded yearly= 21 and half year = 21*2 -1(first deposit after age of 6 month old son) = 41

so,

$A=\frac{600\times[(1+\frac{0.04}{2})^{41}-1]} {\frac{0.04}{2}}$

$A=\frac{600\times[(1.02)^{41}-1]} {0.02}$

$A=\frac{600\times[(2.25220046-1]}{0.02}=37,566.0138$

Thus, the amount after the 21st birthday of Marcia's son is $37,566.0138

But the money after 4 years of Marcia's son 21st birthday without anymore deposits is :

$A=P\times(1+\frac{R}{100})^T$

where,

A = total amount after the compounded interest on the principal amount

P = Principal amount

R = rate of interest compounded annually = $\frac{4}{2\times100}$

T = time in years

$A=37566.0138\times(1+\frac{4}{2\times100})^{4\times2}$

37566.0138(1.17165938) = 44,014.5724Thus, the amount received is : $44,014.5724