Question

A savings account pays interest at the rate of 5%,compounded semi-annually. The amount that should be deposited now so that 250 can be withdrawn at the end of every six months for the next 10 years is

1) 1930.43

2) 3144.47

3) 3897.29

4) 6386.16

5) none of the above

Answer 1

so yu want 6 months interest tu be 250$ at 5%/yeer

thats 2.5% for 6 months

so 0.025*x=250$

x=250/0.025

x=250000/2.5

=10,000$

sept perhaps yu impli a kompleetly differ problem

Comments

It's compound, not simple, interest!

Answer 2

Let the savings deposit be S. After 6 months this amounts to 1.025S because of interest. From this 250 is taken out, leaving 1.025S-250, which continues to accumulate interest. After 12 months this amount accrues further interest: 1.025(1.025S-250) and a further 250 is withdrawn: 1.025(1.025S-250)-250. After 18 months, we have 1.025(1.025(1.025S-250)-250)-250. And so on for 10 years. Let's expand: 1.025(1.025S-250)-250=1.025^2S-250*1.025-250; then 1.025^3S-1.025^2*250-1.025*250-250. So, for 20 6-month periods this expression becomes:

1.025^20S-1.025^19*250-1.025^18*250-...-1.025*250-250

At this point all the savings will have gone, so 1.025^20S-1.025^19*250-...-250=0. So we have a geometric series for the withdrawals:

-250(1+1.025+1.025^2+...+1.025^19). Call the series in brackets s. Multiply s by 1.025: 1.025+1.025^2+...+1.025^20. (Calculate 1.025^20=1.6386 for later.) Subtract s from 1.025s to get 1.025s-s=0.025s=1.025^20-1, so s=0.6386/0.025=25.5447. 

1.025^20S-250s=0, so S=250*25.5447/1.6386=3897.29 (answer 3).

What we can see if we analyse the solution is 3897.29 invested for 10 years at 5% compound interest and taken from this is equivalent to investing 250 every 6 months at the same rate. The first 250 gains the most interest, because the money is in the bank for the whole 10 years, and the next gains six months' less interest and so on, until the last 250 which gains 6 months' interest only.