SOLUTION: If Naomi decides that she will invest ​$4 comma 500 per year in a 4​% annuity for the first ten​ years, ​$9,000 for the next ten​ years, and ​$13,500 for the next tenâ€

Algebra.Com
Question 1142508: If Naomi decides that she will invest ​$4 comma 500 per year in a 4​% annuity for the first ten​ years, ​$9,000 for the next ten​ years, and ​$13,500 for the next ten​ years, how much will she​ accumulate? Treat each​ ten-year period as a separate annuity. After the ten years of an​ annuity, then it will continue to grow at compound interest for the remaining years of the 30 years.
Answer by ikleyn(52775)   (Show Source): You can put this solution on YOUR website!
.

The major idea of the solution is that the FUTURE VALUE of so organized saving plan is the sum of FUTURE VALUES of 3 (three) annuities.


First annuity is 30 years ordinary annuity with the annual deposits of $4500 at 4% APY.


The second annuity is 20 years ordinary annuity with the annual deposits of $4500 at 4% APY.


And the third annuity is 10 years ordinary annuity with the annual deposits of $4500 at 4% APY.


The first annuity starts "this" year (or, better to say, it starts "today" with the first deposit at the end of "this"year); 

the second annuity starts 10 years after; and the third annuity starts 20 years after.


You can EASILY understand why it is so, by splitting each $9000 deposits in two deposits by $4500 each: 

    - then you continue the first annuity with $4500 deposits every year from 11-th year till 20-th year,

    - and you starts the second annuity with $4500 deposits annually from the year 11-th till 20-th.



And then splitting each $13500 deposits in three deposits by $4500 each:

    - then you continue the first annuity with $4500 deposits every year from 21-th year till 30-th year inclusively,

    - then you continue the second annuity with $4500 deposits every year from 21-th year till 30-th year inclusively,

    - and you starts the third annuity with $4500 deposits annually from the year 21-th till 30-th inclusively.



Thus the total future value after 30 years will be


    FV = FV1 + FV2 + FV3,  where


         FV1 = future value of the 4% annuity with $4500 deposits during 30 years;

         FV2 = future value of the 4% annuity with $4500 deposits during 20 years;

         FV3 = future value of the 4% annuity with $4500 deposits during 10 years.


The rest is just a technique.


    FV1 =  =  = 252382.22;


    FV2 =                  = 134001.36


    FV3 =                  =  54027.75.


Now the answer is the sum   252382.22 + 134001.36 + 54027.75 = 440411.33 dollars.


ANSWER.  440411.33 dollars.

Solved.


RELATED QUESTIONS

1. Mike’s Sport Shop deposits $3,600 at the end of each year for 12 years at 7% annual... (answered by edward d,Hyot)
5.) Determine the present value if $15,000 is to be received at the end of eight years... (answered by rfer)
Ashley has $120, 000 to invest and decides to put some in a CD that earns 4% interest per (answered by solver91311,greenestamps)
Find the lump sum that one must invest in an annuity in order to receive R1 000 at the... (answered by ikleyn)
Hello. My name is Rachel. Course Hero is asking me to answer your question or assignment, (answered by robertb)
Ashley has $120, 000 to invest and decides to put some in a CD that earns 9 % interest... (answered by richwmiller)
A person wishes to deposit $5,000 per year in a savings account which earns interest of 8 (answered by ikleyn)
Jane has $10 000 to invest. She invested part of it in a term deposit that pays 4% per... (answered by checkley71)
A woman has a total of $7,000 to invest. She invests part of the money in an account that (answered by josgarithmetic)