A teacher decided when she was 35 to save some money for her retirement. Her objective was to have about $300,000 on her retirement at 65. She had counted on getting 6% compounded annually on her deposits, but due to bad investment choices she had made in the first 15 years, she ended up having savings of only $50,000! She decided to save the balance by obtaining a risk-free interest rate of 5% comp. annually and to double her deposits each and every year for the remaining 15 years to get to her goal of $300,000 at 65. How much does she need to save in the first year, of the remaining 15 years, to accomplish her goal? Thanks a lot for any help.

Guest Mar 10, 2017

#1**0 **

Wow! Messed up is right!.

This problem is quite involved for most students. However, with little bit of thinking, it turns out that it is no that complicated.

The first thing that the student has to keep in mind is that $50,000 that she saved in the first 15 years. You must bear in mind that it will continue to grow for another 15 years at the new rate of 5%. Therefore, if we project its FV for another 15 years @ 5% we will have =$103,946.41. Her total goal is to save $300,000. Therefore, the net amount she needs to save in the next 15 years is=$300,000 - $103,946.41 =**$196,053.59.........(1)**

Since she plans to double her deposits each and every year, will have to find the FV of $1, $2, $4, $8....and so on for 15 years @5%. We can easily do that on any good calculator, such as Wolfram/Alpha, such as I have done here:∑[(2^n * 1.05^(14-n)), n, 0, 14] =**34,490.443233495......(2).**

Now then, all we have to do is divide the amount in (1) above by (2) above and we get:

**$196,053.59 / $34,490.44323495 =$5.68** - which would be her first payment for the remaining 15 years. And, of course, she will double that each and every year: $5.68 , $11.36, $22.72, $45.44...and so on. Here is the link to W/A:

http://www.wolframalpha.com/input/?i=%E2%88%91%5B(2%5En+*+1.05%5E(14-n)),+n,+0,+14%5D

I hope you understand what I have done. If you have any questions, just let me know here. Good luck.

Guest Mar 10, 2017

#2**0 **

Wow! Messed up is right!.

This problem is quite involved for most students. However, with little bit of thinking, it turns out that it is no that complicated.

The first thing that the student has to keep in mind is that $50,000 that she saved in the first 15 years. You must bear in mind that it will continue to grow for another 15 years at the new rate of 5%. Therefore, if we project its FV for another 15 years @ 5% we will have =$103,946.41. Her total goal is to save $300,000. Therefore, the net amount she needs to save in the next 15 years is=$300,000 - $103,946.41 =**$196,053.59.........(1)**

Since she plans to double her deposits each and every year, will have to find the FV of $1, $2, $4, $8....and so on for 15 years @5%. We can easily do that on any good calculator, such as Wolfram/Alpha, such as I have done here:∑[(2^n * 1.05^(14-n)), n, 0, 14] =**34,490.443233495......(2).**

Now then, all we have to do is divide the amount in (1) above by (2) above and we get:

**$196,053.59 / $34,490.44323495 =$5.68** - which would be her first payment for the remaining 15 years. And, of course, she will double that each and every year: $5.68 , $11.36, $22.72, $45.44...and so on. Here is the link to W/A:

http://www.wolframalpha.com/input/?i=%E2%88%91%5B(2%5En+*+1.05%5E(14-n)),+n,+0,+14%5D

I hope you understand what I have done. If you have any questions, just let me know here. Good luck.

Guest Mar 10, 2017