SOLUTION: You deposit $3000 in an account that pays 3.5% interest compounded once a year. Your friend deposits $2500 in an account that pays 4.8% interest compounded monthly. Use this infor

Question 535717: You deposit $3000 in an account that pays 3.5% interest compounded once a year. Your friend deposits $2500 in an account that pays 4.8% interest compounded monthly. Use this information to answer the following questions.
Who will have more money in their account after one year? How much more?
Who will have more money in their account after five years? How much more?
Who will have more money in their account after 20 years? How much more?

Answer by lmeeks54(111) (Show Source): You can put this solution on YOUR website!
There are a couple different ways to solve this problem. I'll talk about two: the one you're probably asked for, that is how to apply compound interest formulas to two separate accounts to compare their performance, or setting it up on a spreadsheet.
...
This is worth noting because very few financial calculations for transactions occuring over time (e.g., regular car or mortgage loan payments, annuity disbursments, principal growth through compound interest, etc.) are done one at a time by hand using formulas. They are almost all done in spreadsheets or other financial software. The reason for this--contrary to this problem, which asks you for comparison between the accounts at only two or three specified time periods--is that analysts and even normal everyday people want to see data about their accounts at more than just one or two discrete time periods.
...
Okay, the long-winded intro into, 'learn how to do spreadsheets someday, they will make your life a LOT easier,' is over. Let's solve this problem for you...
...
The data we care about for problems such as these are:
Principal - the money being borrowed, lent, or saved
Interest rate - usually expressed as an annual percentage
Term - the length of time (usually years or months) of the loan, security, or annuity. This often gets turned in the # of compounding periods
Frequency - for compound interest problems, you care about monthly, quarterly, or annual compounding (it makes a difference...sometimes a BIG difference)
...
Let P = deposit principal
Let F = the future value of the money on deposit
Let r = annual interest rate
Let n = # of compounding periods
...
The basic formula for compound interest is:
F = P*(1 + r)^n
...
Note: we have an interesting situation that is not hard to deal with, but which can be confusing and must be kept track of. Namely, one account compounds annually and one account compounds monthly. But they both have their interest rates expressed in annual terms. For account 1, which has an annual rate and annual compounding, we don't do anything special. For account 2, which has an annual rate and monthly compounding, we have to convert the annual rate to a monthly rate, which is simple: we just divide by 12. Don't forget this step in problems like this or you'll get really squirrelly answers (i.e., the earned (called "accrued") interest will get too big really fast).
...
Let's go back to our variables and formula and change them slightly. Let's add:
...
Let y = year # for calculation purposes (remember, this will be 1, 5, or 20)
Let n = # periods per year for compounding (this will be 1 or 12 in our example, but could be 4 for quarterly compounding, which is common in some places)
...
Let's also amend the basic compounding formula to account for y and the newly defined n:
...
F = P*(1 + r/n)^(y*n)
...
...
That's all the background so we can set up this problem. Now, with the givens for Acct1 and Acct2, we need to calculate F at y = 1, 5, and 20 years for both accounts:
...
P1 = $3,000
r1 = 3.5%
n1 = 1 (for annual compounding)
...
P2 = $2,500
r2 = 4.8% (but we will have to divide it by n2 to use it)
n2 = 12 (for monthly compounding)
...
For both accounts, we will use y = 1, 5, 20...so we have to do 6 calculations.
...
Let's do the first calculation for Acct1 and Acct2 for the end of year 1:
...
F1 = $3,000*(1 + .035/1)^(1*1)
F1 = $3,105.00
...
Note: this represents the $3,000 starting principal + yr 1's one annual interest payment. Giving us a new deposit principal going into year 2
...
F2 = $2,500*(1 + .048/12)^(1*12)
F2 = $2,622.68
...
Note: this also represents the $2,500 starting principal + yr 1's 12 monthly compound interest payments. Also giving us a new principal going into year 2. Of course, with monthly compounding, the principal grows every month.
...
The 1st question was at the end of year 1, which account had more money, and how much more? The answer is account 1 has more money and the difference is $482.32. Note: acct1 started out year 0 with $500 more, so acct2 is already starting to catch up, though it will take a few years to pass acct1.
...
That takes care of y = 1 for year 1. Now all that is left is y = 5 for F1 and F2 again and y = 20 for F1 and F2 again.
...
Note: everything is exactly the same except for the y term. I'll leave you to set that up for yourself, however, I'll give you the correct answers so you can check your work:
...
y = 5
F1 = $3,563.06
F2 = $3,176.60
Acct1 still winning, but now by a lot less: $386.46
...
y = 20
F1 = $5,969.37
F2 = $6,516.75
Acct2 is NOW winning, with a delta of: $547.38
...
Another interesting fact to know is when Acct2 passes Acct1. It turns out because they are compounding differently, between the end of the 11th year and the beginning of the 13th year, the lead for who has more money swaps back and forth. However, after the beginning of the 13th year, Acct2 is in the lead forever, and pulling away more and more as time goes on.
...
Cheers,
Lee
...
Ps. Email me at: w.lee.meeks@gmail.com if you have any more questions about this problem.
RELATED QUESTIONS
You deposit $3000 into an account that earns 7% compounded annually. A friend deposits... (answered by Theo)

You deposit $2500 in an account that pays 3.5% annual interest compounded continuously.... (answered by rfer)

You deposit $6000 into an account that earns 5% compounded annually. A friend deposits... (answered by ikleyn)

You deposit Php 2500 into an account that pays 3.5% annual interest compounded daily. How (answered by ikleyn)

You deposit $4000 into an account that earns 5% compounded annually. A friend deposits... (answered by Theo)

This is a story problem. It reads, "you deposit 2500 dollars in an account that pays 3.5% (answered by lwsshak3)

You deposit $1400 in an account that pays 3% annual interest compounded continuously.... (answered by solver91311)

You deposit 500 in an account that pays 5% annual interest compounded continuously. What... (answered by lwsshak3)

You deposit $1000 into an account that pays 4.8% interest compounded quarterly. A. Write (answered by Alan3354)