Interest problem

[quach123]

Rachael deposits $3,600 into a retirement fund each year. The fund earns 7.5% annual interest,
compounded monthly. If she opened her account when she was 20 years old, how much will she
have by the time she’s 55? How much of that amount was interest earned?

How do I do this problem?

 

[Steven G]

Exactly, how do you solve this problem? In order to receive help from this forum you need to tell us where you are stuck. In the end you will be the one who solves this problem with help from us.

What is the formula for compound interest?
How much is p, the principle?
How much is n, the number of times you get interest per year?
How much is t, the number of years you have the account for?
How much is r as a decimal, where r is the annual interest rate?

Please post back with the answers to my question and any work you have done.

 

[quach123]

P=$3,600
n=12 times/yr
t=35 years
r=0.075

I plugged these numbers into the equation and my teacher responded that it was wrong. I got $49,292.15. He said, "This is ok if she only made one deposit and then let compound interest do the work. But, she made a deposit every year for twenty years. The solution to that involves a little more work." I don't know where to go from here.

 

[Steven G]

So now do the same for t=34 years, and for t= 33 years, and for t= 32 years, .... and t= 1 year. Now add up those numbers. Do you see a pattern? Do you know about geometric series?

 

[quach123]

Yes i see a pattern through the common ratio. I ended up using the geometric sequence formula from there to figure out how much she will save by the time she is 55. Sn=a1(1−r^n)/1−r, where r=1.00625 and n=35 the amount of deposits. a=3,600 the amount she deposits each year and plugged these into the formula and got 140355. Then I multiplied the amount of money she deposited by the amount of years 35 and subtracted that amount she saves, 140355 and got the answer $14335 which is the interest earned.

 

[HallsofIvy]

Here is one way of thinking about it:
Rachel put $3600 in an account when she was 20 and let it accumulate 7.5% interest for 55- 20= 35 year. How much will it be worth after 35 years? (Is that simple interest or compounded annually or compounded quarterly?)

She puts $3600 in the account at 7.5% interest for 34 years. How much will that be worth?

She put $3600 in the account at 7.5% interest for 33 years.

Etc.