Question

Arturo invests $2700 in a savings account that pays 9% interest, compounded quarterly. If there are no other transactions, when will his balance reach $4550? Help please. Thanks..

Answer 1

An explanation would be helpful :)

Answer 2

P(1 +or- %)^time in this case: 2700(1+0.09)^t >= 4550 solve for t

Answer 3

Use A = P(1 + r/n)^nt

A= amount = $4550
P= principal, or starting balance = $2700
r = annual percentage rate = 9% or .09
n = number of compounding periods per year = 4 (quarterly compounding)
t = number of years = quantity you need to find

4550 = 2700(1+ .09/4)^4t

4550/2700 = (1 + .0225)^4t

4550/2700 = 1.0225^4t

Take the log or natural log of both sides of the equation, and use the power rule of logs (ln (x^a) = a ln (x) ) to get your equation into a form you can solve more easily:

ln (4550/2700) = ln (1.0225)^4t

ln (4550/2700) = 4t * ln (1.0225)

Isolate the 4t:

ln (4550/2700) / ln (1.0225) = 4t

Evaluate the log expressions:

0.52187545995257565755587235278274 divided by 0.0222506089348197588430750568433 = 4t


23.454434954177720891530925256033 = 4t

23.454434954177720891530925256033 / 4 = t

5.8636087385444302228827313140083 = t

So, t = about 5.86 years (or, if the problem states to round this to the nearest year, then about 6 years).

You can check by substituting 5.86 for t in the original expression:

A = 2700(1.0225)^(5.86*4)

A = 2700(1.0225)^23.44

A=2700*1.6846

A = $4548 (which is roughly $4550, so it checks out)

Answer 4

Thanks, I get it now. Why do you have to use that equation though?

Answer 5

When you calculate "simple" interest, the formula for the amount you ultimately end up with is this:

A = P + I , where P is the amount you started with, and I is the amount of interest earned.

The interest, I, can be calculated by multiplying the principal amount by the annual interest rate and the number of years you let your investment accrue interest:

I = Prt

So, since A = P + I, and I = Prt, then A = P + Prt which can also be written as A = P(1 + rt)

That is the formula for calculating "simple" interest, where only the beginning balance earns interest.

For your problem, however, you are figuring "compound" interest. That is, after some period, the interest itself also begins to earn interest. The general formula for compound interest is this:

A = P(1 + r/n)^nt

In this case P is the principal, or beginning amount. The annual interest rate is given by r. The variable "n" stands for the number of times you compound your interest per year. So, for example, if you compound it semi-annually (every 6 months), n would be 2; if you compound it quarterly, n = 4; if you compound it daily, n = 365, and so on. The exponent, n times t, gives you the total number of compounding periods you will have over the course of your investment. For example, if you compound semiannually for 7 years, the number of times your interest will compound will be 2 X 7, or 14.
The greater the number of compounding periods, the more interest you earn. However, as the interval between compounding becomes shorter, the amount of interest you earn approaches a limit. When the interval becomes so small that the interest can be said to be "continually" compounding, the formula for calculating interest is given by: A = Pe^rt .

I hope this was helpful!

Answer 6

Yes it was! It makes sense a lot more. Thank you..