# stuck on a present value question...?

"today, brianna has 7424.83 in her bank account. for the last two years, her account has paid 6% compounded monthly. before then, her account paid 6% compounded semi annually, for four years. if she made only one deposit six years ago, determine the original principle."

please walk me through this. thanks!

### 5 Answers

- llafferLv 71 year agoFavorite Answer
The equation for compound interest is:

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

Where A(t) is the amount at "t" years,

P = initial principal (amount when t = 0)

r = annual rate of interest

n = number of times per year interest is calculated

t = time in years

We are told that today's balance is $7424.83 and was earning 6% for the last 2 years compounded monthly.

So we can call A(2) = 7424.83, r = 0.06, n = 12.

Using this we can solve for P for this stage, which will end up being the new A(4) for the second part:

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

A(2) = P(1 + 0.06/12)^(12 * 2)

7424.83 = P(1 + 0.005)²⁴

7424.83 = P(1.005)²⁴

7424.83 = P(1.127160)

While I'm rounding here, I'm not rounding in my calculator to reduce errors due to rounding. One last step:

P = 7424.83 / 1.127160

P = 6587.20 (rounded to nearest penny)

This was the starting balance when the account converted to monthly compounding. Now do this again with n = 2 and t = 4, we can solve for P to get the original deposit amount from 6 years ago:

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

A(4) = P(1 + 0.06/2)^(2 * 4)

6587.2 = P(1 + 0.03)⁸

6587.2 = P(1.03)⁸

6587.2 = P(1.26677)

P = 6587.2 / 1.26677

P = 5200.00 (rounded to nearest penny, rounded up from 5199.9965)

So the starting deposit was $5200.

- SumDudeLv 71 year ago
fyi ... do say 6% PER ANNUM compounded (time frame). < That is better English and better math.

Source(s): accountant (degree and everything) - Mike GLv 71 year ago
If A(t) = Present Value

Ao = Initial Amount invested

r = Annual Interest Rate

n = number of compounding periods in 1 year

t = number of years

A(t) = Ao(1+r/[100n])^(nt)

For the first 4 years

A(4) = Ao(1+6/200)^8 = 1.26677Ao

For the last 2 years

A(2) = 1.26677Ao(1+6/1200)^24 = 1.42785Ao

1.42785Ao = 7424.83

Ao = 5199.999

Ao = $5,200

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- alexLv 71 year ago
the original principle = x

6% compounded semi annually, for four years --->x(1.03)^8

6% compounded monthly , for four years --->[x(1.03)^8](1.005)^24

[x(1.03)^8](1.005)^24 = 7424.83 --->x = $5200