# stuck on a present value question...?

Update:

"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!

Relevance

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.

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• PV = FV / (1 + i)^n

PV = 7424.83/(1+6%/12)^24/(1+6%/2)^8 = 5200.00

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• fyi ... do say 6% PER ANNUM compounded (time frame). < That is better English and better math.

Source(s): accountant (degree and everything)
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• 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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