FINANCIAL MATH: Question on Compounding Interest Semi-annually

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In summary, the conversation discusses the use of a formula to determine how long it will take for $100 to grow to $500 at an interest rate of 6.5%, compounded semi-annually. The formula is A=P(1+i)^n, where A is the final amount, P is the present value, i is the interest rate, and n is the number of compounding periods. The conversation also mentions using logarithms to solve for the unknown time, and provides a formula for continuously compounded interest, A(t) = P(1 + \frac{r}{n})^{(n)(t)}.
  • #1
nicole
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HELP! i know this is an easy question to solve but i need some help. THANKS for any help in advance!

QUESTION: Determine, to the nearest half year, how long it will take $100 to amount to $500 at 6 1/2% compounded semi-annually.

Using the formula A=P(1 + i)*to the power of*n
where A is the final amount, P is the present value, i is the interest rate and n is the number of compounding periods i have gotten to:
A=$500 P=$100 i=6.5%/2 (because it's semi-annually)=3.25% n=2 x n (because it's semi-annually)
In the FORMULA"
A = P(1+i)*to the power of*n
500 = 100(1+0.0325)*to the power of*2n
5 = 1.0325*to the power of*2n
*I am sure the answer is right to this point, I'm just not sure of how to solve for the exponent 2n. thank you for any help!
 
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  • #2
Hello Nicole,

In financial mathematics, the best method in solving for unknown time is by the use logarithms, preferably, the natural logarithm [tex]\ln[/tex].

So far, you're solution is right.

From [tex]5 = 1.0325^{2n}[/tex], apply the natural log to both sides

[tex]
\ln 5 = \ln 1.0325^{2n}
[/tex]


By a property of logarithims,

[tex]
\ln 5 = 2n \cdot \ln 1.0325
[/tex]

I think you can handle it from here :)
 
  • #3
[tex]A(t) = P(1 + \frac{r}{n})^{(n)(t)}[/tex]
 

1. What is compounding interest?

Compounding interest is a method of calculating interest on a loan or investment, where the interest earned in each period is added to the principal amount and then interest is calculated on the new total. This results in the interest earning interest, leading to a higher overall return.

2. How does compounding interest work?

Compounding interest works by using a fixed interest rate and applying it to the initial principal amount, as well as any accumulated interest, at regular intervals. In the case of semi-annual compounding, interest is calculated and added twice a year, resulting in a higher overall return compared to simple interest.

3. What is the formula for calculating compounding interest semi-annually?

The formula for calculating compounding interest semi-annually is: A = P(1 + r/2)^2n, where A is the final amount, P is the principal amount, r is the annual interest rate, and n is the number of years.

4. How does compounding interest affect the final amount?

Compounding interest has a significant impact on the final amount by allowing the interest to earn interest over time. This means that even a small difference in the interest rate or compounding frequency can result in a much larger final amount.

5. How is compounding interest different from simple interest?

The main difference between compounding interest and simple interest is that compounding interest takes into account the accumulated interest and adds it to the principal amount, resulting in a higher overall return. Simple interest, on the other hand, only calculates interest on the initial principal amount and does not consider any accumulated interest.

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