# Solving Interest Problems: Compound Interest & PERT

• Pinu7
In summary, the conversation discusses a problem related to continuously compounded debt and the question of how long it would take to pay off the debt by making constant payments. The attempted solution involves using a formula for compound interest and a cash flow problem with negative and positive cash flows. However, it is mentioned that the solution does not consider inflation and there is uncertainty on how to continue with the problem.

## Homework Statement

This is not a problem in my calculus book.However, I am sure this involves calculus. This is also not a question from an economics class, it is just curiosity.

My question is: If I have a debt that is continually compounded, and I continually pay off the debt at a constant rate, how long will it take to pay off the debt?

## Homework Equations

Compund interest(PERT)

## The Attempt at a Solution

Let:
r=rate on the debt. (assume annually)
y= amount of money I will pay per year.
$$\Delta$$t= an increment of time of which I will pay a quanta of money.

During the time $$\Delta$$ t since I started the debt, I will owe er$$\Delta$$t

At this point I will pay my first quanta of money which would be y$$\Deltat$$. and w

Right before I make my second payment on time 2$$\Delta$$t, I will owe the money f
money owed from last increment AND the compound interest since that time.
ie I will owe (et$$\Delta$$t-y$$\Delta$$t)er$$\Delta$$t=e2r$$\Delta$$t-y$$\Delta$$te$$\Delta$$t

Continuing the pattern, the money I would owe right before my nth payment is:

en$$\Delta$$t-y$$\Delta$$te(n-1)$$\Delta$$tThis is getting a bit tough. Where do I go from here?

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Latex isn't working right now. It's bit hard to read.

But are you referring to a cash flow problem (annuities) i.e. there is a negative cash flow at t =0, and at the end of each year/month, there are equal positive cash flows.

## 1. What is compound interest?

Compound interest is the interest earned on both the initial principal amount and the accumulated interest from previous periods. This means that each time interest is calculated, it is added to the principal amount and the next calculation is based on the new total. This results in a faster growth of the total amount over time compared to simple interest.

## 2. How is compound interest calculated?

The formula for compound interest is A = P(1 + r/n)^(nt), where A is the total amount, P is the principal amount, r is the annual interest rate, n is the number of compounding periods per year, and t is the number of years. This formula takes into account the effect of compounding on the growth of the principal amount.

## 3. What is the difference between simple and compound interest?

The main difference between simple and compound interest is that simple interest is calculated only on the principal amount, while compound interest takes into account both the principal amount and the accumulated interest from previous periods. This means that compound interest results in a higher total amount over time compared to simple interest.

## 4. How can compound interest be applied in real-world scenarios?

Compound interest can be applied in various financial scenarios, such as investments, loans, and savings accounts. For example, when investing money in a savings account with compound interest, the total amount will grow significantly over time due to the compounding effect. On the other hand, when taking out a loan with compound interest, the amount owed will also increase faster over time compared to a loan with simple interest.

## 5. What is PERT and how is it used in solving interest problems?

PERT (Program Evaluation and Review Technique) is a project management tool that uses a weighted average of the best-case, most likely, and worst-case scenarios to estimate the time and cost required to complete a project. In solving interest problems, PERT can be used to estimate the expected interest rate by considering the best-case, most likely, and worst-case scenarios for the interest rate. This helps in making more accurate predictions and decisions regarding investments, loans, and other interest-related scenarios.