A=P*e*^{rt}

**P**=Principal (Original $ Amount Invested)

* e*=Mathematical Constant

**r**=Interest Rate

**t**=Time

Hi everyone and welcome to MathSux! In this post, we are going to go over the continuously compounding interest formula, A=P*e*^{rt}! This is a great topic as it relates to finance and real-world money situations (sort of). We are going to break down everything step by step by understanding what the continuously compounding interest rate formula A=P*e*^{rt} is, identifying each component of the formula, and then applying it to an example. If you’re looking for more, don’t forget to check out the video and the practice questions at the end of this post. Happy calculating! 🙂

**What is Continuous Compounding Interest Formula?**

**Let’s say we have $500 and we want to invest it.**

What if it compounded interest once a year?

Twice a year?

Once a day, or 365 days a year?

What if we compounded interest every second of the day for a total of 86,400 seconds throughout the year!?

And what if we kept going, making the number of times compounded annually more and more often to occur every half second? This is what **Continuous Compounding Interest** is, and it tells us how much we earn on a principle (original amount) if the compound interest rate for the year were to be granted an infinite number of times.

The weird thing is that continuous compounding interest is technically impossible (I’ve yet to see a bank that offers an infinite number of compounding interest!). Even though it is impossible, in math and finance, we look at continuous compounding interest for theoretical purposes, in other words, it’s for money nerds! Luckily, it comes with an easy-to-use formula, let’s take a look:

Now, let’s see this formula in action with the following **Example**:

**Step 1: **First, let’s write out our formula and identify what each value represents based on the question.

**Step 2: **Fill in our formula with the given values and solve.

**Practice Questions:**

1) Sally invested $1000 which was then continuously compounded by 4%. How much money will Sally have after 5 years?

2) Brad invested $1500 into an account continuously compounded by 5%. How much money will he have after 7 years?

3) Fran invests $2000 into an account that is continuously compounded by 1%. How much money will Fran earn by year 5?

## Solutions:

1) $1,221.40

2) $2,128.60

3) $2,102.54

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