How long will it take for a sum of money to triple in value if invested at an annual rate of 3 %

How long will it take for a sum of money to triple in value...

Key Concepts

Continuous Compounding

This concept describes the process where interest is added to the principal continuously, causing the investment to grow at an exponential rate. It is modeled mathematically with the exponential function, reflecting how small intervals of time contribute to overall growth.

Exponential Growth

Exponential growth refers to the increase of a quantity by a fixed percentage rate over time, resulting in growth that accelerates as the base amount increases. In financial applications, it describes how investments grow when interest is compounded continuously.

Natural Logarithm

The natural logarithm is used as the inverse function to the exponential function, making it essential for solving for time in continuous compounding problems. It is especially useful when determining the period required for an investment to reach a certain multiple of its original value.

Transcript

00:01 Okay, so we are given a scenario where we want to find how long it's going to take for a value, a given value to triple in value if invested at an annual rate of 3 % compounded continuously.

00:18 All right.

00:19 So we already know that when you're looking at continuous compounding, the formula that we basically use is the future value is equal to the p.

00:30 Times e are constant to the power of r -t, where your fv is the future value, your p is the principal or the present value.

00:44 And we have your e, which is our constant, r is the interest rate.

00:54 And the t is basically the number of years.

01:01 In this case, that is what we're looking for.

01:07 So in that formula there, you will discover that if the future value has to triple, then it means it's three times should be equal to one if that should be the principle.

01:24 Because that is going to be a constant ratio, whichever value you pick for the principal, the future value, she's going to be triple that.

01:34 So we can as well take 3 is equal to 1.

01:38 E to the power of r.

01:41 T.

01:42 Okay.

01:42 So fortunately we are given the r, which is 3%.

01:47 So we are basically going to substitute in our formula.

01:51 Okay.

01:52 So instead of the r, we're going to put there 3%.

01:56 So basically what we end up having is 3, which is the future value is equal to p.

02:02 Since we are taking one we may not need to write it down e to the power of rt okay we've already established that our rate is 3 % so we are going to replace that with the 0 .03 which is 3 % t now according to the rules of logarithm right are e to the power 0 .0 .0 which is 3 % t now according to the rules of logarithm right our e to the power 0 .03 is simply going to give us 0 .03 t is going to be equal to loan 3.

02:46 And we're going to divide both sides by 0 .03 t is equal to loan 3 divided by 0 .03.

02:59 That's the number of years going to take...

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