How long will it take p18 000 to triple if it is invested at 5% compounded quarterly

Video Transcript

So there's two different types of compound interest that we've talked about. We've talked about compounding continuously which is every second of every hour of every day. And then we've also talked about compounding financially which is daily, monthly etcetera. What am I, how am I compounding here? Well I'm compounded continuously which is a different formula. So A equals P. The original amount E the exponential function Which is a constant. About 2.718 race. There are times t Okay so how long does it take for $6,000? So I start with My original amount is $6,000 and I wanted to triple. So I want my output to be three times 6000 which is about 18,000. He is just a constant. I leave that alone. What's my rate growth? What's my Rate? So do I put 7.6 in for our, no, I'm going to divide by 100 or move the decimal to place to the right to convert it to a decimal and then I'm solving for a time because I'm asking how long will it take? So I use this equation to set up an equation with what I know I'm going to get T by itself is my goal. So I'm gonna divide by 6000 of both sides. This will be three equals e to the 0.0716. Okay, now my question is, how do I get T down from this? Exponents? Well what's the opposite of the exponential function? The opposite of the exponential logarithmic function with the same base? So not that Ellen is the same thing as log base. E. So I can take the Ellen of both sides and since Ellen and er both in verses these will cancel or by the rules of logarithms, this will multiply in front And this will cancel to be one Equals Ln of three. And then finally, I just want to get T by itself. So I'm going to just divide by 0.076. And when I plugged this in my calculator, making sure that I closed the Ellen completely, I'm going to get T is approximately 14.45 Years and I went around to the nearest year. So I went around to the ones place. So does that for one of the four? No, it does not. So it will take approximately 14 years if I compound continuously at 7.6% interest for this investment to triple.

In this section we cover compound interest and continuously compounded interest.

Use the compound interest formula to solve the following.

Example: If a $500 certificate of deposit earns 4 1/4% compounded monthly then how much will be accumulated at the end of a 3 year period?

Answer: At the end of 3 years the amount is $576.86.

Example: A certain investment earns 8 3/4% compounded quarterly.  If $10,000 is invested for 5 years, how much will be in the account at the end of that time period?

Answer: At the end of 5 years the account have $15,415.42 in it.

The basic idea is to first determine the given information then substitute the appropriate values into the formula and evaluate.  To avoid round-off error, use the calculator and round-off only once as the last step.

• Annual  n = 1
• Semiannual n = 2
• Quarterly n = 4
• Monthly n = 12
• Daily n = 365

One important application is to determine the doubling time.  How long does it take for the principal in an account earning compound interest to double?

Example: How long does it take to double $1000 at an annual interest rate of 6.35% compounded monthly?

Answer:  The account will double in approximately 10.9 years.

The key step in this process is to apply the common logarithm to both sides so that we can apply the power rule and solve for time t.  Use the calculator in the last step and round-off only once.

Example: How long will it take $30,000 to accumulate to $110,000 in a trust that earns a 10% return compounded semiannually?

Answer: Approximately 13.3 years.

Example: How long will it take our money to triple in a bank account with an annual interest rate of 8.45% compounded annually?

Answer: Approximately 13.5 years to triple.

Make a note that doubling or tripling time is independent of the principal. In the previous problem, notice that the principal was not given and that the variable P cancelled.

Use the continuously compounding interest formula to solve the following.

Example: If a $500 certificate of deposit earns 4 1/4% annual interest compounded continuously then how much will be accumulated at the end of a 3 year period?

Answer: the amount at the end of 3 years will be $576.99.

Example: A certain investment earns 8 3/4% compounded continuously.  If $10,000 dollars is invested, how much will be in the account after 5 years?

Answer: The amount at the end of five years will be $15,488.30.

The previous two examples are the same examples that we started this chapter with.  This allows us to compare the accumulated amounts to that of regular compound interest.

As we can see, continuous compounding is better, but not by much.  Instead of buying a new car for say $20,000, let us invest in the future of our family.  If we invest the $20,000 at 6% annual interest compounded continuously for say, two generations or 100 years, then how much will our family have accumulated in that time?

The answer is over 8 million dollars. One can only wonder actually how much that would be worth in a century.

Given continuously compounding interest, we are often asked to find the doubling time.  Instead of taking the common log of both sides it will be easier take the natural log of both sides, otherwise the steps are the same.

Example: How long does it take to double $1000 at an annual interest rate of 6.35% compounded continuously?

Answer: The account will double in about 10.9 years.

The key step in this process is to apply the natural logarithm to both sides so that we can apply the power rule and solve for t.  Use the calculator in the last step and round-off only once.

Example: How long will it take $30,000 to accumulate to $110,000 in a trust that earns a 10% annual return compounded continuously?

Answer: Approximately 13 years.

Example: How long will it take our money to triple in a bank account with an annual interest rate of 8.45% compounded continuously?

Answer: Approximately 13 years.

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