In this video we discuss how to find or solve for time in compound interest problems. We also cover how to modify the compound interest formula to solve for time in compound interest problems.

Transcript/notes
The formula for calculating compound interest is A equals P times the quantity, 1 plus r divided by n raised to the n times t.

In this formula, A represents the total compound amount after a certain period of time, P represents the principal or the original amount invested or borrowed, r represents the interest rate, n represents the number of compounding periods per year and t represents the time in years.

If we want to find the time and are given all the values for the other variables, we can modify this formula to solve for T, the time in years.
This can be a bit confusing, so, I will go slow, and we will need to use a calculator with a log function or excel to do the actual calculations for problems.

We first divide both sides by P, the principal, which gives us A divided by P equals the quantity 1 minus r over n raised to the n times t.

Next, we need to eliminate these exponents of n times t. We can do this by using log base 10, taking the log base 10 of the quantity 1 plus r divided by n raised to the n times t. And since we took the log base 10 of the right side of the equation, we must also take the log base 10 of the right side of the equation. So, now we have log base 10 of a over p equals log base 10 of the quantity 1 plus r divided by n raised to the n times t.

Next, we are going to use the power property of logarithms, which states that log base b of a number or quantity, say m, raised to the x, equals x times log base b of m. So, this is a way to remove the exponent.

Using the power property just mentioned, we can remove the n times t exponent, and we have n times t, times log base 10 of the quantity 1 plus r divided by n. And the left side of the equation remains the same, log base 10 of a over p.

The next thing we can do is divide both sides by log base 10 of the quantity 1 plus r divided by n. This will cancel out the log base 10 of the quantity 1 plus r divided by n on the right side. And now we have log base 10 of a over p divided by the log base 10 of the quantity 1 plus r divided by n, equals n times t.

Next, we can multiply both sides by 1 over n. This will cancel out the n on the right side. And after doing this, we have our formula for t, which is t equals, log base 10 of a over p, divided by n times the log base 10 of the quantity 1 plus r divided by n.

Now, lets do an example problem. Lets say that someone invests $5000 into an account that compounds interest monthly. The yearly interest rate is 7.25%. How long will it take to double their money, or have $10,000 in the account?

Looking at the variables, A is the total compounded amount after a certain period of time. In this example that is $10,000, so A equals $10,000. P is the principal or original amount invested, which is $5000. R is the yearly interest rate, which is 7.25% and we need to convert this to a decimal by dropping the percent sign and moving the decimal 2 places to the left to get .0725, and n, the number of compounding periods equals 12, because it is compounded monthly. And, we don’t know what t is.

We have t, the time in years equals, the log base 10 of $10,000 over $5000, divided by the quantity 12 times the log base 10 of the quantity 1 plus .0725 over 12.

First, we can divide $10,000 by $5000, which equals 2, and the dollar sign gets cancelled out. We can also divide .0725 by 12, which equals .00604167 rounded off. We can then add 1 plus .00604167, which equals 1.00604167.

Next, we need to use a calculator to do the calculations for the log base 10 of 2 and log base 10 of 1.00604167. I am going to show you using an online calculator.

A simple google search of basic calculator brings this screen up, which we will use. There are 2 logarithm keys on this calculator, log and ln, make sure you use the log key, not the ln key. This log key automatically applies base of 10. So, we hit the log key and this prompt comes up, then we hit the number 2, and then the equals sign, and we get an answer of .30102999566.

Now for the log base 10 of 1.00604167, we again hit the log key and type in the number 1.00604167, and we get a result of .00261596946.
Now we have t equals .30102999566 divided by 12 times .00261596946. Here are the calculations on the screen for you, and we get a final answer of t equals 9.59 years rounded off.

Here are a couple of more examples of solving for time in compound interest, and the key is to take your time in doing the problem.

Timestamps
0:00 Formula for compound interest
0:25 How to modify formula to solve for time
0:54 Using the log base 10
1:23 Using the power property of logarithms
2:22 Formula for time in compound interest
2:35 Example problem of solving for time in compound interest
4:27 Using online calculator for log base 10

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