**Calculating Compound Growth Rate**

on a simple pocket calculator

on a simple pocket calculator

**If we want to find the answer to a given amount of money compounded over a known number of years, at a constant percentage rate, here's how we do it:**

The Compound Growth Rate formula is

For example: if

The Compound Growth Rate formula is

**(1+i)^n = R**. In this formula,**i**represents our growth rate,**n**represents the number of years we'll be holding our investment, and**R**represents a ratio we will later use to multiply times our beginning value to arrive at our end value five years down the road.For example: if

**i**represented 15%, and**n**represented 5 years, the equation would be verbally expressed as: "1+ 15% to the 5th power equals a Ratio."**Using the same numbers as above, let's do the equation and check our results. We'll start with $100, and let’s use a constant compound growth rate (**

15% expressed as a number is 0.15. We'll add that number to 1, as in (

Spoiler alert: Our answer

Yes, this little equation can be solved with a simple, no frills, pocket calculator.

First, get out your pocket calculator, or smart phone calculator and type in 1.15. Now hit the times (X) key once. DON’T hit the equal (=) key just yet!

Remember, 1.15 to the first power, or 1.15^1, equals 1.15. That number, 1.15, is already displayed on our pocket calculator screen, so no calculations are necessary for this first step. All we have to do is enter the number 1.15 and hit the times (X) key once.

Now, to find 1.15 to the second power, or 1.15^2, we simply hit the equal (=) key once. The 1.15 displayed on the screen will change to 1.3225. Continuing the process, to find 1.15 to the third power, or 1.15^3, we simply hit the equal (=) key again. The answer will change to 1.5208. To find 1.15 to the fourth power, or 1.15^4, we hit the equal (=) key again. Our answer should now be 1.7490. Finally, to find 1.15 to the fifth power, or 1.15^5, we hit the equal (=) key one last time. The answer on the screen should be 2.0114.

That simple series of calculations tells us that 1.15 to the fifth power produces a Ratio of 2.0114. So, if we put $100 into an investment that grows in value at a rate of 15% for 5 years, our investment at the end of that 5 year period will be worth 2.0114 times more that it was when we first started ($100 x 2.0114 = $201.14). We will have doubled our money in 5 years.

Note that in raising 1.15 to the fifth power, we have hit the times (X) key

If we are starting with $50 at 12%, for 8 years, we’ll type in 1.12 on our calculator, then hit the times (X) key

Some pocket calculators and some smart phones will not increase the value of the number displayed on the screen each time we tap the (

The easiest way is to simply multiply each number by itself for the appropriate number of times. In the first example,

In the second example above, we would start out with 1.12, then multiply that number times itself

Most smart phones have two calculators. Holding the phone upright usually gives us a basic calculator. If we rotate the phone 90 degrees left, or right, we’ll sometimes see a calculator that has more functions. Look for the

We’ll use

**i**) of 15% for a period (**n**) of 5 years.15% expressed as a number is 0.15. We'll add that number to 1, as in (

**1+ i**), and that will give us (1+0.15), or (1.15). Next,**n**represents the number of years, so in this case**n**will be 5. Our formula should now look like this: (1.15)^5 = R. Verbally, it's going to sound like this: "1.15 to the fifth power equals a ratio."Spoiler alert: Our answer

**R**, is going to equal 2.0114. If we started with a $100 investment, and it grew at a compounded rate of 15% per year for five years, we would end up with 100 x 2.0114 = $201.14.Yes, this little equation can be solved with a simple, no frills, pocket calculator.

**Believe it or not, this is really simple!****First example:**First, get out your pocket calculator, or smart phone calculator and type in 1.15. Now hit the times (X) key once. DON’T hit the equal (=) key just yet!

Remember, 1.15 to the first power, or 1.15^1, equals 1.15. That number, 1.15, is already displayed on our pocket calculator screen, so no calculations are necessary for this first step. All we have to do is enter the number 1.15 and hit the times (X) key once.

Now, to find 1.15 to the second power, or 1.15^2, we simply hit the equal (=) key once. The 1.15 displayed on the screen will change to 1.3225. Continuing the process, to find 1.15 to the third power, or 1.15^3, we simply hit the equal (=) key again. The answer will change to 1.5208. To find 1.15 to the fourth power, or 1.15^4, we hit the equal (=) key again. Our answer should now be 1.7490. Finally, to find 1.15 to the fifth power, or 1.15^5, we hit the equal (=) key one last time. The answer on the screen should be 2.0114.

That simple series of calculations tells us that 1.15 to the fifth power produces a Ratio of 2.0114. So, if we put $100 into an investment that grows in value at a rate of 15% for 5 years, our investment at the end of that 5 year period will be worth 2.0114 times more that it was when we first started ($100 x 2.0114 = $201.14). We will have doubled our money in 5 years.

Note that in raising 1.15 to the fifth power, we have hit the times (X) key

__once__, and the equal (=) key__four__times.**Second example:**If we are starting with $50 at 12%, for 8 years, we’ll type in 1.12 on our calculator, then hit the times (X) key

__once__, and then the equal (=) key__seven__times. That's (1+ 0.12)^8, or 1.12 raised to the eighth power. Our ratio will be 2.48. If we multiply that ratio times the $50 we started with, we’ll find that 8 years of compounding at 12% will give us a dollar value of $123.80.**Note 1: If this procedure does not work on your pocket calculator or smart phone calculator, please refer to the information below.****Note 2:**Some pocket calculators and some smart phones will not increase the value of the number displayed on the screen each time we tap the (

**=**) key. If that is the case, there are two other ways to go.The easiest way is to simply multiply each number by itself for the appropriate number of times. In the first example,

**1 + i**was 1.15. Using five years as our**n**factor, we'll just multiply 1.15 x 1.15 x 1.15 x 1.15 x 1.15, and we'll get the same answer as in the first example above (2.0114). Notice, the formula started out with 1.15, and then we multiplied that number times itself__four__times. Don't forget to multiply our answer, the ratio of 2.0114, times the starting $ value to arrive at the end $ value.In the second example above, we would start out with 1.12, then multiply that number times itself

__seven__more times. On paper, it should look like this: 1.12 x 1.12 x 1.12 x 1.12 x 1.12 x 1.12 x 1.12 x 1.12. Your answer will be a ratio of 2.48. Again, don't forget to multiply the start value by the Ratio (2.48) to get your end value.**The scientific way, but still, very easy.**Most smart phones have two calculators. Holding the phone upright usually gives us a basic calculator. If we rotate the phone 90 degrees left, or right, we’ll sometimes see a calculator that has more functions. Look for the

**x^y**function. This is the same function we’ll use if we are using a scientific pocket calculator, or the__Scientific__calculator provided by the Microsoft "Office Suite" on your computer. In rare cases, it may be displayed as**(x)^y**.We’ll use

**x**to represent

**(1+i)****, and we’ll use**

To do the calculation using the

In the second example above, we started with $50, at 12% growth rate, for 8 years. On our smart phone, or scientific pocket calculator, type in

**y**to represent the number of years. In our first example above, we started with $100, compounded at 15%, for 5 years.**1+i**was 1+15% and we converted that into 1.15. The number of years was 5.To do the calculation using the

**x^y**function, we’ll first type**1.15**into the display screen, then we’ll tap the**x^y**key. Next, we’ll tap the**5**key. Now, tap the**=**key. That’s it. We’re finished! Our answer should be a ratio of 2.0114. As in the first example, we’ll multiply that ratio by the number we started with ($100), and we’ll get $201.14.In the second example above, we started with $50, at 12% growth rate, for 8 years. On our smart phone, or scientific pocket calculator, type in

**1.12**, now tap the**x^y**key, next tap the**8**key, now tap the equal (**=**) key. The answer should be a ratio of 2.47596. Now, multiply 2.47596 times $50 and the final dollar value at the end of 8 years should be $123.80.**In reference to the first**

The function you see on the right is displayed in the traditional manner, but on some computer programs (TK6) and some smart phones it may also be displayed as

Have fun with this, and if you have any questions, please ask one of our instructors, or go to the

**Note 1**above, here is an example of a smart phone calculator turned sideways. The function we want to use is circled. The**X**in this function represents the**(1+i)**in the Compounded growth rate formula, and the**y**represents the**n**in the compounded growth rate formula.The function you see on the right is displayed in the traditional manner, but on some computer programs (TK6) and some smart phones it may also be displayed as

**x^y**.Have fun with this, and if you have any questions, please ask one of our instructors, or go to the

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