How Do You Calculate the Interest Rate for Compound Interest Problems?

• iksotof
In summary, the conversation was about finding the interest rate in an investment problem given the present value, future value, and compounding periods. The steps involved taking the twelfth root of an equation in order to solve for the interest rate. The conversation also touched on the definitions of cube root and how to find it using a calculator. The summarizer expresses their gratitude for the help and patience.
iksotof
Have this question in relation to some investment exam I am doing, I am a maths novice being some years since leaving school etc, ok enough of the excuses.

Example

FV = future value
PV = present value
R = interest rate
N = number of compounding periods

my PV is 6000 and my FV is 10000. Compounding periods is 12, I need to find the interest rate, thus...

100000 = 6000 (1 + R) to power of 12 (sorry don't know how to represent that on key board).

substitution: 1.67 = (1 + R) power 12

substitution 2: 12√1.67 = 1 + R

1.0435 - 1 = R

R = 4.45.

All very well, but I lose understanding of how the maths is applied at sub 2, my textbook
has given this model answer. For a novice please show me step by step what

12√1.67 means.

Thanks you Darren.

Okay, to "cancel" the operation of cubing a number, you need to to take the cube root in order to get back to your number.

That is we have for any number a:
$$\sqrt[3]{a^{3}}=a$$
(Taking the cube root of a cube leaves you with the original number "a")

Similarly, if a positive number "a" is raised to the twelfth power, you must take the twelfth root to get a back:
$$\sqrt[12]{a^{12}}=a.$$

Now, how to use this in your present case`

WEll, you are to solve for R, but that appears in your equation within the expression $(1+R)^{12}[/tex] Now, you are ALLOWED with an equation to "do" whatever operation you like, AS LONG AS YOU DO THE SAME ON BOTH SIDES!. Thus, in order to remove the twelfth power on the right hand side (increasing your chances to solve for R!), you take the twelfth root of both sides. That's what the second substitution does. iksotof said: Have this question in relation to some investment exam I am doing, I am a maths novice being some years since leaving school etc, ok enough of the excuses. Example FV = future value PV = present value R = interest rate N = number of compounding periods my PV is 6000 and my FV is 10000. Compounding periods is 12, I need to find the interest rate, thus... 100000 = 6000 (1 + R) to power of 12 (sorry don't know how to represent that on key board). Using just the keyboard, 100000= 6000(1+R)^(12). Using "html" tags, 100000= 6000(1+R)12. Using "LaTex" [itex]100000= 6000(1+ R)^{12}$. But you don't mean "100000" on the left, only 10000.

substitution: 1.67 = (1 + R) power 12
Dividing both sides by 6000, 10000/6000= 10/6= 4/3 which is 1.67 to two decimal places.

substitution 2: 12√1.67 = 1 + R

1.0435 - 1 = R

R = 4.45.

All very well, but I lose understanding of how the maths is applied at sub 2, my textbook
has given this model answer. For a novice please show me step by step what

Thanks you Darren.

12√1.67 means.[/quote]
It means the "twelfth root of 1.67", not "square root"! The "twelfth root of x" is defined as the number whose 12 power is x- that's why $^{12}\sqrt{(1+R)^{12}}= !+R$. To actually do the that on a calculator, if you have a "^", power, key, use the "1/12 power": 1.67 ^ (1/12). If your calculator does not have a "^" key you will need to use logarithms. log(x^{1/12})= (1/12)log x= log x/12. To find the 12th root of 1.67, take the logarithm of 1.67, divide by 12, the use the "inverse" to logarithm: 10^ if you are using common logs

$^{12}\sqrt{(1+R)^{12}}$, rather: $\sqrt[12]{(1+R)^{12}}$

Thank you greatly but I am a total novice and whilst I am sure your explanation is clear to a knowledgeable person, I am not, in isolation therefore it doesn't mean a great deal. Could you maybe show the maths in long hand please? For example 2 cubed is 2 x 2 x 2=
8.

Please continue with this long hand through the remainder.

Thanks again Darren.

Yes, and tell me:

What is the definition of the cube root of a number?

iksotof said:
Thank you greatly but I am a total novice and whilst I am sure your explanation is clear to a knowledgeable person, I am not, in isolation therefore it doesn't mean a great deal. Could you maybe show the maths in long hand please? For example 2 cubed is 2 x 2 x 2=
8.

Please continue with this long hand through the remainder.

Thanks again Darren.
Continue with what? 23= 8 exactly as you said. If you want us to now find $\sqrt[3]{8}$ (thanks, arildno)? There is no calculation involved. By the definition of cube root, it is 2.

If you were to ask me, "What is the cube root of 7?", I would whip out my calculator and answer that $\sqrt[3]{7}= 1.913$, approximately.

Thank you, it was the fundamental understanding I was being a total dumb ass with. . Thanks for your patience and inputs, my old school lessons are finally reawkening from the recesses of my mind, 20 years on!

Greatly appreciated, Darren.

What is a square root?

A square root is a mathematical operation that, when applied to a number, gives the value that, when multiplied by itself, results in the original number.

What is the symbol for square root?

The symbol for square root is √, also known as a radical sign.

How do you find the square root of a number?

To find the square root of a number, you can use a calculator or a mathematical formula. The formula is √x = y, where x is the number and y is the square root.

What is substitution in math?

Substitution in math is the process of replacing a variable or expression with another value or expression. This helps simplify equations and solve problems.

How is substitution used in finding square roots?

In finding square roots, substitution is used to replace the radical sign (√) with its equivalent exponential form. This allows for easier calculations and solving of equations involving square roots.

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