Financial Exponential Functions - Compound Interest

Compound Interest and Several Examples of Exponential Functions

Here, we will go over some examples of Compound Interest. It is important to know that Compound Interest is just a exponential function that can be solved using simple algebra. 

Labeled Formula of Compound Interest

P = the principal amount, otherwise known as the original/ initial amount.

r = rate at which the money grows per year; interest rate (written in decimals)

m = number of times compounded in a year (ex. daily, biweekly, weekly, monthly, semimonthly...)

t = after how many years (amount of years)

A = the total amount after 't' years

If 'm' equals the following numbers, that is the way we are compounding in a year.

m = 1 only counted once per year, every year

m = 2 semiannually, because it is only counted 2 times a year; meaning after every 6 months (12/2=6)

m = 4 quarterly, because it is only counted every 4 times a year; meaning after every 3 months (12/4=3)

m = 12 monthly, because there are 12 months in a year, so after every month for 12 total counts in a year.

m = 52 weekly, because there are 52 weeks in a year, so after every week for 52 total counts in a year.

m = 365 daily, because there are 365 days in a year, so after every day for 365 total counts in a year.

Great, let's try an example.

So, we have to solve for x in terms of 'm'.

Let's rephrase the question into a formula.

Original formula: A = P(1 + r/m)^mt

Our equation: A = 200(x)^mt 

where x = (1 + r/m)

We know that 'r' or rate has to be written as a decimal. So the rate HAS to be a decimal.

The answer is A.)

See? No math required on that one!

--

Here's another example:

Peter starts with $5,000. That is his principal value. His rate is written as a percentage; 9.2%. Since we are given percentage, we would need to divide by 100 = (9.2/100). Rate is 0.092. Now we just plug the numbers we have to find total amount (A)

Original formula: A = P(1+r/m)^mt

A = 5000(1+0.092/1)^1*2

P= 5000

r= 0.092

m= 1, it's calculated per year

t= 2, total amount after 2 years

A= 5000(1.092)^2

A = 5000(1.192464)

A = $5962.32

Sine, Cosine, Tangent and their Reciprocals formulas and Trigonometry examples

 

Sine (Sin), Cosine (Cos), Tangent (Tan)



How to remember the sides of a triangle:

This is called "beta"

Simple, the angle with the beta symbol, represents the viewpoint.

The opposite side will always face right across from the angle "beta".

The hypotenuse will always be across from the 90°. 

The adjacent side will always be right below the hypotenuse; the line between the angle "beta" and 90°. 

Diagram for Reference

Awesome! Now we can learn the Sin, Cos, and Tan formulas! They make life a bit more easier.

 

Naming each makes it easier to remember:

Soh for Sin because Sin= Opposite/Hypotenuse

Cah for Cos because Cos=  Adjacent/Hypotenuse 

and Toa for Tan because Tan= Opposite/Adjacent

 

Sin is used for right triangles that relate to the values of opposite side and hypotenuse to find the value of the triangle's "beta" angle.

For reference purposes

Ex.1

o = 5, h = 10 find beta.

Press "2nd" button on calculator.

Following, press the "SIN" button

Next, enter the value of your opposite side. In this case, 5.

Then press the division sign. This makes a ratio 

Lastly, enter the value of your hypotenuse side. In this case, 10.

Your result should be "Sin of 'beta' = 30°."

Ex.2 Only beta and one side are given

  o = 1, "beta"= 90° find hypotenuse.

This is better written and understood as Sin 90° = 1/h, otherwise "1/Sin 90° = h/1"

Cross multiplication.

Start by entering your opposite value. In this case 1.

Then press the division sign

Press the "SIN" button.

Enter the value of "beta". In this case 90.

Your result should be h = 1.

Cos is used for right triangles that relate to the values of adjacent side and hypotenuse to find the value of the triangle's "beta" angle.

For reference purposes

 Ex.1 

a = 5, h = 10 find beta.

Press "2nd" button on calculator.

Following, press the "COS" button

Next, enter the value of your adjacent side. In this case, 5.

Then press the division sign. This makes a ratio 

Lastly, enter the value of your hypotenuse side. In this case, 10.

Your result should be "Cos of  'beta' = 60°."

Ex.2 Only beta and one side are given

  a = 4, "beta"= 60° find hypotenuse.

This is better written and understood as Cos 60° = 4/h, otherwise "4/Cos 60° = h/1"

Cross multiplication.

Start by entering your adjacent value. In this case 4.

Then press the division sign

Press the "COS" button.

Enter the value of "beta". In this case 60.

Your result should be h = 8.

Tan is used for right triangles that relate to the values of opposite side and adjacent side to find the value of the triangle's "beta" angle.

For reference purposes

Ex. 1

o = 10, a = 10 find beta.

Press "2nd" button on calculator.

Following, press the "TAN" button

Next, enter the value of your opposite side. In this case, 10.

Then press the division sign. This makes a ratio 

Lastly, enter the value of your adjacent side. In this case, 10.

Your result should be "Tan of 'beta' = 45

Ex.2 Only beta and one side are given

  o = 10, "beta"= 45° find adjacent side.

This is better written and understood as Tan 45° = 10/a, otherwise "10/Tan 45° = h/1"

Cross multiplication.

Start by entering your adjacent value. In this case 10.

Then press the division sign

Press the "TAN" button.

Enter the value of "beta". In this case 45.

Your result should be a = 10.

Important Symbols to note!

The Various Symbols and Formulas in Statistics:

Prime and Composite Number Theorem (Divisibility Rules)

How can we check if a number is prime or composite?

Let 'x' be the number you check:

x² = x * x

Say √(x) = y 

√(x) = y

y = a * b 

Now if we were to assign a value to x, like x = 123, then: 

√(123) ~ between √(121) and √(144) closer to √(121) because

123 - 121 = 2

144 - 123 = 21

2 < 21

midpoint

144-121= 23

23/2 = 11.5

and so we know that √(123) ~ is close to 11, a little higher somewhere between 11.1 or 11.09.

y = a * b 

Let's say approximate for y is 11.1

So either a or b have to be less than 11.1

1 * 11.1

2 * 5.55

...

We just have to check the numbers less than the whole number 11 to see if 123 is divisible. 

123 / 2 = WE KNOW THIS WON'T WORK BECAUSE LAST DIGIT ISN'T EVEN.

123 / 3 = WE KNOW THIS WILL WORK BECAUSE 1+2+3= 6, AND 6 IS A MULTIPLE OF 3.

WE CAN STOP HERE.

123 / 4 = WE KNOW THIS WON'T WORK BECAUSE DIDN'T SATISFY "MULTIPLE OF 2"

this is literally just 123 / (2*2) if it helps.

123 / 5 = WE KNOW THIS WON'T WORK BECAUSE LAST DIGIT ISN'T 0 NOR 5.  

123 / 6 = SINCE 123 WAS DIVISIBLE BY 3 BUT NOT 2, IT WILL NOT BE DIVISIBLE BY 6.

this is literally just 123 / (2*3) if it helps.

123 / 7 = WILL NOT WORK; CHECK DIVISIBILITY RULE BY 7 BELOW.

123 / 8 = WILL NOT WORK, NOT DIVISIBILE BY BOTH FACTORS OF 8: 2 NOR 4.

this is literally just 123 / (2*4) if it helps.

123 / 9 = WE KNOW THIS WON'T WORK BECAUSE 1+2+3 = 6, AND 6 IS NOT A FACTOR OF 9.

123 / 10 = WE KNOW THIS WON'T WORK BECAUSE LAST DIGIT ISN'T 0.

123 / 11 = WE KNOW THIS WOULDN'T WORK BECAUSE VALUE IS ONLY 2 AWAY FROM NEAREST MULTIPLE OF 11 WHICH IS 121.

Now we just have to check the first one that works. In this case, check 3.

123/3 = 41

Therefore, 123 is NOT a prime, but a composite number. 

This trick will help you find out till what number you should bother checking (dividing it by) to see if x is prime or composite.. :)

Extras:

EXPLORING THE MULTIPLE OF SEVEN RULE (really cool watch!):

https://youtu.be/UDQjn_-pDSs