Mortgage Calculator - Algebra II

Monthly mortgage payments are determined by the principal and interest rate. In fact, a good Algebra II student should be able to understand the derivation of the monthly payment for a standard 30 year mortgage.

Here is an example of a mortgage payment calculator:



The goal of this posting is to show how one could teach Algebra II students the math necessary for creating this widget. Prerequisites:

• Algebraic sophistication (very comfortable with manipulating equations)
  
• Basic understanding of sequences and series
• Understand the telescope method for deriving sequence formulas
• Know the formula for a geometric series

The Proof:

Let

• P_n = principle amount after n months
• r = annual interest rate (compounding monthly)
• m = monthly payment (constant - doesn't change month by month)
• N = the number of monthly payments until the principle is completely payed off (for a 30 year mortgage N = 360)

First derive a recursive definition for the sequence P_n . To get this, notice that the interest that needs to be payed is the principle times the interest percentage for one month. The interest percentage for one month is simply r / 12. Therefore the interest portion of payment n is:

• interest owed on month n = I = P_n • (r / 12)

The principle payed off is the amount of the monthly payment left over after paying the necessary interest. In other words

• Principle Payed on month n = m - I = m - P_n • (r / 12)

Therefore, the principle remaining is given by:

• Priciple Remaining = P_n+1 = P_n - ( m - P_n • (r / 12) )

Simplify this to:

P_n+1 = P_n• ( 1 + r / 12 ) - m

Finally, letting α = ( 1 + r / 12 ) the formula becomes

P_n+1 = α P_n - m

Applying the telescoping method, together with the formula for geometric series we get:

• P_n= αⁿ P₀ - m (αⁿ - 1) / (α- 1)

Finally, since the mortgage is payed off after the N'th payment we have

• P_N = 0
  

or plugging N into the 2nd to last formula:

• 0 = α^N P₀ - m (α^N - 1) / (α- 1)

or solving for m:

• m = α^N (α- 1) / (α^N - 1) P₀

From the "All Your Base Are Belong to Us" Department

From time to time, I see a usage of English so inspiringly bad that I've got to capture it. In case you have no idea what the blog entry title is referring to, that's probably
the most famous and funniest online example of bad English.

First example, Vegan Burdock Ramen. Quoth the preparation instructions:

1. Clean a middle, size of bowl with warm water put rice flour and spice in bowl.
2. Add boiled water in bowl and put cover on it after 3-5minutes, mix rice flour properly.
3. Another method to cook which makes a special taste. Add water 600c.c. into pan, put rice flour and spice into pan when water is boiling it will be ready after 3minutes of boiling.
   

Some deep thoughts to ponder:

• Wouldn't a dirty bowl work just as well?
• If we are starting with the rice flour, why are the instructions for making the rice noodles missing?
• How are we going to boil all that water in a pan?
  

Front of Package



















Preparation Instructions