2 1 Calculating Interest 04 14 2014 The language The principle is equal to what you borrowed or what you have on balance The interest rate i expressed as a percentage ii The compounding period The term length of time that money is invested owed Simple interest interest is a charge for using money For simple interest it is a flat percentage of the principle each year Principle 1 you owe what you borrow interest Example Dave borrows 1 000 from his Uncle Don Dave agrees to pay him back 5 interest per year How much does he owe after 1 year The principle 5 of the principle P 05P P 1 05 P 1000 so he owes 1000 1 05 1 050 What if he couldn t pay back for 2 years Guess 1 100 1 000 1 2 0 05 But is this really fair No Suppose Dave paid him back after one year and then borrowed exactly what he paid back under the same terms How much would he owe at the end of the 2nd year End of the 1st 1 050 What does he owe after a year on that P 1 050 o He owes 1 050 1 0 05 1 102 50 Note in the second example we charged interest on interest o Start Principle of 1 000 o After 1 year 1 050 1000 1 0 05 1 0 05 1 000 1 0 05 2 This is called compound interest After n years Dave would owe o P 1 0 05 n So after 10 years 1000 1 0 05 10 o 1000 1 638 1000 1 0 628 That s a total of 68 interest over the 10 years It would ve doubled in around 15 years However most of the time interest is compounded more often If the interest is I then let i I 100 Common compounding periods Monthly 1 12 of the interest is charged per month compounded Quarterly of the interest in charged every quarter 3 months Yearly The effective interest rate or yield when you compound is the equivalent yearly simple interest rate Example Suppose that you have a credit card at 18 compounded monthly Then you are charged 1 5 per month 18 12 1 5 What would you owe on a balance of P after a year P 1 0 015 12 P 1 015 12 P 1 196 P 1 0 196 The effective yearly rate is 19 6 If George invests 10 000 in a stock that appreciates at 11 per quarter 44 per year How much will his stock be worth in a year 11 per quarter is the same as 44 compounded quarterly P principle i interest after dividing by 100 Then quarterly compounding has the yield P 1 i 12 4 10 000 1 44 4 4 10 000 1 11 4 10 000 1 518 What is my percentage yield 1 518 1 0 518 So 51 8 Credit Cards Deena has 2 300 on her credit card at 19 8 compounded monthly What s her changed interest new balance at the end of the month If she has to make a 2 5 payment then what will her next month balance be at the beginning Start P 2300 I 19 8 i 0 198 The monthly multiplier 1 i 12 1 0 198 12 1 0 165 1 0165 Interest charge 2300 1 0165 2 337 95 Now she owes 2 5 of this or 2 337 95 x 0 025 Change percentage into proportion P principal starting balance I interest rate as i I 100 interest as proportion n number of times per year interest is compounded n 1 yearly n 4 quarterly n 12 monthly N number of years quarters months the principal draws interest B balance at the end of N compounding periods For four years at n 1 N 4 n 4 N 16 n 12 N 48 B P 1 i n N The monthly savings formula for monthly compounding is obtained from the geometric series and is discussed in the book on pages 203 205 If m month are deposited at I i I 100 compounded monthly every month Then the balance after k months will be o D 12 i 1 i12 k 1 M Logic 1 2 be able to identify the difference between inductive prejudice and deductive reasoning conclusions based on observations both be able to identify the basic forms of deductive reasoning direct indirect transitivity or arguments one the other or 3 4 be able to identify basic fallacies mistakes negation of an implication Survey Analysis 1 exact overlap problems 2 variable overlap problems Lines and linear inequalities 1 know how to graph lines and how to check to see if a point is on a line 2 the consequences of the increment property 3 solutions to linear inequalities and system CORNERS 4 Max and min of Ax By on a region