The languageThe principle: is equal to what you borrowed or what you have on “balance”The interest rate:i. expressed as a percentageii. The compounding periodThe term: length of time that money is invested/owedSimple 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 principleP+ .05P = P(1+.05) – P = 1000,so he owes 1000(1.05)= $1,050What 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= $1,050What does he owe after a year on that P=1,050?He owes 1,050 (1+0.05)= $1,102.50Note: in the second example, we charged “interest on interest”Start= Principle of 1,000After 1 year= 1,050 = 1000(1+0.05)(1+0.05)= 1,000(1+0.05)^2This is called compound interest: After n years, Dave would oweP(1+0.05)^nSo after 10 years,1000(1+0.05) ^10= 1000(1.638) = 1000(1+0.628)That’s a total of 68% interest over the 10 yearsIt would’ve doubled in around 15 yearsHowever, most of the time, interest is compounded more often.If the interest is I%, then let i=I/100Common compounding periods:Monthly: 1/12 of the interest is charged per month (compounded)Quarterly: ¼ of the interest in charged every quarter (3 months)YearlyThe 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= 1.5What 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= principlei= interest (after dividing by 100)Then quarterly compounding has the yieldP(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+0.518So, 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= 0.198The monthly multiplier (1+i/12) = (1+0.198/12)= (1+0.165) = 1.0165Interest charge= 2300 (1.0165)= 2,337.95Now, she owes 2.5% of this, or 2,337.95 x 0.025Change percentage into proportionP=principal = starting balanceI= interest rate as %, i= I/100= interest as proportion,n= number of times per year interest is compoundedn=1 (yearly), n=4 (quarterly), n=12 (monthly)N= number of years/quarters/months the principal draws interestB= balance at the end of N compounding periodsFor four years atn=1: N= 4n= 4: N= 16n= 12: N=48B=P (1+ i/n)^NThe 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:D= (12/i) [ (1+i12)^k-1] MLogic:1. be able to identify the difference between inductive (prejudice) and deductive reasoning (conclusions based on observations)2. be able to identify the basic forms of deductive reasoning: direct, indirect, transitivity ‘or’ arguments (one the other or both)3. be able to identify basic fallacies (mistakes)4. negation of an implicationSurvey Analysis:1. exact overlap problems2. variable overlap problemsLines, and linear inequalities:1. know how to graph lines, and how to check to see if a point is on a line2. the consequences of the ‘increment property’3. solutions to linear inequalities, and system. CORNERS!4. Max and min of Ax+By on a region2.1 Calculating Interest 04/14/2014The languageThe principle: is equal to what you borrowed or what you have on “balance”The interest rate: - i. expressed as a percentage - ii. The compounding periodThe term: length of time that money is invested/owed Simple interest: interest is a charge for using money. For simple interest, it isa 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 principleP+ .05P = P(1+.05) – P = 1000,so he owes 1000(1.05)= $1,050What 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)= $1,102.50Note: in the second example, we charged “interest on interest” o Start= Principle of 1,000o 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 oweo P(1+0.05)^nSo after 10 years, 1000(1+0.05) ^10o = 1000(1.638) = 1000(1+0.628)That’s a total of 68% interest over the 10 yearsIt would’ve doubled in around 15 yearsHowever, 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: ¼ of the interest in charged every quarter (3 months)- YearlyThe 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% compoundedmonthly. 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= principlei= interest (after dividing by 100)Then quarterly compounding has the yield P(1+i/12)^4 = 10,000