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Investing $8,000: Understanding Compound Interest and Its Applications

One lucky day, you discover $8,000 on the street and decide to invest it in the Bank of Baker, which offers an interest rate of 10% per year. Learn how to maximize your investment using compound interest over time. Explore the calculations for your growing investment, from the initial payout to projected growth over the years. Understand real-world applications of logarithms and exponentials and the history of crucial mathematical concepts. This guide will help students grasp these financial principles effectively.

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Investing $8,000: Understanding Compound Interest and Its Applications

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  1. One lucky day , you find $8,000 on the street. At the Bank of Baker- that’s my bank, I am offering you an interest rate of 10% a year. Being the smart students you are, you invest your money at my bank.After the first year, your account collects 10% interest, so I would have to payout 8000+8000(.1)= $8,800Or, 8,000(1 + .1) = $8,800 The second year, your $8,800 will collect even more interest and become 8,800(1 + .1) = 8,000(1 +.1)(1+.1)= $9,680

  2. Complete the table below One lucky day , you find $8,000 on the street. At the Bank of Baker- that’s my bank. I am offering you an interest rate of 10% a year. Being the smart students you are, you invest your money at my bank. 9,680 11, 712 12,884 10,648

  3. Today’s objectives: 1.) Understand the real world situations and applications of logarithms and exponentials 2.) Learn the history of the number e and recognize its importance in mathematics

  4. Don’t FORGET The decimal number of 10% = .1 5.8 % = = .058 Move the decimal point over 2 to the left.

  5. Revisit warm-up problem • If you make the initial investment(I) of $8,000 at Bank of Baker, and I offer an interest rate (r) of 10%, how much money will your investment grow to after 20 years? Write an equation of the payout with respect to years. 1st year … 8000(1 +.1) = 8800 2nd year… (8000(1+.1))(1+.1) = 9680 3rd year… (8,000(1 +.1)(1+.1))(1 +.1)= $11, 712.80

  6. Deal or No Deal? • I will compound/apply your interest rate twice in one year. But I am going to cut your interest rate in half.

  7. Deal or No Deal? You come to me with $5000. I have an interest rate of 4.1 %. You want to establish this amount in my bank for 20 years. What if I compound your investment quarterly. I will apply a compounded interest rate 4 times but I will divide the interest rate by 4.

  8. 11,168.24 Interest rate in decimal form Initial investment I will pay 4 times per year for 20 years, but as consequence I will divide interest rate by 4 11,305.21

  9. Compound interest

  10. Suppose Damon only has $3500 to invest but wants $4000 for a hot tub. He finds a bank offering 5.25% interest compounded quarterly. How long will he have to leave his money in the account to have it earn itself $4000. t = 2.56 years

  11. WS 1, 3,5

  12. Compound Interest: • An account starts out with $1, and it pays an interest rate of 100% a year • If the interest is “compounded” once a year, the value is 1(1+1)1 = 2 • If the interest is applied/compounded twice, I will apply interest twice that year, but I will half the interest rate. $1(1 + )2 = 2.25

  13. Compound Interest: • To compound the interest means I will apply the interest as many times as you want, but I will also divide your interest rate by as many times as I compound your investment. • If the interest is “compounded” quarterly …? • If the interest is compounded monthly $1(1 + )4= $2.44 $1(1 + )12 = $2.61

  14. What if you wanted to compound every minute, every second, every millisecond…?

  15. In 1683, mathematician Jacob Bernoulli considered the value of as n approaches infinity. His study was the first approximation of e

  16. e= 2.718281828459045235460287471352662497757246093699959574077078727723076630353547594571382178525166427466391932003059921817413496629043572900338298807531952510190115728241879307….. Comparable to an irrational number like ∏

  17. Continuously Compounded Interest Equation P= Iert I= Initial Investment Amount P= Final Amount/payout r = annual interest rate t= time in years

  18. If Marie invests $2000 and it is compounded continuously for 30 years

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