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# Math 1300: Section 3-1 Simple Interest

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Lecture over Section 3-1 of Barnett's "Finite Mathematics."

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### Math 1300: Section 3-1 Simple Interest

1. 1. Math 1300 Finite Mathematics Section 3.1 Simple Interest Jason Aubrey Department of Mathematics University of Missouri university-logo Jason Aubrey Math 1300 Finite Mathematics
2. 2. Deﬁnition (Simple Interest) I = Prtwhere I = interest university-logo Jason Aubrey Math 1300 Finite Mathematics
3. 3. Deﬁnition (Simple Interest) I = Prtwhere I = interest P = principal university-logo Jason Aubrey Math 1300 Finite Mathematics
4. 4. Deﬁnition (Simple Interest) I = Prtwhere I = interest P = principal r = annual simple interest rate (written as a decimal) university-logo Jason Aubrey Math 1300 Finite Mathematics
5. 5. Deﬁnition (Simple Interest) I = Prtwhere I = interest P = principal r = annual simple interest rate (written as a decimal) t = time in years university-logo Jason Aubrey Math 1300 Finite Mathematics
6. 6. Example: A department store charges 21% for overdueaccounts. How much interest will be owed on a \$650 accountthat is 3 months overdue? university-logo Jason Aubrey Math 1300 Finite Mathematics
7. 7. Example: A department store charges 21% for overdueaccounts. How much interest will be owed on a \$650 accountthat is 3 months overdue?Here we are given that P = \$650, r = 0.21, and t = 3/12. university-logo Jason Aubrey Math 1300 Finite Mathematics
8. 8. Example: A department store charges 21% for overdueaccounts. How much interest will be owed on a \$650 accountthat is 3 months overdue?Here we are given that P = \$650, r = 0.21, and t = 3/12. I = Prt university-logo Jason Aubrey Math 1300 Finite Mathematics
9. 9. Example: A department store charges 21% for overdueaccounts. How much interest will be owed on a \$650 accountthat is 3 months overdue?Here we are given that P = \$650, r = 0.21, and t = 3/12. I = Prt 3 I = 650(0.21) = 34.13 12Therefore \$34.13 is owed. university-logo Jason Aubrey Math 1300 Finite Mathematics
10. 10. Example: A commercial for a loan company states, “You onlypay \$0.16 a day for each \$600 borrowed.” If you borrow \$1800for 240 days, what amount will you repay, and what annualinterest rate is the company actually charging? university-logo Jason Aubrey Math 1300 Finite Mathematics
11. 11. Example: A commercial for a loan company states, “You onlypay \$0.16 a day for each \$600 borrowed.” If you borrow \$1800for 240 days, what amount will you repay, and what annualinterest rate is the company actually charging? You pay a fee of 3 × \$0.16 = \$0.48 per day. university-logo Jason Aubrey Math 1300 Finite Mathematics
12. 12. Example: A commercial for a loan company states, “You onlypay \$0.16 a day for each \$600 borrowed.” If you borrow \$1800for 240 days, what amount will you repay, and what annualinterest rate is the company actually charging? You pay a fee of 3 × \$0.16 = \$0.48 per day. This is a total fee of \$0.48 × 240 = \$115.20 over the life of the loan. university-logo Jason Aubrey Math 1300 Finite Mathematics
13. 13. Example: A commercial for a loan company states, “You onlypay \$0.16 a day for each \$600 borrowed.” If you borrow \$1800for 240 days, what amount will you repay, and what annualinterest rate is the company actually charging? You pay a fee of 3 × \$0.16 = \$0.48 per day. This is a total fee of \$0.48 × 240 = \$115.20 over the life of the loan. So, I = \$115.20, P = \$1800, and t = 240/360. university-logo Jason Aubrey Math 1300 Finite Mathematics
14. 14. Now we apply our simple interest formula... university-logo Jason Aubrey Math 1300 Finite Mathematics
15. 15. Now we apply our simple interest formula... I = Prt university-logo Jason Aubrey Math 1300 Finite Mathematics
16. 16. Now we apply our simple interest formula... I = Prt 240 \$115.20 = (\$1, 800)r 360 university-logo Jason Aubrey Math 1300 Finite Mathematics
17. 17. Now we apply our simple interest formula... I = Prt 240 \$115.20 = (\$1, 800)r 360 r = 0.096 university-logo Jason Aubrey Math 1300 Finite Mathematics
18. 18. Now we apply our simple interest formula... I = Prt 240 \$115.20 = (\$1, 800)r 360 r = 0.096So, you repay a total of \$1,915.20 and the annual interest rateis 9.6%. university-logo Jason Aubrey Math 1300 Finite Mathematics
19. 19. Theorem (Amount: Simple Interest) A = P + Prt = P(1 + rt) A = amount, or future value university-logo Jason Aubrey Math 1300 Finite Mathematics
20. 20. Theorem (Amount: Simple Interest) A = P + Prt = P(1 + rt) A = amount, or future value P = principal, or present value university-logo Jason Aubrey Math 1300 Finite Mathematics
21. 21. Theorem (Amount: Simple Interest) A = P + Prt = P(1 + rt) A = amount, or future value P = principal, or present value r = annual simple interest rate (written as a decimal) university-logo Jason Aubrey Math 1300 Finite Mathematics
22. 22. Theorem (Amount: Simple Interest) A = P + Prt = P(1 + rt) A = amount, or future value P = principal, or present value r = annual simple interest rate (written as a decimal) t = time in years university-logo Jason Aubrey Math 1300 Finite Mathematics
23. 23. Example: What annual interest rate is earned by a 13-weekT-bill with a maturity value of \$1,000 that sells for \$989.37? university-logo Jason Aubrey Math 1300 Finite Mathematics
24. 24. Example: What annual interest rate is earned by a 13-weekT-bill with a maturity value of \$1,000 that sells for \$989.37?In T-bill problems P corresponds to the selling price ( orpurchase price) of the T-bill, and A corresponds to the maturityvalue. Here university-logo Jason Aubrey Math 1300 Finite Mathematics
25. 25. Example: What annual interest rate is earned by a 13-weekT-bill with a maturity value of \$1,000 that sells for \$989.37?In T-bill problems P corresponds to the selling price ( orpurchase price) of the T-bill, and A corresponds to the maturityvalue. Here P = \$989.37, university-logo Jason Aubrey Math 1300 Finite Mathematics
26. 26. Example: What annual interest rate is earned by a 13-weekT-bill with a maturity value of \$1,000 that sells for \$989.37?In T-bill problems P corresponds to the selling price ( orpurchase price) of the T-bill, and A corresponds to the maturityvalue. Here P = \$989.37, A = \$1, 000, university-logo Jason Aubrey Math 1300 Finite Mathematics
27. 27. Example: What annual interest rate is earned by a 13-weekT-bill with a maturity value of \$1,000 that sells for \$989.37?In T-bill problems P corresponds to the selling price ( orpurchase price) of the T-bill, and A corresponds to the maturityvalue. Here P = \$989.37, A = \$1, 000, and t = 13 52 university-logo Jason Aubrey Math 1300 Finite Mathematics
28. 28. Example: What annual interest rate is earned by a 13-weekT-bill with a maturity value of \$1,000 that sells for \$989.37?In T-bill problems P corresponds to the selling price ( orpurchase price) of the T-bill, and A corresponds to the maturityvalue. Here P = \$989.37, A = \$1, 000, and t = 13 52 A = P(1 + rt) university-logo Jason Aubrey Math 1300 Finite Mathematics
29. 29. Example: What annual interest rate is earned by a 13-weekT-bill with a maturity value of \$1,000 that sells for \$989.37?In T-bill problems P corresponds to the selling price ( orpurchase price) of the T-bill, and A corresponds to the maturityvalue. Here P = \$989.37, A = \$1, 000, and t = 13 52 A = P(1 + rt) 13 \$1, 000 = \$989.37 1 + r 52 university-logo Jason Aubrey Math 1300 Finite Mathematics
30. 30. Example: What annual interest rate is earned by a 13-weekT-bill with a maturity value of \$1,000 that sells for \$989.37?In T-bill problems P corresponds to the selling price ( orpurchase price) of the T-bill, and A corresponds to the maturityvalue. Here P = \$989.37, A = \$1, 000, and t = 13 52 A = P(1 + rt) 13 \$1, 000 = \$989.37 1 + r 52 13 1.011 ≈ 1 + r 52 university-logo Jason Aubrey Math 1300 Finite Mathematics
31. 31. Example: What annual interest rate is earned by a 13-weekT-bill with a maturity value of \$1,000 that sells for \$989.37?In T-bill problems P corresponds to the selling price ( orpurchase price) of the T-bill, and A corresponds to the maturityvalue. Here P = \$989.37, A = \$1, 000, and t = 13 52 A = P(1 + rt) 13 \$1, 000 = \$989.37 1 + r 52 13 1.011 ≈ 1 + r 52 13 0.011 ≈ r 52 university-logo Jason Aubrey Math 1300 Finite Mathematics
32. 32. Example: What annual interest rate is earned by a 13-weekT-bill with a maturity value of \$1,000 that sells for \$989.37?In T-bill problems P corresponds to the selling price ( orpurchase price) of the T-bill, and A corresponds to the maturityvalue. Here P = \$989.37, A = \$1, 000, and t = 13 52 A = P(1 + rt) 13 \$1, 000 = \$989.37 1 + r 52 13 1.011 ≈ 1 + r 52 13 0.011 ≈ r 52 r ≈ 0.044 university-logo Jason Aubrey Math 1300 Finite Mathematics
33. 33. Example: What annual interest rate is earned by a 13-weekT-bill with a maturity value of \$1,000 that sells for \$989.37?In T-bill problems P corresponds to the selling price ( orpurchase price) of the T-bill, and A corresponds to the maturityvalue. Here P = \$989.37, A = \$1, 000, and t = 13 52 A = P(1 + rt) 13 \$1, 000 = \$989.37 1 + r 52 13 1.011 ≈ 1 + r 52 13 0.011 ≈ r 52 r ≈ 0.044 or 4.4% university-logo Jason Aubrey Math 1300 Finite Mathematics
34. 34. Example: Many investment ﬁrms charge commissions ontransactions based on the amount of the transaction. Supposethat an investment ﬁrm charges commissions on stock tradesaccording to the following commission schedule: Transaction Size Commission Under \$3,000 \$25+1.8% of principal \$3000-\$10,000 \$37 + 1.4% of principal Over \$10,000 \$107 + 0.7% of principalSuppose an investor purchases 175 shares at \$15.00 a share,holds the stock for 26 weeks, and then sells the stock for\$17.25 per share. Find the annual interest rate earned by thisinvestment. university-logo Jason Aubrey Math 1300 Finite Mathematics
35. 35. Step 1: Find the total cost of the purchase. university-logo Jason Aubrey Math 1300 Finite Mathematics
36. 36. Step 1: Find the total cost of the purchase. \$15.00(175) = \$2,625 - Principal university-logo Jason Aubrey Math 1300 Finite Mathematics
37. 37. Step 1: Find the total cost of the purchase. \$15.00(175) = \$2,625 - Principal \$25 + \$2,625(0.018) = \$72.25 - Commission university-logo Jason Aubrey Math 1300 Finite Mathematics
38. 38. Step 1: Find the total cost of the purchase. \$15.00(175) = \$2,625 - Principal \$25 + \$2,625(0.018) = \$72.25 - Commission \$2,625 + \$72.25 = \$2,697.25 - Total cost of purchase university-logo Jason Aubrey Math 1300 Finite Mathematics
39. 39. Step 2: Find the net revenue from the sale. university-logo Jason Aubrey Math 1300 Finite Mathematics
40. 40. Step 2: Find the net revenue from the sale. \$17.25(175) = \$3,018.75 - Principal university-logo Jason Aubrey Math 1300 Finite Mathematics
41. 41. Step 2: Find the net revenue from the sale. \$17.25(175) = \$3,018.75 - Principal \$37 + (\$3,018.75)(0.014) = \$79.26 -Commission university-logo Jason Aubrey Math 1300 Finite Mathematics
42. 42. Step 2: Find the net revenue from the sale. \$17.25(175) = \$3,018.75 - Principal \$37 + (\$3,018.75)(0.014) = \$79.26 -Commission \$3,018.75 - \$79.26 = \$2,939.49 - Net revenue from sale university-logo Jason Aubrey Math 1300 Finite Mathematics
43. 43. Step 3: Calculate annual interest rate: , university-logo Jason Aubrey Math 1300 Finite Mathematics
44. 44. Step 3: Calculate annual interest rate:Here A = \$2, 939.49, , university-logo Jason Aubrey Math 1300 Finite Mathematics
45. 45. Step 3: Calculate annual interest rate:Here A = \$2, 939.49, P = \$2697.25, university-logo Jason Aubrey Math 1300 Finite Mathematics
46. 46. Step 3: Calculate annual interest rate:Here A = \$2, 939.49, P = \$2697.25, and t = 26/52 university-logo Jason Aubrey Math 1300 Finite Mathematics
47. 47. Step 3: Calculate annual interest rate:Here A = \$2, 939.49, P = \$2697.25, and t = 26/52 A = P(1 + rt) university-logo Jason Aubrey Math 1300 Finite Mathematics
48. 48. Step 3: Calculate annual interest rate:Here A = \$2, 939.49, P = \$2697.25, and t = 26/52 A = P(1 + rt) 2939.49 = 2697.25(1 + r (.5)) university-logo Jason Aubrey Math 1300 Finite Mathematics
49. 49. Step 3: Calculate annual interest rate:Here A = \$2, 939.49, P = \$2697.25, and t = 26/52 A = P(1 + rt) 2939.49 = 2697.25(1 + r (.5)) r = 0.1796 university-logo Jason Aubrey Math 1300 Finite Mathematics
50. 50. Example: Suppose that after buying a new car you decide tosell your old car to a friend. You accept a 270-day note for\$3,500 at 10% simple interest as payment. (Both principal andinterest will be paid at the end of 270 days.) Sixty days lateryou ﬁnd that you need the money and sell the note to a thirdparty for \$3,550. What annual interest rate will the third partyrecieve for the investment? (Express the answer as apercentage, correct to three decimal places). university-logo Jason Aubrey Math 1300 Finite Mathematics
51. 51. Step 1: Find the amount that will be paid at the end of 270days to the holder of the note: university-logo Jason Aubrey Math 1300 Finite Mathematics
52. 52. Step 1: Find the amount that will be paid at the end of 270days to the holder of the note: A = P(1 + rt) university-logo Jason Aubrey Math 1300 Finite Mathematics
53. 53. Step 1: Find the amount that will be paid at the end of 270days to the holder of the note: A = P(1 + rt) 270 = (\$3, 500) 1 + (0.1) 360 university-logo Jason Aubrey Math 1300 Finite Mathematics
54. 54. Step 1: Find the amount that will be paid at the end of 270days to the holder of the note: A = P(1 + rt) 270 = (\$3, 500) 1 + (0.1) 360 = \$3, 762.50 university-logo Jason Aubrey Math 1300 Finite Mathematics
55. 55. Step 2: For the third party, we are to ﬁnd the annual rate ofinterest r required to make \$3,550 grow to \$3,762.50 in 210days (270 - 60). So we need to ﬁnd r given that:. university-logo Jason Aubrey Math 1300 Finite Mathematics
56. 56. Step 2: For the third party, we are to ﬁnd the annual rate ofinterest r required to make \$3,550 grow to \$3,762.50 in 210days (270 - 60). So we need to ﬁnd r given that:A = \$3, 762.50,. university-logo Jason Aubrey Math 1300 Finite Mathematics
57. 57. Step 2: For the third party, we are to ﬁnd the annual rate ofinterest r required to make \$3,550 grow to \$3,762.50 in 210days (270 - 60). So we need to ﬁnd r given that:A = \$3, 762.50, P = \$3, 550,. university-logo Jason Aubrey Math 1300 Finite Mathematics
58. 58. Step 2: For the third party, we are to ﬁnd the annual rate ofinterest r required to make \$3,550 grow to \$3,762.50 in 210days (270 - 60). So we need to ﬁnd r given that: 210A = \$3, 762.50, P = \$3, 550, t = 360. university-logo Jason Aubrey Math 1300 Finite Mathematics
59. 59. Step 2: For the third party, we are to ﬁnd the annual rate ofinterest r required to make \$3,550 grow to \$3,762.50 in 210days (270 - 60). So we need to ﬁnd r given that: 210A = \$3, 762.50, P = \$3, 550, t = 360. A = P(1 + rt) university-logo Jason Aubrey Math 1300 Finite Mathematics
60. 60. Step 2: For the third party, we are to ﬁnd the annual rate ofinterest r required to make \$3,550 grow to \$3,762.50 in 210days (270 - 60). So we need to ﬁnd r given that: 210A = \$3, 762.50, P = \$3, 550, t = 360. A = P(1 + rt) 210 \$3, 762.50 = \$3, 550 1 + r 360 university-logo Jason Aubrey Math 1300 Finite Mathematics
61. 61. Step 2: For the third party, we are to ﬁnd the annual rate ofinterest r required to make \$3,550 grow to \$3,762.50 in 210days (270 - 60). So we need to ﬁnd r given that: 210A = \$3, 762.50, P = \$3, 550, t = 360. A = P(1 + rt) 210 \$3, 762.50 = \$3, 550 1 + r 360 r = 0.10262 university-logo Jason Aubrey Math 1300 Finite Mathematics
62. 62. Step 2: For the third party, we are to ﬁnd the annual rate ofinterest r required to make \$3,550 grow to \$3,762.50 in 210days (270 - 60). So we need to ﬁnd r given that: 210A = \$3, 762.50, P = \$3, 550, t = 360. A = P(1 + rt) 210 \$3, 762.50 = \$3, 550 1 + r 360 r = 0.10262 or 10.262%. university-logo Jason Aubrey Math 1300 Finite Mathematics
63. 63. Example: Many tax preparation ﬁrms offer their clients a refundanticipation loan (RAL). For a fee, the ﬁrm will give the client hisrefund when the return is ﬁled. The loan is repaid when the IRSsends the refund directly to the ﬁrm. Thus, the RAL fee isequivalent to the interest charge for the loan. The schedulebelow is from a major RAL lender. RAL Amount RAL Fee 0-\$500 \$29.00 \$501-\$1,000 \$39.00 \$1,001-\$1,500 \$49.00 \$1,501-\$2,000 \$69.00 \$2,001-\$2,500 \$89.00A client recieves a \$480 RAL which is repaid in 25 days. Whatis the annual interest rate for this loan? university-logo Jason Aubrey Math 1300 Finite Mathematics
64. 64. First, P = \$480. . university-logo Jason Aubrey Math 1300 Finite Mathematics
65. 65. First, P = \$480. So according to the schedule, the amountcharged is I = \$29.00. . university-logo Jason Aubrey Math 1300 Finite Mathematics
66. 66. First, P = \$480. So according to the schedule, the amount 25charged is I = \$29.00. t = 360 . university-logo Jason Aubrey Math 1300 Finite Mathematics
67. 67. First, P = \$480. So according to the schedule, the amount 25charged is I = \$29.00. t = 360 . We now compute r : university-logo Jason Aubrey Math 1300 Finite Mathematics
68. 68. First, P = \$480. So according to the schedule, the amount 25charged is I = \$29.00. t = 360 . We now compute r : I = Prt university-logo Jason Aubrey Math 1300 Finite Mathematics
69. 69. First, P = \$480. So according to the schedule, the amount 25charged is I = \$29.00. t = 360 . We now compute r : I = Prt 25 29 = (480)r 360 university-logo Jason Aubrey Math 1300 Finite Mathematics
70. 70. First, P = \$480. So according to the schedule, the amount 25charged is I = \$29.00. t = 360 . We now compute r : I = Prt 25 29 = (480)r 360 r = 0.87 university-logo Jason Aubrey Math 1300 Finite Mathematics