Math 1300: Section 3-1 Simple Interest
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This document covers simple interest concepts including the simple interest formula of I=PRT, where I is interest, P is principal, R is the annual interest rate, and T is time in years. Examples are provided to demonstrate calculating simple interest for accounts receivable and loans. The amount formula of A=P(1+RT) is also presented, where A is the future value and P is the present value. An example calculates the annual interest rate of a 13-week Treasury bill.
Math 1300: Section 3-1 Simple Interest
- 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. Definition (Simple Interest) I = Prt where I = interest university-logo Jason Aubrey Math 1300 Finite Mathematics
- 3. Definition (Simple Interest) I = Prt where I = interest P = principal university-logo Jason Aubrey Math 1300 Finite Mathematics
- 4. Definition (Simple Interest) I = Prt where I = interest P = principal r = annual simple interest rate (written as a decimal) university-logo Jason Aubrey Math 1300 Finite Mathematics
- 5. Definition (Simple Interest) I = Prt where 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. Example: A department store charges 21% for overdue accounts. How much interest will be owed on a $650 account that is 3 months overdue? university-logo Jason Aubrey Math 1300 Finite Mathematics
- 7. Example: A department store charges 21% for overdue accounts. How much interest will be owed on a $650 account that 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. Example: A department store charges 21% for overdue accounts. How much interest will be owed on a $650 account that 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. Example: A department store charges 21% for overdue accounts. How much interest will be owed on a $650 account that 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 12 Therefore $34.13 is owed. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 10. Example: A commercial for a loan company states, “You only pay $0.16 a day for each $600 borrowed.” If you borrow $1800 for 240 days, what amount will you repay, and what annual interest rate is the company actually charging? university-logo Jason Aubrey Math 1300 Finite Mathematics
- 11. Example: A commercial for a loan company states, “You only pay $0.16 a day for each $600 borrowed.” If you borrow $1800 for 240 days, what amount will you repay, and what annual interest 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. Example: A commercial for a loan company states, “You only pay $0.16 a day for each $600 borrowed.” If you borrow $1800 for 240 days, what amount will you repay, and what annual interest 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. Example: A commercial for a loan company states, “You only pay $0.16 a day for each $600 borrowed.” If you borrow $1800 for 240 days, what amount will you repay, and what annual interest 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. Now we apply our simple interest formula... university-logo Jason Aubrey Math 1300 Finite Mathematics
- 15. Now we apply our simple interest formula... I = Prt university-logo Jason Aubrey Math 1300 Finite Mathematics
- 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. 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. Now we apply our simple interest formula... I = Prt 240 $115.20 = ($1, 800)r 360 r = 0.096 So, you repay a total of $1,915.20 and the annual interest rate is 9.6%. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 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. 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. 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. 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. Example: What annual interest rate is earned by a 13-week T-bill with a maturity value of $1,000 that sells for $989.37? university-logo Jason Aubrey Math 1300 Finite Mathematics
- 24. Example: What annual interest rate is earned by a 13-week T-bill with a maturity value of $1,000 that sells for $989.37? In T-bill problems P corresponds to the selling price ( or purchase price) of the T-bill, and A corresponds to the maturity value. Here university-logo Jason Aubrey Math 1300 Finite Mathematics
- 25. Example: What annual interest rate is earned by a 13-week T-bill with a maturity value of $1,000 that sells for $989.37? In T-bill problems P corresponds to the selling price ( or purchase price) of the T-bill, and A corresponds to the maturity value. Here P = $989.37, university-logo Jason Aubrey Math 1300 Finite Mathematics
- 26. Example: What annual interest rate is earned by a 13-week T-bill with a maturity value of $1,000 that sells for $989.37? In T-bill problems P corresponds to the selling price ( or purchase price) of the T-bill, and A corresponds to the maturity value. Here P = $989.37, A = $1, 000, university-logo Jason Aubrey Math 1300 Finite Mathematics
- 27. Example: What annual interest rate is earned by a 13-week T-bill with a maturity value of $1,000 that sells for $989.37? In T-bill problems P corresponds to the selling price ( or purchase price) of the T-bill, and A corresponds to the maturity value. Here P = $989.37, A = $1, 000, and t = 13 52 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 28. Example: What annual interest rate is earned by a 13-week T-bill with a maturity value of $1,000 that sells for $989.37? In T-bill problems P corresponds to the selling price ( or purchase price) of the T-bill, and A corresponds to the maturity value. Here P = $989.37, A = $1, 000, and t = 13 52 A = P(1 + rt) university-logo Jason Aubrey Math 1300 Finite Mathematics
- 29. Example: What annual interest rate is earned by a 13-week T-bill with a maturity value of $1,000 that sells for $989.37? In T-bill problems P corresponds to the selling price ( or purchase price) of the T-bill, and A corresponds to the maturity value. 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. Example: What annual interest rate is earned by a 13-week T-bill with a maturity value of $1,000 that sells for $989.37? In T-bill problems P corresponds to the selling price ( or purchase price) of the T-bill, and A corresponds to the maturity value. 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. Example: What annual interest rate is earned by a 13-week T-bill with a maturity value of $1,000 that sells for $989.37? In T-bill problems P corresponds to the selling price ( or purchase price) of the T-bill, and A corresponds to the maturity value. 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. Example: What annual interest rate is earned by a 13-week T-bill with a maturity value of $1,000 that sells for $989.37? In T-bill problems P corresponds to the selling price ( or purchase price) of the T-bill, and A corresponds to the maturity value. 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. Example: What annual interest rate is earned by a 13-week T-bill with a maturity value of $1,000 that sells for $989.37? In T-bill problems P corresponds to the selling price ( or purchase price) of the T-bill, and A corresponds to the maturity value. 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. Example: Many investment firms charge commissions on transactions based on the amount of the transaction. Suppose that an investment firm charges commissions on stock trades according 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 principal Suppose 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 this investment. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 35. Step 1: Find the total cost of the purchase. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 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. 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. 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. Step 2: Find the net revenue from the sale. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 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. 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. 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. Step 3: Calculate annual interest rate: , university-logo Jason Aubrey Math 1300 Finite Mathematics
- 44. Step 3: Calculate annual interest rate: Here A = $2, 939.49, , university-logo Jason Aubrey Math 1300 Finite Mathematics
- 45. Step 3: Calculate annual interest rate: Here A = $2, 939.49, P = $2697.25, university-logo Jason Aubrey Math 1300 Finite Mathematics
- 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. 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. 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. 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. Example: Suppose that after buying a new car you decide to sell your old car to a friend. You accept a 270-day note for $3,500 at 10% simple interest as payment. (Both principal and interest will be paid at the end of 270 days.) Sixty days later you find that you need the money and sell the note to a third party for $3,550. What annual interest rate will the third party recieve for the investment? (Express the answer as a percentage, correct to three decimal places). university-logo Jason Aubrey Math 1300 Finite Mathematics
- 51. Step 1: Find the amount that will be paid at the end of 270 days to the holder of the note: university-logo Jason Aubrey Math 1300 Finite Mathematics
- 52. Step 1: Find the amount that will be paid at the end of 270 days to the holder of the note: A = P(1 + rt) university-logo Jason Aubrey Math 1300 Finite Mathematics
- 53. Step 1: Find the amount that will be paid at the end of 270 days 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. Step 1: Find the amount that will be paid at the end of 270 days 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. Step 2: For the third party, we are to find the annual rate of interest r required to make $3,550 grow to $3,762.50 in 210 days (270 - 60). So we need to find r given that: . university-logo Jason Aubrey Math 1300 Finite Mathematics
- 56. Step 2: For the third party, we are to find the annual rate of interest r required to make $3,550 grow to $3,762.50 in 210 days (270 - 60). So we need to find r given that: A = $3, 762.50, . university-logo Jason Aubrey Math 1300 Finite Mathematics
- 57. Step 2: For the third party, we are to find the annual rate of interest r required to make $3,550 grow to $3,762.50 in 210 days (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. Step 2: For the third party, we are to find the annual rate of interest r required to make $3,550 grow to $3,762.50 in 210 days (270 - 60). So we need to find r given that: 210 A = $3, 762.50, P = $3, 550, t = 360 . university-logo Jason Aubrey Math 1300 Finite Mathematics
- 59. Step 2: For the third party, we are to find the annual rate of interest r required to make $3,550 grow to $3,762.50 in 210 days (270 - 60). So we need to find r given that: 210 A = $3, 762.50, P = $3, 550, t = 360 . A = P(1 + rt) university-logo Jason Aubrey Math 1300 Finite Mathematics
- 60. Step 2: For the third party, we are to find the annual rate of interest r required to make $3,550 grow to $3,762.50 in 210 days (270 - 60). So we need to find r given that: 210 A = $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. Step 2: For the third party, we are to find the annual rate of interest r required to make $3,550 grow to $3,762.50 in 210 days (270 - 60). So we need to find r given that: 210 A = $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. Step 2: For the third party, we are to find the annual rate of interest r required to make $3,550 grow to $3,762.50 in 210 days (270 - 60). So we need to find r given that: 210 A = $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. Example: Many tax preparation firms offer their clients a refund anticipation loan (RAL). For a fee, the firm will give the client his refund when the return is filed. The loan is repaid when the IRS sends the refund directly to the firm. Thus, the RAL fee is equivalent to the interest charge for the loan. The schedule below 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 $89.00 A client recieves a $480 RAL which is repaid in 25 days. What is the annual interest rate for this loan? university-logo Jason Aubrey Math 1300 Finite Mathematics
- 64. First, P = $480. . university-logo Jason Aubrey Math 1300 Finite Mathematics
- 65. First, P = $480. So according to the schedule, the amount charged is I = $29.00. . university-logo Jason Aubrey Math 1300 Finite Mathematics
- 66. First, P = $480. So according to the schedule, the amount 25 charged is I = $29.00. t = 360 . university-logo Jason Aubrey Math 1300 Finite Mathematics
- 67. First, P = $480. So according to the schedule, the amount 25 charged is I = $29.00. t = 360 . We now compute r : university-logo Jason Aubrey Math 1300 Finite Mathematics
- 68. First, P = $480. So according to the schedule, the amount 25 charged is I = $29.00. t = 360 . We now compute r : I = Prt university-logo Jason Aubrey Math 1300 Finite Mathematics
- 69. First, P = $480. So according to the schedule, the amount 25 charged 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. First, P = $480. So according to the schedule, the amount 25 charged 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