CHAPTER 11 Simple Interest and Simple Discount Copyright

CHAPTER 11 Simple Interest and Simple Discount Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

11 -1 Learning Outcomes n Find simple interest by using the simple interest formula. n Find the maturity of a loan. n Convert months to a fractional or decimal part of a year. n Find the principal, rate or time using the simple interest formula. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2

Find simple interest using the simple interest formula 11 -1 -1 Section 11 -1 The Simple Interest Formula n The interest formula shows how interest, rate, and time are related. – It gives us a way of finding one of these values if the other three values are known. I=Px. Rx. T Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3

HOW TO: Identify the principal, rate and time Section 11 -1 The Simple Interest Formula n The price paid for using money is called interest. n Principal is the amount borrowed or invested. n Rate is the percent of the principal paid as interest period, usually one year. n Time must be expressed in the same unit of time as the rate. (i. e. one year) Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4

Key Terms… Section 11 -1 The Simple Interest Formula n Interest – An amount paid or earned for the use of money. n Simple interest – Interest earned when a loan or investment is repaid in a lump sum. n Principal – The amount of money borrowed or invested. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5

Key Terms… Section 11 -1 The Simple Interest Formula n Rate – The percent of the principal paid as interest per time period. n Time – The number of days, months or years that the money is borrowed or invested. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6

HOW TO: Find the interest paid on a loan Section 11 -1 The Simple Interest Formula Principal = (P) = $1, 200 Interest rate = 8% (or 0. 08) Time = 1 year Interest = P x R x T Interest = 1, 200 x 0. 08 x 1 Interest = $96 The interest on the loan is $96 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7

Examples… Section 11 -1 The Simple Interest Formula n Find the interest on a 2 -year loan of $4, 000 at a 6% rate. – $480 n Find the interest earned on a 3 -year investment of $5, 000 at 4. 5% interest. – $675 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8

Find the maturity value of a loan 11 -1 -2 Section 11 -1 The Simple Interest Formula n Maturity value is the total amount of money due by the end of a loan period. – The amount of the loan and interest. n If principal and interest are known, add them. – MV = principal + PRT – MV = P(1+RT) Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9

An Example… Section 11 -1 The Simple Interest Formula Marcus Logan can purchase furniture on a 2 -year simple interest loan at 9% interest per year. What is the maturity value for a $2, 500 loan? MV = P(1 + RT); (substitute known values) MV = $2, 500 (1 + 0. 09 x 2) MV = $2, 500 (1 + 0. 18) MV = $2, 500 (1. 18) MV = $2, 950 Marcus will pay $2, 950 at the end of two years. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10

Examples… Section 11 -1 The Simple Interest Formula n Terry Williams is going to borrow $4, 000 at 7. 5% interest. What is the maturity value of the loan after three years? – $4, 900 n Jim Sherman will invest $3, 000 at 8% for 5 years. What is the maturity value of the investment? – $4, 200 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11

11 -1 -3 Convert months to a fractional or decimal part of a year Section 11 -1 The Simple Interest Formula n Write the number of months as the numerator of a fraction. – Write 12 as the denominator of the fraction. n Reduce the fraction to lowest terms if using the fractional equivalent. n Divide the numerator by the denominator to get the decimal equivalent of the fraction. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12

11 -1 -3 Convert months to a fractional or decimal part of a year Section 11 -1 The Simple Interest Formula Convert 9 months & 15 months, respectively, to years, expressing both as fractions & decimals. 9 months = ¾ or 0. 75 of a year 15 months = 1 ¼ or 1. 25 of a year Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13

An Example… Section 11 -1 The Simple Interest Formula To save money, Stan Wright invested $2, 500 for 42 months at 4 ½ % simple interest. How much interest did he earn? 42 months = = 3. 5 I=Px. Rx. T I = $2, 500 x 0. 045 x 3. 5 I = $393. 75 Stan will earn $393. 75. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14

Examples… Section 11 -1 The Simple Interest Formula n Akiko is saving a little extra money to pay for her car insurance next year. If she invests $1, 000 for 18 months at 4%, how much interest can she earn? – $60 n Habib is going to borrow $2, 000 for 42 months at 7%. What will be the amount of interest owed? – $490 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15

11 -1 -4 Find the Principal, Rate or Time Using the Simple Interest Formula Section 11 -1 The Simple Interest Formula Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16

Find the principal using the simple interest formula HOW TO: Section 11 -1 The Simple Interest Formula Judy paid $108 in interest on a loan that she had for 6 months. The interest rate was 12%. How much was the principal? Substitute the known values and solve. P= P= P = $1, 800 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17

Find the rate using the simple interest formula HOW TO: Section 11 -1 The Simple Interest Formula Sam wants to borrow $1, 500 for 15 months and will have to pay $225 in interest. What is the rate he is being charged? Substitute the known values and solve. R= R= R =. 12 or 12% The rate Sam will pay is 12%. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18

Find the time using the simple interest formula HOW TO: Section 11 -1 The Simple Interest Formula Shelby borrowed $10, 000 at 8% and paid $1, 600 in interest. What was the length of the loan? Substitute the known values and solve. T= T= Length of the loan was two years. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19

11 -2 Learning Outcomes n Find the exact time. n Find the due date. n Find the ordinary interest and the exact interest. n Make a partial payment before the maturity date. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20

Key Terms… Section 11 -2 Ordinary and Exact Interest n Ordinary time – Time that is based on counting 30 days in each month. n Exact time – Time that is based on counting the exact number of days in a time period. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 21

Find the exact time 11 -2 -1 Section 11 -2 Ordinary and Exact Interest The ordinary time from July 12 to September 12 is 60 days. To find the exact time from July 12 to September 12, add the following: Days in July (31 – 12) = Days in August Days in September Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19 31 +12 62 days 22

HOW TO: Sequential numbers for dates of the year Section 11 -2 Ordinary and Exact Interest Find the exact time of a loan using the sequential numbers table. (Table 11 -1 in the text) See page 394 If the beginning and due dates of the loan fall within the same year, subtract the beginning date’s sequential number from the due date’s sequential number. Example: From May 15 to October 15 288 – 135 = 153 days, the exact time Copyright © 2014, 2010, 2007 Pearson Education, Inc. 23

An Example… Section 11 -2 Ordinary and Exact Interest Find the exact time from May 15 of Year 1 to March 15 in Year 2. 365 – 135 = 230 + 74 = 304 days The exact time is 304 days. Note: If Year 2 is a leap year, the exact time is 305 days. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 24

Find the due date 11 -2 -2 Section 11 -2 Ordinary and Exact Interest Subtract the beginning date’s sequential number from 365. Add the due date’s sequential number to the result from the previous step. If February 29 falls between the two dates, add 1. (Is it a leap year? ) Copyright © 2014, 2010, 2007 Pearson Education, Inc. 25

An Example… Section 11 -2 Ordinary and Exact Interest A loan made on September 5 is due July 5 of the following year. a) ordinary time Find: b) exact time in a non-leap year c) exact time in a leap year Ordinary time = 300 days Exact time (non-leap year) = 303 days Exact time (leap year) = 304 days Copyright © 2014, 2010, 2007 Pearson Education, Inc. 26

11 -2 -3 Find the ordinary and exact interest Section 11 -2 Ordinary and Exact Interest n An interest rate is normally given as rate per year. n If the time period of the loan is in days, using the simple interest formula requires that the rate also be expressed as a rate per day. – Ordinary interest: assumes 360 days per year. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 27

Key Terms… Section 11 -2 Ordinary and Exact Interest n Ordinary interest – A rate per day that assumes 360 days per year. n Exact interest – A rate per day that assumes 365 days per year. n Banker’s rule – Calculating interest on a loan based on ordinary interest which yields a slightly higher amount of interest. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 28

Find the ordinary interest HOW TO: Section 11 -2 Ordinary and Exact Interest For ordinary interest rate per day, divide the annual interest rate by 360. Ordinary interest rate per day = Interest rate per year 360 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 29

Find the exact interest HOW TO: Section 11 -2 Ordinary and Exact Interest For exact interest rate per day, divide the annual interest rate by 365. Exact interest rate per day = Interest rate per year 365 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 30

Use ordinary time to find the ordinary interest on a loan HOW TO: Section 11 -2 Ordinary and Exact Interest A loan of $500 at 7% annual interest rate. The loan was made on March 15 and due on May 15. (Principal = $500) I=Px. Rx. T Length of loan (ordinary time) = 60 days Rate = Interest = (ordinary interest) = $5. 83 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 31

Use exact time to find the ordinary interest the same loan HOW TO: Section 11 -2 Ordinary and Exact Interest A loan of $500 at 7% annual interest rate. The loan was made on March 15 and due on May 15. (Principal = $500) I=Px. Rx. T Length of loan (exact time) = 61 days Rate = Interest = (ordinary interest) = $5. 93 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 32

Use exact time to find the exact interest the same loan HOW TO: Section 11 -2 Ordinary and Exact Interest A loan of $500 at 7% annual interest rate. The loan was made on March 15 and due on May 15. (Principal = $500) I=Px. Rx. T Length of loan (exact time) = 61 days Rate = Interest = (ordinary interest) = $5. 84 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 33

11 -2 -4 Make a Partial Payment Before the Maturity Date Section 11 -2 Ordinary and Exact Interest To find the adjusted principal and adjusted balance due at maturity for a partial payment made before the maturity date: STEP 1 Determine the exact time from the date of the loan to the first partial payment. STEP 2 Calculate the interest using the time found in Step 1. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 34

11 -2 -4 Make a Partial Payment Before the Maturity Date Section 11 -2 Ordinary and Exact Interest STEP 3 Subtract the amount of interest found in Step 2 from the partial payment. STEP 4 Subtract the remainder of the partial payment (Step 3) from the original principal. This is the adjusted principal. STEP 5 Repeat process for additional partial payments. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 35

11 -2 -4 Make a Partial Payment Before the Maturity Date Section 11 -2 Ordinary and Exact Interest STEP 6 At maturity, calculate interest from the last partial payment and add to adjusted principal. This is the adjusted balance due at maturity. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 36

An Example… Section 11 -2 Ordinary and Exact Interest Tony borrows $5, 000 on a 10%, 90 day note. On the 30 th day, Tony pays $1, 500 on the note. If ordinary interest is applied, what is Tony’s adjusted principal after the partial payment, and adjusted balance due at maturity? $5, 000(0. 1)(30 ÷ 360) = $41. 67 $1, 500 - $41. 67 = $1, 458. 33 $5, 000 - $1, 458. 33 = $3, 541. 67 (Adj. Principal) $3, 541. 67(. 1)(60 ÷ 360) = $59. 03 (Interest) $3, 541. 67 + $59. 03 = $3, 600. 70 (Adj. Balance) Copyright © 2014, 2010, 2007 Pearson Education, Inc. 37

11 -3 Learning Outcomes n Find the bank discount and proceeds for a simple discount note. n Find the true effective interest rate of a simple discount note. n Find the third-party discount and proceeds for a third-party discount note. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 38

11 -3 -1 Find the bank discount and proceeds for a simple discount note Section 11 -3 Promissory Notes For the bank discount, use: Bank discount = face value x disc. rate x time I=Px. Rx. T For the proceeds, use: Proceeds = face value – bank discount A=P-I Copyright © 2014, 2010, 2007 Pearson Education, Inc. 39

A Promissory Note Section 11 -3 Promissory Notes Copyright © 2014, 2010, 2007 Pearson Education, Inc. 40

11 -3 -2 Find the True or Effective Interest Rate of a Simple Discount Note Section 11 -3 Promissory Notes To find the true or effective interest rate of a simple discount note: STEP 1 Find the bank discount (interest). I = PRT STEP 2 Find the proceeds: proceeds = principal – bank discount. STEP 3 Find the effective interest rate: R = I/PT (using the proceeds as the principal) Copyright © 2014, 2010, 2007 Pearson Education, Inc. 41

An Example… Section 11 -3 Promissory Notes What is the effective interest rate of a $5, 000 simple discount note, at an ordinary bank discount rate of 12%, for 90 days? I = PRT; I = $5, 000(0. 12)(90 ÷ 360) I = $150 (Bank discount) Proceeds = $5, 000 – $150 = $4, 850 R= R= R = 0. 1237113402 R or the effective interest rate = 12. 4% Copyright © 2014, 2010, 2007 Pearson Education, Inc. 42

11 -3 -3 Find the Third Party Discount and Proceeds for a Third Party Discount Note Section 11 -3 Promissory Notes For the bank discount, use: Third party discount = maturity value of the original note x discount rate x discount period For the proceeds, use: Proceeds = maturity value of original note – third-party discount A=P-I Copyright © 2014, 2010, 2007 Pearson Education, Inc. 43

An Example… Section 11 -3 Promissory Notes Mihoc Trailer made a note for $10, 000 with Darcy Mihoc, owner, at 9% simple interest based on exact interest and exact time. The note is made on August 12 and due November 10. Since Mihoc Trailer needs cash, the note is sold to a third party on September 5. The third-party agrees to accept the note with a 13% annual discount using the banker’s rule. Find the proceeds of the note. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 44

An Example… Section 11 -3 Promissory Notes Find the proceeds of the note: To find the proceeds, we find the maturity value of the original note, then the third-party discount. Exact time is 90 days (314 -224) I = PRT; I = $10, 000(0. 09)(90 ÷ 365) = $221. 92 MV = P + I = $10, 000 + $221. 92 MV = $10. 221. 92 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 45

An Example… Section 11 -3 Promissory Notes Find the proceeds of the note: Exact time of the discount period is 66 days (314 - 248) between Sept. 5 and Nov. 10. Ordinary discount rate is 0. 13 ÷ 360. Third party discount: I = PRT Third party discount = $10, 221. 92(0. 13 ÷ 360)(66) Third party discount = $243. 62 Proceeds: A = P – I Proceeds = $10, 221. 92 - $243. 62 = $9, 978. 30 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 46

Exercises Set A Copyright © 2014, 2010, 2007 Pearson Education, Inc. 47

EXERCISE SET A 2. Find the simple interest. Round to the nearest cent when necessary. Principal $3575 Interest 11% Time 3 years Interest ? ? I = 3, 575(0. 11)(3) = 1, 179. 75 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 48

EXERCISE SET A 4. Find the rate of annual simple interest. Principal $800 Interest $124 Time 1 year Copyright © 2014, 2010, 2007 Pearson Education, Inc. Rate ? ? 49

EXERCISE SET A 6. Find the time period of the loan using the formula for simple interest. Principal $450 Annual Rate 10% Interest Time $135 ? ? Copyright © 2014, 2010, 2007 Pearson Education, Inc. 50

EXERCISE SET A 8. Find the principal, based on simple interest. Interest $300 Annual Rate 3% Time Principal 2 years ? ? Copyright © 2014, 2010, 2007 Pearson Education, Inc. 51

EXERCISE SET A 10. A loan for three years with an annual simple interest rate of 9% costs $486 interest. Find the principal. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 52

EXERCISE SET A 12. Write a fraction expressing each amount of time as a part of a year ( 12 months = 1 year ). 14. Use the banker’s rule to find the interest paid on a loan of $1, 200 for 60 days at a simple interest rate of 6% annually. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 53

EXERCISE SET A 18. Use Table 11 -1 to find the exact time from the first date to the second date for non–leap years unless a leap year is identified. January 27, 2008, to September 30, 2008 Leap year (add 1 to total) September 30 = day 273 January 27 = day 27 273 - 27 = 246 days; 246 days + 1 day = 247 days Copyright © 2014, 2010, 2007 Pearson Education, Inc. 54

EXERCISE SET A 20. If a loan is made on the given date, find the date it is due. August 12 for 60 days August 12 = 224 th day 60 days + 224 days = 284 days = October 11 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 55

EXERCISE SET A 22. Find the discount (ordinary interest) and proceeds on a promissory note for $2, 000 made by Barbara Jones on February 10, 2011, and payable to First State Bank on August 10, 2011, with a discount rate of 9%. August 10 = day 222 February 10 = day 41 Exact time = 222 - 41 = 181 days Discount = 2, 000(0. 09)(181/360) = 90. 50 Proceeds = 2, 000 - 90. 50 = 1, 909. 50 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 56

EXERCISE SET A 24. Malinda Levi borrows $12, 000 on a 9. 5%, 90 -day note. On the 30 th day, Malinda pays $4, 000 on the note. If ordinary interest is applied, what is Malinda’s adjusted principal after the partial payment? What is the adjusted balance due at maturity? What is the amount of interest saved by making the partial payment? 12, 000(0. 095)(30/360) = 95 4, 000 - 95 = 3, 905 12, 000 - 3, 905 = 8, 095(0. 095)(60/360) = 128. 17 8, 095 + 128. 17 = 8, 223. 17 The adjusted principal after 30 days is $8, 095 and the adjusted balance due at maturity is $8, 223. 17. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 57

EXERCISE SET A 24. Malinda Levi borrows $12, 000 on a 9. 5%, 90 -day note. On the 30 th day, Malinda pays $4, 000 on the note. If ordinary interest is applied, what is Malinda’s adjusted principal after the partial payment? What is the adjusted balance due at maturity? What is the amount of interest saved by making the partial payment? Interest with no partial payment: $12, 000(0. 095)( 90/360) = $285 Interest with partial payment: $95 + $128. 17 = $223. 17 Interest saved: $285 - $223. 17 = $61. 83 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 58

EXERCISE SET A 26. Bennett Sales holds a 180 -day note of $7, 500 that has an interest rate of 8% annually. After 60 days, the note is sold to a bank at a discount rate of 7% annually. Find the proceeds on the third-party discount note. I = $7, 500(0. 08)(180/360) = $300 MV = $7, 500 + $300 = $7, 800 Bank discount time: 180 - 60 = 120 days Bank discount interest: $7, 800(0. 07)(120/360) = $182 Proceed to original payee: $7, 800 - $182 = $7, 618 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 59

Practice Test Copyright © 2014, 2010, 2007 Pearson Education, Inc. 60

PRACTICE TEST 2. How much money was borrowed at 12% annually for 6 months if the interest was $90? Copyright © 2014, 2010, 2007 Pearson Education, Inc. 61

PRACTICE TEST 4. A loan of $5, 000 at 12% annually requires $1, 200 interest. For how long is the money borrowed? Copyright © 2014, 2010, 2007 Pearson Education, Inc. 62

PRACTICE TEST 6. Find the exact time from October 12 to March 28 of the following year (a leap year). December 31 = day 365 October 12 = day 285 80 days March 28 = day 87 87 + 80 = 167 days + 1 leap day 168 days Copyright © 2014, 2010, 2007 Pearson Education, Inc. 63

PRACTICE TEST 8. Sondra Davis borrows $6, 000 on a 10%, 120 -day note. On the 60 th day, Sondra pays $2, 000 on the note. If ordinary interest is applied, what is Sondra’s adjusted principal after the partial payment? What is the adjusted balance due at maturity? What is the amount of interest saved by making the partial payment? The adjusted principal after 60 days is $4, 100 and the adjusted balance due at maturity is $4, 168. 33 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 64

PRACTICE TEST 8. Sondra Davis borrows $6, 000 on a 10%, 120 -day note. On the 60 th day, Sondra pays $2, 000 on the note. If ordinary interest is applied, what is Sondra’s adjusted principal after the partial payment? What is the adjusted balance due at maturity? What is the amount of interest saved by making the partial payment? Interest with no partial payment: Interest with partial payment: $100 + $68. 33 = $168. 33 Interest saved: $200 $168. 33 = $31. 67 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 65

PRACTICE TEST 10. A bread machine with a cash price of $188 can be purchased with a one-year loan at 10% annual simple interest. Find the total amount to be repaid. I = $88(0. 1)(1) = $18. 80 Total repaid = $188 + $18. 80 = $206. 80 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 66

PRACTICE TEST 12. Find the exact interest on a loan of $850 at 11% annually. The loan was made January 15 and was due March 15 = day 74 January 15 = day 15 74 15 = 59 days I = $850(0. 11)(59/365) = $15. 11 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 67

PRACTICE TEST 14. Find the duration of a loan of $3, 000 if the loan required interest of $213. 75 and was at a rate of 91/2 % annual simple interest. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 68

PRACTICE TEST 16. A promissory note using the banker’s rule has a face value of $5, 000 and is discounted by the bank at the rate of 14%. If the note is made for 180 days, find the amount of the discount. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 69

PRACTICE TEST Jerry Brooks purchases office supplies totaling $1, 890. 18. He can take advantage of cash terms of 2/10, n/30 if he obtains a short-term loan. If he can borrow the money at 10. 5 % annual simple ordinary interest for 20 days, will he save money if he borrows to take advantage of the cash discount? How much will he save? Find discount amount: 1, 890(0. 02) = 37. 80 Discount price = 1, 890 37. 80 = 1, 852. 20 I = 1, 852. 20(0. 105)(20/360) = 10. 80 1, 852. 20 + 10. 80 = 1, 863 1, 890 1, 863 = 27 Yes, he saves $27. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 70

PRACTICE TEST 20. Find the exact interest on a loan of $1, 510 at 7 3/4% annual interest for 27 days. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 71