1. Unit 4 Seminar: Simple Interest
Welcome to MM255!
Unit 4 Seminar: Simple Interest

2. Topics for today Define Simple Interest Formula
Compute simple interest using formula
Compute Maturity Value of an investment or loan
Ordinary interest vs. Exact Interest
Adjusted Maturity Value – Early Payment

3. Key Terms Interest: an amount paid or earned for the use of money.
Simple interest: interest earned when a loan or investment is repaid in a lump sum.
Principal: the amount of money borrowed or invested.
Rate: the percent of the principal paid as interest per time period.
Time: the number of days, months or years that the money is borrowed or invested.

4. The Simple Interest Formula
Shows how interest, rate, and time are related and gives us a way of finding one of these values if the other three values are known.
I = P x R x T
Interest = Principal x Rate x Time

5. Identify the principal, rate and time
James borrowed $8500 from his parents to purchase his first car. He agreed to pay them 6.5% simple interest per year over 5 years.
The interest is always expressed as a percentage (6.5%)
Principal is the amount borrowed or invested. ($8500)
Rate of interest is a percent for a given time period (5 years).
Be Careful! Time (T) must be expressed in the same unit of time as the rate (R).
5 ye

6. Find the interest paid
James borrowed $8,500 from his parents to purchase his first car. He agreed to pay them 6.5% simple interest per year over 5 years.
How much interest did James pay?
Principal = (P) $8,500 Interest = P x R x T
Interest rate = 6.5% (or 0.065) = 8,500 x x 5
(When not stated assume interest rate is per year.)
Time = 5 years = $2,762.50
The interest on the loan is $2,762.50

7. Find the interest paid on a loan
Henry borrowed $1,200 from his parents to purchase his a dishwasher. He agreed to pay them 8% simple interest per year for 12 months.
How much interest did Henry pay?
Principal = (P) = ?
Interest rate = ?
Time = ? (Remember this must be in the same unit as the rate.)
Interest = P x R x T
The interest on the loan is ?

8. Find the interest paid on a loan
Henry borrowed $1,200 from his parents to purchase his a dishwasher. He agreed to pay them 8% simple interest for 12 months.
How much interest did Henry pay?
Principal = (P) $1,200 Interest = P x R x T
Interest rate = 8% (or 0.08) = 1,200 x 0.08 x 1
Time = 12 months = 1 year = $96
The interest on the loan is $96

9. Maturity Value
Maturity value: the total amount of money due by the end of a loan or investment period - includes the principal and interest Maturity Value of Loan = Principal + Interest

10. Find the Maturity Value of a Loan
MV = principal + interest
But interest = I = P*R*T
So, MV = principal + PRT
This is the same as MV = P + PRT
We can factor out the P to get MV = P(1+RT)
Remember to multiply R*T before adding 1.

11. An example of how this works
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)
Verify time units match. Substitute known values.
MV = $2,500 [ 1 + (0.09 x 2)]
MV = $2,500 [ ]
MV = $2,500 [ 1.18 ]
MV = $2,950
This is what Marcus will owe at the end of the two years.

12. Now your turn to try
Terry Williams is going to borrow $4,000 at 7.5% annual simple interest. What is the maturity value of the loan after three years?
MV = P [ 1 + (R*T)]

13. Answer MV = P [ 1 + (R*T)] P = $4,000 R = 7.5% per year T = 3 years
= $4,000 [ ]
= $4,000 [ 1.225]
= $4,900

14. Interest Using Partial years
What if we are faced with this problem?
To save money, Stan Wright invested $2,500 for 42 months at 4 ½ % simple interest. How much interest did he earn?
How do we calculate the problem when we have a weird number of months?

15. Convert Months to a Fractional or Decimal Part of a Year
Write the number of months as the numerator of a fraction.
Write 12 as the denominator of the fraction because there are 12 months in a year.
Reduce the fraction to lowest terms if using the fractional equivalent.
Divide the numerator by the denominator to get the decimal equivalent of the fraction

16. Convert the following to fractional or decimal part of a year
Convert 9 months and 15 months, respectively, to years, expressing both as fractions and decimals
9/12 = ¾ = 0.75
9 months = 0.75 of a year
15/12 = 1 3/12 = 1 ¼ = 1.25
15 months = 1.25 of years

17. Back to our problem
To save money, Stan Wright invested $2,500 for 42 months at 4 ½ % simple interest. How much interest did he earn?
First convert the time to the same units.
42 months = 42/12 = 3.5 years
Convert the interest rate to a decimal.
4 ½% = 4.5% Divide by 100 to get 0.045
Then substitute known values into our formula I = PRT.
I = $2,500 x x 3.5 = $393.75
Stan will earn $ in interest

18. Find the Principal, Rate or Time Using the Simple Interest Formula
We can find any of the 4 values if we know the other three …

19. Find the principal using the simple interest formula
I = PRT Use formula operations to solve for P
P = I / RT
Judy paid $108 in interest on a loan that she had for 6 months. The interest rate was 12%. How much was the principal?
P = 108/ (0.12 x 0.5) We converted 6 months to 0.5 years
P = 108/0.06
P = $1,800

20. Find the rate using the simple interest formula
I = P R T Use formula operations to solve for R
R = I / PT
Sam wants to borrow $1,500 for 15 months and will have to pay $225 in interest. What rate is charged?
Rate = $225/ ($1,500 x 1.25) months = 1.25 years
R = 225/1,875
R = .12 or 12%
The rate Sam will pay is 12%.

21. Find the time using the simple interest formula
I = P R T Use formula operations to solve for T
T = I / RP
Shelby borrowed $10,000 at 8% and paid $1,600 in interest. What was the length of the loan?
Time = $1,600/(0.08 x $10,00o)
T= $1,600/800
The loan period was 2 years

22. Ordinary and Exact Interest
Ordinary time: time that is based on counting 30 days in each month.
30 days per month * 12 months a year = 360 days in a year.
Exact time: time that is based on counting the exact number of days in a time period.

23. Example The ordinary time from July 12 to September 12 is 60 days.
July 12 to August 12 = 30 days
August 12 to Sept. 12 = 30 days
To find the exact time from July 12 to September 12, add the following:
Days in July ( = 19)
Days in August:
Days in Sept
= 62 days

24. Using the Sequential Numbers Table … for due dates within the same year
Subtract the beginning date’s sequential number from the due date’s sequential number.
What is the exact time between May 5th and August 5th?
On the table Aug 5 = 217 and May 5 = 125
= 92 days is the exact time

25. Beginning and due dates in different years
1. Subtract the beginning date’s sequential number from 365.
(With 365 days in a year, how many are left that we need?)
2. Add the due date’s sequential number to the result from the previous step.
(Now add in how many days in this year we will use.)
3. If February 29 falls between the two dates, add 1.
(Having 2/29 makes it a leap year, so we have 366 days in that year.)

26. Example
Find the exact time from May 15 on Year 1 to March 15 in Year 2.
Step 1: May 15 is day 135
365 – 135 = 230
Step 2: March 15 is day 74
= 304 days
The exact time is 304 days.
Note: If Year 2 is a leap year, the exact time is 305 days.

27. Try these to be sure we got it
A loan made on September 5 is due July 5 of the following year.
Find: a) ordinary time b) exact time in a non-leap year c) exact time in a leap year.

28. Example
A loan made on September 5 is due July 5 of the following year.
a. Ordinary time = 300 days
Count the months, 10 between Sept and July. Ordinary assumes 30 days per month. 30 * 10 = 300 days.
b. Exact time (non-leap year) = 303 days
Get the days remaining in the first year – Sept.5 = X
Look up the days used in the next year. Value for July 5 = Y
Add those together = X + Y = 303
c. Exact time (leap year) = 304 days
We know exact time with out a leap year is We also know that July 5th is after February 29th which means there is an extra day unaccounted for = 304 days.

29. Ordinary Interest vs. Exact Interest
Ordinary interest: a rate per day that assumes 360 days per year.
Exact interest: a rate per day that assumes 365 days per year

30. Find the ordinary interest (using ordinary time) on a loan
A loan for $500 was made on March 15 and due on May 15. The annual interest rate is 7%.
Length of loan (ordinary time) =
30 days * 2 months = 60 days
Using ordinary time, we want 60 days of 360 in a year.
Rate = 0.07 for the year
Interest = PRT = $500 * 0.07 * (60/360)
Interest = $5.83

31. Find the ordinary interest (using exact time)
A loan for $500 was made on March 15 and due on May 15. The annual interest rate is 7%.
Length of loan (exact time) = 61 days {Use the table}
Using exact time, we want 61 days of an ordinary year or 360.
Rate = 0.07
Interest = $500 * 0.07 * (61/360)
Interest = $5.93

32. Find the exact interest (using exact time)
A loan for $500 was made on March 15 and due on May 15. The annual interest rate is 7%.
Length of loan (exact time) = 61 days
Using exact time, we want 61 days of an exact year or 365.
Rate = 0.07
Interest = $500 * 0.07 * (61/365)
Interest = $5.84

33. Comparison Time Rate Interest Ordinary interest using ordinary time
60 days 7%
$5.83
Ordinary interest using exact time
61 days
$5.93
(highest)
Exact Interest using exact time
$5.84

34. Banker’s Rule
Financial institutions typically calculate interest on a loan based on ordinary interest and exact time
…. which yields a slightly higher amount of interest.
This is because you get a higher per day interest rate dividing by 360 instead of 365 and you use it over the exact days rather than only 30 days in a month.

35. Partial Payments Before the Maturity Date
Tony borrows $5,000 on a 10%, 90 day note. On the 30th day, Tony pays $1,500 on the note. If ordinary interest is applied using exact time (banker’s rule), what is Tony’s adjusted principal after the partial payment, and adjusted balance due at maturity?
We split the loan length into two periods – before the partial payment and after.

36. Adjusted Maturity Value when a partial payment made before the maturity date
1. Determine the exact time from the date of the loan to the first partial payment.
2. Calculate the interest for period up to the payment. Subtract that interest from the partial payment.
3. Reduce the initial Principal by the remaining amount of the partial payment. This is the adjusted principal.
4. At maturity, calculate interest from the last partial payment and add to adjusted principal. This is the adjusted balance due at maturity.

37. Work step by step
Tony borrows $5,000 on a 10%, 90 day note. On the 30th day, Tony pays $1,500 on the note. If ordinary interest/exact time is applied, what is Tony’s adjusted principal after the partial payment, and adjusted balance due at maturity?
Determine the exact time from the date of the loan to the first partial payment.
30th Day. Using ordinary interest, our time for interest will be 30/360.
Calculate the interest for period up to the payment.
I = PRT
I = $5,000(.1)(30/360) = $41.67

38. Work step by step
Tony borrows $5,000 on a 10%, 90 day note. On the 30th 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?
Reduce the initial Principal by the amount of the partial payment. This is the adjusted principal.
The interest we calculated was $41.67.
We pay the interest off first. So reduce your partial payment by the interest. P – I = $1,500 - $41.67 = $1,458.33
Then you subtract the rest of the partial payment to reduce the principal. $5,000 - $1, = $3, This is the adjusted principal, the amount you will calculate interest on in the future.

39. Work step by step
Tony borrows $5,000 on a 10%, 90 day note. On the 30th 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?
At maturity, calculate interest from the last partial payment and add to adjusted principal. This is the adjusted balance due at maturity.
Our new principal is $3,541.67
On our 90 day note, 30 days have past so we have 60 days left. Using ordinary interest our time is now 60/360.
Our new interest amount is now I = PRT = $3,541.67(.1)(60/360) = $59.03
This is our new interest due so we add this to the principal balance due on our loan.
Our adjusted balance or balance due is
$3, $59.03 = $3,600.70

40. Quick review of the steps we took
1. Determine the exact time from the date of the loan to the first partial payment.
2. Calculate the interest for period up to the payment. Subtract that interest from the partial payment.
3. Reduce the initial principal by the remaining amount of the partial payment. This is the adjusted principal.
(You could repeat the above steps for as many partial payments as you need to.)
4. At maturity, calculate interest from the last partial payment and add to the adjusted principal. This is the adjusted balance due at maturity.

41. Any final questions on seminar?
Reminder of what to complete for Unit 4:
Discussion = initial response to one question + 2 reply posts
MML assignment
Instructor graded assignment (download from doc sharing)
Seminar quiz if you did not attend, came late, or left early