BAHAN AJAR MATEMATIKA BISNIS
SIMPLE INTEREST AND SIMPLE DISCOUNT 1.1 The Simple Interest Formula
Find simple interest by using the simple interest formula. Find the maturity of a loan.
Convert months to a fractional or decimal part of the year. Find the principal, rate or time using the simple interest formula. 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. 1.1.1 The Simple Interest Formula
The 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
Find the simple interest using the simple interest formula
Identify the principal, rate and time
The interest is a percentage.
Principal is the amount borrowed or invested.
Rate of interest is a percent for a given time period, usually one year.
Time must be expressed in the same unit of time as the rate. (i.e. one year) Find the interest paid on a loan
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. Try these examples
Find the interest on a 2-year loan of $4,000 at a 6% rate. $480
Find the interest earned on a 3-year investment of $5,000 at 4.5% interest. $675
11.1.2 Find the Maturity Value of a Loan
Maturity value: the total amount of money due by the end of a loan period; the amount of the loan and interest.
If the principal and the interest are known, add them. MV = principal + PRT
MV = P(1+RT) Look at this example
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)
What is the maturity value?
MV = $2,500 ( 1 + 0.09 x 2) MV = $2,500 (1 + 0.18) MV = $2,500 (1.18) MV = $2,950
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
Jim Sherman will invest $3,000 at 8% for 5 years. What is the maturity value of the investment?
$4,200
1.1.3 Convert Months to a Fractional or Decimal Part of a Year 1.1.4 Find the Principal, Rate or Time Using the Simple Interest Formula
Find the principal using the simple interest formula
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?
Substitute the known values and solve. P = 108/ 0.12 x 0.5
P = $1,800
Find the rate using 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 = 225/ $1,500 x 1.25
R = .12 or 12%
The rate Sam will pay is 12%.
Find the time using the simple interest formula
T = I / RP
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 = $1,600/0.08 x $10,000 T = 2
The length of the loan was two years. 1.2 Ordinary and Exact Interest
Find exact time. Find the due date.
Find the ordinary interest and the exact interest. Make a partial payment before the maturity date.
11.2.1 Find Exact Time
Ordinary time: time that is based on counting 30 days in each month.
Exact time: time that is based on counting the exact number of days in a time period. Examples
The ordinary time from July 12 to September 12 is 60 days.
Days in August 31
Days in September 12 +
62 days Sequential Numbers for Dates of the Year
Find the exact time of a loan using the sequential numbers table.
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.
Ex.: From May 15 to October 15 288-135 = 153 days is the exact time
Beginning and due dates in different year
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?)
Look at this example
Find the exact time from May 15 on Year 1 to March 15 in Year 2. 365 – 135 = 230
230 + 74 = 304 days The exact time is 304 days.
Note: If Year 2 is a leap year, the exact time is 305 days. Try this example
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.
Ordinary time = 300 days
Exact time (leap year) = 304 days
1.2.3 Find the Ordinary Interest and the 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.
Banker’s rule: calculating interest on a loan based on ordinary interest and exact time which yields a slightly higher amount of interest.
Find the ordinary interest per day
For ordinary interest rate per day, divide the annual interest rate by 360. Ordinary interest rate per day = Interest rate per year/360
Find the exact interest per day
For exact interest rate per day, divide the annual interest rate by 365.
Exact interest rate per day = Interest rate per year 365
Use ordinary time to find the ordinary interest on a loan
A loan of $500 at 7% annual interest rate. The loan was made on March 15 and due on May 15. (Principal = $500) I = P x R x T
Length of loan (ordinary time) = 60 days Rate = 0.07/360 (ordinary interest) Interest = $500 x 0.07/360 x 60 Interest = $5.83
Find the ordinary interest using exact time for the previous loan
A loan of $500 at 7% annual interest rate. The loan was made on March 15 and due on May 15. (Principal = $500) I = P x R x T
Interest = $500 x 0.07/360 x 61 Interest = $5.93
Find the exact interest using exact time for the previous loan
A loan of $500 at 7% annual interest rate. The loan was made on March 15 and due on May 15. (Principal = $500) I = P x R x T
Length of loan (exact time) = 61 days Rate = 0.07/360 (ordinary interest) Interest = $500 x 0.07/360 x 61 Interest = $5.93
Find the exact interest using exact time for the previous loan
A loan of $500 at 7% annual interest rate. The loan was made on March 15 and due on May 15. (Principal = $500) I = P x R x T
Length of loan (exact time) = 61 days Rate = 0.07/365 (exact interest) Interest = $500 x 0.07/365 x 61 Interest = $5.84
1.2.4 Make a Partial Payment Before the Maturity Date
To find the adjusted principal and adjusted balance due at maturity for 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 using the time found in Step 1.
3. Subtract the amount of interest found in Step 2 from the partial payment.
4. Subtract the remainder of the partial payment (Step 3) from the original principal. This is the adjusted principal.
5. Repeat process for additional partial payments.
Look at this example
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?
$5,000(.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) 1.3 Promissory Notes
Find the bank discount and proceeds for a simple discount note. Find the true effective interest rate of a simple discount note.
Find the third-party discount and proceeds for a third-party discount note. 1.3.1 Find the Bank Discount and Proceeds for a Simple Discount Note
For the bank discount, use:
Bank discount = face value x disc. rate x time [I = P x R x T]
For the proceeds, use:
Proceeds = face value – bank discount A = P - I
1.3.2 Find the True of Effective Interest Rate of a Simple Discount Note
To find the true or effective interest rate of a simple discount note: 1. Find the bank discount (interest). I = PRT
2. Find the proceeds: proceeds = principal – bank discount. 3. Find the effective interest rate: R = I/PT using the proceeds as the principal
Try this example
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(.12)(90/360) I = $150 (Bank discount)
Proceeds = $5,000 - $150 = $4,850 R = I/PT; R = $150/$4,850(90/360) R = .1237113402
R or the effective interest rate = 12.4%
1.3.3 Find the Third Party Discount and Proceeds for a Third Party Discount Note
For the bank discount, use:
For the proceeds, use:
Proceeds = maturity value of original note – third-party discount A = P - I
Look at this example
Mihoc Trailer made a note of $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 taken 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.
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) Exact interest rate is .09/365 MV = P(1+ RT)
MV = $10,000 ( 1 + 0.09/365 x 90) MV = $10.221.92
Find the proceeds of the note
Exact time of the discount period is 66 days. (314 - 248) period 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
