Simple Discount

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Simple Discount

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MATH INVS – Simple Discount:

MATH INVS – Simple Discount Arnie Dris

Discounting an amount A:

Discounting an amount A The process of finding the present value P of an amount A , due at the end of t years, is called discounting A . The difference between A and its present value P , or A – P , is called the discount on A . From A = P + I , I = A – P ; thus, I which is the interest on P , also is the discount on A .

Example 1:

Example 1 If PhP 1150.00 is the discounted value of PhP 1250.00, due at the end of 7 months, then P = 1150.00 A = 1250.00 I = A – P = 100.00 The discount on the amount A = 1250.00 is I = 100.00 The interest on the principal P = 1150.00 is I = 100.00

Discount rates:

Discount rates In considering I as the interest on a known principal P , we computed I at a certain rate r per cent of P : I = Prt In considering I as the discount on a known amount A , it is convenient to compute I at a certain rate per cent per year, of A . If i is the discount on A , due at the end of 1 year, and if d is the discount rate expressed as a decimal, then by definition i = Ad , or d = i/A .

Simple discount vs Simple interest:

Simple discount vs Simple interest Simple discount , like simple interest, is discount which is proportional to the time. If Ad is the discount on A , due in 1 year, the discount on an amount A due at the end of t years is t ( Ad ), or I = Adt. From P = A – I , P = A – Adt or P = A (1 – dt ). Compare with simple interest: A = P (1 + rt ).

The main differences:

The main differences

Board Work 1:

Board Work 1 Note: If the time t is given in days, we may use either exact or ordinary simple discount, according as we use one year as equal to 365 or to 360 days, in finding the value of t . Problem 1 : Find the discount rate if PhP 340.00 is the present value of PhP 350.00, due after 60 days. Use ordinary simple discount. Problem 2 : If the discount rate is 6%, find the present value of PhP 300.00, due at the end of 3 months.

Interest to discount rates, and vice-versa:

Interest to discount rates, and vice-versa The use of a discount rate in finding the present value P of a known amount A is equivalent to the use of some interest rate, which is always different from the discount rate. From the equation for simple discount: P = A (1 – dt ) From the equation for simple interest P = A /(1 + rt )

Interest to discount rates, and vice versa (continued):

Interest to discount rates, and vice versa (continued) Equating, cancelling A , and cross-multiplying, we get (1 – dt )(1 + rt ) = 1 t ( r – d ) = t 2 ( rd ) t = ( r – d )/ rd

Interest to discount rates, and vice-versa (continued – part 2):

Interest to discount rates, and vice-versa (continued – part 2) This last equation is valid for all t – in particular, it holds for t = 1, so that rd = r – d d = r (1 – d ) r = d /(1 – d ) = –1 + 1/(1 - d ) d ( r + 1) = r d = r /( r + 1) = 1 – 1/( r + 1)

Interest to discount rates, and vice-versa (continued – part 3):

Interest to discount rates, and vice-versa (continued – part 3) In general , for all t : rdt = r – d d = r (1 – dt ) r = d /(1 – dt ) d (1 + rt ) = r d = r /(1 + rt )

Example 2:

Example 2 What is the discount of PhP 10,000 for 3 months at (a) 6% discount rate? (b) 6% interest rate? (a) F = 10,000, t = 3 months = ¼ year = 0.25 year, d = 0.06. Solving for P , we have: P = F (1 – dt ) = 10000(1 – (0.06)(0.25)) P = PhP 9,850.00

Example 2 (continued):

Example 2 (continued) What is the discount of PhP 10,000 for 3 months at (a) 6% discount rate? (b) 6% interest rate? (b) F = 10,000, t = 3 months = ¼ year = 0.25 year, r = 0.06. Solving for P , we have: P = F /(1 + rt ) = 10000/(1 + (0.06)(0.25)) P = PhP 9,852.22

Example 3:

Example 3 In availing discount, what interest rate is equivalent to 6% discount rate in 2 years? d = 0.06, t = 2 years. Solving for r : r = d /(1 – dt ) = 0.06/(1 – (0.06)(2)) r = 0.06/(1 – 0.12) = 0.06/0.88 = 0.06818 r is approximately 6.82%

Board Work 2:

Board Work 2 What equivalent discount rate can a man avail for a loan of 6% interest rate for 8 months? What discount rate should the lender charge to earn a simple interest of 12 ½ % in an 8-month loan? Find the discount rate equivalent to simple interest at the rate of 12% from July 15 to October 13 of the same year.

Promissory Notes:

Promissory Notes This section aims to: Enumerate, define, and illustrate the features of promissory notes in simple interest and simple discount; and Compute proceeds and maturity values in simple notes

Promissory Notes (continued):

Promissory Notes (continued) A promissory note is a written promise done by a borrower to pay a certain sum to the lender within a specified time. The proceeds is the sum or amount the borrower receives.

Features of Promissory Notes:

Features of Promissory Notes Date of the note – date when the note is done Maturity date – date when the note is due Term of the note – length of time from the date of the note to the maturity date Face value – principal amount stated on the note Maturity value – principal plus interest

A simple interest note:

A simple interest note PhP 20,000 Manila April 30, 2001 Ninety days after the date specified above, the undersigned promise to pay to the order of Mark Gutierrez, the principal sum of Ten Thousand Five Hundred Pesos at 12% per annum payable at Bank of Pilipinas, Manila. (Signed) Jose Cruz

A simple interest note (continued):

A simple interest note (continued) Jose Cruz – the person who owes the money and who executes the note PhP 20,000 – the amount borrowed or loaned April 30, 2001 – date when the note is done July 30, 2001 – date when the note is due (the months of May, June and July each has 30 days) To compute the final amount, we have: Interest I = Prt = 20000(0.12)(90/360) = PhP 600.00 Maturity Value F = 20000 + 600 = PhP 20,600.00

Example 4:

Example 4 In October 10, 1999, the following note is discounted at 12%: PhP 15,000 Manila July 4, 1999 Six months after the date stated above, the undersigned promise to pay to the order of Mr. Ernesto Zarate, Fifteen Thousand Pesos. (Signed) Irma dela Torre Give the following: (a) term of the note, (b) maturity date, (c) final amount, (d) term of the discount, and (e) proceeds.

Example 4 (continued):

Example 4 (continued) Jul 4, 1999==Oct 10, 1999==Jan 4, 2000 The term of the note is 6 months. The maturity date for six months is January 4, 2000, i.e. from July 4, 1999 to January 4, 2000. The final amount is PhP 15, 000.

Example 4 (continued – part 2):

Example 4 (continued – part 2) The term of the discount gives the number of days from October 10, 1999 to January 4, 2000, as follows: October 10 to 31 - 21 November - 30 December - 31 January 1 to 4 - 4 ================================= 86 days The proceeds is the money received, that is computed as follows: Proceeds = F (1 – dt ) = 15000(1 – (0.12)(86/360)) = PhP 14,570.00

Assignment:

Assignment On July 20, 1998, Mrs. Estella Rosales discounted the note stated below at a bank which charges a discount rate of 10%. How much did she receive? PhP 25,000 Quezon City April 10, 1998 One hundred fifty days after the date stated above, the undersigned promise to pay to the order of Mrs. Estella Rosales the amount of Twenty Five Thousand Pesos. (Signed) Jun Jimenez

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