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Learning Competency

• Define interest and discounts.

• Derive formula from interest to find the Principal, Rate
and Time.
• Find the interest, principal, rate, time and final amount.
• Solve problems involving interest and discount.

1.0 SIMPLE INTEREST AND DISCOUNT

Interest is defined as an income derived from an invested capital or as a payment
for the use of money.
Interest depends upon three factors:
➢ principal, the sum barrowed or invested ;
➢ time (term), the period for which the money is to be used ;
➢ interest rate, the ratio for which the money between the interest earned in a certain
unit of time and the principal
• SIMPLE INTEREST FORMULA

I = Prt
Where I = interest in pesos
P = sum invested or borrowed
r = rate of interest expressed as an annual rate
t = time expressed in years

EXAMPLE 1. A man borrowed P 15,000 at the rate of 8% per annum payable at the
end of one year. Find the interest.

EXAMPLE 2: What is the simple interest on P 8750 at 9 ½ % for 2 ½ years?

• ORDINARY AND EXACT SIMPLE INTEREST

If time (t) is given in months, t can be expressed as

𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑚𝑜𝑛𝑡ℎ𝑠
𝑡=
12
If time is given in days, t can be expressed as
-For exact simple interest ( Ie)

𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑦𝑠
𝑡=
365

-For ordinary simple interest ( Io)

𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑦𝑠
𝑡=
360
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EXAMPLE 1: Dr. Hannah Alexandra P. Mariano invested P 150, 000 at 10 % simple

interest for 9 months. Find the simple interest.

EXAMPLE: Sky borrows P 12, 900 from a community credit union which charges her 7 ½
% ordinary simple interest for 86 days. The credit union uses the ordinary simple interest
because it is the prevailing interest.
a. ) Compute the interest on Sky’s loan.
b. ) Show why ordinary and not exact simple interest is favored by lenders.

The PRINCIPAL, INTEREST RATE AND TIME

𝐼 𝐼 𝐼
𝑃= , 𝑟= , 𝑡=
𝑟𝑡 𝑃𝑡 𝑃𝑟

1. How much should Mr. Hanz Alexander invest today so that his money earns P 125 at
6 % for 10 months?
2. Mr. Sky Garcia borrows P 3200 payable after five months. If his money earns P 160
simple interest, find the rate of interest.
3. How long will it take P 5, 800 to earn P 348 if the money is invested at 8% simple
interest?

Activity 1
A. Complete the following using the interest formula.
I P r t Final Amount
1 _______ P 9 500 12% 2 ½ years _______
2 P 900 _____ 7¼% 10 mos ________
3 P 3000 P 9000 _____ 240 days ________
4 P 750 P 3500 6% _______ ________
5 ______ P8000 ____ 11 ½ mos P 18, 200

B. Solve the following problems:

6. Ms. Hannah Alexandra invested P 28, 500 in a bank which yields 10 ½ %
simple interest for 10 months. Find the interest.

7. How long will it take a principal to earn half of itself if invested at 12 % simple
interest?

8. After eighteen months, an amount invested at 12 % earned P 10, 800. How

much was invested?
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9. Mr. Hanz Alexander borrowed P 3 500 to buy feeds for his poultry. After 6
months, he paid P 1 525.00 as interest. Find the rate of interest.

10. Find the interest earned if half of P 10, 000 was invested at 12 % simple
interest for 1 year.

11. Find the interest earned by investing P 45 000 at 6 % simple interest for 145
days

12. At what rate was P 1550 invested if it earned P 70.00 for 150 days?

13. Marco borrowed P 7500 at 7 ½ % for 265 days. How much interest did he
pay for the use of the money?

14. Ms. Amor invested P 115,000 for her jewellery shop. After a year, her total
money (final amount) of jewellery was P 235, 000.00. What is the rate of interest?

15. How long will it take P 2400 to accumulate to P 5000 if money is invested at 8
%?

Learning Competency

• Find the exact and approximate numbers of days.

• Differentiate Ordinary interest and exact interest for
exact and approximate time.
• State the Bankers’ Rule

1.2 FINDING THE ORDINARY AND EXACT INTEREST FOR APPROXIMATE AND
EXACT NUMBER OF DAYS

HB.1. Determine the time from May 16, 2006 to September 22, 2006.(Exact and
Approximate Days)
To compute the approximate time, subtract serial numbers of the first date from
the last date.

Year Month Day

2006 9 22
2006 5 16
4 6
AT = 4(30) + 6 = 126 days
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To compute the exact number of days , determine the number of days from
month to month.
May ( 17-31 ) 15
June 30
July 31
August 31
September 22
Total 129 days
*** For the exact day, add 1 day for the leap year.
HB2. Find all four types of simple interest on P 5000 at 6% from April 8, 2006 to August
31, 2006.
A. Ordinary Interest for exact number of days. ( 𝐼𝑂𝐸 )
145
𝐼𝑂𝐸 = 5000(0.06) ( ) = 𝑃120.83
360
B. Ordinary Interest for Approximate number of days ( IOA).
143
𝐼𝑂𝐴 = 5000 (0.06) ( ) = 𝑃 119.17
360
C. Exact Interest for Exact Number of days. ( IEE ).
145
𝐼𝐸𝐸 = 5000(0.06) ( ) = 𝑃 119.18
365
d. Exact Interest for Approximate number of days. ( IEA).
143
𝐼𝐸𝐴 = 5000(0.06) ( ) = 𝑃117.53
365
Notice that among the four types of interest, ordinary interest for exact number of
days yields the highest interest. This is known as Banker’s Rule.
Activity 2. A. Find the exact number of days between the following dates.
_______1. January 5, 2006 to November 16, 2007.
_______2. March 29, 2006 to November 17, 2007.
_______3. November 11, 2006 to June 25, 2007.
_______ 4. June 27, 2016 to November 11, 2016
_______5. January 27, 2016 to August 27, 2016
B. Find the approximate number of days.
_______ 6. June 7, 2007 to December 11, 2007
_______7. July 11, 2011 to January 28, 2012
_______8. July 7, 2007 to April 15, 2011
_______9. January 2, 1993 to September 27, 2016
_______10. January 27, 1991 to September 27, 2016
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Activity 3
A. Find the ordinary interest for approximate time and exact time.
1. P 4650 at 6% from January 10, 2006 to September 16, 2007.
IOA =_______
IOE = _______

2. P10 200 at 8 ½ % from December 8, 2006 to October 16, 2007.

IOA =_______
IOE = _______

B. Find the exact interest for approximate time and exact time.
3. P 6000 at 4 ¼% from August 15, 2006 to December 5, 2007.
IEE = _______
IEA = _______
4. P 10,000 at 12 % from March 15, 2006 to February 12, 2007.
IEE = _______
IEA = _______
C. Solve the following problems.
5. On September 16, 2006, Ms.Pakumbo borrowed 15,0000 pesos at 8 ½% simple
interest. If the loan was paid on May 6, 2007, how much exact interest did she pay?

6. Using approximate time, find the exact interest on P 15, 480 from November 25, 2006
to April 19, 2007 at 7 ½ simple interest.

Activity 4
Solve the following problems.
1. Using the Banker’s Rule, find the simple interest on P 24, 900 at 8% from November
29, 2006 to July 27, 2007.

2. Assuming 365- day a year, find the simple interest rate for which P 9500 earned P
237. 30 from July 15 to November 6 of the same year. Use Exact Time.

3.Matubato borrowed P3 million from Ms. Delima on February 23, 2014, at 19% simple
interest. How much interest must he pay on September 14 of the same year?

1.3 SIMPLE INTEREST ON INSTALLMENT PAYMENTS

The Merchant’s Rule
Oftentimes, people prefer to buy household appliances and accessories, like
refrigerator, TV sets, and washing machine on instalment basis. The interest on money
is computed on the balance of the principal before each payment.
HB1. Mrs. Dela Cruz bought a washing machine worth P 5,000.00 on installment basis,
money is worth 6% payable in one year. She has given the following payments P 1,000.00
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was paid after 2 months, P 1,500 after 5 months and P 2 000.00 after 9 months. How
much more must be paid at the end of the year?

1.4 THE FINAL AMOUNT

The final amount ( S ) is the amount due from the investment of the principal (P)
for a given period of time ( t ) at a given rate ( r ). It is an accumulated amount which
represents the principal plus the interest earned. By definition,
S=P+I

S = P + Prt

S = P( 1 + rt)

HB. 1 Mrs. Bah Nil loaned from Mrs. Puh Tolk the sum of P 5 500 and promised to pay
the sum interest at the end of 9 months. If money is worth 12 %, how much will Mrs. Bah
Nil pays Mrs. Puh Tolk at the end of the term?

HB 2. Mrs. Claud Mariano loaned P 50,000 to buy additional raw materials for her
business. The Marianos bank charged 12 % on the loan which is due at the end of one
year. How much is the accumulated amount?

HB. 3. Mrs. Klawdita Maldita borrowed P45,000 to open a shop. If the interest rate is
16%, how much will she have to pay back at the end of the year?

1.5 THE PRESENT VALUE

If a sum of money is invested at a given interest rate ( r ) for a given period of time
( t ) in order to accumulate to S, then this sum is called the present value ( P ) of the
Final Amount ( S ). To discount S means to find its present value (P). To compute the
present value ( P ) at simple interest, the formula,

S = P (1 + rt ) is modified
𝑆
Hence, 𝑃 =
1+𝑟𝑡
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• Compare the basic concepts of simple and compound

interest.
• Develop mathematical skills on correct calculations of
savings, purchases and investment.
• Introduce different nominal rates on making correct
decisions for investments and credit
• Enhance the ability to manage resources.

2. COMPOUND INTEREST
All investors want to gain more earnings out of their capital. This is realized when
they use compound interest as the basis of computing the profits and gains.
The result of simple interest is constant thorough out the investment period. But when
such result is added to the principal at regular intervals and the sum becomes the new
principal, the interest is said to be the compound interest.
-Definition and Related Terminologies
The interest increases periodically and the total interest at the end of transaction
period is called compound interest. Loans, mortgage, bank deposits, insurance
premiums and sinking funds are mostly paid using compound interest. The difference
between the compound amount and the original principal is called the compound interest.
Conversion Period or Interest Period is the time between two successive conversions of
interest. It may be any convenient length of time. The symbol used to denote the
conversion period m.The conversion period is usually taken.
as an exact division of the year. Semi-annually is denoted by m = 2; quarterly, m = 4 ;
monthly, m = 12 and annually , m = 1.

NOTATION
Aside from m which represents the number of conversions per year the following
letters will be used:
P – Original principal or the present value
S – accumulated value of the principal or the compound amount of P.
n – total number of conversion periods in the entire transaction period
J – nominal rate or yearly rate
i – interest rate per conversion period
𝑗
𝑖= 𝑎𝑛𝑑 𝑛 = 𝑚(𝑡)
𝑚
Thus, to calculate the compound amount.
𝑺 = 𝑷(𝟏 + 𝒊)𝒏

HB 1. Find the compound interest and compound amount of 10, 600 invested at 12 %
compounded semi-annually for 5 years.
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Solution:
𝑗
a. 𝑖 = 𝑚 b.) n = mt
12%
𝑖= n = (2)(5)
2
i = 6% n = 10
n
c. S = P ( 1 + i ) d. I = S - P
= 10, 600 ( 1+ 0.06 ) 10 = 18, 982.99 – 10, 600
=10, 600(1.06)10 = P 8, 382.99
= P 18, 982.99
Activity 6.
Answer the following:
1. Accumulate P 13, 700 at 12 %, m= 12 for 10 years.

2. Accumulate P 20, 800 at 8% , m = 2 for 3 mos.

3. Accumulate P 23, 049 at 6%, compounded quarterly for a year.

Activity 7.

Find the interest rate per conversion period ( i ) and the total number of
conversion periods ( n ) at the end of the indicated time.

1. 5 years at 10 % compounded monthly.

2. 3 years at 12 % compounded quarterly.

3. 2 ½ years at 6 % compounded semi-annually.

4. 3 years and 3 mos at 8 % compounded quarterly.

5. 8 years and 6 months at 12 % compounded quarterly.

6. 12 years at 6 % compounded annually.

7. 33 months at 18 % compounded monthly.

8. 1 ½ years at 5% compounded semi-annually.

9. 11 years at 16% compounded quarterly

10. From April 24, 2006 to July 24, 2010 at 18% compounded quarterly.
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Activity 8.

Solve for the value of the unknown:

1. Find the amount and interest of P 7500 invested for 3 years and 6 months at the rate
of 12% compounded:
a. monthly

b. semi-annually

c. quarterly

2. Find the amount of P 11, 800 invested for 2 years and 3 months at 18%
Compounded:
a. quarterly

b. monthly

3. Accumulate P 26,000 for 8 years

2.1 PRESENT VALUE

Present value refers to the value of the amount at an earlier date before
the amount is due. The present value (P) is the principal invested at an earlier date at a
given rate will accumulate to a specified amount S at some later. The process of finding
the present value is called discounting and the difference between S and P is called
discount.

𝑷 = 𝑺(𝟏 + 𝒊)−𝒏
HB1. P 22, 500 is due in 6 yrs at 12% , m = 12. Determine the present value.
Solution:
i = j/m= 12%/12
i = 1%
n = 6(12) = 72
𝑷 = 𝑺(𝟏 + 𝒊)−𝒏
P = 22,500(1.01)-72
P = 22500(0.488496)
P = P 10 991.16
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Activity 9. Find the present value of each of the following amounts:

FREQUENCY OF
AMOUNT NOMINAL TERM
CONVERSION

1. P 7500 8% Semi-annually 5 yrs.

2. P 5260 12 % Monthly 3 yrs and 7 mos

3. P 8390 10% Quarterly 6 yrs and 9 mos

4. P 10 250 7% Semi-annually 12 years

5. P 50, 490 6% Annually 15 yrs