1 “in the name of allah the most beneficient, the most merciful”
TRANSCRIPT
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“IN THE NAME OF ALLAH THE MOST BENEFICIENT, THE MOST MERCIFUL”
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SCIENTIFIC CALCULATORS
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Success is a ladder, on which you cannot climb with both hands in your pockets.
Course Objectives
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In the course the main concentration is on the applications of mathematics to businesses.
These may be seen in the form of stated problems from the text book & other reference material.
Our focus will be on the procedures to find the solutions to these problems.
Time to time some business terminologies will be used in the course.
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A WARM
WEL COME
PERCENTAGE
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Percentage is another aspect of fraction, specifically numbers expressed as fractions of 100. by expressing two or more numbers in the same terms – as parts of 100 – we can compare them directly. Thus 100 is the denominator of the fraction, and the number is said be so many hundredths or so many percent of the total. Percent is the Latin expression for hundredths.
CHANGING A DECIMAL TO A PERCENT:
Because percent means hundredths, we can change the expression 0.04 (read as 4 hundredths) to 4% (read as 4 percent). To change a decimal to a percent:
Move the decimal point two places to the right (two places stands fro hundredths), and place the percent sign to the right of the number.
CHANGING A PERCENT TO A DECIMAL
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To convert a percent to a decimal, move the decimal point two places to the left and remove the percent sign. Remember that whole number percents between 1% and 99% are written in the first two decimal places.
CHANGING A PERCENT TO A FRACTION1. Change the percent to the decimal equivalent.
2. Read the value of the decimal as “ – hundredths ,” and write this value as a fraction.
3. Reduce the fraction to lowest terms.
4. The only exception to this procedure is with a percent containing a fraction. Because percent means hundredths (1/100), take the percent containing a fraction and multiply by 1/100.
5. Reduce the resulting fraction to the lowest terms.
Examples
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1. 4% = 4 hundredths (% sign means hundredths) = 0.04
2. 85% = 85.% = 0.85
3. 46.7% = 46.7% = 0.467
4. 243.2% = 243.2% = 2.432
5. 0.8% = 00.8% = 0.008
All the above examples tell how to convert a percentage to the decimals.
6. 94% = 0.94 = 94 hundredths =
7. 150% = 1.50 = one and 50 hundredths
8. 25.5% = 0.255 = 255 thousands
50
47
100
94
2
11
100
501
200
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1000
255
PROJECT 1 PERCENTS
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A. Express each of the following as percent.
1. 0.89 = 4. 0.0037=
2. 6.93= 5. 0.045=
3. 0.0007= 6. 0.508=
B. Express each of the following as a decimal.
7. 35%= 9. 0.03%=
8. 72.6%= 10. 125.04%=
C. Express each of the following as common fraction in the lowest terms.
11. 36%= 14. 0.035%=
12. 224%= 15. 0.0062%=
13. 6.3%=
APPLICATIONS OF PERCENTS
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Percents are always expressed in some variation of the formR B = P where R is the rate
B is the baseP is the percentage
The base is the number or quantity taken as a whole, the total, or 100%. It may be total students in KARDAN, gross or net sales for the year, the population of a country in a particular year, etc.The rate is the number of hundredths of the total (the base) under consideration. Without a base, the rate is meaningless. If one says that 50 students represent 10%, the question immediately arises, 10% of what?The percentage is the product of the rate timesthe base. It is easy to compute anyone ofthe three parts of the formula if you know theother two parts.
Examples
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1. What is 35% of $756?P = R BP = 0.35 756 (of means multiply)P = $264.60
2. What percent of $428 is $64.20.
R B = PR $428 = 64.20
[Note: when finding the rate remember that the final answer must be expressed in percent from.]
%1515.0428
20.64428
20.64
428
428
R
R
Divide by 428
Examples (cont…)
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3. 12.5% of what is 625?
R B = P 0.125 B = 625
There are many uses of this percentage formula. One is finding the rate of increase or decrease (some times call percent of change). When computing rate of change always use the original , or beginning, quantity as the base.
000,5125.0
625125.0
625
125.0
125.0
B
BDivide by 0.125
Examples (cont…)
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The formula R B = P may be restated as
Rate x Base = Amount of increase or decrease
4. The price of a pound of ground beef was $1.00 last year and $1.50 this year. What was the rate of increase?
R B = Amount of increase
Amount of increase = 1.50 – 1.00 = 0.50
R 1.00 = 0.50
%502
100.1
50.0
00.1
00.1
R
RDivide by 1.00
Examples (cont…)
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5. A portable radio bought for $24.50 is sold for $32.
What percent of the cost is profit?
R 24.50 = (32.00 – 24.50) Profit is the amount of increase
R 24.50 = 7.50
%60.3050.24
50.7
50.24
50.24
R
R Divide by 24.50
PROJECT 2 PERCENTS: APPLICATIONS
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1. What is 45% of $850?
2. Rida bought a calculator for $75. She made a down
payment of 20% of the price. (a) How much was the
down payment? (b) what percent of the price was the
unpaid balance?
3. What percent of $180 is $15?
4. $31.25 is what percent of $375?
5. If 4% of a number is $3.56, what is the number?
6. 17% of what number is 20.23?
PROJECT 2 (cont…)
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7. $525 is what percent more than $500?
8. $500 is what percent less than $525?
9. There were 500 freshman last year and 496 this year in a
small Midwestern college. Find the percentage decrease.
10. Sarah Gold received a dividend of $450, which is 5% of
her investment. What is the size of the investment?
11. Last year’s taxes on a house were $1,550. this year’
taxes are $1,250. What percent are this year’s taxes of
last year’s taxes?
PROJECT 2 (cont…)
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12. Five out of 25 students in Professor Ford’s class received an”A”
grade. What percent of the class did not receive “A” grade?
13. In 2003, Dunkin’ Donuts Company had $300,000 in doughnuts
sales. In 2004, sales were up 40%. What are Dunkin’ Donuts sales
for 2004?
14. The price of an Apple computer dropped from $1,600 to $1,200.
What was the percent decrease?
15. In 1982, a ticket to the Boston Celtics cost $ 14. In 2003, a ticket
cost $50. What is the percent increase to the nearest hundredth
percent?
PROJECT 2 (cont…)
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16. At a local McDonald’s, a survey showed that out of 6,000 customers
eating lunch, 1,500 ordered diet coke with their meal. What percent of
customers ordered Diet Coke?
17. Out of 6,000 college students surveyed, 600 responded that they do not
eat breakfast. What percent of the students do not eat breakfast?
18. What percent of the college students in problem 17 eat breakfast?
19. Borders bookstore ordered 80 Marketing books but received 60 books.
What percent of the order was missing?
20. The sales tax rate is 6%. If Jim bought a new Buick and paid a sales tax of
$1,920. what was the cost of the Buick before the tax?
PROJECT 3 PERCENTS APPLICATIONS
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1. John bought a new car and made a down payment of $1,080. after
the payment, he owes 85% of the purchase price. What is the price
of the car?
2. Salma bought a piece of jewelry for her friend for $28.00, which
includes a federal tax of 10% and a state sales tax of 2%. What is
the price excluding the taxes?
3. Nancy needs 352.8 inches of cloth to complete her project of home
economics class. The cloth will shrink 2% after washing. How
long should the piece be before washing?
PROJECT 3 (cont…)
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4. Fancy furniture company sold a bedroom set for $960, for a loss of
$163.20 on the original price. What was the percent of loss on the
original purchase price?
5. Jim Hill works for a salary of $285 per week plus a commission of
6% of his sales. What were his sales during a week in which he
earned $407.85?
6. Handy Office Supply bought a manual typewriter for $220 from
the factory and sold it to customer for $420. What percent of the
selling price is the cost?
PROJECT 3 (cont…)
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7. The Sewing Center Shop cut the price of a dress fabric from $6 a
yard to $4. what percent price cut did they advertise?
8. Sam’s Suit Shop is selling for one day a man’s three-piece suit for
$232.50, a loss of 7% on the cost from the manufacturer. What was
the cost of each suit?
9. Mountain Peak Community College decided to cut the size of the
entering freshman class by 15% because of the 15% budget cut.
Last year’s class had 1,180 students. How large is this year’s class?
PROJECT 3 (cont…)
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10. The value of the inventory of Johnny’s Corner Variety Store
increases from $90,000 to $97,200 during one-year period. What
percent rise in value did this represent?
11. In Kalamazoo, the number of building permits issued during April
dropped from 720 last year to 576 this year. What percent decrease
in permits did this represent?
12. The cost of a table was increased by 15% of itself to get the selling
price. What was the cost of the table if the selling price was $161?
Mathematics of Finance 1
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CHAPTER NO 9
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Profit:The difference between the selling price and
the cost price of an item is called the profit or mark-up. Thus if “S” is the selling price and “C” is the cost price the mark-up can be calculated as:
P = S – CThe mark-up is expressed as the percentage
of the cost price or the selling price.Often the mark-up is based on the cost.The selling price can be determined by
adding the mark-up to the cost.
Continue…
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If “C” and “S” represents the cost price and the sale price respectively and “r” is the percentage mark-up, then:
S = C + Cr = C(1+r)
Example:If the Cost price of an item is $ 2,400 and the mark-up on cost is 23%. Find the sale price.
Solution:Using the formula
S = C(1+r)Putting the values
S = 2,400(1+0.23) = 2,952
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When mark-up is stated as a percentage on sale
If “d” is the percentage mark-up on sales. Then d.S would represent the disount or mark-down.Now
Profit = Sales price – Cost price d.S = S – C
Or C = S - d.S C = S(1- d)
Example:After a mark-up of 30% on sales a watch sells for $
225.i) What is its cost price?ii)What is the %age mark-up on sales if the cost price
of watch would have been $ 153?
Simple Interest and Present Value
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o Interest is a fee which is paid for having the use of money.
o The amount of money that is lent or invested is called principal.
o Interest is usually paid in proportion to the principal and the period of time over which the money is used.
o The interest rate specifies the rate at which interest accumulates.
o The interest rate is typically stated as a percentage of the principal per period of time.
Continue…
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o The interest paid only on the principal is called simple interest.
o Simple interest is usually associated with loans or investments which are short-term in nature.
ComputationSimple interest = (principal ) * ( interest rate) * (number of time period)
Or I = PrtWhere I = simple interest
P = principal ( also denoted by PV )
r = interest rate ( also denoted by i )
t = number of time period of loan( denoted by n also )
Continue…
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The total amount “A” to be repaid is the principal plus the accumulated interest, or
A = FV = P + I = P + P r t = P(1 + r t) or FV=PV(1 + r t )
If the future value “A” is known, the present value of an amount “P” at simple interest “r” can be written as :
(1 ) (1 )
A FVP
r t r t
Important
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Note that in computing the interest it is customary to consider a 360-day year instead of a 365-day year. Thus 30 days will be considered as of an ordinary year and so on. The interest thus obtained is called “ordinary interest” but if it is based on 365 days, it is called the “exact interest”.
12
1
360
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Application of the formula
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Example:A credit union has issued a 3-year loan of $5,000. Simple interest is charged at the rate of 10% per year. The principal plus interest is to be paid at the end of third year. Compute the interest for the 3-year period. What amount will be repaid at the end of the third year?
Solution: On board
Simple Discount
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If “A” is the amount to be paid at maturity after a time “t” at the simple interest rate of “r percent” per annum. Then the simple discount “D” on maturity value “A” in time “t” is given by
To get the present value “P”, we must subtract this from “A”.
ArtD
)1( rtAP
ArtAP
Discounting Negotiable Instrument
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A written promise to pay money at a certain specific date is called Negotiable Instrument.They are of two types,
Non interest bearingInterest bearing
The basic principles of discounting a bill of exchange or short term note at a bank or at any other party are the same as those of obtaining a loan from a bank which deducts interest in advance.
Discounting non-Interest-Bearing Note
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Example:After Khalid accepted a bill for $ 4,500. Hanif
discounted it at National Bank Karachi on April 15. The maturity date of the bill was May 15. How much did Hanif receive if the bill was discounted at 8%?
Solution:Period of discount = 30 days or t = 1/12 year.A = 4,500 & r = 8%the discounted value
P = A(1-rt) = $ 4,470
Discounting Interest-Bearing Note
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We follow the following two steps in discounting an interest-bearing note.
i. Find the maturity value from the face value of the note after adding the interest which would have been earned up to the maturity date at the given rate.
ii. Find the proceeds by discounting the maturity value obtained in step (i) at the discounting rate.
Example
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Mohsin had a note for $ 15,000 with an interest rate of 6%. The note was dated January 12, 1983 and the maturity date was 90 days after date. On January 27, 1983 he took the note to his bank which discounted it at a rate of 7%. How much did he receive?
Solution:Step-1: Find maturity value FVStep-2: Discount the maturity value at 7%
Equivalent Values of Different Debts and their Payments
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Sometimes a situation arises when a single debt or a set of debts are to be paid on different dates by means of a single payment or a set of payments. To satisfy both the creditor and the debtor, the values of payments should be equivalent to the values of the original debts on a certain date called the comparison date.Due to interest, a sum of money has different values at different times. Therefore a comparison date should first be chosen to equate the sum of the values of the original debts with the values of the desired payments on the same date. This process will bring the different debts and the subsequent payments on the same footing.
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Example:A man owes $ 800, $ 1,000, and $ 200 due in 30 days, 60 days and 90 days respectively. If the rate of interest is 8%, Find the amount of the single payment, if the payment is made:
1.after 75 days from now2.after 110 days from now3.after 30 days from now
Getting More Involved
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SolvingPRACTICE SET 9, p # 110-111&PROBLEM SET 9,p # 112-113
CHAPTER 10
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MATHEMATICSOFFINANCE-II
Compound Interest
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Compounding involves the calculation of interest periodically over the life of the loan (or investment).
After each calculation the interest is added to the principal.
Future calculations are on the adjusted principal (old principal plus interest).
Compound interest is the interest on the principal plus the interest of prior periods.
Future value, or compound amount, is the final amount of the loan or investment at the end the last period.
Lets see
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How $1 will grow if it is calculated for 4 years at 8% annually?
Present value$ 1.00
After 1 period $1
$ 1.08
After 2 periods
$ 1.1667
After 3 periods$1.2597
After 4 periods$1.3605
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 1 2 3 4
Some term to understand
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Compounded annually: Interest calculated on the balance once a year.
Compounded semiannually: Interest calculated on the balance every 6 months or every ½ yeas
Compounded quarterly: Interest calculated on the balance every 3 months or every ¼ yeas
Compounded monthly: Interest calculated on the balance each month.
Compounded daily: Interest calculated on the balance each day.
Number of periods: Number of years multiplied by number of times the interest is compounded per year.
For Example
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If we compound $1 for 4 years at 8% annually, semiannually, or quarterly, the following periods will result:
Annually: 4 years * 1 = 4 periods
Semiannually: 4 years * 2 = 8 periods
Quarterly: 4 years * 4 = 16 periodsRate for each period: Annual interest rate divided by
the number of times the interest is compounded per year. Compounding changes the interest rate for annual, semiannual, and quarterly periods as follows:
Continue…
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Annually: 8% / 1 = 8%Semiannually: 8% / 2 = 4%Quarterly: 8% / 4 = 2%
Note:Both the number of periods (4) and the
interest rate (8%) for the annual example do not change. The rate and periods (not years) will always change unless the interest is compounded yearly.
Compound Amount Formula
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Let P = Principal i = interest rate per compounding periodn = number of compounding periods (number of periods in which the
principal has earned interest)S = compound amount
The compound amount after one period isS = P + iPS = P(1 + i)
Continue…
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Similarly, the compound amount after three periods is given by:
compound amount
after two periods
compound amount after one period
+interest earned
during the secondperiod
=
2)1(
)1)(1(
)]1([i1PS
iPS
iiPS
iPi
3)1( iPS
Continue…
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Thus we have the following definition:If an amount of money “P” earns interest compounded at a rate of “i” percent per period, it will grow after “n” periods to the compound amount “S”, where
This equation is often referred as the compound interest formula.
The compound interest is given by:Compound interest = S - P
(1 )nS P i
Examples
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1. Find out the compound amount and the compound interest at the end of three years on a sum of $ 20,000 borrowed at 6% compounded annually.
2. If $ 3,000 are invested at 6% interest compounded semi-annually, what would it amount to at the end of 8 years?
Effective Interest Rates
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Interest rates are typically stated as the annual percentages. The stated annual rate is usually referred to as the nominal rate.
When interest is compounded semiannually, quarterly, and monthly, the interest earned during a year is greater than if compounded annually.
When compounding is done more frequently than annually, an effective annual interest rate can be determined.
Continue…
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Definition: The effective interest rate is the interest rate compounded annually which is equivalent to a nominal rate compounded more frequently than annually. The two rates would be considered equivalent if both will result in the same compound amount.
Example:The nominal rate of 8% compounded quarterly is equivalent to the effective rate of interest 8.2432%.
Formula for finding the Effective rate of interest
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11 fieWhere
e = the effective ratei = interest rate per conversion periodf = number of conversion periods in one year (frequency)
Formula for Finding the Equivalent Rates of Interest
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1 2
1 2
1 2
1 1f f
r r
f f
Unknown nominal interest rate
# of times the compounding isrequired per year (for nominal rate)
# of periods fornominal rate(frequency)
# of periods forthe given rate(frequency)
Given interest rate
# of times the compounding isrequired per year (for given rate)
Example
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At what nominal rate compounded quarterly will a principal accumulate to the same amount as at 8% compounded semi-annually?SOLUTION ON BOARD
Depreciation by Reducing Balance Method
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If ‘C’ is the original cost of a machinery and ‘T’ is the trade in or scrap value of the machinery after ‘n’ years of useful life and ‘r’ is the percentage rate of depreciation on the reduced balance each year.Then,Depreciation for 1st year = CrResidual value after 1st year = C – Cr = C(1-r)Depreciation for the 2nd year = C(1-r)rResidual value after 2nd year = C(1-r) – C(1-r)r = C(1-r)2 Continuing in this way we get:Residual value after ‘n’ years = C(1-r)n
Continued….
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Since ‘n’ is the useful life of the machinery when its residual value is ‘T’, therefore,
n
n
n
n
n
C
Tr
C
Tr
C
Tr
C
Tr
TrC
1
1
1
1
)1(
)1(
or
Getting More Involved
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Discussion on Practice Set 10-A, (P# 121/122)
PRESENT VALUE(AT COMPOUND INTEREST)
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The compound amount formula is given by:
A slight rearrangement of the formula gives:
This formula is called the Present value or discounting formula.
The factor or iscalled the present value factor or the discounting factor.
niPS 1
n
n iSi
SP
1
1
ni1
1 ni 1
Examples
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i. Find the present value of $ 4,814.07 due at the end of 8 years if money is worth 6% compounded semi-annually.
ii. What sum of money invested at 6% compounded annually will amount to $ 500 in 4 years?
iii. Find the present value of $ 4,958.54 due at the end of 8½ years if money is worth 6% compounded semi-annually.
Application on Discounting Interest bearing and Non-interest bearing Notes
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Example:A non-interest bearing note of $ 3,000 is due in 5 years from now. If the note is discounted now at 6% compounded semi-annually, what will be the proceeds and the compound discount?(SOL ON BOARD)
Example: (interest bearing note)
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An interest bearing note of $ 5,000 dated January 1,1980 at 6% compounded quarterly for 10 years was discounted on January 1, 1984. what were the proceeds and the compound discount if the note was discounted at 8% compounded semi-annually?
Examples
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1. Mr. Ahmad owes Mr. Bashir $ 5,000 in three years and $ 10,000 in 5½ years. How much should Mr. Ahmad pay at the end of 4 years which may be acceptable to Mr. Bashir if money is worth 8% compounded semi-annually?
2. Mr.Zahir owes to Mr. Mohmood $ 4,000 due in 2 years and $ 5,000 due in 4 years. If he agree to pay $ 4,500 now, how much he will have to pay to Mr. Mahmood three years from now to settle his two debts, if money is worth 6% compounded semi-annually?
Continued…
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Discussion onPractice Set 10-B, P#(127-
128)
&Problem Set 10,
P# (128-129)
CHAPTER # 11
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MATHEMATICS OF FINANCE – III
ANNUITIES
DEFINITIONS
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An annuity is a series of periodic payments (usually equal in amounts).
The payments are made at regular intervals of time such as annually, semi-annually, quarterly or monthly.
Examples of annuities include Regular deposits to a savings account Monthly car Mortgage Insurance payments Periodic payments to a person from a retirement fund
Continue…
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If the payments are made at the end of the payment periods the annuity is called an “ordinary annuity”
If the payments are made at the beginning of each interval the annuity is called “annuity due”
The time between two successive payment dates is called “payment period”.
The time between the beginning of the 1st payment & the end of the last payment period is called the “term of the annuity”
Sum of an annuity
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LetR= payment per periodi= interest rate per periodn= number of annuity payments
(also number of periods)S= sum (future value) of the
annuity after n periods (payments)The sum of the annuity is given by:
i
iRS
n 11
Example
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1. Find the amount of an annuity of $ 500 payable at the end of each year for 10 years, if the interest rate is 6% compounded annually.
Finding R when S is Known2. A father at the time of birth of his daughter
decides to deposit a certain amount at the end of each year in the form of an annuity. He wants that the sum of $ 20,000 should be made available for meeting the expenses of his daughter’s marriage which he expects to be solemnized just after her 18th birthday. If the payments accumulate at 8% compounded annually, how much should he start depositing annually?
Use of the annuity table
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In table 5 at book page 309 values of the sum of an ordinary annuity of Re.1 are given. We use the symbol read as “s angle n at i ” is
used for the factor .
To search an entry in the table we consult the table against i and n, and then multiply by the payment per period.
The use of the table will be discussed in the solution of problems.
ins
i
i n 1)1(
Finding n when S is known
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Example:How many semi-annual payments of $ 100 each at an account in the form of an ordinary annuity will accumulate $ 3,000 if the interest rate is 8% semi-annually?
Present Value of an Ordinary Annuity
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The present value of an annuity is an amount of money today which is equivalent to a series of equal payments in the future.
For exampleIf a loan has been made, we may be interested in determining the series of payments (annuity) necessary to repay the loan with interest.
Finding the Present Value of an Annuity
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LetR = amount of an annuityi = Interest rate per compounding
periodsn = number of annuity payments (also,
the number of compounding)P = Present Value of the annuity
Then,
i
iRP
n)1(1
Using table
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Table – 6 (book page 311) gives the present value of an ordinary annuity of Rupee 1 per period. The factor is denoted by the
symbol ( a angle n at i ). Thus the general formula for the present value of an annuity becomes:
i
i n)1(1
ina
i
iRaRP
n
in
)1(1
Examples
74
1. A loan of $ 94 is to be paid back in monthly installments in one year the first one starting after one month of the starting of the loan. If the interest is charged at the rate of 24% monthly on the unpaid principal, what will be the amount of the monthly installment?
1. Find the present value of an annuity of $ 600 payable at the end of each year for 15 years if the interest rate is 5% compounded annually.
Finding R when P is known
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Mr. Ahmad borrows $ 21,000 from a bank to build a house with the condition that he would pay back the loan in the semi-annual equal installments in four years with interest rate at 6% compounded semi-annually. If the first payment is to start at the end of first six monthly period, what would be amount of each installment?
Finding n when P is known
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Mr. Ashraf wants to deposit his savings of $ 50,000 in a bank which offers 8% interest compounded semi-annually so as to withdraw $2,500 at the end of each six months from the date of deposit. How many withdrawals will he or his heir (in case of his death) be able to make before the entire amount is exhausted?
Annuity Due
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When the periodic payments of an annuity starts at the beginning of an interval rather than at the end of interval the annuity is called an annuity due. Its term begins on the date of the first payment and ends on one interval after the last payment is made.
The annuity due has a payment at the beginning of each interest period but none at the end of the term. Therefore the formula for calculating an annuity due is given as:
1
1
( )
( 1)n i
n i
S due R s R
R s
1(1 ) 1( ) 1
niS due R
i
or
Present Value of an annuity due
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For the present value of an annuity due, we find out the present value of (n-1) periods of an ordinary annuity and then add the 1st payment which has the same present value. The formula for calculating the present value of an annuity due is given by:
1
1
( )
( 1)n i
n i
P due R a R
R a
11 (1 )( ) 1
niP due R
i
or
Examples
79
1. If $ 250 are deposited at the beginning of each quarter in a fund which earns interest at the rate of 8% compounded quarterly what will it amount to after the end of the year?
2. Mrs. Ahmad bought a sewing machine by paying $ 50 each month for 10 months, beginning from now. If money is worth 12% compounded monthly, what was the selling price of the machine on cash payment basis?
PERPETUITY
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An annuity whose payments starts on certain date and continues indefinitely is called perpetuity. As the payments continues for ever, it is impossible to compute the amount of the perpetuity but its present value can be determined easily.
The formula for calculating the present value of the perpetuity is given as:
where R is the size of periodic payment and i is the interest rate per period.
RP
i
Examples
81
1. Pakistan Manufacturing Co. is expecting to pay $ 4.80 every 6 months on the share of its stocks. What is the present value of a share if money is worth 8% compounded semi-annually?
2. Find the present value of Karachi Toy Company share which is expected to earn $ 5.60 quarterly, if money is worth 8% compounded quarterly.
82
Discussion on Selected Exercises fromPractice sets 11-A (p# 139), 11-B(p#157-159) &Problem set 11 (p# 160-162)
83
Chapter 16Matrix Algebra
Matrix
84
A matrix is a rectangular array of numbers enclosed in brackets or in bold face parenthesis. Matrices are represented by capital letters such as A, B, C, X, and Y etc.
Examples of matrices are:
1 4
8 0A
5 1
3 3
5 4
B
67
9C
Continued...
85
A matrix is described by 1st stating its number of rows and then its number of columns. This description of a matrix is known as order of a matrix.
In the above examples matrix A has the order 2 x 2 (2 by 2), B has the order 3 x 2 (3 by 2), and the order of C is 2 x 1 (2 by 1).
Generally if there are m rows in a matrix and n columns, the order of the matrix would be m x n and we may call it as m x n matrix.
If m = n, the matrix is called a square matrix.
General Form of m x n Matrix
86
Generally an m x n matrix may be in the form given below:
11 12 13 1
21 22 23 2
31 32 33 3
1 2 3
n
n
n
m m m mn
a a a a
a a a a
a a a a
a a a a
Operation with Matrices
87
(i) Addition / Subtraction Matrices of the same order can be added/Subtracted. While adding/subtracting two matrices of the same order we
add/subtract their corresponding elements.Given two matrices:
then
11 12 13
21 22 23
31 32 33
a a a
A a a a
a a a
11 12 13
21 22 23
31 32 33
b b b
B b b b
b b b
&
11 11 12 12 13 13
21 21 22 22 23 23
31 31 32 32 33 33
a b a b a b
A B a b a b a b
a b a b a b
Continued...
88
(ii) Multiplicationa) Multiplication of a matrix by a real number (Scalar)
When a matrix is multiplied by a real number, each element of the matrix is multiplied by that real number.
The product obtained is a matrix of the same order.
Example Let
then
1 6 9
0 1 2
1 5 3
D
1 6 9
3. 3. 0 1 2
1 5 3
3*1 3*6 3*9
3*0 3*( 1) 3*2
3*1 3*5 3*3
3 18 27
0 3 6
3 15 9
D
89
b) Multiplication of a matrix by another matrix Multiplication of two matrices is only possible if the number of
columns in the first matrix is equal to the number of rows in the second matrix.
If this condition is not satisfied multiplication will not be possible.
If the order of the first matrix is m x n and the order of the second matrix is n x p multiplication will be possible and the order of the resultant matrix will be m x p.
To obtain any element in the product matrix, 1st determine the row and column (in which the element lies) in the product matrix.
Multiply the row of the first matrix with that column of the second matrix, this value will give us that element.
Further the method is explained in the following examples.
Examples
90
If possible multiply the following matrices.
2 5 3
7 8 4
0 2 8
1 5 6 9 3 5
4 1 1
1 3 5 70 8 1
2 4 6 83 4 2
5 3 1 74 6 5
9 7 6 2
A B
C D
M N
with
with
with
Continued…
91
In the given matrices
perform the following operations.
(a)A + B (b) C - D
(c)A . B (d) A . E
(e)E . C (f) C . F
135
742
,
851
364
,
63
72
15
48
93
21
,71
65,
42
93
FED
CBA
Example
92
A bakery makes three types of bread using the ingredients listed in the given table in convenient units per loaf of bread.
these ingredients can be put in the matrix form as:
If an order is placed for 60 loaves of type I, 75 loaves of type II, and 50 loaves of type III which can be shown in the matrix form as
1.Find the number of units of each ingredient required by the bakery to fill the order.
2.If per unit costs to the bakery of the ingredients A, B, C, D, and E are given by the matrix given below, then find the cost for each type of bread
Ingredients Required
Type of bread
A B C D E
IIIIII
312
211
112
111
011
07.0$
05.0$
06.0$
08.0$
10.0$
507560
11212
11111
01123
DETERMINANT
93
A determinant is a rectangular arrangement of numbers in rows and columns in two vertical lines. It is written in a manner similar to its associated form of square matrix except that the bracket of the matrix is replaced by two vertical lines.
For Example: The determinant of the matrix
is represented as
and the determinant of the matrix
is
37
51A
37
51A
645
301
972
M
645
301
972
M
How to evaluate the determinant?
94
The value of a determinant can be evaluated by two methods
i. By cross multiplication
ii. By finding the minors
Evaluation of Determinant by Cross Multiplication
Consider a 2 x 2 square matrix
The determinant of A is given by:
2221
1211
aa
aaA
2221
1211
aa
aaA
p1s1Primary DiagonalSecondary Diagonal
Continue…
95
The determinant of a 3 x 3 matrix has 3 primary diagonals p1, p2, p3 and 3 secondary diagonals s1, s2, s3
The numerical value of a determinant is found out as:
a)Multiply the elements on each primary diagonal and add their products
b)Multiply the elements on each secondary diagonal and add their products
c)Subtract the results.
3231
2221
1211
333231
232221
131211
bb
bb
bb
bbb
bbb
bbb
B
s2s1 s3 p1 p2 p3
The 1st & 2nd columns are copied
Exercises
96
Find the value of the following determinants:
430
152
124
)(
210
306
038
)(
012
212
022
)(
987
654
322
)(
58
35)(
40
09)(
79
54)(
94
82)(
h
g
fe
dc
ba
Example
97
The National Center for Higher Education Management System uses matrices as model to study college management. The elements of an important matrix in this model, the induced course-load matrix, are the average numbers of units taken in each field by students classified according to majors.
5.15.15.15.15.15.1
5.10.35.15.15.15.1
3.05.30.00.00.04.0
2.10.00.54.11.02.0
0.20.19.10.68.16.1
5.20.26.26.18.15.2
2.35.25.15.15.46.3
8.25.10.15.18.35.3History
English
MathematicsBiology
Chemistry
Accounting
Economics
Physical Ed.
Histor
y
Englis
h
Biolo
gyChe
mist
ryBus
ines
sUnd
ecid
ed
Total 15 15 15 15 15 15
From Past
98
ALGEBRAIC EXPRESSIONEQUATION
(Statement which indicates that two algebraic expressions are equal)
SYSTEM OF LINEAR EQUATIONS
(We have discussed only the systems consisting of two equations and two variables and the elimination method for finding the solution of these systems).
In this chapter we will do the same but this time with a different method known as the Cramer’s Rule.
Cramer’s Rule
99
Given a system of linear equations of the form
A X = B
Where A is an (n x n) square matrix of coefficients, X is the column matrix of the variables, and B is the column matrix which contains the values from the right sides of the equations in the system.
Cramer’s Rule provides a method of solving the system by using determinants. To solve for the value of the jth variable, form the matrix Aj by replacing the jth column of A with column vector B.
If we denote the determinant of Aj by the value of the jth variable is determined as:
j
jjx
Continue…
100
If , the given system of equations has a unique solution.If , the computation of is undefined. If and
, the system has infinitely
many solutions.If , and any , then the system has no solution.Further explanation of the method is given in the following
example.
Examples:
Using Cramer’s Rule, determine the solution to the given system .
00 jx 0
021 n
0 0 j
8042
8023
21
21
xx
xx
Exercises
101
Determine the solution to the given systems using Cramer’s Rule. If the system has no solution, or infinitely many solutions, so state.
16963
1242
2432
.6
223
5423
1
.5
3435
484.4
1842
42.3
4753
845.2
064
1323.1
zyx
zyx
zyx
zyx
zyx
zyx
yx
yx
yx
yx
yx
yx
yx
yx
Exercises Continue…
102
91263
252
342
.9
30129
1043.8
5284
132.7
zyx
zyx
zyx
yx
yx
yx
yx
Problem:A store has three different mixes of nuts in sixteen kilogram bags marked A, B,and C. The contents of each type of bag are given in the following table
Bags Ground Nuts Almonds Apricots A 8Kg 5Kg 3Kg B 6Kg 4Kg 6Kg C 10Kg 2Kg 4KgAn order is received for a mixture of 132 Kg of ground nuts, 54 Kg of almonds and 70 Kg of apricots. How can the store fill this order.