I i BASIC FORMULAE & DEFINITIONS FOR AN INTRODUCTORY COURSE IN BUSINESS MATHEMATICS INTEREST Amount required to cover operating expenses and make a profit • cost of goods is what the r
Trang 1
I
i
BASIC FORMULAE & DEFINITIONS FOR AN INTRODUCTORY COURSE IN BUSINESS MATHEMATICS
INTEREST
Amount required to cover operating expenses and make a
profit
• cost of goods is what the retailer pays the supplier or manufacturer
for goods
• selling price of goods is what the customer pays the retailer for
goods (i.e., the buyer's purchasing price)
• cost + markup = selling price
• markup (or margin) = selling price - cost
• percent markup on cost = markup + cost
• percent markup on cost = percent markup on selling price +
(I - percent markup on selling price)
• percent markup on selling price = markup + selling price
• percent markup on selling price = percent markup on cost +
(1 + percent markup on cost)
• markdown = original selling price - marked-down price
• markdown percent = markdown + original selling price
• markdown = markdown percent X original selling price
• selling price per unit = [total cost of all units - percent markup
on cost (total cost of all units)] + [total units - total spoiled
units]
DISCOUNTS
Discounts: Reduction to a basic price
Trade Discounts: Discounts given to partners in the distribution
channel of goods; also called functional discounts, and are given to
distribution channel members to perform specific tasks
• net price = list price - trade discount amount
Cash Discount: Discounts given to customers for paying with cash;
cash discount is also called sales discount by seller or purchase
discount by buyer
• cash discount = selling price (or invoice amount) x cash
discount rate
• amount buyer pays = selling price - cash discount
Terms: Cash discount rate is usually stated in credit terms; credit
term 2/10 on an invoice means that a 2% cash discount is allowed if
the payment is made within 10 days of the date of the invoice; cash
discount period can a/so begin when the buyer receives the goods
(ROG = receipt of goods)
PAYROLL
• gross pay = (number of regular hours x regular hourly rate) +
(number of overtime hours x overtime hourly rate)
of work produced
• gross pay = total number of acceptable pieces produced x
piece work rate
usually paid to the person generating the sales
• gross pay = commission = sales x commission rate
Exact Number of Days: Interest is calculated over a time period, from when a loan is given to the end date; day the loan is due mayor may not be included in the period
• To determine loan period, the actual number
of days ill each month should be known Tip: 30 days hath September, Ap r il, Jun e, and November; all the rest
have 31, excepting February alone, which hath but 28, in fine, till leap year gives it 29
• Leap years are divisible by 4, which is why 2004 was a leap year and also why an extra day is added to February in every leap year
Simple Interest: Time is expressed in the
same units as the rate; if r is an annual rate, then t is in years
• simple interest (i) = principal (p) x rate
in percent (r) x time (t)
• maturity value = principal + interest
• p is principal of the loan (or borrowed) amount or invested amount (face value);
i is the amount of interest paid for the loan or earned on the investment; r is the percent rate of interest paid for the use
of someone else's money or earned for lending the money
• exact interest: time = t =
exact number of days
365
• ordinary interest (Banker's Rule):
_ _ exact number of days
Partial Payment Rule: Any partial loan payment is first used to pay the interest that has accrued (total interest to date), and the remainder is used to reduce the principal of the loan
DEPRECIATION
Depreciation: Loss in value of tangible business assets or property (excluding land) over its useful life due to deterioration, obsolescence, etc.; also called depreciation expense; periodically charged to operating expenses; total depreciation limited to cost of property
• accumulated depreciation = total amount of depreciation
to date
• cost of asset = cost paid for asset, including freight
• book value of asset = cost of asset - accumulated depreciation
1
Compound In erest: Interest that C
is paid on both the principal and the 1
interest accumulated from past periods; interest gained is added to "
• maturity value = p x (I + i)n
• wherc p = principal; i = interest rate in ,
percent per period; n = number of periods
• alternative formula: maturity value (future value) = p x (I + rln)nt
• where t = time in years; II = number of
periods per year; r = interest rate in percent per year
compounding period is calculated every annual
IDGIIdl
Present & Future Value
• present value (PV) = value of the ,
• future value (FV) or maturity value
= final amount of the loan or l investment at the end of the last ~
Simple Interest Future Value 2
• FV = PV x (I + (i x n)]
• where i = interest rate per period; n = , number of periods
Compound Interest Future Value
• FV = PV x ( I + i )"
• where i = interest rate in percent per
period; n = number of periods Interest Earned = FV - PV
• residual value (or salvage value, scrap value) = cash value of asset at end of useful life Straight line Method: Depreciation expense is equal over eaeh year of its useful life
• depreciation expense per year =
• partial-year depreciation expense = depreciation expense per year x ~ number of months of useful life in the year C
Units of Production Method: Depreciation III expense based on the usage of the asset II
(asset cost - residual value)
estimated total number of units ,
produced over useful life of asset
Trang 2• depreciation expense per year = depreciation rate per unit X
number of units produced per year
Service Hours Method: Depreciation expense based on hours of
useful service
• depreciation rate per service hour =
(asset cost - residual value)
• depreciation expense per year = depreciation rate per service hour
x number of service hours per year
Sum of Years-Digits Method: Depreciation expense is greater for
earlier years than for later years
• sum of years-digits = sum of the digits representing year of useful life
OR
sum of years-digits = N (~ +1), where N is the number of years of
useful life
• for an asset with six years of useful life, sum of years-digits = 1 + 2 +
3 + 4 + 5 + 6 = 21, or 6(;+1) =4{=21
• depreciation expense per year = (asset cost - residual value) x
Declining Balance Method: Depreciation expense declines steadily
over the useful life of the asset
• depreciation rate for double declining balance method = (100% 7
estimated number of years of useful life of the asset) x 2
• depreciation expense per year = book value of asset at the
beginning of the year x depreciation rate
• book value of asset at the beginning of a year = book value of asset
at the end of the previous year
• book va~ue at the end of a year = asset cost x (1 - depreciation
rate)n; n = estimated number of years of useful life of the asset
Sales Tax: Tax paid on purchase of most goods and services, though
some are exempt from sales tax; it is applied to the net price (selling
price - trade discounts) but not to shipping charges; sales tax varies
between states; collected by the business and paid to the state
government
• sales tax = net price x sales tax rate
• purchase price = net price (1 + sales tax rate)
• actual sales = total sales
1+sales tax rate
Excise Tax: Tax paid on specific goods and services, such as luxury
automobiles, gasoline and air travel
• excise tax = net price x excise tax rate
Property Tax: Levied on the assessed value of property by local
government to pay for services such as schools, fire and police services;
assessed value is a fraction of actual market value of the property that is
used for tax purposes
• property tax rate = estimated revenue from tax
total taxable assessed value
property tax rate =
• assessed value = market value x assessment rate
• property tax = assessed value x property tax rate
• mill rate: a mill is 1/1000 of a dollar or 0.001 dollar; tax rate in mills
is the tax per $1,000.00 of assessed value
Cost of Goods Sold
• cost of goods sold = cost of goods available for sale - cost of ending inventory Weighted Average Method: Used to calculate cost of ending inventory when the goods available for sale were purchased at different costs at different points in time cost of goods available for sale
• weIghted average cost per umt = number of units available for sale
• cost of ending inventory = units in ending inventory x weighted average cost per unit
First In-First Out (FIFO) Method: Used to calculate cost of ending inventory when the goods available for sale were purchased at different costs and at different points in time; assumption is that goods purchased earliest into inventory are the ones that ar e
sold first; goods in ending inventOlY are those that were purchased most recent~v
• cost of ending inventory = units in ending inventory X their corresponding costs
Last In-First Out (LIFO) Method: Used to calculate cost of ending inventory when the goods available for sale were purchased at different costs and at different points in time; assumption is that goods purchased most recently into inventory are the ones that are sold first; goods in ending inventory are those that were purchased the earliest
• cost of ending inventory = units in ending inventory x their corresponding costs
Inventory Turnover: How often a business sells and replaces its inventory; usually over a year
• inventory turnover at retail = net sales
average IOventory at retaIl
• average inventory at retail = beginning inventory at retail +
• mventory turnover at cost = average IOven orya cos t t t
• average inventory at cost = beginning inventory at cost +
BASIC FINANCIAt REPORTS
In o me Statement (Profit and Loss [P & L! Statement): A financial report of
a business that shows net profit or loss for a specific period by reporting revenue and expense items during that period of operations
Income Statement Items
• revenue from sales (or revenues, sales, income, turnover)
• sales = number of items x Oist price - trade discount)
• net sales = sales - sales discount or cash discount
• cost of goods sold COGS (or cost of sales) is the amount a product cost to produce
• COGS = net purchase price + cost of acquiring, preparing and placement of goods for sale
• gross profit on sales (or gross profit) = net sales - cost of goods sold
gross profit
• gross margm percent = net sales x 100
• operating expenses (including general and administrative expenses IG & AI) =
expenses to manage the business, and include salaries, legal and professional fees, utilities, insurance, stationery supplies, property and payroll taxes
• sales and marketing expenses = expenses needed to sell products, and include sales, salaries and commissions, advertising, freight and shipping
• R&D expenses = expenses incurred in research and development
• operating expense = G & A expense + sales & marketing expense + R&D expense
• earnings before interest, taxes, depreciation and amortization (EBITDA)
OR operating income = gross profit - operating expense
• operating margins percent = ;!IJa?:S x 100
• earnings before interest and taxes (EBIT) = EBITDA - depreciation and amortization expenses
2
Trang 3• earning before taxes (EBT) or pretax net income = EBIT - interest
expenses
• taxes include federal, state and local government taxes on income
• net income (or earnings) = EBT - taxes
• profit margin = net income x 100
net sales Balance Sheet
• assets - liabilities = owner's equity or shareholder's equity
• assets are items on a company's books that have a positive monetary
value; they typically include items of obvious value, such as cash or
equivalent investments (treasuries, CDs, money market), accounts
receivable, prepaid expenses, inventory of finished goods that are ready
for sale, depreciated real estate and equipment, and other intangibles,
such as goodwill, copyrights, trademarks and patents
• liabilities are monies owed; they typically include accounts payable, bank
and bond short-term debt (to be paid off within a year), and long-term debt
Basic Financial Statement Ratios
• liquidity ratios: measures of ability jilr a business to meet short-term
obligations
• current ratio = current assets -;- current liabilities
• quick ratio = cash + accounts receivable -;- current liabilities
• activity ratios: measures ofefficiency in generating sales with assets
lOventory turnover
• collection period = acco~nts receivable
credIt sales per day
• mventory turnover = average lOven ory t
• asset turnover = net sales
total assets
• profitabili!y ratios: measures ofreturns
.return on sales = net income
net sales .return on assets (ROA) = net income
total assets
• return on equity (ROE) = net in~ome
eqUIty earnings available to common stockholders
• earnings per share = -
=-n~u=-=-m~b-"'e':":r =o"-f:;;=sh;="'a-"'re-"s~o"'f';C' price per share of common stock
• price to earnings (P/E) ratio = earnings per share
LIFE INSURANCE
Life Insurance: Insurance that pays a specified sum to the policyholder's
beneficiary at the time of the policyholder's death
• insured: person covered by policy
• policyholder/policy owner: person who owns policy
• premium: periodic payments made for insurance coverage
• face amount: proceeds received on the death o(the insured
• beneficiary(ies): person(s) who receivers) the face amount
Types of Life Insurance
• term life is life insurance coverage for a specified period oftime; can be
at a guaranteed rate or a guaranteed rate for a period of time and then a
projected rate; no cash value except face amount in event of death of
insured within the period of the insurance
• whole life is life insurance that has a guaranteed level premium (i.e., no
increases in premium) and a guaranteed cash value; also called
straight life or ordinary life
• universal life is life insurance that is permanent; premiums are not
guaranteed (i.e., may go up or down)
Calculating Premiums: Using insurance tables; read tables according to
age and gender of insured; insurance rates are generally per $1,000.00 of
coverage
• premium = (coverage amountll,OOO) x insurance rate
Series of periodic payments usually made in equal amounts; payments
Period of time between two successive payment dates
Time between the beginning ofthe first payment period and the end of the last payment period
Future dollar amount
of a series of annuity payments and the accrued interest
• annuity certain: term of annuity begins and ends on definite dates; has a specified number of payments
• contingent annuity: term of annuity begins on a definite date, but ending date
is dependent on a future or uncertain event; no fixed number of payments
• perpetual annuity: term of annuity begins on a definite date, but has no ending date; length of term is infinite
D
• ordinary annuity: periodic payments are made at the end of each payment period
• deferred annuity: periodic payments are made at the end of each payment period, but the term of the annuity begins after a specified period of time
• annuity due: periodic payments are made at the beginning of each payment period
Je of an Ordinary Annuity
• using annuity tables
• future value of ordinary annuity = annuity payment per period x ordinary annuity table factor
• using formula
• future value of ordinary annuity = annuity payment amount per period x
[<1+ij"-1 J
• where i = interest rate per period; n = number of payments during term of annuity
Je of an Annuitv Due
• using annuity tables
• add 1 to the number of periods, and then read the table
• future value of annuity due = (annuity payment per period x ordinary annuity table factor) - (one annuity payment amount)
• using formula
• future value of annuity due = annuity payment amount per period x
[O+i)O+I-IJi - (one annuity payment amount)
• where i = interest rate per period; n = number of payments during term of annuity
Ordinary Annuity
• using annuity tables
• present value of ordinary annuity = annuity payment per period x present value of ordinary annuity table factor
• using formula
• present value of ordinary annuity = annuity payment amount per period x
[t-(IiO-oJ
• where i = interest rate per period; n = number of payments during term of annuity
Fund into which periodic deposits are made so that the principal
is repaid on the maturity date (i.e., the amount of the annuity is the value of the principal of the debt on the maturity date); deposits need /lot b e of e qual amount s
or made at equal intervals oftime ; interest for the debt is not paid from the fund
• using sinking fund tables
• sinking fund payment per period = future value x sinking fund table factor
• using formula
• sinking fund payment per period = futUre value x [( i)1I J
• where i = interest rate per period; n = number of payments during term of annuity
Trang 46
of payments in one year, n = total number of scheduled payments in life of loan, C
• monthly payment = [rate + ( r~~~n1h' 1X principal
1+ rate -1 = finance charges per payment period, P = principal or original loan amount
Ratio of the jinance charge • constant ratio method: APR = P~:~O
to the average amount 0.( credit in use during the life ofthe loan;
ted by expressed as a percentage rate per year; is a true cost of a loan; • direct ratio method: APR = 3P (n + ~H-CC(n
meant to prevent lenders from advertising a low rate by hiding
fees; rules to compute APR are not clearly defined • n ratio method: APR = 12n(n+l)(4P+C)
Stocks: Shares of ownership in a company
• common stock gives the holder voting rights
• preferred stock does not allow the holder to have voting rights, but
instead, otTers preference in dividend payments
• dividends are payments to shareholders from projits
h earnings available to shareholders
• earnmgs per s are = total number of shares outstanding
• ratIO = pnce earnmg ratIO = earnings per share
Id d"d d Id yearly dividends per share
k
• stoc Yle = IVI en Yle = common s oc prIce t k
- ending price+total dividend income received _ 1
I
• tota return - b egmnmg prIce
Bonds: Promises of payment for monies loaned
• bondholders are creditors
Whole Numbers: Set of all positive integers (1, 2, 3, ), zero (0), and
negative integers (-I, -2, -3, ); integers are whole numbers
• numeric representation: $8,614,757,210,943.36 was the U.S National
Debt on 12/27/06; National Debt is the amount of money that the U.S
Treasury Department has borrowed to date in order to meet Congress's
expenditures beyond its income
• in words: eight trillion six hundred and fourteen billion, seven hundred
and fifty-seven million, two hundred and ten thousand, nine hundred and
forty-three dollars and thirty-six cents
8 614 757 210 9 4 3 3 6
~~"f,{Jj'!N 1 ~ ":\l'4~t!!t!~ H'w,' I\',f,j, I(.~ :: ~~(J!,(i!tt -J" t "fi,?l r",m, !" ~'~1\'1(~~ 'IJ' )i~"'IC ;"':-1-' ';1;.t~'J -," ',' :;" N;l
(, ( I ~ '!H J"i i1(J#IJ."',"A.m'~" i~~~k.W.~'-icf"'1!· ';"-'i."r"GiJ",,'i'1.YlJ!lII"~if.B~JitJuS {.l.'Jti'-G1l;',::iJ!tii:mt
numerator (number written above the line)
• denominator (number written below the line)
Types of Fractions
• proper fraction: numerator is less than the denominator
• EX: tori
• improper fraction: numerator is greater than or equal to the denominator
• EX: ~ori
• complex fraction: either the numerator, the denominator, or both are a
fraction
• EX' Ys
'%
Mixed Number: Consists of a whole number and a fraction
• the sum of the two numbers (whole number + fraction)
• EX: A sum of 5 and 3/4 is written as the mixed number 53/4
Converting Fractions: Improper fractions may be turned into whole
numbers or mixed numbers
• divide the numerator by the denominator; if there is a remainder, then the
result is a mixed fraction; if the remainder is zero, then the result is a
whole number
'fS are
• yearly interest = face value of bond x yearly interest rate
Id - yearly interest
• current Yle - bond price
1 6.7,
b d Id total yearly interest
• on Yle = bond price Mutual Funds: Monies invested in multiple entities (shares = ownership, similar to stocks)
• net asset value (NAV) is dollar cost of one share of the mutual fund or
ur the
price per share ofthe mutualfund
• hen
• NAV =
• If d Id _ income distribution per share
• t I t
to a re urn
-• EX: 5/ 2 is 5 resulting in a mixed number
• EX: 313 is 3 divided by 3, which is 1 with a remainder of 0, resulting in a whole number 1
Mixed Number to Improper Fraction
• improper fraction =
eq ual (denominator of fraction part X whole number part)
(6X3)+5 23
• EX: 3~ 6 6
• reduction: converting the fraction to higher or lower terms by multiplying d has
or dividing the numerator and denominator by the same number (any number other than zero); value ofthe fraction does not change by S2
• EX' 1-3x4_!1
4-4X4 -16
• EX' 21-2177_1 28 - 2877 - 4
• lowest terms: when the numerator and denominator of a fraction do not have a common divisor; also called simplest form
• EX: i or 1~
• GCD or HCF: divide the numerator and denominator of a fraction by their greatest common divisor (GCD) to reduce it to its lowest terms;
GCD is also called the highest common factor (HCF)
• EX: GCD or HCF of 63 and 294 is 21
• to calculate:
• step 1: divide the larger number in the fraction by the smaller number (divide 294 by 63, quotient 4, remainder 42)
• step 2: if there is a remainder in step I, then divide the smaller
• step 3: ifthere is a remainder in step 2, then divide the remainder in step
I by the remainder in step 2 (divide 42 by 21, quotient 2, remainder 0)
• step 4: continue dividing each remainder by its succeeding
-9
Trang 5Review Skills Basics
• 63 = 63721 ~21-l
• LCD or LCM: lowest common denominator (LCD) of a group of
those fractions
of th e numb er s; LCM is 8 X 24 X 45 = 8,640
• EX: Calculate the LCD or LCM of 6, 15, 42; when there are
th e pri m e n umbers and the final quotie n ts:
2) 6, 15,42
3) 3,5,21
1,5,7
Adding Fractions: The denominator of the sum is the least common
mUltiple (LCM) of the individual denominators, and the numerator of
the sum is the sum of the individual numerators
• add the integers and the fractions separately when adding mixed
numbers
(LCM of 4,3, 7 = 84)
Subtracting Fractions: The denominator of the difference is the least
common mUltiple (LCM) of the individual denominators, and the
numerator of the difference is the difference of the individual numerators
• subtract the integers and the fractions separately when subtracting
mixed numbers
• convert the mixed number into an improper fraction before subtracting
when the fractional part of the number you are subtracting is larger
than the fractional part of the number you are subtracting from
Multiplying Fractions: The numerator of the product is the product of
the individual numerators, and the denominator of the product is the
product of the individual denominators
• convert mixed numbers into improper fractions and then
multiply:
~x~ =
b d
Dividing Fractions
• ~ -i- ~ = ~ X ~ = aXd
Decimals
Format: 0.2368
place you want to round (here, it is 3), then identify the next digit to the
right (here, it is 6); if this digit is greater than or equal to 5, the digit at the
• division by multiples of 10: move the decimal to the left by the same
• multiplication by multiples of 10: move the decimal to the right by the
digits to the right
Decimals]
• EX: 2 is 200%, 0.15 is 15%,0.2846 is 28.46%
Conversions
From To Rule Example
Fractions Decimals Div ide and round as needed
to
~
Move decimal poiat two
DecimaIa Percents the riabt aDd add za'OI
tMa Idd perceat
Move decimal point two pl Percents Decimals the left and add zeros when needed, 64.48% is 0
then delete percent symbol ( %
Basic Algebra Basic Terms: While arithmetic operations use numbers and fractions based
on the 10 Arabic numerals 0 through 9, algebra uses letters, symbols, numerals and equations
Signs: Plus (+) sign is used to represent positive numbers (greater than zero);
Absolute Value: Value of any number, disregarding its sign
• absolute value is denoted by the sign II
• EX: 1+51 = I-51 = 5
has many parts, the parts are connected by + and - signs, and each such part, together with its sign, is called a term; a monomial is an expression with
more than one term Factors & Coefficients: When two or more numbers are mUltiplied, each
of the numbers or their product is called a factor of the resulting term
• any individual factor in a term is the coefficient of the remaining factors
of that term
• EX: 5x is a term, 5 is the (numerical) coefficient of x
coefficient of Rx Power: Product of equal factors is called a power of that factor
Basic Algebraic Rules: Consider the numbers a, b, c, d
- (- a) = + a
(-a) (+b) = - ab a+b=b+a
a + (b + c) = (a + b) + c axb=bxa
jf a = band c = b, then a = c
If a = band c = d, then a - c = b - d
Trang 6Review Skills Basics
Adding Numbers with Same Sign: Add the absolute
values of the numbers to get the sum and then prefix the
common sign
• EX: (+5) + (+6) = +11; (-3) + (-5) =-8
Adding Numbers with Opposite Signs: Add the
absolute values of the numbers with like signs, then
subtract smaller absolute value from the larger absolute
value, and prefix the sign of the larger value
Adding & Subtracting Algebraic Expressions: Terms
adding or subtracting polynomials is done by adding or
subtracting the numerical coefficients of like terms
• EX: (2a + 5b - 6) + (3a - 2b + 8) - (a + 2b - 4)
= (2a + 3a - a) + (5b - 2b - 2b) + (-6 + 8 + 4)
= 4a + b + 6
Multiplying Algebraic Expressions
coefficients x product of literal factors
partial products
add each of the partial products
Dividing Algebraic Expressions
coefficients x quotient of literal coefficients
quotients
= (24ab/6abc) + (6 a c/6abc) + (42bc/6abc)
= 4/c + I/b + 7/a
Author: Ravi Behara, PhD,
without written permission from the publisher
Basic Statistics
Measures of Central Tendency are mean, median and mode
• mean (arithmetic mean or average)
• mean is very sensitive to extreme values among the set of numbers; it is usually represented by
sample (subset of those numbers)
• median
arranged according to their magnitude or size
• for an even number of items, the median is the average of the two middle numbers
• EX: Median value of 6,4,3, 2, 7, 9 is the average of the two middle numbers of 2,3,4,6,7,
• mode
• EX: Mode of 6,4,2,7,7,6,7,4,7,9 is 7 as it occurs the most times
there is no mode
Measures of Dispersion are range, percentile, quartile, variance and standard deviation
• range of a set of numbers is the difference between the highest and lowest values
of items, and then multiplying this by 100
• quartile
• QI is the first quartile (25th percentile) when 1/4 of the items are below the value QI
• Q2 is the second quartile (50th percentile) when 1/2 of the items are below the value Q2; Q2
corresponds to the median
• Q3 is the third quartile (or 75th percentile) when 3/4 of the items are below the value Q3
• EX: What are the quartiles for the set of numbers 2,4,7,3,5,8,9, 10?
Arranging the numbers in ascending order: 2, 3, 4, 5, 7, 8, 9, 10; arranging them into four equal parts: 2, 3; 4, 5; 7, 8; 9,10
QI = (3+4)/2 = 3.5, value under which there are 1/4 of the items
Q3 = (8+9)/2 = 8.5, value under which there are 3/4 of the items
mean of those numbers; the standard deviation is equal to the square root of the variance, and has
the same units as the original numbers
n
;=1
• EX: Calculate the standard deviation of 3, 6, 15, 19, 27
when a set of numbers are grouped into several groups, the groups are called classes, the size of the class is called the class interval; the number of items in each class is called its frequency, and the grouped data is called a frequency distribution
NOTE: This QuickStudy " guide is intended for infonnational purposes only, Due to its cond e nse d fonnat, this guide cannot cover every aspect of the subject;
mther, it is intended for u se in conjunction with course work and assigned texts
Neither BarCharts , Inc., its writers , editors nor design staff, are in any way 11111=111 free n n re dfwn~adS o ! titles at &
qUlc 5 uuy.com
respon sible or liable for the use or misuse of the infonnation contained in this guide
All rights reserved No part of this publication may be reproduced or transmitted in any form, or by any means , electronic or mechanical, including photocopy recording , or any information storage and retrieval system ,