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
r-------------------__ ~
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, April,
June, 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
tIme - t 360
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
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
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)
DEPRECIATION
PAYROLL
Pay Based on Hourly Wages: Pay is based on hours of work done
• gross pay = (number of regular hours x regular hourly rate) +
(number of overtime hours x overtime hourly rate)
Pay Based on Piece Work: Pay is based on acceptable pieces
of work produced
• gross pay = total number of acceptable pieces produced x
piece work rate
Pay Based on Commission: Commission is a percentage of sales;
usually paid to the person generating the sales
• gross pay = commission = sales x commission rate
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 Int erest: Interest that C
is paid on both the principal and the
interest accumulated from past
periods; interest gained is added to "
the original investment
• 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
1
2
compounding
period is
interest is calculated every
n
annual
quarter 4
IDGIIdl
day
day
365
Present & Future Value
..
• present value (PV) = value of the ,
loan or investment today
• future value (FV) or maturity value .
= final amount of the loan or
investment at the end of the last ~
period
II I
C
l
2
Simple Interest Future Value
• 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 =
(asset cost - residual value) estimated useful life of asset in years • partial-year depreciation expense =
depreciation expense per year x
number of months of useful life in the year
12
Units of Production Method: Depreciation
expense based on the usage of the asset
• depreciation rate per unit =
(asset cost - residual value)
estimated total number of units
produced over useful life of asset
~
C
1
III
II i
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)
estimated total number of hours of useful service over useful life of asset • 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)
remaining useful life in years sum of years-digits 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
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 are
sold first; goods in ending inventOlY are those that were purchased most recent~v
• cost o f 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 o f 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 + ending inventory at retail 2
cost of goods sold
.
• mventory turnover at cost = average .IOven t orya t cos t
• average inventory at cost =
beginning inventory at cost + ending inventory at cost 2
BASIC FINANCIAt REPORTS
Inco me St ateme nt (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
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
OR budgeted need of local government property tax rate =
total taxable assessed value • 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
Income St atement 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 o f 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
• 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
Series of periodic payments usually made in equal amounts; payments
computed by compound interest methods; payments made at equal intervals oftime
• profit margin = net income x 100
net sales
payment dates
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 Financi al St at ement Ratio s
• 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
• days inventory
365
lOventory turnover
= .
• collection period
=
acco~nts receivable
credIt sales per day
cost of goods sold
.
• mventory turnover = average lOven
.
t ory
• 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'---common stock outstanding • price to earnings (P/E) ratio =
price per share of common stock
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)
Period of time between two successive
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-IJ
i
- (one annuity payment amount)
• where i
annuity
=
interest rate per period; n
=
number of payments during term of
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
annuity
=
interest rate per period; n
=
number of payments during term of
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 be of equal amounts
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
Calculating Premiums: Using insurance tables; read tables according to
• sinking fund payment per period = futUre value x [(
age and gender of insured; insurance rates are generally per $1,000.00 of
coverage
• premium = (coverage amountll,OOO) x insurance rate
• where i
annuity
=
interest rate per period; n
=
i)1I
J
1+1 -I
number of payments during term of
A
Used mainly for consumcr loans; where m = number
of payments in one year, n = total number of scheduled payments in life of loan, C
= finance charges per payment period, P = principal or original loan amount
• monthly payment = [rate + (
r~~~n1h' -1 1X principal
1+ rate
Ratio of the jinance charge
to the average amount 0.( credit in use during the life ofthe loan;
expressed as a percentage rate per year; is a true cost of a loan;
meant to prevent lenders from advertising a low rate by hiding
fees; rules to compute APR are not clearly defined
• constant ratio method: APR =
• direct ratio method: APR
=
•
P~:~O
3P (n + ~H-CC (n + I ) ted by
a
.
mC(95n+9)
• n ratio method: APR = 12n(n+l)(4P+C)
'fS
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
•
PIE'
. I
.
.
ratIO = pnce earnmg ratIO =
closing common stock price
earnings per share
• yearly interest = face value of bond x yearly interest rate
. Id - yearly interest
• current Yle bond price
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
price per share ofthe mutualfund
. Id d"d d . Id yearly dividends per share
.
• stoc k Yle = IVI en Yle =
common s t oc k prIce
• NAV
- ending price+total dividend income received _ 1
• tota I return b egmnmg
. . prIce
.
•
Bonds: Promises of payment for monies loaned
• bondholders are creditors
• tIt
_ ending NAV +total distribution per NAV _ I to a re urn initial NAV =
8
614
~~"f,{Jj'!N 1 ~ ":\l'4~t!!t!~
(, (. I
~ '!H....J"i.
757
H'w,'
210
r"
943.
1\'1(
3
6
I\',f,j, I(.~ :: ~~(J!,(i!tt
t "fi,?l
,m, !" ~'~
~~ 'IJ' )i~"'IC ;"':-1-' ';1; .t~'J -," ',' :;" N;l
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.
-J"
numerator (number written above the line)
• denominator (number written below the line)
tori
~ori
5/ 2
is 5 divided by 2, which is 2 with a remainder of 1, resulting in a mixed number 21/2 • 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 =
(denominator of fraction part X whole number part)
+ numerator of fraction part denominator of fraction part (6X3)+5 23
6
6
Fraction Operations • reduction: converting the fraction to higher or lower terms by multiplying
or dividing the numerator and denominator by the same number (any
number other than zero); value ofthe fraction does not change
• EX:
eq ual
3~
the
d has
It
by S2
• lowest terms: when the numerator and denominator of a fraction do
not have a common divisor; also called simplest form
• EX:
• complex fraction: either the numerator, the denominator, or both are a
fraction
Ys
'%
• EX'
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
ur the
• then
• EX' 21- 2177_1
. 28 - 2877 - 4
• improper fraction: numerator is greater than or equal to the denominator
• EX:
1. 6.7,
• EX' 1- 3x4_!1.
. 4-4X4 -16
Types of Fractions
• proper fraction: numerator is less than the denominator
• EX:
total value of portfolio number of shares outstanding in the fund 6
If d . Id _ income distribution per share
mutua un Yle NA V
• EX:
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
are
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
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
number in the fraction by the remainder in step I (divide 63 by 42, quotient I, remainder 21) • 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
remainder until the remainder is zero
-9
lilill
Review Skills Basics
• step 5: the last divisor (the last non-zero remainder) is the GCD or
HCF (which is 2 1)
• 63 = 63721 ~21-l
294
294' -14
• LCD or LCM: lowest common denominator (LCD) of a group of
fractions is the least common multiple (LCM) of the denominators of
those fractions
• EX: Calculate the L CD or L CM of 8, 24 and 45; when there is no
common factor in a group of numbers, then the LCM is the product
of the numbers; LCM is 8 X 24 X 45 = 8,640
• EX: Calculate the LCD or LCM of 6, 15, 42; when there are
common factors in a group of numbers, then the numbers are
repeatedly divided by their common prime factor; at least two
numbers should be divided in each step; the LCM is the product of
the prime numbers and the final quotients:
2) 6, 15,42
3) 3,5,21
1,5,7
LCM = 2 X 3 X 1 X 5 X 7 = 210
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
• EX: 3/4 + 2/3 + 6/7 3/4 = 3
X
21/4 X 21 = 63/84 , 2/3 = 2
=6
X
12/7 X 12 =
617
X
191/
84
= 223/
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
• EX: 4/5 - 1/7 = 28/ 35 - 5135 = 23/35 (LeM of 5 and 7 is 35) 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:
d
aXe bxd • EX.. 3/7 X 2/9 = 7x9
2x3 = 6/63 = 2/21
Dividing Fractions
• dividend -i- divisor = dividend
•
~
b
-i-
~= ~
d
b
• EX: 3/7 -i-
X
2/9
X
reciprocal of the divisor
~ = a Xd
e
bXe
= 3/7 X
Rule From
To
Example
Fractions
Decimals Div ide and round as needed
Move decimal poiat two P DecimaIa Percents
Percents
to
the riabt aDd add za'OI if ~
tMa Idd perceat symbol ( Move decimal point two pl aces to Decimals the left and add zeros when needed, 64.48% is 0.6448 then delete percent symbol (% ) 28/3 X 28 = 56/84 , (LCM of 4,3, 7 = 84)
b
Conversions
72/84
3/ + 2/ + 6/ = 63/ + 56/ + 72/ = 63+56+72 =
4
3
7
84
84
84
84
~x~ =
• division by multiples of 10: move the decimal to the left by the same
number of spaces as the number of zeros
• EX: 0.27 -i- 1,000 = 0.00027
• multiplication by multiples of 10: move the decimal to the right by the
same number of spaces as the number of zeros; add zeros if there are no
digits to the right
• EX: 0.27 X 1,000 = 270
Percent: To convert any whole number or decimal number to a
percentage, move the decimal point two places to the right (adding zeros
if necessary) and add a percentage symbol (% ) at the end of the num ber
• rounding percents follow s the same rules as rounding decimals [see
Decimals]
• EX: 2 is 200%, 0.15 is 15%,0.2846 is 28.46%
9/ 2 =
27/14 = 113/ 14
Decimals
Format: 0.2368
• rounding: to round 0.2368 to two places, first identify the digit at the
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
place of rounding is increased by I- if not, it remains the same (because
6 is greater than 5, the digit 3 is increased by I); the decimal 0.2368 is
rounded to 0.24; similarly, rounding 0.1239 to two places is 0.12
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);
the minus (- ) sign is used to represent negative numbers (less 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
Expressions & Terms: Any symbol or combination of symbols that
represents a number is called an algebraic expression ; when an expression
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
one term, a binomial has two terms, and a polynomial 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
• EX: Rxy is a term, 8 is the (numerical) coefficient of xy, y is the (literal)
coefficient of Rx
Power: Product of equal factors is called a power of that factor
• EX: 2 X 2 X 2 = third power of 2 = 23
• EX: a X a X a X a x a = fifth power of a = as
Basic Algebraic Rules: Consider the numbers a, b, c, d
- (- a) =
+a
(-a) (-b) = + ab
(-a) (+b) = - ab
a+b=b+a
a + (b + c) = (a + b) + c
axb=bxa
a x (b x c) = (a x b) x c
jf a = band c = b, then a
If a = band c = d, then a
If a = band c = d, then a
If a = band c = d, then a
=c
+c = b +d
- c=b- d
x c=bx d
Ifa = band c = d, then ale =
bId'
when c is not equal to zero
Review 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
Basic Statistics
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
• EX: (+5) + (-4) + (+3) + (-2) = (+8) + (-6) = +2
Measures of Central Tendency are mean, median and mode
• mean (arithmetic mean or average)
• mean of a set of numbers = sum of the numbers/number of items
• mean is very sensitive to extreme values among the set of numbers; it is usually represented by
the lowercase Greek letter mu (/1) for a set of numbers (population) and x-bar (x) for a
sample (subset of those numbers)
• EX: Mean of 6,4,2,7,9 = (6 + 4 + 2 + 7 + 9) + 5 = 28/5 = 5.6
Adding & Subtracting Algebraic Expressions: Terms
in an expression with the same factors are called like 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
• median
• median of a set of numbers is the central or middle number in the set when the numbers are
arranged according to their magnitude or size
• EX: Median value of 6,4,2,7,9 is the middle value among 2, 4, 6, 7, 9; so, median = 6
• 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,
9, which is the average of 4, 6; so, median = 5
Multiplying Algebraic Expressions
• monomial x monomial = product of the numerical
coefficients x product of literal factors
• mode
• mode of a set of numbers is the number that occurs most frequently in the set
• EX: Mode of 6,4,2,7,7,6,7,4,7,9 is 7 as it occurs the most times
• if two numbers occur the most number of times, the set is bimodal; if many numbers occur the
most number of times, the set of numbers is multimodal; if all numbers appear only once, then
there is no mode
• EX: 6ab x 8c = 48 abc
• polynomial x monomial = each term of the
polynomial x the monomial, then add the resulting
partial products
• EX: (5a + 8b) x 2c = lOac + 16bc
• polynomial x polynomial = each term of one
polynomial x each term of the other polynomial; then
add each of the partial products
• EX: (6a + 4b) x (2c + 5d) = 12ac + 8bc + 30ad + 20bd
Dividing Algebraic Expressions
• monomial + monomial = quotient of numerical
coefficients x quotient of literal coefficients
• EX: 36ac + 6c =
e6/6) x
(ac/c)
= 6a
• polynomial + monomial = each term of the
polynomial + the monomial, then add the partial
quotients
• EX: (24ab + 6ac + 42bc) + (6abc)
+ (6ac/6abc) + (42bc/6abc)
+ I/b + 7/a
= (24ab/6abc)
= 4/c
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
• EX: Range of 2, 6, 4, 8,3,9, 7 is (9 - 2) = 7
• percentile for a value n is found by dividing the number of items less than n by the total number
of items, and then multiplying this by 100
• quartile
• QI is the first quartile (25 th percentile) when 1/4 of the items are below the value QI
• Q2 is the second quartile (50 th percentile) when 1/2 of the items are below the value Q2; Q2
corresponds to the median
• Q3 is the third quartile (or 75 th percentile) when 3/4 of the items are below the value Q3
• EX: If 45 out of 50 students in a class have scores less than 85 on an exam, then a student with a score of 85 is in the 90 th percentile 1(45/50) x 100 = 90% I • 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
Q2 = (5+7)/2 = 6, value under which there are 1/2 of the items
Q3 = (8+9)/2 = 8.5, value under which there are 3/4 of the items
• variance & standard deviation of a set of numbers measures the ,\pread of" the data ahout the
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
• standard deviation is usually represented by the lowercase Greek letter sigma (0'), and variance by S2
• for a set of n numbers (population):
variance
= S2=-k'i(Xi-jJ.)2; standard deviation = ( I =
;=1
i=1
n
• EX: Calculate the standard deviation of 3, 6, 15, 19, 27
n=5,jJ.= 7%=14,(I=j3~O =8.72
• Frequency Distribution: A set of numbers or data arranged in ascending order is called an array;
the number of times a specific number is repeated in a data set or array is called its frequency;
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
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Author: Ravi Behara, PhD,
NOTE: This QuickStudy" guide is intended for infonnational purposes only,
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