Business Mathematics

AKLAN STATE UNIVERSITY
COLLEGE OF INDUSTRIAL TECHNOLOGY
TEACHER EDUCATION DEPARTMENT
Andagao, Kalibo, Aklan
Math- BUSINESS MATHEMATICS
In Partial Fulfillment
for the Requirements in the Course
BUSINESS MATHEMATICS (Math)
Content Speaker/s:
“GROUP IV” Julie Ann T. Poblador
Nancy D. Francisco Melissa P. Roldan
Dana Mae T. Patricio Shanaine C. Lago
Vea D. Leocario Cris L. Garcia
BSED III-A
OCTOBER 2017

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CHAPTER 1
Fundamental Arithmetic
Operations
Rounding off Numbers
Fundamental Operations with Decimals
Fractions
Scientific Notation and Electronic Calculation

Rounding off Numbers
We are so familiar of numbers but we do
not realize how important they are to us.
We owe these numbers to our ancestors
who had their ways of expressing
numbers. The numbers that we usually use
are called Hindu-Arabic numbers or the
decimal systems which use the principle of
place value and utilizes zero as a place
holder.

A number written in Hindu-Arabic system of
notation should be read by everyone. The digits
are groups of three digits, divided into three
orders as shown in the table below.
BILLIONS MILLIONS THOUSANDS UNITS
Hundreds Hundreds 4 Hundreds 0 Hundreds
1
Tens Tens 6 Tens 9 Tens
8
Ones Ones 4 Ones 7 Ones
5

Example:
The numeral is read as “four hundred sixty-four
million ninety-seven thousand one hundred
eihty-five.” following the billion are trillion,
quadrillion, quintillion, sectillion, septilion,
octillion, nonillion, decillion, and the like.
When a number has both integral and decimal
parts, the decimal point is used as separarix like
108.63. if the number is pure decimal, a zero is
placed in the units portion, e.g., 0.82. the zero
emphasizes the position of the decimal point.

0.0 6 3 4 5 7
Tenths
Hundredths
Thousandths
Tenthousandths
hundredthousandths
Millionths
Tenmillionths
Hundredmillionths

Examples:
Round off 6,486.4525 to the nearest ---
1. Unit 6,486
2. Tens 6,490
3. Thousands 6,000
4. Tenths 6,486.5
5. Thousandths 6,486.453
Note that:
(1) When the portion to be dropped begins with 5 or
above (6,7,8,9), the last figure retained is increased by
1.
(2) When the number to be dropped is less than 5
(4,3,2,1,0), the preceeding

FUNDAMENTAL OPERATIONS with
DECIMALS
A. The process of combining two or more like
numbers (addends) and representing their
amount by a single number (sum) is ADDITION.
It is easier to add when numbers are arranged
in column, decimal points are aligned, with the
digits of the same place value and order in a
vertical line. The numbers are then added from
top to bottom.

Example:
Find the sum of 486.73, 1,495.6, 78.615, 22.3, and
0.007.
486.730
1,495.600
78.615
22.3
0.007
2,083.252
Notice that the number of decimal places are made
the same.

B. The process of finding the difference between
two numbers is called
SUBTRACTION. The difference or remainder is
found by taking the subtrahend from the minuend.
Example:
Subtract 29.846 from 30.79
30.79 minuend 30.790
--29.486 subtrahend --29.486
0.944
Just like the addition, decimal points are in vertical
line and only digits of the same place value can be
subtracted.

C. Short cut for long addition is MULTIPLICATION.
The numbers to be multiplied are called factors and
the result is product
Example:
Find the cost of six pencils at P5.50 each. One way
of finding the solution is by addition.
P5.50
5.50
5.50
5.50
5.50
5.50
P33.00

An easier way of finding the cost is –
P5.50
x 6
P33.00
Note that multiplying decimals is just like
multiplying whole numbers. The product must have
as many decimal places as there in the factors.
Example:
65.65 2 decimal places
x 0.7 1 decimal place
45.955 3 decimal places

D. Dividing decimals is much like dividing
whole numbers. A decimal divisor is made
a whole number by multiplying both the
divisor and the dividend by a power of ten
corresponding to the number of decimal
places the divisor has. This is also the
same as moving the decimal point in the
divisor until after the last digit. The
number of decimal places the point in the
divisor has been moved, the same
movement is done in the dividend.

CHAPTER 2
Fundamentals of Algebra in
Business
 Algebraic Symbols and Expressions
Writing Basic Expressions and Equations
Solving Verbal Problems

Fundamentals of Algebra in Business
Algebraic Symbols and Expressions
Algebraic symbols such as the numerals, the
variables and the 4 basic operations comprise an
algebraic expression. The whole numbers,
fractions and decimal numbers are the numerals
while the variables can be any letter such as x
that reserves a place for a number. Any
expression such as 2(300 – 50) +100 that gives a
specific number when evaluated is called a
numerical expression.

• It should be noted also that the numerical
expression 9x8 can be written as 9 • 8 or 9(8)
while 72 ÷ 8 can be written as 72 / 8. On the
other hand, an expression such as 2(x + 7) that
contains at least one variable is called an
algebraic expression. To evaluate an algebraic
expression, replace each variable with a given
number and perform the indicated operations.
In the example 2(x + 7), if x is replaced by 5 the
numerical expression 2(5 + 7) is evaluated by
combining first those terms inside the
parenthesis ( ). Hence, 2(5 + 7) = 2(12) = 24.

Basic Laws of Algebra
Associative Property of Addition
(a + b) + c = a + (b + c)
Example: (300 + 50) + 100 = 300 + (50 + 100)
350 +100 = 300 + 150
450 = 450
Associative Property of Multiplication
(a • b)c = a(b • c)
Example: (2 • 12)3 = 2(12 • 3)
(24)3 = 2(36)
72 = 72

Commutative Property of Multiplication
a • b = b • a
Example: 2 • 12 = 12 • 2
24 =24
Distributive Property of Multiplication Over
Addition
a(x + y + z) = ax + ay + az
Example: 3(5 + 3 + 7) = 3 • 5 + 3 • 3 + 3 • 7
3(15) = 15 + 9 + 21
45 = 45

The grouping symbols such as the parenthesis ( ),
the brackets [ ], or the braces { } are used to
indicate the order in which operation should be
done. For instance, the numerical expression
3(2000 – 5[2(100 – 50) + 100] – 100) should be
evaluated as follows:
3{2000 – 5[2(100 – 50) + 100] – 100}
= 3{2000 – 5[2(50) + 100] – 100}
= 3{2000 – 5[100 +100] – 100} = 3{2000 -5[200]
– 100}
= 3{2000 – 1000 – 100} = 3{900}
= 2700 First evaluate expression
inside ( ), then, inside the
brackets [ ], then, inside
the { }.

The fraction bar /, which serves as a division
symbol is also considered a grouping symbol. If
there is no given grouping symbol in a
numerical expression such as 5 – 6 • 12 ÷ 9 +
20 the M-D-A-S Rule must be applied. This
means the order of operations must be
multiplication, then division, then addition and
finally, subtraction. Hence, 5 – 6 • 12 ÷ 9 + 20 =
5 – 72 ÷ 9 + 20
= 5 – 8 + 20
= 25 – 8
= 17

Writing Algebraic Expressions and Equations
Business and accounting problems are usually
presented in a narrative form, which means they
are written in verbal phrases. These verbal
phrases must be translated into numerical or
algebraic expressions to make a mathematical
solution possible. These are verbal phrases that
suggest addition, subtraction, multiplication or
division operation.

Examples of phrases with their corresponding
numerical expressions or algebraic expressions.
Phrases Numbers/Variables Expressions
1a 5 increased by 3 5 5 + 3
1b a number increased by 3 n n + 3
2a 7 more than 5 5 5 + 7
2b 7 more than a number m m + 7
3a 6 decreased by 4 6 6 – 4
3b a number decreased by 4 p p – 4
4a 5 less than 8 8 8 – 5
4b 5 less than a number q q – 5
4c a number less than 8 t 8 - t

5a the difference of 8 and 5 8 8 – 5
5b the difference of a number and 5 b b – 5
5c the difference of 8 and a number c 8 – c
6a 5 subtracted from 8 5 and 8 8 – 5
6b 5 subtracted from a number d d – 5
6c a number subtracted from 8 f 8 – f
7a the product of 7 and 8 8 7(8)
7b the product of 7 and a number x 7x
8a thrice of 10 3 and 10 3 (10)
8b thrice of a number y 3y
8c a number times 10 m 10m

Solving Verbal Problems
When solving a worded or verbal problem, one can use the
following guidelines or the so-called Problem-Solving Checklist:
STEP1. Understand the question. Read the problem carefully and
decide what question is asked.
STEP2. Find the needed data. Decide what information is needed
to solve the problem.
STEP3. Plan what to do. Choose a strategy o a series of
operations that will help you solve the problem. Other Problem-
Solving strategies are ‘table construction', 'look for a pattern’,
‘draw a picture’, ‘write an equation’, ‘simplify the problem’, and
‘guess-check-devise’.
STEP4. Find the answer. Complete the reasoning or computation
needed to find the answer.
STEP5. Check Back. Reread the problem. Examine your answer
whether or not it seems reasonable. Estimation or backward
computation are suggested strategies in checking your answers.

CHAPTER 3
Income Statement
Basic Terminologies
Basic Formulas

Income Statement
In a single business activity, one of the major
concerns of any individual who wants to
engage in business is the return of investment
of profit. Profit is the difference between the
amount of money invested in a certain
business and what one gets in return.

Basic Terminologies
Basic terminologies are used to identify the entries in
an income statement problem such as:
Gross – means total, whole, without deductions
Net – remaining amount, after deductions have been
made
Returns, Allowances and Discounts – repayment given
to customer for returned goods or defective
merchandize or for pricing adjustments
Cost of Goods Sold – refers to the actual amount of
goods sold, paid or unpaid including freight charges.

Operating Expenses or Period Cost – pertains to day to
day expenses such as salaries, rentals, utilities,
depreciation, tax, insurances, and other miscellaneous
expenses
Overhead or Product Costs – costs that are part of
manufacturing certain product such as direct labor,
and direct materials
Gross Sales – refer to the net invoice price of good sold
(Note: Gross sales include cash sales and credit sales)
Net Sales – remaining amount after deducting returns,
allowances, and discounts from the gross sales.

Basic Formulas
Case I. Profit if Net Sales > Cost of Goods Sold
Cost of Goods Available for Sale(COGAS) = Beginning
Inventory + Net Purchases
Cost of Goods Sold(CGS) = COGAS – Ending Inventory
Net Sales = Gross Sales – Returns, Allowances and
Discounts
Gross Income(GI) = Net Sales – CGS
Net Income(NI) = Gross Income – Operating Expenses
Case II. Loss if Net Sales < Cost of Goods Sold
Gross Loss = Cost of Goods Sold – Net Sales
Net Loss = Goss Loss + Operating Expenses

CHAPTER 4
Bank Reconciliation
Checkbook/Saving account
Bank Statement

Checkbook/Savings Account
In bank, a depositor may arrange to have either
a savings account or account. A savings account
depositor is issued a passbook by the bank,
where deposits, withdrawals and interest are
recorded. A checking account depositor, on the
other hand, is given a checkbook, which contains
the checks and the check stubs. A check is a
written order instrument made by the depositor
directing his bank to pay a person, or a business
firm a specified sum of money. A sample check is
shown below with explanation on the various
entries.

The check stub is where the depositor
keeps a record of checks issued,
deposits made and service charges in
order to keep track of the balance in a
checking account. An example of a
check stub with recorded transactions
is shown.

Date
Check
Number
Particulars Deposit Withdrawal Balance
Balance
Brought
Forward
P10,695.55
3/19/03 0119347
for MRI
2,469.60
1/6/04 0119348
for MRI and fire
6,639.19
2/20/03
Mrs. Grace
Macapulary
4,000
2/16/03 √ 12,047.06
2/15/03/ √ 6,639.19
2/18 √ 5,729.18
24,415.43
2/17/03 Cash 25,000
2/18/03 Cash 20,000
2/18/03 PSBank 10,695
2/20/03 P20,585.12

Example:
Mr. Aris Arce writes check #119266 on April
3, 2003, for P2, 360.00 payable to LESI, and
#119267 on April 18, 2003, for P10, 695
payable to PSBank. The balance brought
forward from the last checkstub is P12, 981.91,
and since that entry he has made a cash
deposit of P37, 000 on April 15, 2003. Record
this on the checkstub and compute the new
balance.

Solution:
Date Check
Number
Particulars Deposit Withdrawa
l
Balance
P12,981.9
1
4/3/03 119266 LESI 2,360 10,621.91
4/15/03 Cash 37,000 47,621.91
4/18/03 119267 PSBank 10,695 36,926.91
(New
Balance)

Bank Statement
Every month the bank prepares a report or short
statement for every checking account depositor
which is called a bank statement. This statement
shows the following:
1. the amount of deposit at the beginning of the
period;
2. the checks charged to the account during the
period;

3. any other amounts charged to the account,
such as service charges, charges for checks
printed, charges for returned checks, and so on;
4. any amounts credited to the account during the
period;
5. the account balance at the end of the period.
The common practice is for banks to mail the
depositor’s statement and include the cancelled
checks, deposit slips, and any other credit or debit
memoranda that have affected the account.

Upon receiving the bank statement,
the depositor should compare the
bank statement balance with his
checkbook balance. This is very
important in order to ascertain the
actual cash balance one has in his
checking account.

The difference in the checkbook balance and bank
statement balance may be due to the following:
1. Miscellaneous debits. These are charges such as
service charges, deductions for items deposited that
are uncollectable, charges for printing checks, and
other bank charges that may not be shown in the
checkbook.
To reconcile: Deduct from the checkbook balance.
2. Outstanding checks. These are checks that have been
issued but are still in the possession of the payee or
have not reached the bank for payment.
To reconcile: Deduct the total from the bank
statement balance.

3. Deposit in transit. Made too late to be included
in the bank statement.
To reconcile: Add the total to the bank
statement balance.
4. Deposit written twice on the stub.
To reconcile: Deduct from the checkbook
balance.
5. Check issued written twice on the stub.
To reconcile: Add to the checkbook balance.
6. Deposit omitted on the stub.
To reconcile: Add to the checkbook balance.

7. Check issued omitted on the stub.
To reconcile: Deduct from the checkbook balance.
8. Deposit wrongly recorded on the stub.
To reconcile:
a. If the amount recorded on the stub is more than the actual
amount deposited, deduct the difference from the checkbook
balance.
b. If the amount recorded on the stub is less than the actual
amount deposited, add the difference to the checkbook balance.
9. Check issued wrongly recorded on the stub.
To reconcile:
a. If the amount recorded is more than the actual amount
issued, add the difference to the checkbook balance.
b. If the amount is less than the actual amount issued, deduct
the difference to the checkbook balance.

Example:
On September 1, Tony Mananquil was given his monthly
bank statement showing his balance to be P17,063.50. On
that date, his checkbook showed a balance of P25,194.50.
Upon examining the bank statement and his checkbook, he
discovered the following:
a. Outstanding check was P4,035
b. Deposit of P7,500 was not recorded on the stub
c. Deposit of P6,185 was entered as P6,851 on the stub
d. Check deposited for P25,930 was entered as P2,493 on
the stub

e. Deposit of P10,240 was recorded twice on
the stub
f. Deposit of P12,560 was late to be included in
the bank statement
g. Service fee of P200 was deducted by the
bank
h. Check issued for P15,645 was written on the
stub as P18,645
Prepare a reconciliation statement.

a. Outstanding check in the amount of P4,
035.00 should be deducted from the bank
statement balance.
b. Upon seeing the bank statement, there
was a deposited of P7, 500.00 but in the
checkstub it is not yet recorded. Therefore,
the amount is not yet added to the
checkbook balance. To reconcile, P7,
500.00 must be added to the checkbook
balance.

d. Any deposit should be added to your balance
whether it is a cash or a check deposit not unless
the check deposited is a bounced check. In this
case, the amount recorded is less than the
amount of the check deposited. Therefore, the
difference of P1, 000.00 must be added to the
checkbook balance.
e. Deposit in the amount of P10, 240.00 was added
twice in the checkbook balance. Therefore,
deduct P10,240.00
f. The deposit of P12, 560.00 is known as deposit in
transit. This is a late deposit which is not yet
included in the bank statement so, it should be
added to the bank statement balance.

g. The amount deducted by the bank for the
services rendered to its checking account
depositor must be deducted also from
the checkbook balance. In this case, it is
P200.00
h. Another error is incurred in recording a
check. The difference of P3, 000.00 must
be added to the checkbook balance since
the amount of the check issued is less
than the amount the amount recorded.

CHAPTER 5
AVERAGE
Simple Average
Weighted Average

AVERAGES
An average is a single value that is used
to represent a group of value. It is commonly
called the arithmetic mean. The use of
averages has a wide application in business
affairs such as the computation of average
prices, average sales, average costs,
average inventories and the like. There are
two types of averages, namely simple
average and weight average.

CHAPTER 6
Depreciation
*Straight-Line Method
*Steps to Prepare Depreciation
Schedule by the Straight-line
Method.

Where:
Depreciation or depreciation expense = the
decrease in the value from the original cost of a long-
term asset over its useful life.
Book Value = the value of an asset at any given time.
It is the original cost less the accumulated
depreciation to the point.
`= Original Cost – ( Accounting Period x Annual
Depreciation)

Total cost or Original Basis = the total amount a
company pays for an asset, including freight,
handling, and setup charges.
Residual, scrap, salvage or trade-in value = the
value of an asset at the time it is taken out of
service.
Useful life = the length of time an asset is
expected to generate revenue.

Since May 1 through December 31
encompasses 8 months, we multiply
the monthly depreciation by 8 to find
the depreciation applicable to the first
year.
P300/mo. X 8 mos. = P2,400
depreciation for the first year

3. If bought on April 28 9 Example 2), the end of
the year (December 31) book value of the
machine in Example 1
Book Value = Original Cost – Depreciation
= P22,000- P2,400
= P19,600
At the end of the fifth year, the book value of
this machine is its salvage value, P3,600.

CHAPTER 7
Ratio, Proportion
and Percent

Introduction:
• A ratio is a comparison of one
number to another. The comparison
can be of a part to the whole, or one
part to another part. A ratio shows
how large or small one number is
relative to another number.

• Just like ratios, a percent also
represents a part of the whole.
Accountants use many ratios to
analyze the financial conditions of
business firms and these will be
discussed in the next chapter. After
studying this chapter, students will be
ready to do applications and
interpretations of ratio, proportion
and percent to business problems.

R A T I O
• A ratio is the relation between two
like numbers or quantities expressed
as the quotient. It is a method of
comparing one quantity with
another quantity.

• Like common fractions, ratios are
usually reduced to lowest terms. The
fraction 9 may be reduced to the
5 lowest term of 3.
5
In ratio form, the notation is 9:15 (read
as 9 is to 15), which equals to 3:5.

• Ratios should be expressed in terms before
being reduced. It should be noted that the
ratio of 1 ton to 5,000 kilos is not the ratio 1
is to 5,000. The terms should be converted to
the same unit of measurement. In changing
ton to kilos, the solution is as follows:
1 ton is to 5,000 kilos
1 ton = 1,ooo kilos
1 ton x 1,000 kilos = 1,000 kilos
ton
1,000 kilos : 5,000 kilos or 1:5 (1 is to 5)

• Ratios are very useful in business
computations like analysis of
financial statements, allocation
of costs and expenses,
distribution of income and
looses, and the like.

EXAMPLES:
• Mr. Julian Ayala and Henry Tontoco
formed a partnership. Ayala invested
P4,500,000 whereas Tontoco contributed
P7,500,000 to the business. Find the ratio
of the following:
a.) Ayala’s investment to Tontoco’s
b.) Tontoco’s investment to Ayala’s
c.) Ayala’s investment to the total capital
d.) Tontoco’s investment to the total capital

Solutions:
a.) P4, 500,000: P7,500,000
= 3:5
b.) P7, 500,000: P4,500,000
=5:3
c.) P4,500,000: (P4,500,000+ P7,500,000)
=P4,500,000: P12,000,000
=3:5
d.) P7,500,000: P12,000,000

ALLOCATION OF A NUMBER
ACCORDING TO R A T I O

To allocate or divide a number into
parts according to ratio, consider the
following steps:
1. Add the terms of the ratio
2. Write the terms as the numerator
with their sum as the denominator.
3. Multiply the fraction to the whole to
get the part.

Suppose that P2,700,000
is to be allocated for the
sales department,
marketing department,
and advertising
department in the ratio of
2:3:4. How much money
will be allocated for each
department?
Given:
Whole:
P7,200,000
Ratio:
2:3:4
Examples

Solution:
1st : 2+3+4= 9
2nd : 2 ; 3 or 1 ; 4
9 9 3 9
3RD : 2 x P2, 700,000 = P600,000
9
1 x P2,700,000 = P900,000
3
4 x P2,700,000 = P1,200,000
9

Sales department will have P600,000, Marketing
department will have P900,000 and Advertising
Department will have P1,200,000
Examples:
Joy Espinosa bought a house and lot that was
advertised as follows:
Land…………………………………….……….…P1,500,000
House…..…………………………………………P3,000,000
Appliances………………………………………P1,200,000
Rugs and Curtains…………………………..P1500,000
TOTAL………………………………….………….P5,850,000

Joy was able to buy the property for
P5,265,000. For the record purposes,
the buyer wanted tom allocate the
P5,265,000 to the various
components in the same ratio as the
component were advertised. Find
the cost allocation to each
component.

The sum of the terms and the
denominator is P5, 265,000
Allocation is then (P5,265,000/ P5,850,000 =
0.9)
Land -0.9 x P1,500,000 = P1,350,000
House -0.9 x P3,000,000 = P2,700,000
Appliances -0.9 x P1,200,000 = P1,080,000
Rugs & Curtains -0.9 x P150,000 = P135,000
TOTAL - - - P5,265,000

• When ratio contains a set of two
or more fractions, the fractions
should be converted to fractions
with a common denominator and
then use the ratio of their
numerators.

Examples:
• Rica, Gemma, and Rose formed a business
venture sharing gains and losses on the ratio
of 1/2, 3/10, 1/5 respectively. After Rose
retired, Rica and Gemma agreed to continue
using the ratio ½ is to 3/10 respectively. The
financial statement of the business shows a
profit of P24,000,000. Determine the profit
share of:
A.) Rica
B.) Gemma

Solution:
1 + 3 = 5+3
2 10 10
Use the ratio of the numerators 5 and 3 of the
common denominator 10.
a.) 5+3=8
5 x P24,000,000 = P15,000,000 share of Rica
8
b.) 5 x 24,000,000 = P9,000,000 share of 10
Gemma

Proportion
• Proportion is the relation of two
things in size, number, amount of
degree. It is a statement of equality
between two ratios. The ratio 2/4
and 5/10 are two equal ratios and
therefore, constitute a proportion.

A proportion is written as 2/4 = 5/10 or
2:4::5:10.
The double :: is read equals. In a
proportion there are four terms: the inner
terms are the means, while the outside
terms are the extremes.
means
2 : 4 = 5 : 10
extremes

If the terms in a proportion is
missing, the rule of proportion can
be applied, that is, the product of the
means equals the product of the
extremes and vice versa.

Examples:
8 : 12 + x : 9
Solution:
12 (x) = 8 (9)
12 x = 72
x= 6

Solving word Problems
by Proportion

• In solving word problems by proportion,
determine the type of relationship that
exist between two ratios. For directs
proportion problems, set up the ratio in
the form of fractions. If b is directly
proportional to a , and b1, and b2
correspond and respectively to a1 and a2,
then,
b1 a1
b2 a2

• For inverse proportion problem, set
up the ratio in the form of fraction. If b
is indirectly or inversely proportional
to a1 and b2 corresponds respectively
to a1 and a2, then
b1 a2
b2 a1

• Note that the numerator of the
fraction at the left corresponds to
the denominator of the fraction at
the right.

Examples:
5 ½ 1 ¾
x P210
1 ¾ (x) = (5 ½) (P210)
1 ¾ x = P1,155
1 ¾ x P1,155
1 ¾ 1 ¾
X = P660 coat 0f 5 ½ kilos of pork

or
5 ½ : x = a ¾ : P210
(x) (1 ¾) = (5 ½) (P210)
1 ¾ x P1155
1 ¾ 1 ¾
X = P660 cost of 5 ½ kilos of port

Example:
Four painters need 15 days to paint a house.
If the job is to be completed in 5 days,
how many painters will be needed?
x (5) = 4(15) x:4 = 4(15)
5x = 60 x (5) = 60
5x 60 or 5x 60
5 5 5 5
x= 12 painters x= 12 painters

PERCENT %
-Aliquot Parts of 100%
-Formulas on Percentage, Base and
Rate
Percentage of Increase or Decrease

PERCENT
Percent is the number of hundredths of a
certain number included in a number
particular other number. Percents are
fractions or decimals with the denominator
100. In other words, the % sign is
hundredths. To use percent in arithmetic
application, they must first be changed to
decimal or fraction.

To change percent to a decimal, simply drop the
% symbol and move the decimal point two places to
the left (divide by 100). If the percent is in fractional
units, first we change fraction to decimal before
moving the decimal point.
Examples: a. 20% = .20
b. .003% = .00003
c.
1
8
% = .125% = .00125
A. Changing Percent to a Decimal

B. Changing Decimal to a Percent
To change decimal to a percent, move the
decimal point two places to the right and
annex the percent sign.
Examples: a. .18 = 18%
b. 34.5 = 3.450%
c. .002 = .2%

C. Reduce Decimal to a Common
Fraction
To reduce a decimal to a common
fraction, write the given decimal number
disregarding the decimal point as the
numerator of a common fraction with a
denominator of the power of 10 of the given
decimal.

Examples:
a. .9 =
9
10
, for 1 decimal point the denominator
is 10.
b. .25 =
25
100
, for 2 decimal points the
denominator is 100.
c. .358 =
358
1,000
for 3 decimal points the
denominator is 1,000.

Divide the numerator by the denominator.
Examples: a.
3
5
= .6
b.
1
4
= .25
c.
18
55
= .3272727
D. Reduce a Common Fraction to a Decimal

Drop the percent symbol, then multiply the
number by 1/100 and reduce to lowest terms.
Examples: a. 76% = 76 ×
1
100
=
76
100
=
19
25
b.
1
8
% =
1
8
×
1
100
=
1
800
c. 156% = 156 ×
1
100
=
156
100
= 1
14
25
E. Convert Percent (or a fractional percent) to a
Fraction

F. Convert Fraction to a Percent
Divide the numerator by the denominator to
convert fraction to a decimal, then move the
decimal point two places to the right, and add the
percent symbol.
Examples: a.
1
20
= .05 = 5%
b.
1
5
= .20 = 2%
c.
3
4
= .75 = 75 %

Aliquot Parts of 100%
A number that exactly divides (with zero
remainder) anoyher number is called an aliquot
part of the number. Thus, 3 is an aliquot part of
6 (because 3 is an exact divisor of 6); 5 is an
aliquot part of 15 (15 ÷ 5 is 3 remainder 0); 12
1
2
is an aliquot part of 100 (divides 100 exactly 8
times).

Percent
equivalent of
Common
FractionS

Multiplying by an Aliquot Part of 100%
To multiply any number by an aliquot part of
100%, multiply the given number by the fractional
equivalent and by the corresponding 100%.
Examples:
a. 394 × 50% = 394 ×
1
2
× 100% = 394 ×
1
2
× 1 = 197
b. 126 by .25 = 126 ×
1
4
× 100% = 126 ×
1
4
× 1 = 31.5
c. price 78 items at .0375 = 78 × 3/8 × 1 = 29.25

Dividing by an Aliquot Part of 100%
In any number by an aliquot part of 100%,
divide the given number by the fractional
equivalent.
Examples:
a.
540
30%
= 540
3
10
= 540 ×
10
3
=
5,400
3
= 1,800
b. Divide 228 by 33
1
3
% =
228
1
3
= 228 ×
3
1
= 684

Formulas on Percentage, Base, and
Rate
P = B × R; R = P/B; B = P/R

Percentage (P) is a number expressed as a
percent of some other number. It is the
product of the base multiplied by the rate.
The base (B) is the number upon which the
percent is computed and it is usually
preceded by the preposition “of” in word
problems. The word “of” means
multiplication. The rate (R) is usually in the
form of decimal or in fraction.

Finding the Percentage
Percentage = Base × Rate
Examples:
1. What number is 30% of 520?
P = B × R
= 520 × 30%
= 520 × .30
P = 156

2.
1
2
of P80 is how much?
P = B × R
= 80 ×
1
2
P= P40

3.
5
8
as great as 48 is how much?
P = B × R
= 48 ×
5
8
=
240
8
P= 30
P= 30 + 48
P= 78

Finding the Base
Base =
Percentage
Rate
Examples:
a. What number multiplied by 6% equals P90?
B =
P
R
=
P90
.60
= P150

b. 220 is
2
5
of what amount?
B =
P
R
=
20
2
5
= 20 ×
2
5
=
100
52
= 50
c.
1
5
% of what sum is 24?
B =
P
R
=
24
1
5
%
=
24
.2%
=
24
.002
= 12,000

Rate of Increase or Decrease
To determine the rate of increase or
decrease, get the difference between the given
number and the base, then divide it by the base
or original number. If the result is fraction
express to percent.
Rate of increase or decrease = Larger number−Smaller number
Base or Original number

Examples:
a. What percent is more than 48 is 32?
Solution:
48−41
32
=
16
32
=
1
2
= .5 = 50%
b. What part less than 75 is 25?
Solution:
75−25
75
=
50
75
=
2
3
c. What percent less than 64 is 24?
Solution:
64−24
64
=
40
64
= 62.5%

Base of Increase or Decrease
To determine the base on the given number that
is fractional part or percent greater than or smaller
than that which is missing, divide the given number
by the sum (if greater than) or difference (if smaller
than) between 1 and the given fraction or 100%
and the given rate.
Base of Increase =
P
1+Fraction
Base of Increase =
P
100%+Given %
Base of Decrease =
P
1−Fraction
Base of Decrease =
P
100%−Given %

Examples:
a. 96% is
1
3
greater than what number?
Solution:
96
3
3
+
1
4
=
96
4
3
=
96×3
4
=
228
4
= 72
b. 30% is more than what amount is P120?
Solution:
P120
100%+30%
=
P120
130%
=
P120
1.3
= P92.31

c.
3
4
less than what number is P56?
Solution:
56
1−
3
4
=
56
4
4
−
3
4
=
56
1
4
=
54 ×4
1
= 224
d. 80% less than what amount is P42?
Solution:P42
P42
100%+80%
=
P42
20%
=
P42
.20
= P210

d. Finding the Rate
Rate = Percentage/Base
Examples:
a. What percent of 120 is 10?
R =
P
B
=
10
120
= .0833 = 8.33% or 8
1
3
%
b. P75 is what part of P225?
R =
P
B
=
P75
P225
=
75
P225
=
3
9

c.
1
2
of 60 is what part of 1/5 of 450?
P = B × R P = B × R
= 60 ×
1
2
= 450×
1
5
P = 30 (percentage) P = 90 (base)
R =
P
B
=
30
90
=
1
3

Percentage of Increase or Decrease
To determine the percentage of increase and
decrease, multiply the base by the percent of
fraction, then add or subtract the product of the
base or the given number.
Percentage of Increase = Base + (Base × Rate)
Percentage of Decrease = Base - (Base × Rate)

Examples:
a. P350 greater than 15% is how much?
x = Base + (Base × Rate)
= P350 + (350 × .15)
= P350 + 52.50
= P402.50

b. How much is
2
5
less than 450?
x = Base + (Base × Rate)
= 450 - (450 ×
2
5
)
= 450 - 180
x = 270

CHAPTER 8
Profit Ratio Analysis

Example 1:
Jolly motors works
In come statement for the year ended Dec. 31, 2003
Amount Percent (%)
Net sales P 995,000 100.00%
Cost of Good sold P 450,000 45.23%
Gross Profit P 545,00 54.77%
Selling & Administrative Expenses P 175,000 17.59%
Advertising P 30,00 3.02%
Lease Payments P 24,000 2.41%
Depreciation & Amortization P 15,000 1.51%
Repairs and Maintenance P 7,500 0.75%
Total Operating Expenses P 251,500 25.28%
Operating Profit P 293,500 29.50%
Other Income (expenses) P 4,500 0.45%
Interest Income (P 7,200) -0.72%
Interest Expenses P 290,800 29.23%

Earnings Before Income taxes P 290,800 29.23%
Income Taxes P 16,350 1.64%
Net income P 247,450 27.58%

• Conclusion:
The fact that the net profit margin of Jolly
Motor is positive, that is , 27.28%, it can be
concluded that its operation for the current
year is profitable.
But it is better if this profit margin is
compared with previous year’s ratio or with the
profitability ratio of its competitors.

Comparative Income Statement
Comparative Income Statement are income
statement that shows the result of operation in two ore
more successive year. These statement are prepared to
further analyze the entity’s operating efficiency and
profitability.
A base year has to be defined, normally as the earlier
year. It is the basis for determining increases or
decreases in the elements of income statement. The
increase or decrease indicated in the comparative
income statement should be related with qualitative
factors to enhance the evaluation of a company’s
profitability and efficiency.

J & N grill’s income statement has
information for both the current and
preceding year. Frequently, comparative
statements include these differences since
the percent of net change, especially if it is
significantly lower, affects future business
decisions.

Example 2
Comparative income statements for the years ended dec. 31 2002 & 2003
Revenues 2003 2002
Amount of
increase
(decrease)
Percent of
increase
(decrease)
Sales P 895,450 P655,000 P 240,450 36.71%
Less returns P 12,600 P6,500 P6,100 93.85%
Net sales P822,850 P648,500 P234,350 36.14%
Cost of Goods sold: - -
Inventory, January 1 P192,00 P165,000 P27,000 16.36%
Purchases P450,000 P420,000 P30,000 7.4%
Available for sale P642,000 P585,000 P57,000 9.74%
Inventory for, December 31 P205,000 P202,000 P3,000 1.49%
Cost of good sold P437,000 P383,000 P54,000 14.10%
Gross profit on sales P445,000 P265,500 P18,350 67.93%

Operating expenses
Salary & Benefits P158,000 P92,000 P66,000 71.74%
Rent & utilities P47,600 P40,500 P7,100 17.53%
Advertising P30,400 P24,500 P5,900 24.08%
Depreciation P35,000 P25,000 P10,000 40.00%
Equipment & supply P14,100 P10,700 P3,400 31.78%
Administrative P9,600 P8,000 P1,600 20.00%
Total Operating Expenses P294,700 P200,700 P94,000 46.84%
Operating profit P151,150 P64,800 P86,350 133.26%
Other income (expenses)
Interest Income P1,500 P900 P600 66.67%
Interest expenses (P2,300) (P1,250) (P1,050) 84.00%
Earnings before income taxes P150,350 P64,450 P85,900 133.28%
Income taxes P26,500 P19,500 P7,000 35.90%
Net income P123,850 P44,950 P78,900 175.53%

It should be noted that there is an increase in
the gross profit margin, net profit margin, and
operating profit margin. This has to be analyzed
further by looking at every element of the items
directly related to the determination of gross
profit margin , operating margin, and net profit
margin. That is, there is probably a decrease in
the cost of purchase or increase in selling price.

CHAPTER 9
Payroll , commission
and taxes

Payroll
The most important activity to perform in a business
firm is to prepare the compensation or income of the
employees and making sure workers are paid on time and
in the correct amount. The word compensate means the
renumeration for services performed by an employee for
his employee such as salaries, wages, bonuses, allowances,
honoraria which constitute compensation income. Payroll
is a sheet of information containing the total wages or
salaries of the employee for a specific period. The types of
compensation are salary, time wages , price-rate
compensation and commission.
Employees and workers usually use sign-in system, time
card, punch card and bundy clock to record the number of
hours worked.

A. Salary is a payment for managerial,
administrative, and for other similar services.
Salaries may be computed as weekly, biweekly,
monthly, semi- monthly and annually.
Computations of salaries:
1. Weekly salary = annual salary / 52
2. Biweekly salary = annual salary / 26
3. Monthly salary = annual salary/ 12
4. Semi-monthly = annual salary /24

Examples:
1. Grace Suralbo, an employee of Kristen
International Co, received an annual salary of
P118,502.30. What is her monthly salary?
Solution:
P118,502.30 / 12= P9,875.19 monthly salary
2. Find miss Suraldo’s biweekly salary if her
annual salary is P118,502.30.
Solution:
P118,502.30/26 = P4,557.78

Computations of overtime rates: o.t
hourly rate
1. Time and fourth: O.T = 1.25 (regular hourly
rate)
2. Time and a half: O.T= 1.5 ( regular hourly
rate)
3. Time and three forts: O.T= 1.75 (regular
hourly rate)
4. Double time: O.T = 2(regular hourly rate)

Examples
Cinderella worked 44 hours in a week. If her regular wage rate was
P32.00 an hour and an overtime rate of time and a half, find her
total earnings for a week.
Solution:
Regular pay = no. of hours worked x hourly rate
= 40 x P35.00
= P1,400
Overtime pay =no. of overtime worked x hourly rate
= 4 x (P35.00x 11/2)
= 4 x (P35.00x 1.5)
= 4 x P52.50

Gross pay = regular pay + overtime pay
= P1,400 + P210
= P1,610

Commission is the payment received by a person
doing business services. Persons paid on
commission basis are called sales agents or brokers.
Common types of commissions are straight
commission, commission plus override, commission
and bonus and salary and commission.
Formula:
Commission= amount of sales x rate of commission
Rate of commission= commission / Amount of sales
Amount of sales= commission / rate of commission

• Straight commission
-the sales representative who has paid
on a commission has no fixed income.
His earnings are usually based on the
amount of his sales in a given period.

Taxes are very much a part of our daily
lives and will continue to be. In this topic,
you will learn how the various taxes that you
pay or can expect to pay in the future are
computed.
A tax is a sum of money levied on persons
or corporations by the government to
finance its various projects. Some forms of
taxes are real property tax, personal property
tax, income tax, sales tax, and residence tax.

CHAPTER 10
-BUYING-
Trade discounts
Discount Series
Retail Discounts
Cash Discounts
Partial Payments

• Manufacturers and wholesalers wanting
to encourage customers to purchase,
goods in large quantities frequently offer
trade discount and grant a deduction
large enough to enable their customers
to generate a satisfactory net profit. This
deduction or trade discount is printed in
a catalogue or price sheet. The list price
or catalogue price for merchandise being
sold to various buyers contain prices that
the retailers should charge their
customers.

• The price offer when the trade discount
has been subtracted from the list price
is called the invoice price or the net
price. To compute the trade discount or
discount and the net price, the
following formulas should be used:
1. Amount of discount= trade
discount rate x list price or discount=
discount rate x list price
- 2. Net price = list price – discount

Other formulas
3. Discount rate = amount of discount
list price
4. List price = net price____
100% - discount

Example 1.
• Compute the net price and the
discount for a P25, 000 stereo set offered at
a 25% discount rate.
Solution:
Discount = discount rate x list price
= 25% x P25, 000
= .25 x 25, 000
Discount = P6, 250

Net price = list price – discount
= 25, 000 – 6, 250
Net price = P 18, 750
The discount is P6, 250 and the
net price for the stereo set is P18, 750.

Example 2.
• A sala set had a list price of P35, 000 and was
sold to a customer for P29, 750. What was
the trade discount rate?
Solution:
Discount rate = amount of discount
list price
Amount of discount = list price – net price
= 35,000 – 29,750
= P5, 250

Discount rate = 5, 250
35, 000
Discount rate = .15 or 15%

Example 3.
• Rex paid P10, 200 for a refrigerator on the sale
that had 20% off the list price. What was the list
price of the refrigerator?
Solution:
List price = net price
100% - discount rate
= 10, 200
100% - 20%

= 10, 200
80%
= 10, 200
8
= P12, 750
Thus, the list price of the refrigerator is P12,
750.

• When two or more trade discounts are
offered on the same article, these discounts
are known as discount series.
In solving for the discount series, the
following steps must be followed:
1. Express all sales in decimals.
2. Subtract each of the results in step (1) from 1.
3. Multiply results in step (2) together.
4. Subtract from 1 the product obtained from
step (3). The remainder must be changed to
percent.

Example 1.
• Find the single discount sale equivalent to
the discount series of 30% and 10%.
Solution:
Step 1. 30% is 0. 30
10% is 0.10
Step 2. 1 - 0.30 = 0.70
1 – 0.10 = 0.90
Step 3. (0.70) (0.90) = 0.63
Step 4. 1 – 0.63 = 0.37 or 37%

• Another method of solving of solving
discount series follows very closely the
suggested procedure for solving the net
price given the list price less a series of
discounts. All the data interpreted in
terms of percent, 100% representing
the list price, each discount is
successively applied to the list price
and the resulting remainder.

Illustration:
Consider Example 1:
Less: 30% (1st discount Sale)
70% (1st remainder)
Less : 7% (10% of the 1st remainder)
63% (net price)
Therefore, the equivalent single sale is 37%.
To check: 100% - 63% = 37%

The third method is used when
the discounts are only two at a time.
The following steps are followed:
1. Add the two discounts.
2. Multiply the two discounts
3. Subtract the product from the sum to
get the single equivalent rate.

Illustration:
From Example 1:
Step 1. 30% + 10% = 40%
Step 2. 30% (10%) = 3%
Step 3. 40% - 3% = 37%

Example 2.
• Which is a better offer, a discount series of
20%, 15%, and 10%, or 44%?
Solution:
Method 1:
Step 1. 20% = 0.20
15% = 0. 15
10% = 0.10

Step 2. 1 - 0 .20 = 0.80
1 – 0.15 = 0. 85
1 - 0.10 = 0.90
Step 3. (0. 85) (0. 80) (0.90) = 0.612
Step 4. 1. – 0.612 = 0.388 or 38.8%

Method 2:
100%
-20%
80%
-12% (15% of 80%)
68%
-6.8% (10% of 68%)
61.2%
• Then: 100% - 61.2% = 38.8 %

• Retail discounts and trade discounts
are essentially the same. A retail discount,
whether a single rate of discount or a discount
series, is a deduction from a list price or
catalogue price and is used in determining the
actual selling price of the goods, and is offered
by retailers to consumers. The list price or
marked price or quoted price are commonly
known as the markdown. The price which the
customer actually pays for the merchandise is
the selling price or net price.

Example 1:
Aaron bought a component set at an
appliance center. The list price of the
component set is P65, 000 with 8%
discount. How much did he actually pay?
Given:
List price = P65, 000
Discount sale = 8%

Solution:
Retail discount = P65, 000 x 8% = P5, 200
Net price = P65, 000 – P5, 200 = P59, 800

Example 2:
A discount of P5, 000 was given to
Eric when he bought a TV set with a list price
of P30, 000. Find the discount rate.
Given:
Retail discount = P5, 000
List Price = P30, 000

Solution:
Rate of discount = retail discount
list price
= P 5, 000
P 30, 000
= 16 2/3%

Example 3.
How much must a student pay for a
school bag marked P250 less 15% discount?
Given:
List price = P250
Discount rate = 15%

Solution:
Retail discount = P250 x 15% = P 37. 50
Net price = P250 – P37. 50 = P112. 50

Cash discounts are discounts offered by large
companies to encourage customers to pay bills quickly
within a specified period of time. Cash discount are
often listed under the heading terms on a bill or
invoice. The terms of the sale are said to be the
conditions of the payment. The terms of a cash
discount are usually noted in various ways as follows:
for example, 4 3/10, net 30 (sometimes written 4,
3/10, n/30) means that the customer is entitled to
receive a 4% deduction for immediate payment; if the
bill is paid within 10 days, a 3% is allowed and the
amount due is to be paid in full, not later than the 30th
day. After that, an interest may be charged on the
balance.

Figure A shows an invoice from ABC Automotive
Supply, Inc. which sold car wax to Manuel Auto Supply
for P24, 000. Manuel will pay the delivery charges.
The invoice data is June 23 and the invoice term is
3/10, n/30.
Figure A
ABC Automotive Supply, Inc. INVOICE NO.
728355
Sold to Manuel Auto Supply Date June 23,
2003
67 Quirino Avenue, Terms 3/10, n/30
Las Penas City Ship Via LBC

Quantity Descripti
on
Unit
Price
Gross
Amount
Net
Amount
24 Gallons Car wax P1,000 P24, 000 P24, 000

The expression 3/10, n/30 means that
Manuel can get a 3% discount if he pays the
invoice within 10 days of the invoice date. Ten
days after June 23 is July 3, the discount
date. The 10-day discount period is between
June 23 to July 3. Then n/30 is short for net
30, which means that if Manuel does not pay
within 30 days. ABC Auto Supply will charge an
interest for penalty. The due date is July 23, 30
days after June 23.

Example:
• Zeta company receives a bill for P35, 200
dated June 25 with terms 7, 5/10, 3/30, n/60.
What cash discount will Zeta receive if Zeta pays
a) immediately b) in 10 days, c) on the 23rd day, d)
on the 38th day?
Solution:
a) Cash discount = P35, 200 x .07 = P2,464
b) Cash discount = P35, 200 x .05 = P1, 760
c) Cash discount = P35, 200 x .03 = P1.056
d) No discount

If Zeta Company pays immediately,
Zeta receives a 7% discount on
P2,464.Therefore, Zeta Company
receives a P2, 464 discount and paid
only P35, 200 - P2, 464 = P32, 736. If
Zeta pays in 10 days, Zeta receives a 5%
discount on P1, 0760. If Zeta pays on
the 23rd day it receives a 3% discount,
or P1, 056. If Zeta pays on the 38th day,
Zeta Company is entitled to no discount.

Partial Payments
• There are some instances that a
buyer of a certain product wants to pay his bill
early in order to get a discount, but lacks
sufficient amount of money to pay for the
entire amount of the billed good. In this
situation, one person is to make partial
payment and get a discount on the fractional
part of the bill. The buyer`s obligation will be
credited and the maintaining amount will be
owed.

Example:
• An invoice of P65, 000 is sent to
a buyer on May 5 with terms 6/10, net
30. The buyer, failing to pay the full
amount within 10 days, makes a partial
payment of P30, 000 on May 10. How
much is the balance due on the
account after the payment is made?

Solution:
Since the buyer makes the
payment within 10 days of the invoice
date, he is entitled to a 6% discount. The
amount paid, therefore, represents 94%
(100% - 6%) of the amount to be
credited. If we let n represent the
amount to nbe credited to his account,
94% of n = P30, 000.

.94n = P30, 000 = P31, 914. 89
.94 .94
Thus, P31, 914. 89 is credited to the
buyer`s account. The amount owed is
P65, 000 – P31, 914. 89 = P33, 085. 11.

Business Mathematics

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