Math in Math in BusinessBusiness

Presents……….Presents……….

Contents……Contents……

Chapter I. Chapter I. Basic operationBasic operation Chapter 2. Operation Chapter 2. Operation

with Decimalswith Decimals Chapter 3. FractionsChapter 3. Fractions Chapter 4 Percent in Chapter 4 Percent in

BusinessBusiness Chapter 5: Employee’s Chapter 5: Employee’s

CompensationCompensation

Chapter 6: BUYING AND Chapter 6: BUYING AND SELLINGSELLING

Chapter 7:SIMPLE INTERESTChapter 7:SIMPLE INTEREST Chapter 8: COMPOUND Chapter 8: COMPOUND

INTERESTINTEREST Chapter 9: DEPRECIATIONChapter 9: DEPRECIATION Chapter 10: INCOME Chapter 10: INCOME

STATEMENTSTATEMENT

Chapter I. Chapter I. Checking Checking answers answers of the of the Basic Basic

OperationOperationss

Addition, subtraction, multiplication, Addition, subtraction, multiplication, and division are the basic operations in and division are the basic operations in mathematic. Users confronted with mathematic. Users confronted with different problems in business require different problems in business require thorough knowledge of these basic thorough knowledge of these basic operations. With the invention of operations. With the invention of calculators, a person can solve a calculators, a person can solve a mathematical problem with greater mathematical problem with greater speed. Since calculators are mere speed. Since calculators are mere instrument in performing the instrument in performing the operations, accuracy is largely operations, accuracy is largely dependent on mastery of the operations dependent on mastery of the operations by the user and on his skills in checking by the user and on his skills in checking answersanswers..

Basic Math OperationsBasic Math Operations AdditionAddition Commutative Property of Commutative Property of

Addition · Associative Property Addition · Associative Property of Addition · Distributive of Addition · Distributive Property · Additive Identity: Property · Additive Identity: ZeroZero

We'll assume the reader can add digits, so that We'll assume the reader can add digits, so that 2 + 2 = 4 2 + 2 = 4

is not a surprise. In addition, we'll also assume that the is not a surprise. In addition, we'll also assume that the fundamentals of "carrying" are not that big of a problem, and so fundamentals of "carrying" are not that big of a problem, and so most readers will immediately know how we didmost readers will immediately know how we did

We're more interested here in the general properties of addition We're more interested here in the general properties of addition that impact algebra. With that in mind, let a, b and c be three that impact algebra. With that in mind, let a, b and c be three real numbers. Then the following properties of addition turn out real numbers. Then the following properties of addition turn out to be usable and important:to be usable and important:

Commutative Property of AdditionCommutative Property of Addition

(Addition is the same regardless of the order (Addition is the same regardless of the order one adds the numbers, i.e., forwards addition is one adds the numbers, i.e., forwards addition is the same as backwards addition).the same as backwards addition).

ExamplesExamples

Note that negative numbers are sometimes Note that negative numbers are sometimes enclosed in parentheses to avoid confusion enclosed in parentheses to avoid confusion between the sign of the number and the addition between the sign of the number and the addition operation. This is merely a matter of style – operation. This is merely a matter of style – many textbook writers use spacing to set off the many textbook writers use spacing to set off the difference instead.difference instead.

Associative Property of AdditionAssociative Property of Addition

(Addition of a list of numbers is the same regardless (Addition of a list of numbers is the same regardless of which are added together first, i.e., grouping does of which are added together first, i.e., grouping does not matter)not matter)

ExamplesExamples

MultiplicationMultiplication

Commutative Property of Commutative Property of Multiplication · Associative Multiplication · Associative Property of Multiplication · Property of Multiplication · Distributive Property · Distributive Property · Multiplicative IdentityMultiplicative Identity

Again, we’ll assume that the basics of Again, we’ll assume that the basics of multiplication are well known, so that 2x2=2· multiplication are well known, so that 2x2=2· 2=2*2=4. There are obviously several different 2=2*2=4. There are obviously several different notations in use, depending when one learns it notations in use, depending when one learns it and what context one learns it. We will use all of and what context one learns it. We will use all of these notations, as well as another: when using these notations, as well as another: when using variables, multiplication is assumed when variables, multiplication is assumed when symbols are merely written next to each other.symbols are merely written next to each other.

Multiplication follows the same two laws just Multiplication follows the same two laws just described for addition.described for addition.

Commutative Property of MultiplicationCommutative Property of Multiplication

ExamplesExamples

We are following the same convention as with We are following the same convention as with addition. Note that just writing numbers next to addition. Note that just writing numbers next to each other is a poor idea because, for example, each other is a poor idea because, for example, 32 can be confused with 3· 2. Thus, we need 32 can be confused with 3· 2. Thus, we need some sort of symbol to make the two distinct. some sort of symbol to make the two distinct. There are several correct ways to do this – in There are several correct ways to do this – in other articles, we’ll make a lot of use of other articles, we’ll make a lot of use of parentheses, so that 3· 2 will be written 3(2) or parentheses, so that 3· 2 will be written 3(2) or (3)(2). Experience with the notation will help (3)(2). Experience with the notation will help make it very clear what is meant.make it very clear what is meant.

Associative Property ofAssociative Property of MultiplicationMultiplication

ExamplesExamples

Please note that we do not have to limit ourselves Please note that we do not have to limit ourselves to parentheses; the last computation could have to parentheses; the last computation could have been writtenbeen written

This looks a bit neater. Again, such things are a This looks a bit neater. Again, such things are a matter of style, and the reader is encouraged to use matter of style, and the reader is encouraged to use whatever bracketing makes the most sense and whatever bracketing makes the most sense and allows clear, proper ordering of calculations.allows clear, proper ordering of calculations.

Distributive PropertyDistributive Property

When both addition and multiplication appear in a When both addition and multiplication appear in a single mathematical expression, this distributive law single mathematical expression, this distributive law controls the operation. This is probably one of the controls the operation. This is probably one of the most important laws in mathematics; getting it most important laws in mathematics; getting it wrong guarantees bad calculations! The "reverse" is wrong guarantees bad calculations! The "reverse" is NOT CORRECT:NOT CORRECT:

ExamplesExamples

SubtractionSubtraction

We’re going to define subtraction in terms of We’re going to define subtraction in terms of addition of the negativeaddition of the negative::

This means that subtraction is a shortcut or an This means that subtraction is a shortcut or an abbreviation of the above addition operation. The abbreviation of the above addition operation. The "triple equals" sign used here means "triple equals" sign used here means definitiondefinition, , and is meant to signify an and is meant to signify an operation that is always operation that is always truetrue. It is a stronger statement than a simple equals . It is a stronger statement than a simple equals sign.sign.

ExamplesExamples

Subtraction is not Subtraction is not in generalin general commutative: commutative:

ExampleExample

Nor is it Nor is it in generalin general associative: associative:

ExampleExample

DivisionDivision

We’re going to define division in terms of We’re going to define division in terms of multiplication multiplication of the inverseof the inverse. Please see the discussion of multiplicative . Please see the discussion of multiplicative inverse below for more info. For now, suppose that a > 1 inverse below for more info. For now, suppose that a > 1 and thatand that

The number b has a very important function – it is called The number b has a very important function – it is called the multiplicative inverse of a, also known as the inverse the multiplicative inverse of a, also known as the inverse of a. Naively, we can writeof a. Naively, we can write

but it should be pointed out that this is not really a but it should be pointed out that this is not really a definition. In order to properly define division, we would definition. In order to properly define division, we would have to discuss have to discuss rational numbersrational numbers and how they work. and how they work. Instead of doing that now, we’ll simply take it as given that Instead of doing that now, we’ll simply take it as given that the reader intuitively understands fractions. With this the reader intuitively understands fractions. With this definition of inverse, we can tackle division:definition of inverse, we can tackle division:

ExampleExample Suppose a = 2. ThenSuppose a = 2. Then

Note that the decimal expansion 0.5000… is the Note that the decimal expansion 0.5000… is the result of long division, a subject we are avoiding result of long division, a subject we are avoiding here (for a discussion, see the division article). We here (for a discussion, see the division article). We can "prove" that 0.5000… is the inverse of 2 by can "prove" that 0.5000… is the inverse of 2 by multiplying:multiplying:

To the extent that this is rather unsatisfying, we To the extent that this is rather unsatisfying, we must ask the reader to suspend disbelief. The must ask the reader to suspend disbelief. The mechanics of division are complicated, and deserve mechanics of division are complicated, and deserve a separate article.a separate article.

Now we’ll use this notion to do a division.Now we’ll use this notion to do a division.

ExampleExample

Note that we expanded 6 into 2· 3 and canceled the Note that we expanded 6 into 2· 3 and canceled the 3’s. This is a bit sloppy, but again, our definition of 3’s. This is a bit sloppy, but again, our definition of division is intuitive rather than precise.division is intuitive rather than precise.

Chapter Chapter 2. 2.

OperatioOperation with n with

DecimalsDecimals

Our money representation is based on a decimal Our money representation is based on a decimal system. The peso is composed of 100 centavos. system. The peso is composed of 100 centavos. Since we rarely deal with amounts less than a Since we rarely deal with amounts less than a centavo, the lowest value we use is P.01. Thus, in any centavo, the lowest value we use is P.01. Thus, in any mathematical computation, the final answer should mathematical computation, the final answer should be exact to the nearest centavo. An answer of P be exact to the nearest centavo. An answer of P 135.256 should be written as P 135.26. Usually, it 135.256 should be written as P 135.26. Usually, it serves no purpose to express an answer id peso serves no purpose to express an answer id peso value with fractional part of a centavo. For discussion value with fractional part of a centavo. For discussion purpose, the money value P 135.26 is read as “one purpose, the money value P 135.26 is read as “one hundred thirty five pesos and twenty six centavos.” hundred thirty five pesos and twenty six centavos.” Note that the decimal point is read as “and” to Note that the decimal point is read as “and” to separate peso value from the centavo. Sometimes, it separate peso value from the centavo. Sometimes, it is necessary to indicate fractional parts of a centavo is necessary to indicate fractional parts of a centavo in such cases as statistical figures, indices, real in such cases as statistical figures, indices, real estate tax, and surveys.estate tax, and surveys.

In performing addition and subtraction, the numbers In performing addition and subtraction, the numbers containing decimal components are arranged containing decimal components are arranged according to their place value. In multiplication and according to their place value. In multiplication and division, the numbers with decimal components are division, the numbers with decimal components are treated as whole numbers. Care must be taken intreated as whole numbers. Care must be taken in placing the decimal points. The basic operations will placing the decimal points. The basic operations will be discussed in detail in the following sections.be discussed in detail in the following sections.

Addition with DecimalsAddition with Decimals

Addends with decimal components should be arranged Addends with decimal components should be arranged according to their proper order (column). The decimal according to their proper order (column). The decimal points should fall in one vertical column in order to align points should fall in one vertical column in order to align the figures according to their values. Hence, units of equal the figures according to their values. Hence, units of equal value will fall under the same column such as tens, value will fall under the same column such as tens, hundreds, thousands, millions, etc.hundreds, thousands, millions, etc.

Example 1. Example 1. Add 3.085; 12.314 and 94.65Add 3.085; 12.314 and 94.65 3. 085 3. 085 12. 31412. 314 94. 6594. 65 110. 0492110. 0492 Same columnSame column

Subtraction with Subtraction with DecimalsDecimals

The rule used in addition applies equally to The rule used in addition applies equally to subtraction. Decimal points of the minuend and subtraction. Decimal points of the minuend and subtrahend are placed in the same vertical column.subtrahend are placed in the same vertical column.

Example 2. Example 2. Subtract 125.2684 from 436.1052Subtract 125.2684 from 436.1052

436. 1052436. 1052 -- 125. 2684125. 2684 310. 8368310. 8368 Same columnSame column

Multiplication with Multiplication with DecimalsDecimals

The number of the decimal places of the product is The number of the decimal places of the product is equal to the total number of decimal places (total digit equal to the total number of decimal places (total digit numbers to the right of the decimal points) of the numbers to the right of the decimal points) of the factors.factors.

Example 3. Example 3. Determine the product of factors 3.248 and 1.26Determine the product of factors 3.248 and 1.26

Note: For convenience, use the shorted factor ( lesser digit Note: For convenience, use the shorted factor ( lesser digit numbers) as multiplier (1.26)numbers) as multiplier (1.26)

3.248 multiplicand3.248 multiplicand (3 decimal places) (3 decimal places) 1.26 multiplier1.26 multiplier (2 decimal places)(2 decimal places) 1948819488 6497664976 32483248 4.09248 product4.09248 product (3+2 = 5 decimal places) (3+2 = 5 decimal places)

Division with decimalsDivision with decimals When the dividend contains decimal fractions and the When the dividend contains decimal fractions and the

divisor is a whole number, the decimal point of the divisor is a whole number, the decimal point of the quotient is aligned vertically above (same vertical column) quotient is aligned vertically above (same vertical column) the decimal point of the dividend.the decimal point of the dividend.

Example 4. Divide 125.265 by 25Example 4. Divide 125.265 by 25 Same columnSame column 5. 0106 quotient5. 0106 quotient Divisor 25/ 125. 265Divisor 25/ 125. 265 125125 22 00 2626 2525 1515 00 150150 150150 0 remainder 0 remainder

Rounding off NumbersRounding off Numbers

Generally, numbers are rounded off Generally, numbers are rounded off in making estimates or forecasts. In in making estimates or forecasts. In monetary computations, it is the monetary computations, it is the practice to round off the final answer practice to round off the final answer to the nearest centavo because the to the nearest centavo because the third decimal place (thousandths) is third decimal place (thousandths) is often immaterial.often immaterial.

Chapter Chapter 3. 3.

FractionFraction

1/41/4

While our exposure to the numeration system dealt While our exposure to the numeration system dealt mostly with whole numbe5rs, we also come across mostly with whole numbe5rs, we also come across figures wherein the use of fractions is inescapable. figures wherein the use of fractions is inescapable. Such terms as “halves,” “fourths” or “quarters” are Such terms as “halves,” “fourths” or “quarters” are used with reference to measurement like “quarter to used with reference to measurement like “quarter to twelve,” “half a mile” “three fourths full” and so on. twelve,” “half a mile” “three fourths full” and so on. Like negative numbers, fractions `and computation. Like negative numbers, fractions `and computation. If a unit is divided equally into two or more parts, If a unit is divided equally into two or more parts, each equal part is the fractional part of the unit each equal part is the fractional part of the unit which is usually expresses in a fraction. For which is usually expresses in a fraction. For example, a parent bought a whole pizza pie and example, a parent bought a whole pizza pie and would want to share the pie equally among four (4) would want to share the pie equally among four (4) children. The parent would cut the pie into four children. The parent would cut the pie into four equal parts and one part is given to each which is equal parts and one part is given to each which is one-fourth of the whole pie. As illustrated belowone-fourth of the whole pie. As illustrated below

1/41/4

1/4 1/41/4 1/4

¼ means ¼ means oneone part of the whole which was divided into 4 part of the whole which was divided into 4 equal parts.equal parts.

The number written above the line (numerator The number written above the line (numerator indicates the number of parts and the number written indicates the number of parts and the number written below the line (denominator) indicates the total units or below the line (denominator) indicates the total units or parts the cake was divided into. A fraction may be used to parts the cake was divided into. A fraction may be used to represent a ratio.represent a ratio.

Fractions are classified into:Fractions are classified into: proper fractionsproper fractions, ½, ¾, 5/6, 7/8, etc., are fractions, ½, ¾, 5/6, 7/8, etc., are fractions whose numerators are less than the denominators. These whose numerators are less than the denominators. These

fractions indicate values less than one (1).fractions indicate values less than one (1). Improper fractionsImproper fractions. 3/2, 4/3, 5/4, 7/6, etc., are fractions . 3/2, 4/3, 5/4, 7/6, etc., are fractions

whose numerators are greater than or equal to the whose numerators are greater than or equal to the denominator. These fractions indicate values equal to or denominator. These fractions indicate values equal to or greater than (1).greater than (1).

Mixed numbersMixed numbers. 1 ½ 2 ¼ 3 1/3 4 ¼ etc., are fractions . 1 ½ 2 ¼ 3 1/3 4 ¼ etc., are fractions written as the sum of integer or whole number and a written as the sum of integer or whole number and a proper fraction. An improper fraction can also be proper fraction. An improper fraction can also be expressed as a mixed number or vice versa.expressed as a mixed number or vice versa.

Fundamental Rules in Fundamental Rules in Dealing with FractionsDealing with Fractions

One should understand the fundamental rules in One should understand the fundamental rules in dealing with fractions. These rules are very important dealing with fractions. These rules are very important in the basic operations of addition, subtraction, in the basic operations of addition, subtraction, multiplication and division of fractions. The rules are multiplication and division of fractions. The rules are the following:the following:

The value of a fraction is not changed when the The value of a fraction is not changed when the numerator and the denominator are both multiplied numerator and the denominator are both multiplied by the same number other than zero.by the same number other than zero.

Example 1. 1/3 multiplied both by 3Example 1. 1/3 multiplied both by 3

1/3 x 3/3 = 1/3 x 3/3 = 1 x 31 x 3 3 x 1 = 3/33 x 1 = 3/3

Note: 3/3 is the same as 1.Note: 3/3 is the same as 1. 2. The value of a fraction is not changed if we 2. The value of a fraction is not changed if we

divide both the numerator and the denominator by divide both the numerator and the denominator by the same number other than zero. The result is like the same number other than zero. The result is like dividing the fraction by 1.dividing the fraction by 1.

Changing Improper Changing Improper Fraction to a Whole or Fraction to a Whole or Mixed NumberMixed Number Mathematically, the line between the numerator and Mathematically, the line between the numerator and

denominator indicates division. In solving problems, it is denominator indicates division. In solving problems, it is more convenient or easier to work with an improper more convenient or easier to work with an improper fraction than mixed number.fraction than mixed number.

To change an improper fraction to a whole or To change an improper fraction to a whole or mixed number, divide the numerator by the denominator. mixed number, divide the numerator by the denominator. Examples are:Examples are:

Example 3:Example 3:

Change the improper faction 2/2 to a whole or mixed Change the improper faction 2/2 to a whole or mixed number.number.

Solution:Solution: 2/2 = 2 divided by 2 = 12/2 = 2 divided by 2 = 1

Changing Mixed Changing Mixed Numbers to Improper Numbers to Improper FractionsFractions This conversion is very convenient when we perform most This conversion is very convenient when we perform most

of the basic operations with fractions. Below are the ways of the basic operations with fractions. Below are the ways in converting a mixed number to an improper fraction:in converting a mixed number to an improper fraction:

Multiply the whole number and the denominator;Multiply the whole number and the denominator; Add the numerator to the product ; andAdd the numerator to the product ; and

Make the result of (2) as the numerator of the new Make the result of (2) as the numerator of the new fraction and use the same denominator as that of the fraction and use the same denominator as that of the original mixed number.original mixed number.

Example 5Example 5. Change 4 . Change 4 2 2 to improper fractionto improper fraction 33

Solution:Solution: follow the steps given above follow the steps given above 4 x 3 (whole number x denominator) = 124 x 3 (whole number x denominator) = 12 12 + 2 (add numerator to product) = 1412 + 2 (add numerator to product) = 14 14/3 (improper fraction with same denominator as the 14/3 (improper fraction with same denominator as the

original mixed number).original mixed number).

Reduction of FractionsReduction of Fractions The process of converting fractions to other The process of converting fractions to other

equivalent forms of either higher or lower terms equivalent forms of either higher or lower terms without changing the value of the fraction is called without changing the value of the fraction is called reduction. As in division, multiplying or dividing both reduction. As in division, multiplying or dividing both the numerator and denominator by the same the numerator and denominator by the same number other than zero does not affect the quotient. number other than zero does not affect the quotient.

Examples are:Examples are:

Reduce ¾ to higher terms.Reduce ¾ to higher terms. 33 = = 3 x 33 x 3 = = 99; or ; or 33 = = 3 x 5 3 x 5 = = 1515 4 4 x 3 124 4 x 3 12 4 4 x 5 20 4 4 x 5 20 Reduce 25/625 to lower terms.Reduce 25/625 to lower terms. 2525 = = 2525 ÷÷ 55 = = 5 5 or or 11

625625 625 ÷ 5 125 25625 ÷ 5 125 25

When the numerator and the denominator have no When the numerator and the denominator have no more common divisor except 1 the fraction has more common divisor except 1 the fraction has been reduced to its lowest term. To reduce a been reduced to its lowest term. To reduce a fraction to its lowest term, both the numerator and fraction to its lowest term, both the numerator and the denominator should be divided by their greatest the denominator should be divided by their greatest possible common divisor. possible common divisor.

For example, For example, in the proper fraction in the proper fraction 46 46 23 is the greatest possible 23 is the greatest possible

161 161 common divisor of 46 and 161, hence,common divisor of 46 and 161, hence,

4646 = = 46 46 ÷ ÷ 23 23 = = 2 2 161 161161 161 23 7 23 7

Finding the greatest Finding the greatest Common Divisor (g.c.d)Common Divisor (g.c.d) The greatest common divisor is called The greatest common divisor is called highest highest

commoncommon factorfactor. If the greatest common divisor is not . If the greatest common divisor is not readily apparent, the following steps are readily apparent, the following steps are recommended to determine the desirable divisor:recommended to determine the desirable divisor:

Divide the larger number of the numerator and Divide the larger number of the numerator and denominator by the smaller number.denominator by the smaller number.

If there is a remainder in step 1, divide the smaller If there is a remainder in step 1, divide the smaller number by the remainder.number by the remainder.

If there is still a remainder in step 2, divide the If there is still a remainder in step 2, divide the remainder in step 1 by the remainder in step 2.remainder in step 1 by the remainder in step 2.

Continue dividing each remainder by its succeeding Continue dividing each remainder by its succeeding remainder until the remainder is 0. The last divisor is remainder until the remainder is 0. The last divisor is the greater common divisor.the greater common divisor.

Example: Find the g.c.d. of Example: Find the g.c.d. of 6969 184184 22 69/184 step 1. 69/184 step 1. 69 is less than 18469 is less than 184 138138 4646 1 1 46 /69 step 2.46 /69 step 2. 4646 2323 22 23/46 step 3 and 423/46 step 3 and 4 4646 0 remainder0 remainder

Changing Fractions to Changing Fractions to DecimalDecimal There are two ways of expressing parts of a whole, There are two ways of expressing parts of a whole,

namely, common fractions and decimal fractions. namely, common fractions and decimal fractions. Most of the problems in business require converting Most of the problems in business require converting fractions into decimals. For example, if a fractions into decimals. For example, if a mathematical operation with fractions is performed mathematical operation with fractions is performed on a pocket calculator, all fractions must be on a pocket calculator, all fractions must be converted to decimals before entering them in the converted to decimals before entering them in the calculator.calculator.

Fraction indicates division, which means Fraction indicates division, which means that the numerator is to be divided by the that the numerator is to be divided by the denominator. An example is ¼ which is same as 1 denominator. An example is ¼ which is same as 1 divided by 4.divided by 4.

Example:Example: 0.40.4 2 = 2 ÷ 5 = 5/ 2.02 = 2 ÷ 5 = 5/ 2.0 2.02.0 00 The answer is 0.4The answer is 0.4

Changing Decimal to Changing Decimal to FractionsFractions

When a decimal fraction is written in a common When a decimal fraction is written in a common fractions, the decimal fraction without the decimal fractions, the decimal fraction without the decimal point is used as the numerator, and the point is used as the numerator, and the denominator is 1 with as many zeros annexed as denominator is 1 with as many zeros annexed as there are decimal places in the original decimal there are decimal places in the original decimal fraction. Then the common fraction is reduced to its fraction. Then the common fraction is reduced to its lowest term.lowest term.

Example:Example: express .5 to fractionexpress .5 to fraction

.5 = .5 = 55 = = 11 10 210 2 lowest termlowest term

Addition of fractionsAddition of fractions Two different things cannot be added. The same is true with Two different things cannot be added. The same is true with

numbers and fractions. numbers and fractions. To add fractions, all denominators must be converted to a To add fractions, all denominators must be converted to a

common number without changing the values of the common number without changing the values of the fraction.fraction.

When two or more fractions with different denominators are When two or more fractions with different denominators are to be added, we must change the fractions to their to be added, we must change the fractions to their equivalent fractions with the same denominator. The equivalent fractions with the same denominator. The procedure involves the convertion of the fractions to its procedure involves the convertion of the fractions to its higher or lower terms as discussed in the preceding topics.higher or lower terms as discussed in the preceding topics.

Example: Example: 11 & & 11 2 42 4

11 = = 1 x 21 x 2 = = 22 2 2 x 2 42 2 x 2 4 After converting 1/2 to ¼, lets add this to ¼ w/c has same After converting 1/2 to ¼, lets add this to ¼ w/c has same

denominatordenominator 2 2 ++ 1 1 = = 33 4 4 44 4 4

Solution:Solution:

1st step – Determine the 1st step – Determine the “lowest possible”“lowest possible” number number that can be divided by the different denominators that can be divided by the different denominators (this number is called least common denominator). In (this number is called least common denominator). In the example, the least common denominator is 12.the example, the least common denominator is 12.

2nd step – Determine the equivalent fractions of each 2nd step – Determine the equivalent fractions of each fraction with the same denominator as 12.fraction with the same denominator as 12.

11 == 1 x 61 x 6 == 66 22 2 x 6 2 x 6 12 12

3rd step – Add the equivalent fractions with the same 3rd step – Add the equivalent fractions with the same denominators.denominators.

11 + + 11 + + 1 1 = = 6 6 + + 44 + + 33 2 32 3 4 12 12 12 4 12 12 12

= = 6 + 4 + 36 + 4 + 3 1212

= 13/12 or 1 ½= 13/12 or 1 ½

Addition of Mixed Addition of Mixed NumbersNumbers When mixed numbers are added, When mixed numbers are added,

it is unnecessary to convert the it is unnecessary to convert the numbers to their equivalent numbers to their equivalent improper fractions. Add the whole improper fractions. Add the whole integers and the fractions integers and the fractions separately. The fractional part of separately. The fractional part of the answer should reduce to its the answer should reduce to its lowest term.lowest term.

Subtraction of FractionsSubtraction of Fractions

Fractions subtracted must first Fractions subtracted must first be converted to their equivalents be converted to their equivalents with the with the least commonleast common denominatordenominator (l.c.d.). then the (l.c.d.). then the numerators of the converted numerators of the converted fractions are subtracted. The fractions are subtracted. The answer will have the numerator answer will have the numerator difference as the numerator and difference as the numerator and the least common denominator the least common denominator as the denominator.as the denominator.

Subtracting a Mixed Subtracting a Mixed Number from a Whole Number from a Whole NumberNumber By way of an example, the By way of an example, the

following are the steps in following are the steps in subtracting a mixed from a subtracting a mixed from a whole number:whole number:

Step 1. Convert one unit of the minuend into an Step 1. Convert one unit of the minuend into an improper fraction with the same denominator as the improper fraction with the same denominator as the fraction in the subtrahend. Reduce the whole fraction in the subtrahend. Reduce the whole number by one so as not to change the value of the number by one so as not to change the value of the minuend.minuend.

12 = 11 8/812 = 11 8/8

Step 2. SubtractStep 2. Subtract

11 8/8 – 5 1/8 = (11-5)+(8/8 – 1/8)11 8/8 – 5 1/8 = (11-5)+(8/8 – 1/8)

= 6 + 7/8= 6 + 7/8

= 6 7/8= 6 7/8

Subtracting Mixed Subtracting Mixed NumbersNumbers When necessary, convert the fractional parts of When necessary, convert the fractional parts of

both the minuend and the subtrahend so that they both the minuend and the subtrahend so that they have a common denominator. If the fraction in the have a common denominator. If the fraction in the subtrahend is smaller than the fraction in the subtrahend is smaller than the fraction in the minuend, you can subtract, but if the fraction is minuend, you can subtract, but if the fraction is greater than the fraction in the minuend, follow the greater than the fraction in the minuend, follow the steps below.steps below.

Convert one unit of the minuend into an improper Convert one unit of the minuend into an improper fraction with “correct’ denominator, and add this unit fraction with “correct’ denominator, and add this unit to the existing fraction in the minuend.to the existing fraction in the minuend.

Reduce the whole number in the minuend by one Reduce the whole number in the minuend by one (the unit which is now a fraction).(the unit which is now a fraction).

You can now subtract the fraction.You can now subtract the fraction.

Multiplication of Multiplication of FractionsFractions There is no need to determine the common There is no need to determine the common

denominator of the fractions in multiplication. denominator of the fractions in multiplication. The basic principle in multiplication of fraction The basic principle in multiplication of fraction is to determine the product of both numerators is to determine the product of both numerators and both denominators and the products and both denominators and the products become the numerator and denominator of the become the numerator and denominator of the answer.answer.

In multiplying a whole number by a mixed In multiplying a whole number by a mixed number, change the mixed number to an number, change the mixed number to an improper fraction and determine the product.improper fraction and determine the product.

If both factors are mixed numbers, convert If both factors are mixed numbers, convert each mixed number to an improper fraction each mixed number to an improper fraction before multiplying.before multiplying.

Example: Multiply 1/2 by 2/3 Example: Multiply 1/2 by 2/3

1/2 x 2/3 = 1/2 x 2/3 = 1 x 2 1 x 2 == 2 2 or or 11 reduce to lowest term reduce to lowest term

2 x 3 = 6 32 x 3 = 6 3

Divisions of FractionsDivisions of Fractions

The simplest way to divide fractions is to The simplest way to divide fractions is to multiply the dividend by the reciprocal multiply the dividend by the reciprocal (inverted form) of the divisor. The (inverted form) of the divisor. The numbers 3 and 1/3 are called reciprocal numbers 3 and 1/3 are called reciprocal to each other. Likewise, 4/5 is the to each other. Likewise, 4/5 is the reciprocal of 5/4.reciprocal of 5/4.

When dividing a mixed number by When dividing a mixed number by another mixed number, change each another mixed number, change each mixed number to an improper fraction mixed number to an improper fraction and then multiply after inverting the and then multiply after inverting the divisor.divisor.

Example:Example:

Divide: 5/8 by 2/3Divide: 5/8 by 2/3

5 5 ÷÷ 22 = = 5 5 x x 3 3 88 3 8 2 3 8 2

= = 5 5 x x 3 3 = = 1515

8 2 168 2 16

Chapter 4 Chapter 4 Percent Percent

in in BusinessBusiness

The use of percent is gaining The use of percent is gaining wide acceptance in all spheres wide acceptance in all spheres of activity. When relating the of activity. When relating the parts to a whole, relationship is parts to a whole, relationship is expressed in percent.expressed in percent.

Percent is also frequently Percent is also frequently used in presenting accounting used in presenting accounting and statistical data relationship.and statistical data relationship.

Percent to FractionPercent to Fraction Percent numbers are not really “new” since Percent numbers are not really “new” since

they are in reality fractions.they are in reality fractions. To express 80% as a fraction in its lowest To express 80% as a fraction in its lowest

term, convert 80% in its fractional form first term, convert 80% in its fractional form first and perform the operation of division of and perform the operation of division of proper fractions as discussed in Chapter proper fractions as discussed in Chapter III.III.

80% = 80% = 8080 100100

Then = Then = 80 ÷ 2080 ÷ 20 100÷20100÷20 = = 44 5 5 thus, 80% is 4/5 in fractionthus, 80% is 4/5 in fraction

Percent to DecimalPercent to Decimal In any mathematical computation, percent is In any mathematical computation, percent is

converted to decimal before multiplying of dividing it converted to decimal before multiplying of dividing it with other quantifying numbers. There are several with other quantifying numbers. There are several percent values encountered in business percent values encountered in business transactions that do not convert into fractions transactions that do not convert into fractions making such percent values as decimals rather making such percent values as decimals rather than as fractions.than as fractions.

The basic principle in converting percent to The basic principle in converting percent to decimal is to move the decimal point two places to decimal is to move the decimal point two places to the left and drop the percent sign.the left and drop the percent sign.

Example 4. Change 15 % to a decimalExample 4. Change 15 % to a decimal

Solution:Solution:

15% 15% = = 15 15 = .15= .15 100100

Representing Decimals Representing Decimals as Percentsas Percents The fastest way to convert decimals to percent is to The fastest way to convert decimals to percent is to

the decimal point two places to the right. This the decimal point two places to the right. This method is actually the reverse order of converting method is actually the reverse order of converting percent to its decimal equivalent as discussed in the percent to its decimal equivalent as discussed in the preceding sections.preceding sections.

Example:Example:

0.25 = 0.25 = 0.25 x 100 = 25 %0.25 x 100 = 25 %

Representing Fractions Representing Fractions as Percentsas Percents In converting a fraction to percent, change In converting a fraction to percent, change

the fraction to its decimal equivalent as the fraction to its decimal equivalent as discussed in the previous chapter. Then discussed in the previous chapter. Then multiply the decimal equivalent by 100 or multiply the decimal equivalent by 100 or simply move the decimal point two places simply move the decimal point two places to the right and suffix the percent symbol.to the right and suffix the percent symbol.

Example: Convert 1 7/8 to its equivalent percent Example: Convert 1 7/8 to its equivalent percent form,form,

Solution:Solution:

1 1 77 = = (8 x 1) + 7(8 x 1) + 7 = = 1515 8 8 88 8 8 = 1.875 x 100= 1.875 x 100 = = 187.5 %187.5 %

Fraction-Decimal-Fraction-Decimal-Percent EquivalentsPercent Equivalents The equivalents discussed in the previous chapters The equivalents discussed in the previous chapters

can be tabulated into what are known as aliquot can be tabulated into what are known as aliquot parts of 100. An aliquot part is a portion of a number parts of 100. An aliquot part is a portion of a number by which the number may be divided leaving no by which the number may be divided leaving no remainderremainder

Example: Example:

Multiply 60 by 33 Multiply 60 by 33 1 1 % % 33

60 by 33 60 by 33 1 1 % % = 60 x= 60 x 33

= 20= 20

Determining the Determining the PercentagePercentage In the statement, 40 is 40/700 0f 700 the In the statement, 40 is 40/700 0f 700 the

denominator of the fraction which is the denominator of the fraction which is the basis of comparison is called the base. The basis of comparison is called the base. The rate is the percent indicating the number or rate is the percent indicating the number or quantity for every 100. The percentage quantity for every 100. The percentage refers to the number of items in the desired refers to the number of items in the desired situation or condition.situation or condition.

Based on the above concept, the principle Based on the above concept, the principle is written. is written.

P= b x r Equation 4.1P= b x r Equation 4.1

When:When: P = percentage P = percentage B = baseB = base r = rate, usually expresses in %r = rate, usually expresses in %

Stated as a principle, percentage is the product of Stated as a principle, percentage is the product of the base times the rate. In any mathematical the base times the rate. In any mathematical computation, before we add, subtract, multiply or computation, before we add, subtract, multiply or divide, we must convert the percent to its decimal or divide, we must convert the percent to its decimal or fractional equivalent whichever is convenient to fractional equivalent whichever is convenient to use. use.

Example: An employee who earned P 1,500 spent Example: An employee who earned P 1,500 spent 30 % of his earnings to buy a wrist watch. How 30 % of his earnings to buy a wrist watch. How much is the wrist watch?much is the wrist watch?

Given:Given: B = 1,500 (base) B = 1,500 (base) r = 30 %r = 30 %

Solution: P = B x rSolution: P = B x r = 1,500 x 30 %= 1,500 x 30 % = 1,500 x .3 = 1,500 x .3 = 450.00= 450.00

Finding the RateFinding the Rate

One of the most commonly used representations of One of the most commonly used representations of a number in business is the rate. Although a number in business is the rate. Although representation of number based on rate involves representation of number based on rate involves different interpretations in business or surveys.different interpretations in business or surveys.

Based on the preceding discussion, a Based on the preceding discussion, a formula can be derived from Equation 4.1 as formula can be derived from Equation 4.1 as follows:follows:

P = B x r Equation 4.1P = B x r Equation 4.1

r = P ÷ Br = P ÷ B

r= P/b Equation 4.2r= P/b Equation 4.2

Finding the BaseFinding the Base

Basically, the base quantity can be determined by Basically, the base quantity can be determined by applying the same concept in Equation 4.2. the applying the same concept in Equation 4.2. the formula is written below.formula is written below.

B = P ÷ r = P/r Equation 4.3B = P ÷ r = P/r Equation 4.3

Ratio and ProportionRatio and Proportion As mentioned earlier, percent is another way of As mentioned earlier, percent is another way of

comparing a number with the whole, while its comparing a number with the whole, while its equivalent in decimal is comparing with one unit.equivalent in decimal is comparing with one unit.

Ratio Ratio is a relation between two numbers expressed is a relation between two numbers expressed in terms of a quotient.in terms of a quotient.

The result of reducing ratios to lowest terms is The result of reducing ratios to lowest terms is called equivalent reduced ratio.called equivalent reduced ratio.

The equality of two ratios is called The equality of two ratios is called proportionproportion..

Chapter Chapter 5: 5:

EmployeeEmployee’s ’s

CompensCompensationation

Among other expenses, a business enterprise pays Among other expenses, a business enterprise pays the salaries to its personnel. Salary is the the salaries to its personnel. Salary is the renumeration received by an employees for renumeration received by an employees for services rendered. The labor code of the services rendered. The labor code of the Philippines defines wages or salaries as “ the Philippines defines wages or salaries as “ the renumeration or earning, however designated, renumeration or earning, however designated, capable of being expressed in terms of money, capable of being expressed in terms of money, whether fixed or ascertained on a time, task, piece whether fixed or ascertained on a time, task, piece or commissions basis or other method of calculating or commissions basis or other method of calculating the same which is payable by an employer to an the same which is payable by an employer to an employee and includes the fair and reasonable employee and includes the fair and reasonable value of board, lodging or other facilities customarily value of board, lodging or other facilities customarily furnished by the employer to the employee.”furnished by the employer to the employee.”

Salaries Salaries may be based on period of time, may be based on period of time, production, or percentage on an agreed basis. production, or percentage on an agreed basis. Employees may be classified according to the Employees may be classified according to the bases of their salary payments. There are the fixed bases of their salary payments. There are the fixed wags earners, the piece-workers, the commission wags earners, the piece-workers, the commission earners and the salary-plus-commission earners.earners and the salary-plus-commission earners.

Computing Wages Computing Wages Based on Time (period)Based on Time (period)

Fixed wage earnersFixed wage earners are paid based on the number are paid based on the number of hours worked. For work done beyond the regular of hours worked. For work done beyond the regular eight (8) working hours a day, the employee is eight (8) working hours a day, the employee is given additional pay called given additional pay called overtime pay.overtime pay.

The Labor Code fixes the minimum rates of The Labor Code fixes the minimum rates of overtime pay, night shift differential, and work on overtime pay, night shift differential, and work on regular and special holidays. The rates range from regular and special holidays. The rates range from 25 %, 30% to 100% over and above the regular pay 25 %, 30% to 100% over and above the regular pay rate. For purposes of discussion, overtime pay as rate. For purposes of discussion, overtime pay as herein used will be regular pay plus 50% of said pay herein used will be regular pay plus 50% of said pay which means that for every one hour overtime, the which means that for every one hour overtime, the pay will be equivalent to 1.5 of the regular hourly pay will be equivalent to 1.5 of the regular hourly rate.rate.

Computing Wages on Computing Wages on Piecework BasisPiecework Basis The minimum rates set by the Labor Code The minimum rates set by the Labor Code

do not apply to employees whose salaries do not apply to employees whose salaries are not based on the number of working are not based on the number of working hours. Employees may be paid by results hours. Employees may be paid by results like like “pakyaw” (pakiao),“pakyaw” (pakiao), “takay”“takay” and by and by the pieces of product.the pieces of product.

Workers in selected industries such Workers in selected industries such as those engaged in manufacturing, as those engaged in manufacturing, packing, and handicrafts are paid on packing, and handicrafts are paid on straight pieceworkstraight piecework basis. The wage of a basis. The wage of a straight pieceworker depends solely on straight pieceworker depends solely on number of units completed each period of number of units completed each period of time.time.

Computing Wages on Computing Wages on Commission BasisCommission Basis Most of the commission wage earners are staff in Most of the commission wage earners are staff in

the sales of a business. They are paid on the basis the sales of a business. They are paid on the basis of number of units sold. Since sales is the bloodline of number of units sold. Since sales is the bloodline of the company, the sales people are doubly of the company, the sales people are doubly motivated to increase sales output. The three most motivated to increase sales output. The three most commonly used compensation schemes in sales commonly used compensation schemes in sales are the straight commission, commission and are the straight commission, commission and bonus, and salary plus commission.bonus, and salary plus commission.

The regular sales force of a company is The regular sales force of a company is usually paid a fixed monthly rate plus commission usually paid a fixed monthly rate plus commission earned from sales.earned from sales.

To compute for the commission, multiply To compute for the commission, multiply the sales (in number of units or peso value) by the the sales (in number of units or peso value) by the commission rate.commission rate.

Incentive or BonusesIncentive or Bonuses

Incentive pay and bonuses in money or in Incentive pay and bonuses in money or in kind are given to employees to motivate kind are given to employees to motivate them. The incentive pay may be based on them. The incentive pay may be based on the number of units completed or sold in the number of units completed or sold in excess of a required minimum. excess of a required minimum.

DeductionsDeductions

The earnings of an employee are subject to The earnings of an employee are subject to deductions which are required by law of by deductions which are required by law of by agreement between the employer and the agreement between the employer and the employee. Deductions required by law are employee. Deductions required by law are income taxes, social security taxes, income taxes, social security taxes, medicare contributions, contributions to medicare contributions, contributions to pension plans and contributions to pag-pension plans and contributions to pag-ibig.ibig.

Chapter Chapter 6: Buying 6: Buying

and and SellingSelling

Buying and selling are the essential Buying and selling are the essential functions of a trading concern. Manufacturers functions of a trading concern. Manufacturers or producers buy raw materials or parts to or producers buy raw materials or parts to produce finished products. A trading concern produce finished products. A trading concern tries to buy merchandise at the lowest possible tries to buy merchandise at the lowest possible cost in order to maximize its profits. A cost in order to maximize its profits. A manufacturer endeavors to acquire materials manufacturer endeavors to acquire materials and/or finished parts in order to minimize its and/or finished parts in order to minimize its cost of production which is a way of increasing cost of production which is a way of increasing the income upon the sale of the finished the income upon the sale of the finished products.products.

In a trading business, the trader of the In a trading business, the trader of the buyer should determine the goods needed by buyer should determine the goods needed by the customers and the price the customers are the customers and the price the customers are willing to pay for the goods. He should also willing to pay for the goods. He should also know where to obtain the goods at the lowest know where to obtain the goods at the lowest price to minimize cost and maximize profit. price to minimize cost and maximize profit.

In general, a manufacturer sells its products In general, a manufacturer sells its products in bulk to a wholesaler. In turn, the wholesaler in bulk to a wholesaler. In turn, the wholesaler sells in smaller lots to retailers who sell the sells in smaller lots to retailers who sell the product to the end users or consumers.product to the end users or consumers.

DiscountsDiscounts Discounts are of two types, namely:Discounts are of two types, namely:

Trade discountTrade discount. It is a discount offered by a seller . It is a discount offered by a seller to induce trading. This is usually offered by a to induce trading. This is usually offered by a manufacturer or wholesaler. This type is usually manufacturer or wholesaler. This type is usually encountered in catalogs where a list price is printed encountered in catalogs where a list price is printed together with the trade discount thereon. The list together with the trade discount thereon. The list price less the trade discount is the suggested price less the trade discount is the suggested selling price upon release of the product.selling price upon release of the product.

Cash discountCash discount. . It is a reduction on the selling It is a reduction on the selling price offered to a buyer to induce him to pay price offered to a buyer to induce him to pay promptly.promptly.

Finding the Net PriceFinding the Net Price There are two methods in determining the net price There are two methods in determining the net price

to the buyer when trade discount is given. These to the buyer when trade discount is given. These are the are the Discount Rate MethodDiscount Rate Method and the and the Net Price Net Price Rate Method.Rate Method.

Method A. Method A. Discount Rate Method.Discount Rate Method. The formula The formula for discount is the same for percentage where the for discount is the same for percentage where the terms percentage (P), base (b) and rate (r) in terms percentage (P), base (b) and rate (r) in Equation 4.1 on page 56 are substituted with the Equation 4.1 on page 56 are substituted with the terms discount (D), selling or list price (L.P) and terms discount (D), selling or list price (L.P) and discount rate (d) respectively. Thus,discount rate (d) respectively. Thus,

D = L.P. x d Equation 6.1D = L.P. x d Equation 6.1 L.P.L.P. = Selling price or list price = Selling price or list price DD = Discount rate usually expressed in = Discount rate usually expressed in

percentpercent

The net price then is the difference between The net price then is the difference between the selling price and the discount which is shown the selling price and the discount which is shown below.below.

N.P. = L.P. − D Equation 6.2N.P. = L.P. − D Equation 6.2 N.P. = L.P. x (net price rate) N.P. = L.P. x (net price rate) = L.P. x (100% - d in%) Equation 6.3= L.P. x (100% - d in%) Equation 6.3

Example 1. The list price of a cassette tape Example 1. The list price of a cassette tape recorder at AVESCO is P1, 850. The Trade recorder at AVESCO is P1, 850. The Trade discount rate is 12 %. How much will the buyer pay discount rate is 12 %. How much will the buyer pay for the recorder? Use for the recorder? Use Discount Rate MethodDiscount Rate Method and and the the Net Price Rate Method.Net Price Rate Method.

Given: Given: L.P. = P1,850L.P. = P1,850 d = 12% d = 12%

Solutions:Solutions: Method A.Method A. Discount Rate MethodDiscount Rate Method PhPPhP1,850 (L.P.)1,850 (L.P.) X .12X .12 (d) (d) P 222 (D)P 222 (D)

Determining the net price:Determining the net price: PhPPhP1,850 (L.P.)1,850 (L.P.) ─ ─ 222 222 (D)(D) Php1,628 (N.P) Php1,628 (N.P) ============== Method B.Method B. Using Equation 6.3Using Equation 6.3 N.P. = L.P. x N.P.RN.P. = L.P. x N.P.R = = PhpPhp1,850 x (100%-12%)1,850 x (100%-12%) = = PhpPhp1,850 x 88% = 1, 850 x .881,850 x 88% = 1, 850 x .88 = = PhpPhp1, 628.001, 628.00

Finding the Discount Finding the Discount RateRate A supplier may list only the net price and list price of A supplier may list only the net price and list price of

the products being sold. The buyer should know the products being sold. The buyer should know how to determine the discount rate in order to how to determine the discount rate in order to evaluate the reasonableness of the discount evaluate the reasonableness of the discount offered.offered.

Example 2. What is the discount rate if the Example 2. What is the discount rate if the list price is P1,500 and the discount is P450.00?list price is P1,500 and the discount is P450.00?

Given: L.P. = P1,500, D = P450Given: L.P. = P1,500, D = P450 Solution: Solution: Substituting the values in Equation 6.1Substituting the values in Equation 6.1 D = L.P. x dD = L.P. x d 450450PhpPhp = 1,500 = 1,500Php Php x dx d d = d = 450 450 = .3 or 30%= .3 or 30% 1,5001,500

Example 3.Example 3. What is the discount rate, if the What is the discount rate, if the net price is 246 and the discount is P83.00?net price is 246 and the discount is P83.00?

Given: N.P.= P246Given: N.P.= P246 D = P83 D = P83 Solution: Solution: N.P. = L.P. ─ DN.P. = L.P. ─ D N.P. = is net priceN.P. = is net price L.P. = N.P. + DL.P. = N.P. + D = P246 + P83= P246 + P83 = P 329.00= P 329.00 Substituting the list price in Equation Substituting the list price in Equation

6.16.1:: D = L.P. x dD = L.P. x d d = D/L.P.d = D/L.P. = 83/329 = 83/329 = .252 or 25.2%= .252 or 25.2%

Discounts in SeriesDiscounts in Series To dispose their goods quickly, wholesalers may To dispose their goods quickly, wholesalers may

offer successive trade discount rates which we call offer successive trade discount rates which we call discounts in series.discounts in series. If a manufacturing concern If a manufacturing concern would give 15% and 10% trade discounts on its would give 15% and 10% trade discounts on its product, this does not mean that the total discount product, this does not mean that the total discount rate is 25%. As a discount series, it simply means rate is 25%. As a discount series, it simply means that the first discount of 15% is applied to the original that the first discount of 15% is applied to the original list price and the second discount of 10% to the list price and the second discount of 10% to the balance of the original list price and the first discount balance of the original list price and the first discount price. price.

Example 4. Example 4. A home appliance dealer was A home appliance dealer was offered television set with a list price of P5,680 less 15% offered television set with a list price of P5,680 less 15% and 10%. What is the net price?and 10%. What is the net price?

Given:Given: List Price (L.P)List Price (L.P) = P5,680 = P5,680 First discount rate (d1)First discount rate (d1) = 15% = 15% Second discount rate(d2Second discount rate(d2) = 10%) = 10% Solution:Solution: Method A. Method A. Discount Rate MethodDiscount Rate Method 1rst Step1rst Step. Using Equation 6.1. Using Equation 6.1 D1 = L.P.1 x d1D1 = L.P.1 x d1 = P5,680 x .15= P5,680 x .15 = P852.00= P852.00

2nd Step. To get the balance (L.P2) on which the 2nd Step. To get the balance (L.P2) on which the second discount rate (d2) will be applied, subtract second discount rate (d2) will be applied, subtract the first discount rate from the list price–the first discount rate from the list price–

PhPPhP 5,680 (L.P1) 5,680 (L.P1) ─ ─ 852 852 (d1)(d1) PhpPhp 4,828 (L.P2) 4,828 (L.P2) ======= ======= 3rd Step. Solve for D2 using Equation 13rd Step. Solve for D2 using Equation 1 D2 = L.P2 x d2 D2 = L.P2 x d2 = 4,828 x .10= 4,828 x .10 = P482.80= P482.80 4rth Step. Solve for the net price (N.P.) by 4rth Step. Solve for the net price (N.P.) by

subtracting the second discount (D2) from L.P2subtracting the second discount (D2) from L.P2

PhpPhp 4,828 (L.P2) 4,828 (L.P2) 482482 (D2) (D2) PhpPhp 4,345.20 (N.P.) 4,345.20 (N.P.) ==============

Method B. Method B. Net Price Net Price Rate MethodRate Method 1rst Step1rst Step. The net price rate is the difference . The net price rate is the difference

between 100% and the discount rate (See Equation between 100% and the discount rate (See Equation 6.3). The same formula is also applied when two or 6.3). The same formula is also applied when two or more discounts are given. The final net price is the more discounts are given. The final net price is the product of all the net price rates found by using product of all the net price rates found by using Equation 6.3.Equation 6.3.

Final net price rate = (100% ─ d1) x (100% - d2)Final net price rate = (100% ─ d1) x (100% - d2) = (100% ─ = (100% ─

15%) (100% - 10%)15%) (100% - 10%) = 85% x 90%= 85% x 90% = .85 x .90= .85 x .90

= .765= .765 2nd Step2nd Step. The net price is the product of the list price . The net price is the product of the list price

and the final net price rate.and the final net price rate.

Net price = Php 5,680 x .765Net price = Php 5,680 x .765 = Php 4,345.20= Php 4,345.20

Obviously, the net price rate method is Obviously, the net price rate method is relatively shorter and simpler than the discount relatively shorter and simpler than the discount method, more so when there are more than two method, more so when there are more than two series of discounts. In such cases, the formula to series of discounts. In such cases, the formula to use in determining the net price isuse in determining the net price is::

N.P. = L.P. x (100% ─ d1) (100% ─ d3) N.P. = L.P. x (100% ─ d1) (100% ─ d3) (100% ─ dn) Equation 6.4 (100% ─ dn) Equation 6.4

Where dn indicates the last discount rateWhere dn indicates the last discount rate

Note: Note: It is immaterial as to which order the It is immaterial as to which order the discounts are taken.discounts are taken.

Single-Discount Rate Single-Discount Rate EquivalentEquivalent

There are two practical ways to evaluate discount There are two practical ways to evaluate discount series offered by two or more sellers. Let us series offered by two or more sellers. Let us assume that in terms of services, neither firm held assume that in terms of services, neither firm held any advantage over the other. Hence, the buyer’s any advantage over the other. Hence, the buyer’s only concern is to determine which firm’s net selling only concern is to determine which firm’s net selling price was smaller. As earlier discussed, the price was smaller. As earlier discussed, the discount series of, for example, 10% and 15% are discount series of, for example, 10% and 15% are not equivalent to the total discount rate of 25%.not equivalent to the total discount rate of 25%.

One way of evaluating discount series is to One way of evaluating discount series is to determine the final net price rate which was determine the final net price rate which was illustrated in Example 4, Method B. From the illustrated in Example 4, Method B. From the buyer’s point of view, the smaller the final net price buyer’s point of view, the smaller the final net price rate, the lesser is the net price.rate, the lesser is the net price.

An easier way to evaluate the discount An easier way to evaluate the discount rates in a discount series is therates in a discount series is the equivalent single equivalent single discount ratediscount rate.. From the buyer’s point of view, the From the buyer’s point of view, the bigger the single discount rate equivalent, the lesser bigger the single discount rate equivalent, the lesser is the net price.is the net price.

Example 5Example 5. Charing’s Beauty Parlor was offered beauty . Charing’s Beauty Parlor was offered beauty product with trade discounts of 15% and 10%. Find the product with trade discounts of 15% and 10%. Find the single discount rate equivalent.single discount rate equivalent.

Given: d1 = 15% Given: d1 = 15% d2 = 10%d2 = 10%

Solution:Solution: First StepFirst Step. Derive the net price rate (NPR) of . Derive the net price rate (NPR) of

each discount rate.each discount rate. Since NPR = 100% ─ dSince NPR = 100% ─ d a. NPR1a. NPR1 = 100% ─ d1 = 100% ─ d1 = 100% ─ 15%= 100% ─ 15% = 85%= 85% b. NPR2 = 100% ─ 10%b. NPR2 = 100% ─ 10% = 90%= 90% Second Step.Second Step. Determine the single discount rate Determine the single discount rate

equivalent (SDRE).equivalent (SDRE). SDRE = 100% ─ (NPR1 x SDRE = 100% ─ (NPR1 x

NPR2)NPR2) = 100% ─ (85% x 90|%)= 100% ─ (85% x 90|%) = 100% ─ (.85 x .90)= 100% ─ (.85 x .90) = 100% ─ (.765)= 100% ─ (.765) = 23.5%= 23.5%

Example 6.Example 6. A product is being offered in the market by A product is being offered in the market by Supplier A and Supplier B. The trade discounts being Supplier A and Supplier B. The trade discounts being offered by Supplier A are 20% and 10% while that of offered by Supplier A are 20% and 10% while that of Supplier B are 15% and 15%. If you were the buyer, Supplier B are 15% and 15%. If you were the buyer, which is the better offer?which is the better offer?

Given:Given: Supplier A =Supplier A = 20% and 10% (d1 and d2) 20% and 10% (d1 and d2) Supplier B =Supplier B = 15% and 15%. (d1 and d2) 15% and 15%. (d1 and d2) Solution:Solution:

Supplier A. Solving for the single discount rate equivalent Supplier A. Solving for the single discount rate equivalent of Supplier A, determine first the net price rate applicable of Supplier A, determine first the net price rate applicable

NPR1 = 100% - 20% = 80%NPR1 = 100% - 20% = 80% NPR2 = 100% - 10% = 90%NPR2 = 100% - 10% = 90% Therefore, Therefore, SDRE = 100% - (80% x 90%)SDRE = 100% - (80% x 90%) = 100%- (.80 x .90)= 100%- (.80 x .90) = 100% - (.72)= 100% - (.72) = 100% - 72%= 100% - 72% = 28%= 28%

Supplier B. Solve for the net price applicable to the Supplier B. Solve for the net price applicable to the single discount rate equivalent of Supplier B.single discount rate equivalent of Supplier B.

NPR1NPR1 = 100% - 15% = 85%= 100% - 15% = 85% NPR2 = 100% - 15% = 85%NPR2 = 100% - 15% = 85% Therefore,Therefore, SDRE = 100% - (85% x 85%)SDRE = 100% - (85% x 85%) = 100%- (.85 x .85)= 100%- (.85 x .85) = 100% - (.7225)= 100% - (.7225) = 100% - 72.25%= 100% - 72.25% = 27.75%= 27.75%

Based on the net price rates being offered, Based on the net price rates being offered, the offer of a lower net price rate (72%) by Supplier the offer of a lower net price rate (72%) by Supplier A is the better offer. But, based on the single A is the better offer. But, based on the single discount rate equivalent, the offer of a higher discount rate equivalent, the offer of a higher discount (28%) by Supplier A is also the better offer.discount (28%) by Supplier A is also the better offer.

Cash DiscountsCash Discounts To encourage wholesalers or retailers to buy is one thing To encourage wholesalers or retailers to buy is one thing

and to induce the buyers to pay their purchases is another and to induce the buyers to pay their purchases is another thing. While sellers give buyers credit lines in that the thing. While sellers give buyers credit lines in that the buyer are given several days after delivery within which to buyer are given several days after delivery within which to pay their purchases, sellers encourage buyers to pay pay their purchases, sellers encourage buyers to pay promptly their purchases by offering cash discounts. promptly their purchases by offering cash discounts. AA cash discountcash discount should not be confused with a trade should not be confused with a trade discount. The former is a reduction in the discount. The former is a reduction in the net pricenet price or or invoice priceinvoice price upon payment on a specified period of time upon payment on a specified period of time while the latter is a deduction from the list price.while the latter is a deduction from the list price.

The cash discount rates offered are clearly The cash discount rates offered are clearly written on the invoice. An example of a cash discount written on the invoice. An example of a cash discount is”is”5,5, n ”n ”

10 3010 30 which means that a cash discount of 5% will be deducted which means that a cash discount of 5% will be deducted from the net price if the invoice paid within 10 from the net price if the invoice paid within 10

days from the date of the invoice. The "n/30" means that days from the date of the invoice. The "n/30" means that after 10 days, no discount is given and that the invoice after 10 days, no discount is given and that the invoice price should be paid not later than 30 days of the invoice price should be paid not later than 30 days of the invoice date.date.

If the retailer or buyer would avail of the cash If the retailer or buyer would avail of the cash discount offered, the price to be paid is the invoice price discount offered, the price to be paid is the invoice price less the amount of cash discount.less the amount of cash discount.

Example 7. An invoice with a list price of P18,200 Example 7. An invoice with a list price of P18,200 dated October 12, has trade discounts of 15% and dated October 12, has trade discounts of 15% and 10% and the terms are 4/10 ,2/30 ,10% and the terms are 4/10 ,2/30 , n/60. How n/60. How much must be paid if the invoice is settled on (a) much must be paid if the invoice is settled on (a) October 22? (b) November 11?October 22? (b) November 11?

Given:Given: List price = P 18,200List price = P 18,200 Trade Discount series = 15% and 10%Trade Discount series = 15% and 10% Cash discounts = Cash discounts = 4 4 , , 2 2 , and , and nn 10 30 6010 30 60 SolutionSolution: : (a.) If the amount is settled on October 22(a.) If the amount is settled on October 22 1rst Step. Solve first the net price (N.P.)1rst Step. Solve first the net price (N.P.) N.P. = L.P. x (NPR1) (NPR2)N.P. = L.P. x (NPR1) (NPR2) = P 18,200 x (100% - 15%) = P 18,200 x (100% - 15%)

(100% - 10%)(100% - 10%) = 18,200 x (85%) (90%)= 18,200 x (85%) (90%) = 18,200 x (.85) (.90)= 18,200 x (.85) (.90) = P 13,923.00= P 13,923.00

2nd Step.2nd Step. Having solve for the N.P., which is Having solve for the N.P., which is P13,923.00, we can now proceed to solved the cash P13,923.00, we can now proceed to solved the cash discount (Dc). If the purchaser pays on October 22 which discount (Dc). If the purchaser pays on October 22 which is 10 days from the invoice date, the cash discount term is is 10 days from the invoice date, the cash discount term is 4/10. Thus, the cash discount rate is 4%4/10. Thus, the cash discount rate is 4%

Dc = P13, 923 x 4%Dc = P13, 923 x 4% = 13, 923 x .04= 13, 923 x .04 = P556.92= P556.92 3rd Step3rd Step. Determine the net invoice price (NIP) by simply . Determine the net invoice price (NIP) by simply

subtracting cash discount from the net price.subtracting cash discount from the net price. NIP = N.P. - Dc NIP = N.P. - Dc = P13, 923 - P556.92= P13, 923 - P556.92 = P13,366.08= P13,366.08

Alternatively, the computation may be as follows:Alternatively, the computation may be as follows: NIP = N.P. x (100% - dc)NIP = N.P. x (100% - dc) = (100% - 4%)= (100% - 4%) = 13,923 x (96%)= 13,923 x (96%) = 13,923 x (.96)= 13,923 x (.96) = P 13, 666.08= P 13, 666.08 If the invoice is paid on November 11.If the invoice is paid on November 11. First StepFirst Step. Same as solution (a.).. Same as solution (a.). N.P. = P 13,923.00N.P. = P 13,923.00

2nd Step.2nd Step. If the Purchaser pays on November 11 If the Purchaser pays on November 11 which is 30 days from the invoice date, the cash which is 30 days from the invoice date, the cash discount rate is 2%.discount rate is 2%.

Dc = 13,923 x 2% = P278.46Dc = 13,923 x 2% = P278.46 NIP = 13,923 – P278.46NIP = 13,923 – P278.46 = P 13,644.54= P 13,644.54 Alternatively, the computation may be as Alternatively, the computation may be as

follows:follows: NIP = N.P. x (100% - dc)NIP = N.P. x (100% - dc) = P 13, 923 x (100% - 2%)= P 13, 923 x (100% - 2%) = 13, 923 x (98%)= 13, 923 x (98%) = 13, 923 x .98= 13, 923 x .98 = P 13,644.54= P 13,644.54

Mathematics of PricingMathematics of Pricing The price of a commodity is determined by many The price of a commodity is determined by many

factors obtaining in the market. It is greatly affected factors obtaining in the market. It is greatly affected by the need of the consumers for the commodity, its by the need of the consumers for the commodity, its availability, its quality level and the number of availability, its quality level and the number of different lines with similar products. If there is a different lines with similar products. If there is a great demand for the commodity and the stock is great demand for the commodity and the stock is limited, the seller could dictate his own selling price. limited, the seller could dictate his own selling price. But if the price demanded is prohibitive such that But if the price demanded is prohibitive such that the average consumer could not afford it, the the average consumer could not afford it, the consumers may shift to other available similar consumers may shift to other available similar products and the seller may stand to lose.products and the seller may stand to lose.

The business should understand pricing The business should understand pricing terms such as cost, mark-up, margin and the like to terms such as cost, mark-up, margin and the like to enable himself to price his commodities reasonably. enable himself to price his commodities reasonably. Cost Cost refers to the amount the purchaser acquired refers to the amount the purchaser acquired the good. the good. MarginMargin (gross profit) is the excess of (gross profit) is the excess of selling price over cost from viewpoint of the seller. selling price over cost from viewpoint of the seller. If an original selling price is adjusted upward, the If an original selling price is adjusted upward, the addition is called addition is called Mark-upMark-up.. The reverse is called The reverse is called Mark-down.Mark-down.

To illustrate clearly the above terminologies, let To illustrate clearly the above terminologies, let us take an example. A car dealer bought a car us take an example. A car dealer bought a car for P14,500.00 and sold it for P17,200.00.for P14,500.00 and sold it for P17,200.00.

P14,500 is the cost of the article, P14,500 is the cost of the article, assuming he did not spend any amount for assuming he did not spend any amount for recondition the car before selling it.recondition the car before selling it.

P 17,200 is the selling price, tag price, P 17,200 is the selling price, tag price, or retail price of the article. (P17,200 – P14,500) or retail price of the article. (P17,200 – P14,500) = P2,700 is the gross profit or margin.= P2,700 is the gross profit or margin.

In the above illustration, the margin In the above illustration, the margin P2,700 can be expressed in percent based in P2,700 can be expressed in percent based in either the selling price or cost. This is called either the selling price or cost. This is called percent margin.percent margin. If the selling price is used as If the selling price is used as the base, the the base, the percent margin on selling pricepercent margin on selling price is is P2,700P2,700 = .15697 = 15.7%. = .15697 = 15.7%.

If the cost is P17,200 used as the base, the If the cost is P17,200 used as the base, the percent percent margin margin on cost is on cost is P2,700 P2,700 = .1862 =1 =18.62% = .1862 =1 =18.62%

P14,500P14,500

Formula-wise, the equations for the above are the Formula-wise, the equations for the above are the following:following:

Cost + Margin = Selling priceCost + Margin = Selling price OrOr Margin = Selling price – CostMargin = Selling price – Cost Percent Margin(%) = Percent Margin(%) = Margin Margin x 100, based on x 100, based on

selling priceselling price Selling priceSelling price Percent Margin(%) = Percent Margin(%) = Margin Margin x 100, based on cost x 100, based on cost CostCost

Example 8.Example 8. The merchandise that cost a dealer The merchandise that cost a dealer P4,150 was sold for P5,200. Determine the percent P4,150 was sold for P5,200. Determine the percent margin (a) based on selling price, (b) based on margin (a) based on selling price, (b) based on cost.cost.

Given:Given: Selling price = P5,200Selling price = P5,200 Cost = P4,150Cost = P4,150 Solution: Solution: Immediately subtract P4,150 Immediately subtract P4,150

from P5,200 to get the margin.from P5,200 to get the margin. Margin Margin = Selling price - Cost = Selling price - Cost = P5,200 -= P4,150= P5,200 -= P4,150 = P 1,050= P 1,050 If the percent margin is required based on selling If the percent margin is required based on selling

price divide the margin by the selling price.price divide the margin by the selling price.

Percent Margin (%)Percent Margin (%) = = P1,050P1,050 x 100 x 100 P5,200P5,200 = .202 x 100= .202 x 100 = 20.2%= 20.2%

If percent margin is required based on cost, If percent margin is required based on cost, divide the margin by the cost.divide the margin by the cost.

Percent Margin (%)Percent Margin (%) = = P1,050P1,050 x 100 x 100 P4,150P4,150 = .253 x 100= .253 x 100 = 25.3%= 25.3% NoteNote: : Another term used for gross margin is Another term used for gross margin is

mark on.mark on.

Chapter Chapter 7: Simple 7: Simple InterestInterest

Money borrowed usually bears a cost called Money borrowed usually bears a cost called interest.interest. The amount borrowed is called The amount borrowed is called principal.principal.

Sources of loans are the banks, investment Sources of loans are the banks, investment houses, savings and loan associations, cooperatives, houses, savings and loan associations, cooperatives, credit unions, and other financing companies.credit unions, and other financing companies.

Determining the InterestDetermining the Interest The interest or interest charge is usually expressed in The interest or interest charge is usually expressed in

percent, such as 6%,which means a charge of P6.00 percent, such as 6%,which means a charge of P6.00 for every P100.00 for a definite period of time. Unless for every P100.00 for a definite period of time. Unless otherwise stated, quoted interest rate is for one year.otherwise stated, quoted interest rate is for one year.

The formula for computing interest or interest charge The formula for computing interest or interest charge is:is:

I = P x r x t Equation 7.1I = P x r x t Equation 7.1 Where:Where: I = interest expressed in monetary I = interest expressed in monetary

value.value. P = principal or the amount borrowed.P = principal or the amount borrowed. R = interest rate on the principal, R = interest rate on the principal,

usually stated in percent.usually stated in percent. T = time or duration of the loan, T = time or duration of the loan,

expressed in number of years, months, expressed in number of years, months, days, etc.days, etc.

To illustrate, if Mr. X borrowed P2,600.00 from the To illustrate, if Mr. X borrowed P2,600.00 from the bank at an interest of 5%, what is the interest in 3 bank at an interest of 5%, what is the interest in 3 years?years?

Given:Given: P = P2,600.00P = P2,600.00 r = 5% (per year)r = 5% (per year) t = 3 yearst = 3 years I = to findI = to find

Solution:Solution: Substituting the values in Equation 7.1Substituting the values in Equation 7.1

I = P x r x tI = P x r x t = P 2,600.00 x 5% x 3= P 2,600.00 x 5% x 3 = 2,600 x .05 x 3= 2,600 x .05 x 3 = P390.00= P390.00

The amount to be paid by the borrower upon maturity of the loan The amount to be paid by the borrower upon maturity of the loan is – is –

WhereWhere:: A = sum or total amount to be paid.A = sum or total amount to be paid. P = principalP = principal I = interestI = interest Thus,Thus, A = P 2,600 + P390.00A = P 2,600 + P390.00 = P2,990.00= P2,990.00 Example 1. Pedro borrowed from Jose P5,200 with an Example 1. Pedro borrowed from Jose P5,200 with an

interest of 6%. How much should Pedro pay after 16 interest of 6%. How much should Pedro pay after 16 months?months?

Given:Given: P = P5,200P = P5,200 r = 6% (per year)r = 6% (per year) t = 16 monthst = 16 months Solution:Solution: Method A. Convert 16 months to number of years.Method A. Convert 16 months to number of years. I = P x r x tI = P x r x t = P 5,200 x 6% x = P 5,200 x 6% x 1616 1212 = 5,200 x .06 x = 5,200 x .06 x 44 33 = P 416.00= P 416.00

Method B. Convert interest rate per month.Method B. Convert interest rate per month. I = P x r x tI = P x r x t = P 5,200 x 6%/12n x 16 months= P 5,200 x 6%/12n x 16 months 1212 = 5,200 x .5% x 16= 5,200 x .5% x 16 = 5,200 x.005 x16= 5,200 x.005 x16 = P 416.00= P 416.00

The amount to be paid after 16 months is –The amount to be paid after 16 months is – A= P + IA= P + I = P5,200 + P416= P5,200 + P416 = P5,616.00= P5,616.00

Determining the Rate Determining the Rate of Interest (r)of Interest (r) This particular problem is encountered in evaluating two or This particular problem is encountered in evaluating two or

more alternative choices. A lender wants a higher rate of more alternative choices. A lender wants a higher rate of interest for his money, while a borrower prefers a lesser rate of interest for his money, while a borrower prefers a lesser rate of interest on money borrowed.interest on money borrowed.

Assume that Mr. Y would like to borrow P5,000 payable after Assume that Mr. Y would like to borrow P5,000 payable after one (1) year from a bank whose interest charge is 18% per one (1) year from a bank whose interest charge is 18% per year. On the other hand, Mr. Z is willing to lend him the same year. On the other hand, Mr. Z is willing to lend him the same principal and interest charge is P950 per year. Since the rate principal and interest charge is P950 per year. Since the rate of interest on the bank loan is already know, Mr. Y must know of interest on the bank loan is already know, Mr. Y must know the rate of interest on Mr. Z’s to enable him to determine which the rate of interest on Mr. Z’s to enable him to determine which is the better offer.is the better offer.

Solution:Solution: r = r = I I Equation 7.4 Equation 7.4 PtPt == 950950 = 0.19= 0.19 5,0005,000 = 0.19 x 100%= 0.19 x 100% = 19%= 19% Thus, as compared to Mr. Y’s proposal of a 19% interest Thus, as compared to Mr. Y’s proposal of a 19% interest

rate per year, the bank loan with 18% interest rate per rate per year, the bank loan with 18% interest rate per year is definitely the better offer.year is definitely the better offer.

Determining the Determining the Time (t)Time (t) As derived from Equation 7.1As derived from Equation 7.1 t = t = I I Equation 7.5 Equation 7.5 PrPr Example 3. How long will it take for a deposit of Example 3. How long will it take for a deposit of

P1,500.00 to earn P186.00 invested at the rate of 7- P1,500.00 to earn P186.00 invested at the rate of 7- 1/2%?1/2%?

Given:Given: I = P186.00I = P186.00 P = P1,500.00P = P1,500.00 r = 7- 1/2%r = 7- 1/2% Solution: Solve for t, using Equation 7.5Solution: Solve for t, using Equation 7.5 t =t = I_ I_ = = Php186.00Php186.00 PrPr Php1,500 x 7 Php1,500 x 7 11%% 22 == Php186.00Php186.00 Php1,500 x .075Php1,500 x .075 == 1.65 years1.65 years

The time may be converted to its appropriate units The time may be converted to its appropriate units of time measure like months or daysof time measure like months or days

(Months) t(Months) t == 1.65 years x 12 1.65 years x 12 months per yearmonths per year

== 19.8 months19.8 months OrOr (Days) t(Days) t == 1.65 years x 360 1.65 years x 360

days per yeardays per year == 594 days594 days If it is required to express the time (t) in If it is required to express the time (t) in

combination of two or more units such as in years; combination of two or more units such as in years; months, and days the procedure is as follows:months, and days the procedure is as follows:

In the above example, to convert t = 1.65 In the above example, to convert t = 1.65 years to years, months and days.years to years, months and days.

1rst Step.1rst Step. The period of 1.65 indicates one(1) year The period of 1.65 indicates one(1) year and a fraction of year expressed in decimal part as and a fraction of year expressed in decimal part as 0.65, thus number of years is 1.0.65, thus number of years is 1.

2nd Step2nd Step. The fractional part of a year a (0.65) is . The fractional part of a year a (0.65) is multiplied by the conversion factor of 12 months per multiplied by the conversion factor of 12 months per year.year.

Thus, number of months is 7.Thus, number of months is 7.

3rd Step3rd Step. The fractional part of month (0.8) is . The fractional part of month (0.8) is multiplied by 30 days per month (approximate).multiplied by 30 days per month (approximate).

Therefore, the final answer is:Therefore, the final answer is:

t =t = 1 year, 7 months and 24 days.1 year, 7 months and 24 days.

Determining the Determining the Principal, Given (A) Principal, Given (A) (r) (t)(r) (t) This particular problem involves Equations 7.1 and This particular problem involves Equations 7.1 and

7.2. In any mathematical computation, solving two 7.2. In any mathematical computation, solving two unknown variables using one equation or formula is unknown variables using one equation or formula is not possible. However, two equations with at least not possible. However, two equations with at least one common variable can be combined to derive a one common variable can be combined to derive a new equation.new equation.

If A = P + I (Equations 7.2) and I = Prt If A = P + I (Equations 7.2) and I = Prt (Equation 7.1), by substituting (I) using its (Equation 7.1), by substituting (I) using its equivalent (Prt) in Equation 7.2, in the following equivalent (Prt) in Equation 7.2, in the following manner.manner.

A = P + IA = P + I A = P + PrtA = P + Prt And factoring out P, we haveAnd factoring out P, we have A = P (I + rt) Equation 7.6 A = P (I + rt) Equation 7.6

P = P = A A Equation 7.7 Equation 7.7 I + rtI + rt

Example 4.Example 4. If a bank offers 8 ½% interest rate and If a bank offers 8 ½% interest rate and a depositor wants to have P3,878.20 after 5 years, a depositor wants to have P3,878.20 after 5 years, how much should he deposit now?how much should he deposit now?

Given:Given: r = 8 r = 8 11%% 22 A = P3,878.20A = P3,878.20 t = 5 yearst = 5 years Solution:Solution: Using Equation Using Equation P = P = A___ A___ 1 + r x t1 + r x t P P = = P3,878.20 P3,878.20 1 + 8 1 + 8 11% x 5 years% x 5 years = = P3,878.20P3,878.20 1 + .085 x 51 + .085 x 5 = = P3,878.20P3,878.20 1 + .4251 + .425 = = P3,878.20P3,878.20 1.4251.425 = P 2,721.54= P 2,721.54

Determining the Sum Determining the Sum (A)(A) Example 5. The bank offers an annual Example 5. The bank offers an annual

rate of 10 rate of 10 11 % interest, for 3 ½ years, % interest, for 3 ½ years, 2 2

how much will the accumulated sum be if how much will the accumulated sum be if the initial deposit is P 1,350.00?the initial deposit is P 1,350.00?

Given:Given: P = P 1, 350P = P 1, 350 r = 10r = 10 1 1 % % 22 t = 3 ½ yearst = 3 ½ years

In this particular problem, two methods In this particular problem, two methods may be used to determine the amount (A).may be used to determine the amount (A).

1rst Method.1rst Method. Determine the interest by using Determine the interest by using

Equation 7.1, I = Prt and then, add the interest Equation 7.1, I = Prt and then, add the interest and the principal to determine the amount A, using and the principal to determine the amount A, using 7.27.2

2nd Method2nd Method.. By A = P(1 + r x t)By A = P(1 + r x t) A = P 1,350 (1 + 10A = P 1,350 (1 + 10 1 1 % x 3 % x 3 1 1)) 2 22 2 = 1,350 (1 + .105 x 3.5)= 1,350 (1 + .105 x 3.5) = 1,350 (1.3675)= 1,350 (1.3675) = P 1, 846.13= P 1, 846.13

Promissory NotesPromissory Notes A promissory note may be used for raising money. The A promissory note may be used for raising money. The

borrower , in exchange for a loan, may issue his own borrower , in exchange for a loan, may issue his own promissory note or an immature note received by him promissory note or an immature note received by him form another person. A form another person. A promissorypromissory notenote is a written is a written promise to pay to a person a certain sum of money promise to pay to a person a certain sum of money on a specified date.on a specified date. The person who signs the The person who signs the promissory note is thepromissory note is the maker.maker. The person to whom his The person to whom his payment is promised is thepayment is promised is the payee. payee. The The period, period, or or termterm of of the note is the duration or length of time from the date of the note is the duration or length of time from the date of the note to the date of maturity. the note to the date of maturity. Maturity dateMaturity date is also is also referred to as the expiration date or due date of the note. referred to as the expiration date or due date of the note. The The maturity valuematurity value is the total amount to be paid on the is the total amount to be paid on the maturity or due date. The maturity or due date. The face valueface value of a promissory of a promissory note is the amount specifically mentioned in the note. If note is the amount specifically mentioned in the note. If the interest is mentioned in the note, it is an the interest is mentioned in the note, it is an interest- interest- bearing notebearing note. . If no interest is mentioned, then it is a If no interest is mentioned, then it is a non-interest bearing notenon-interest bearing note..

To illustrate, let us assume thatTo illustrate, let us assume that Mr. Darwin borrowed P10,000 from his friend Carlos on Mr. Darwin borrowed P10,000 from his friend Carlos on

November 8, Mr. Darwin issued a promissory note November 8, Mr. Darwin issued a promissory note (shown below) due on February 8 worth P10,000 plus (shown below) due on February 8 worth P10,000 plus 10% interest.10% interest.

The following is an example of promissory note:The following is an example of promissory note:

P 10,000P 10,000 November 8 November 8

Ninety days Ninety days after after I I promise to pay to the order of promise to pay to the order of Mr. CarlosMr. Carlos the sum of the sum of Ten Thousand Pesos and Ten Thousand Pesos and no/100no/100For value received with interest at 10%.Due For value received with interest at 10%.Due

Date: Date: February 8February 8 Mr. Mr.

DarwinDarwin

SignedSigned Where:Where: Date of the note is November 8.Date of the note is November 8. Term of the note is 90 daysTerm of the note is 90 days Maker is Mr. Darwin]Maker is Mr. Darwin] Face value o the note is P10,000Face value o the note is P10,000 Interest rate is 10% per yearInterest rate is 10% per year Maturity date is February 8.Maturity date is February 8. To compute for the Maturity Value (a) of the To compute for the Maturity Value (a) of the

promissory note in the above example –promissory note in the above example –

1rst Step. Determine the interest by using the 1rst Step. Determine the interest by using the formula I = P x r x tformula I = P x r x t

II == P10,000 x 10% x P10,000 x 10% x 9090

360360

= 10,000 x .1 x .25= 10,000 x .1 x .25 = P 250.00= P 250.00 2nd Step. Determine the amount or maturity value 2nd Step. Determine the amount or maturity value

by using formula A = P + Iby using formula A = P + I AA == P10,000 + P250P10,000 + P250 == P10,250P10,250

Alternatively, the amount or maturity value can be Alternatively, the amount or maturity value can be computed as follows using formula A = P(1 + r x t)computed as follows using formula A = P(1 + r x t)

AA == P10,000 (1 +10% x 90/100)P10,000 (1 +10% x 90/100) == 10,000 (1 + .1 x.25)10,000 (1 + .1 x.25) == 10,000 (1.025)10,000 (1.025) == P10,250.00P10,250.00

On the other hand, if no interest is mentioned in the On the other hand, if no interest is mentioned in the note, the face value is the maturity value. To note, the face value is the maturity value. To illustrate, convert above interest-bearing note to a illustrate, convert above interest-bearing note to a non-interest bearing note as shown below:non-interest bearing note as shown below:

P 10,250.00P 10,250.00 November 8 November 8

Ninety days Ninety days after after I I promise to pay to the order of promise to pay to the order of Mr. Mr. CarlosCarlos the sum of the sum of Ten Thousand Two hundred fifty Ten Thousand Two hundred fifty Pesos and no/100Pesos and no/100For value received with interest at For value received with interest at 10%.10%.

Due Date: Due Date: February 8February 8 Mr. Darwin Mr. Darwin

SignedSigned

The omission of a stated interest rate does not The omission of a stated interest rate does not necessarily mean that the original debt bears no necessarily mean that the original debt bears no interest. The interest, if any, might have been added interest. The interest, if any, might have been added to the original debt when the face value was to the original debt when the face value was determined. As illustrated, the original debt was P determined. As illustrated, the original debt was P 10,000 but the face value indicated in the note was 10,000 but the face value indicated in the note was P10, 250.00 which is also the maturity value.P10, 250.00 which is also the maturity value.

Discounting of NotesDiscounting of Notes Discounting a noteDiscounting a note is based on the maturity value while is based on the maturity value while

simple interest is based on the principal. Nonetheless, simple interest is based on the principal. Nonetheless, the basic principles of discounting a note or any the basic principles of discounting a note or any unmatured drafts or negotiable instruments at a bank are unmatured drafts or negotiable instruments at a bank are the same as those for obtaining a loan from a bank from the same as those for obtaining a loan from a bank from which the interest is deducted in advance. For example, which the interest is deducted in advance. For example, a businessman was granted a one (1) year loan by a bank a businessman was granted a one (1) year loan by a bank worth P10,000. If the bank charges 10% interest the worth P10,000. If the bank charges 10% interest the amount the businessman will receive is P10,000 less amount the businessman will receive is P10,000 less interest in advance worth (I= P x r x t = 10,000 x .1 x 1) interest in advance worth (I= P x r x t = 10,000 x .1 x 1) P1,000. Therefore, the businessman will receive only the P1,000. Therefore, the businessman will receive only the discounted amount of P9,000 but the maturity value after discounted amount of P9,000 but the maturity value after one year is still P10,000.one year is still P10,000.

The The discount rate (d)discount rate (d) is usually expressed in is usually expressed in percent or its equivalent decimal and is generally quoted percent or its equivalent decimal and is generally quoted on a yearly basis. Its is the ration of the discount for the on a yearly basis. Its is the ration of the discount for the period to the maturity value. If the discount rate is not period to the maturity value. If the discount rate is not mentioned in any given problem, it is assumed that the mentioned in any given problem, it is assumed that the interest rate is also the discount rate. The interest rate is also the discount rate. The discount (discount (D)D) is the amount to be deducted to theis the amount to be deducted to the maturity value. The maturity value. The proceedsproceeds (Pd) (Pd) is the amount a creditor is willing to pay a is the amount a creditor is willing to pay a note before its maturity or due date. note before its maturity or due date.

Term of the discount Term of the discount (td),(td), is the number of days from is the number of days from the maturity date to the discount date (date the note the maturity date to the discount date (date the note was sold), Otherwise specified, the Banker’s Rule will was sold), Otherwise specified, the Banker’s Rule will be used in computing the term of the discount. Hence, be used in computing the term of the discount. Hence, to determine the discount (D), the formula isto determine the discount (D), the formula is

D = A x d x td Equation 7.8D = A x d x td Equation 7.8

Such formula is also referred to as the bank discount. Such formula is also referred to as the bank discount. The formula to determine the proceeds is The formula to determine the proceeds is

Pd = A - D Equation 7.9Pd = A - D Equation 7.9 Combining Equations 7.8 and 7.9 and Combining Equations 7.8 and 7.9 and

factoring out A, the resulting equation isfactoring out A, the resulting equation is

Pd = A (1 – d x td ) Equation 7.9Pd = A (1 – d x td ) Equation 7.9

Discounting an Discounting an Interest-Bearing Interest-Bearing NoteNote Example 8. Mr. Dennis had a note for P8,000 with an Example 8. Mr. Dennis had a note for P8,000 with an

interest rate of 6%. The note was dated November 15, interest rate of 6%. The note was dated November 15, 1981, and the maturity date is after 90 days. On 1981, and the maturity date is after 90 days. On November 30, 1981, he took the note to his bank, November 30, 1981, he took the note to his bank, which discounted it at the rate of 7%. How much did which discounted it at the rate of 7%. How much did he receive from the bank?he receive from the bank?

Given:Given: P P == Php 8,000 face valuePhp 8,000 face value rr == 6% (per year)6% (per year) Date of the noteDate of the note == November 15, 1981November 15, 1981

Term of the noteTerm of the note == 90 days = t90 days = t

Discount date Discount date == November 30, 1981November 30, 1981 Discount rate (d)Discount rate (d)== 7% (per year)7% (per year)

PP == to findto find

Solution:Solution: 1rst Step1rst Step. Find the maturity value according to the face value, . Find the maturity value according to the face value,

the rate of interest and the time stipulated on the note. the rate of interest and the time stipulated on the note. Substituting the values in the formula: A = P (1 + r x t)Substituting the values in the formula: A = P (1 + r x t)

AA == P 8,000 ( 1 + 6% x P 8,000 ( 1 + 6% x 9090)) 360360 == 8,000 (1 + .015)8,000 (1 + .015) == 8,000 (1.015)8,000 (1.015) == P 8, 120.00P 8, 120.00 2nd Step2nd Step. Determine the discount by using formula . Determine the discount by using formula D = A x d x D = A x d x

td).td). DD == A x d x tdA x d x td == 8,120 x 7% x 8,120 x 7% x 7575 360360 == P 118.42P 118.42 Therefore, the proceeds is Therefore, the proceeds is PdPd == A – DA – D == P 8, 120 – P 118.42P 8, 120 – P 118.42 == P 8, 001.58P 8, 001.58 Or, using Equation 7.10 after determining the maturity Or, using Equation 7.10 after determining the maturity

value and the term of the discount.value and the term of the discount. PdPd == A (1 – d x td)A (1 – d x td) == 8,120 (1 – 7% x 75/360)8,120 (1 – 7% x 75/360) == 8,120 (1 - .0145833)8,120 (1 - .0145833) == P8,001.58P8,001.58

Example 9Example 9. . Cruz pays his account with Santos on Cruz pays his account with Santos on October 1, 1985 with a P8,000 60-day, 5% note. On October 1, 1985 with a P8,000 60-day, 5% note. On October 26, 1985, Santos offered to discount Cruz’ note October 26, 1985, Santos offered to discount Cruz’ note with the bank at 6%.with the bank at 6%.

Given:Given: Date of the note = October 1, 985Date of the note = October 1, 985 Face value of note = P 8,000Face value of note = P 8,000 Term of the note = 60 daysTerm of the note = 60 days Interest rate = 5% (per year)Interest rate = 5% (per year) Discount dateDiscount date = October 26, = October 26,

19851985 Discount rate = 6% (per year)Discount rate = 6% (per year) Solution:Solution: Analyzing the problem, the note of Mr. Analyzing the problem, the note of Mr.

Cruz in the possession of Santos has already earned an Cruz in the possession of Santos has already earned an interest on October 26. On the other hand, the bank will interest on October 26. On the other hand, the bank will be discounting at 6% the same note whose maturity value be discounting at 6% the same note whose maturity value will be 35 days from October 26 to November 30, 1985.will be 35 days from October 26 to November 30, 1985.

October 1, 1985 October 26October 1, 1985 October 26 November 30, November 30, 1985__1985__

Date of the note Date of the Maturity date Date of the note Date of the Maturity date

discount discount Discounted for P 8, 000 Discounted for P 8, 000

35 days35 days

Maturity value is Maturity value is AA == 8,000 ( 1 + 5% x 8,000 ( 1 + 5% x 6060)) 360360 == 8,000 (1 + .008333)8,000 (1 + .008333) = = 8,000 (1 .008333)8,000 (1 .008333) == P 8,066.66P 8,066.66 Therefore, the proceeds is Therefore, the proceeds is Pd =Pd = A (1 – d x td)A (1 – d x td) == P 8,066.66 ( 1 - 6% x P 8,066.66 ( 1 - 6% x 35)35) 360360 == P 8,066.66 ( 1 - .0058333)P 8,066.66 ( 1 - .0058333) == P 8,019.60P 8,019.60

Chapter Chapter 8: 8:

CompounCompound Interestd Interest

In loan and saving transactions, interest In loan and saving transactions, interest arrangements may be arrangements may be simplesimple or or compoundedcompounded. .

In the former the periodic interest is based only on In the former the periodic interest is based only on the the original principaloriginal principal. . In the later, the periodic In the later, the periodic uncollected interest are added to the principal and uncollected interest are added to the principal and their sum is the basis of the interest for the their sum is the basis of the interest for the succeeding interest period.succeeding interest period.

The process of accumulating a principal to obtain a The process of accumulating a principal to obtain a compound amount is called compound amount is called compound compound accumulation.accumulation. Compound interest is the difference Compound interest is the difference between the original principal and the between the original principal and the compound compound amountamount (principal plus total interest). The period (principal plus total interest). The period for computing interest, usually at regularly stated for computing interest, usually at regularly stated intervals such as annually, semi-annually, quarterly, intervals such as annually, semi-annually, quarterly, or monthly, is called the or monthly, is called the conversion period.conversion period. The The stated interest rate is called stated interest rate is called nominal rate.nominal rate. Therefore, if the nominal rate of interest is 8% Therefore, if the nominal rate of interest is 8% compound semi-annually, the interest rate is compound semi-annually, the interest rate is expressed as 8% compound semi-annually, the expressed as 8% compound semi-annually, the interest rate per conversion period is interest rate per conversion period is 8 8 % or 4%. % or 4%.

2 2

Thus,“6% compounded annually” means that the Thus,“6% compounded annually” means that the interest per year is 6% and the interest is paid annually. interest per year is 6% and the interest is paid annually. Therefore, the conversion period is 1 and the interest Therefore, the conversion period is 1 and the interest per conversion periods is 6% also.per conversion periods is 6% also.

““6% compounded semi-annually” means that interest 6% compounded semi-annually” means that interest per year is 6% but the interest is paid every six(6) per year is 6% but the interest is paid every six(6) months or semi-annually. The interest per conversion is months or semi-annually. The interest per conversion is 66% or3%.% or3%.

22 ““6% compounded quarterly” means that the interest per 6% compounded quarterly” means that the interest per

year is 6% but the interest is paid every 3 months or year is 6% but the interest is paid every 3 months or quarterly. To determine the interest per quarter divide quarterly. To determine the interest per quarter divide 6% by the conversion factor of 4 (four 3-months for 6% by the conversion factor of 4 (four 3-months for every year), thus, interest per conversion period is every year), thus, interest per conversion period is 6/4% or 16/4% or 1 1 1%.%.

To illustrate, let us assume that a loan of P10,000 for 5 To illustrate, let us assume that a loan of P10,000 for 5 years was taken on January 1, 1982. If the years was taken on January 1, 1982. If the arrangement is for simple interest, for every year, the arrangement is for simple interest, for every year, the annual interest is annual interest is

P 1,000.00 ( P1,000 x .10). If the interest is P 1,000.00 ( P1,000 x .10). If the interest is compounded, the annual interest for 1983 is −compounded, the annual interest for 1983 is −

(P 10,000 x.10) −P1,000; 1984 − (P 10,000 + P1,000) x (P 10,000 x.10) −P1,000; 1984 − (P 10,000 + P1,000) x .10) − P1,000; 1985 − (P12,000 x .10) −P1,210; 1986 .10) − P1,000; 1985 − (P12,000 x .10) −P1,210; 1986 − (P13,310 x .10) −P1,321; and 1987− (P14, 461.10) − (P13,310 x .10) −P1,321; and 1987− (P14, 461.10) −P1,464.−P1,464.

Simple Interest Simple Interest FormulaFormula Example 1. Adorada deposited P1,000 with the Example 1. Adorada deposited P1,000 with the

Pacific Bank and allowed it to remain for 4 years. If Pacific Bank and allowed it to remain for 4 years. If the bank paid its depositors 5% on their savings once the bank paid its depositors 5% on their savings once a year, how much money did she have in the bank at a year, how much money did she have in the bank at the end of 4 years?the end of 4 years?

Solution:Solution: Using simple interest Using simple interest formula.formula.

1rst year interest on P 1,000.001rst year interest on P 1,000.00

II == P x r x tP x r x t == P 1,000.00 x 5% x 1P 1,000.00 x 5% x 1 == P 1,000.00 x .05 x 1P 1,000.00 x .05 x 1 == P 50.00P 50.00

At the end of the first year, Adorada has accumulated At the end of the first year, Adorada has accumulated P 1,000.00 and P50.00 or P1,050.00P 1,000.00 and P50.00 or P1,050.00

2nd year interest on P1,050.002nd year interest on P1,050.00 II == P x r x tP x r x t == P 1,050.00 x 5% x 1P 1,050.00 x 5% x 1 == P 1,050.00 x .05 x 1P 1,050.00 x .05 x 1 == P 52.50P 52.50

When this idea is extend, the compound When this idea is extend, the compound amount at the end of the amount at the end of the nthnth period may be period may be expressed as:expressed as:

A = P (+i)n Equation 8.1A = P (+i)n Equation 8.1 Where:Where: AA == Compound amount usually in PCompound amount usually in P PP == Original value of the principal in Original value of the principal in

P P ii == Rate of interest per conversion Rate of interest per conversion

periodperiod nn = Number of conversion periods= Number of conversion periods

Thus, in the above example, if P = P 1,000, r = I = Thus, in the above example, if P = P 1,000, r = I = 5% per year, n = 4 years, using the equation5% per year, n = 4 years, using the equation

A=A= P ( 1 + 1) n P ( 1 + 1) n == 1,000 (1 + 5%)41,000 (1 + 5%)4 == 1,000 (1.21550625)1,000 (1.21550625) == A,215.50625 or P 1,215.51A,215.50625 or P 1,215.51

The table on the Compound Amount of 1 in the The table on the Compound Amount of 1 in the index shows the compound interest accumulation at index shows the compound interest accumulation at different rates and periods. The formula using this different rates and periods. The formula using this Table for the computation of the 3rd year interest Table for the computation of the 3rd year interest on P 1,102.50on P 1,102.50

I =I = P x r x tP x r x t == P 1,102.50.00 x 5% x1P 1,102.50.00 x 5% x1 == P 1,102.50.00 x .05 x 1P 1,102.50.00 x .05 x 1 == P 55.13P 55.13 At the end of the 3rd year, Adorada have At the end of the 3rd year, Adorada have

accumulated P 1,102.50 and P55.13 or P 1,157.63.accumulated P 1,102.50 and P55.13 or P 1,157.63. 4rth year interest on P1,157.634rth year interest on P1,157.63

II == P x r x tP x r x t == P 1,157.63 x 5% x 1P 1,157.63 x 5% x 1 == P 1,157.63 x .05 x 1P 1,157.63 x .05 x 1 == P 57.88P 57.88

Therefore, at the end of the 4th year, Adorada has Therefore, at the end of the 4th year, Adorada has accumulated P1,157.63 and P 57.88 or P1,215.51.accumulated P1,157.63 and P 57.88 or P1,215.51.

Summarizing the computations:Summarizing the computations:

At the end of At the end of Interest Interest PrincipalPrincipal 1rst year1rst year P 50.00 Php1, 050.00 P 50.00 Php1, 050.00 2nd year 52.502nd year 52.50 1,102,50 1,102,50 3rd year 55.133rd year 55.13 1,157.63 1,157.63 4rth year 57.8814rth year 57.881 1,215.51 (final 1,215.51 (final

answer)answer)

As shown in the preceding example, the As shown in the preceding example, the computation using simple interest method is quite computation using simple interest method is quite tedious and time consuming.tedious and time consuming.

Compound amount isCompound amount is

AA == Principal x equivalent in the Principal x equivalent in the indexindex

The index A provides eight (8) decimal places for The index A provides eight (8) decimal places for each equivalent. It would be a waste of time if we each equivalent. It would be a waste of time if we use 8 decimal places in multiplying with a small use 8 decimal places in multiplying with a small amount. All interest rates used on Index A are in amount. All interest rates used on Index A are in percent per conversion period. Therefore, if the percent per conversion period. Therefore, if the computed or given of interest is expressed in computed or given of interest is expressed in decimal form, compute its equivalent in percent decimal form, compute its equivalent in percent before using the index. Then, a column indicates before using the index. Then, a column indicates the number of conversion periods.the number of conversion periods.

Example 3.Example 3. If the rate of interest is 5% compounded If the rate of interest is 5% compounded annually and the period is 5 years, what is its annually and the period is 5 years, what is its equivalent in index A?equivalent in index A?

Solution:Solution: Since the rate of interest is in the percent per year Since the rate of interest is in the percent per year

and period is 5 years, determine its equivalent in the and period is 5 years, determine its equivalent in the Index A by using their original values (n=5) and (1 Index A by using their original values (n=5) and (1 =5%). On the “n” column, locate 5 and move =5%). On the “n” column, locate 5 and move horizontally to the right until reaching “I” column with horizontally to the right until reaching “I” column with 5%. Thus the Index equivalent = 1.27281565%. Thus the Index equivalent = 1.2728156

Example 4Example 4. Using example 3, get the Index . Using example 3, get the Index equivalent if the rate of interest is 5% compounded equivalent if the rate of interest is 5% compounded semi-annually.semi-annually.

Solution:Solution: If the rate of interest is 5% If the rate of interest is 5% compounded semi-annually and the interest per compounded semi-annually and the interest per conversion period is every 6 months, interest per conversion period is every 6 months, interest per conversion period is 5/2% = 2 ½ %. The period conversion period is 5/2% = 2 ½ %. The period given in 5 years which must be converted into 6-given in 5 years which must be converted into 6-month periods. 5 x 2 = 10(6-month period). Using month periods. 5 x 2 = 10(6-month period). Using now is = 2 ½% and n = 10. Index equivalent = now is = 2 ½% and n = 10. Index equivalent = 1.280084541.28008454

Example 3. Using example 3, if the rate of interest Example 3. Using example 3, if the rate of interest is 5% compounded quarterly.is 5% compounded quarterly.

ii == 5%/4 = 1 5%/4 = 1 11%% 44 nn == 5 x 4 = 20 (interest Period)5 x 4 = 20 (interest Period)

Index equivalent = 1.28203723Index equivalent = 1.28203723

Chapter Chapter 9: 9:

DepreciatioDepreciationn

The long-lived properties of business are of two The long-lived properties of business are of two types, the types, the non-depreciablenon-depreciable and and depreciabledepreciable. . Non-depreciable assets have an almost perpetual Non-depreciable assets have an almost perpetual life. An example is land. Depreciable assets life. An example is land. Depreciable assets have limited economic (useful) lives. Examples have limited economic (useful) lives. Examples are building, machines, equipment, furniture, and are building, machines, equipment, furniture, and the like.the like.

The use of long-lived assets is The use of long-lived assets is indispensable in the earning of revenues. If this indispensable in the earning of revenues. If this is so, then the value of this use is a cost of is so, then the value of this use is a cost of earning the revenue.. as part of the costs of earning the revenue.. as part of the costs of operations, the costs of depreciable assets are operations, the costs of depreciable assets are allocated over their useful years. This is allocated over their useful years. This is allocated cost is called allocated cost is called depreciation expense depreciation expense or or simply simply depreciation.depreciation.

There are different methods of calculating There are different methods of calculating periodic depreciation but only the Averages and periodic depreciation but only the Averages and the Reducing Charge Methods will be discussed.the Reducing Charge Methods will be discussed.

Average MethodsAverage Methods Straight line MethodStraight line Method Service Hour MethodService Hour Method

Average MethodsAverage Methods In the In the straight line methodstraight line method, the depreciation expense is , the depreciation expense is

distributed in equal amounts over the estimated useful life of the distributed in equal amounts over the estimated useful life of the asset. The formula is:asset. The formula is:

R = R = C − SC − S Equation 9.1 Equation 9.1 nn

Where:Where: RR == depreciation charge per yeardepreciation charge per year nn == number of period in years, months number of period in years, months

and the likeand the like CC == original cost of the assetoriginal cost of the asset SS == estimated scrap value or salvage estimated scrap value or salvage

valuevalue C – S =C – S = Total Total depreciable costdepreciable cost Example 1Example 1. A car purchased for P69,000 has an . A car purchased for P69,000 has an

estimated useful life of three (3) years and a scrap value of estimated useful life of three (3) years and a scrap value of P9,000. Use the straight line method to find the depreciation P9,000. Use the straight line method to find the depreciation expense per year and construct a depreciation schedule.expense per year and construct a depreciation schedule.

Solution:Solution: Given:Given: C = P69,000C = P69,000 S = P 9,000S = P 9,000 n = 3 yearsn = 3 years

Substituting the values in Equation 9.1Substituting the values in Equation 9.1 RR == C − SC − S nn == P 69,000 – P9,0000P 69,000 – P9,0000 33 == 60,000 60,000 = P 20,000= P 20,000 33 Depreciation Schedule − Straight Line MethodDepreciation Schedule − Straight Line Method End of Year Annual Depreciation AccumulatedEnd of Year Annual Depreciation Accumulated

Book Value of Assets Book Value of Assets

ExpenseExpense DepreciationDepreciation 00 − − − − − − − − 00 P 90,000P 90,000 11 P 20,000 P 20,000P 20,000 P 20,000 49,000 49,000 22 20,000 20,000 40,00040,000 29,000 29,000 33 20,000 60,000 20,000 60,000 9,000 9,000

TotalTotal Php 60,000Php 60,000

The The Service hours MethodService hours Method estimates depreciation on estimates depreciation on the productive capacity of the asset per service hour. the productive capacity of the asset per service hour. It assumes that the number of productive hours It assumes that the number of productive hours decreases as the property becomes older.decreases as the property becomes older.

Example 2. A sewing machine purchased Example 2. A sewing machine purchased for P4,500 has an estimated life of four years and a for P4,500 has an estimated life of four years and a scrap value of P 300.00. Use the straight line scrap value of P 300.00. Use the straight line method equation to find the depreciation. Assume method equation to find the depreciation. Assume that the useful life of the machine is estimated to be that the useful life of the machine is estimated to be 15, 000 hours and the actual number of hours spent 15, 000 hours and the actual number of hours spent in production each year is as follows:in production each year is as follows:

1rst year:1rst year: 4,500 service hours4,500 service hours 2nd year:2nd year: 4,100 service hours4,100 service hours 3rd year:3rd year: 3,500 service hours3,500 service hours 4rth year:4rth year: 2,900 service hours2,900 service hours Use the service hours method to find the Use the service hours method to find the

depreciation expenses for each year and construct a depreciation expenses for each year and construct a depreciation schedule.depreciation schedule.

Given:Given: CC == P 4,500P 4,500 SS == P 300P 300 NN == 15,000 service hours15,000 service hours

Solution:Solution: Substituting the value in equation 9.1Substituting the value in equation 9.1 RR == P 4,500 – P 300P 4,500 – P 300 15,00015,000 == P 0.28 per hourP 0.28 per hourThe annual depreciation as computed are:The annual depreciation as computed are:

1rst year:1rst year: 4,500 x P 0.28 = P P 1,260.004,500 x P 0.28 = P P 1,260.00 2nd year:2nd year: 4,100 x P 0.28 = P 1,148.004,100 x P 0.28 = P 1,148.00 3rd year: 3rd year: 3,500 x P 0.28 = P980.003,500 x P 0.28 = P980.00 4rth year:4rth year: 2,900 x P 0.28 = P812.002,900 x P 0.28 = P812.00

Depreciation Schedule − Straight Line MethodDepreciation Schedule − Straight Line Method End of Year Annual Depreciation End of Year Annual Depreciation Accumulated Book Accumulated Book

Value of Value of AssetsAssetsExpenseExpense Depreciation Depreciation 00 − − − −− − − − − − − − − − − − P 4,500.00P 4,500.00 11 P 1,260.00 P 1,260.00 P1,260.00P1,260.00 3,240.00 3,240.00 22 1,148.00 1,148.00 2,408.00 2,092.0 2,408.00 2,092.0 33 980.00 980.00 3,388.00 3,388.00 1,112.00 1,112.00 44 812.00 812.00 4,200.00 4,200.00 300.00 300.00

TotalTotal Php 60,000 Php 60,000

Chapter Chapter 10: 10:

Income Income statemenstatemen

tt

Businessmen invest their money in business Businessmen invest their money in business primarily to make profit. Profit is realized (earned) primarily to make profit. Profit is realized (earned) when revenues exceeds costs and expenses when revenues exceeds costs and expenses (outgoing property). Say for example, a car bought (outgoing property). Say for example, a car bought for P30,000 was sold for cash of P35,000. The for P30,000 was sold for cash of P35,000. The revenue of P 35,000 exceeds cost of P 30,000. A revenue of P 35,000 exceeds cost of P 30,000. A profit of P 5,000 is realized.profit of P 5,000 is realized.

In order to record intelligently the various In order to record intelligently the various activities of a business one should understand the activities of a business one should understand the important terminologies used for bookkeeping.important terminologies used for bookkeeping.

Sales.Sales. The revenue from the selling of goods or The revenue from the selling of goods or services is called sales. A trading concern sells services is called sales. A trading concern sells goods that he buys in the same form. Example is a goods that he buys in the same form. Example is a bookstorebookstore..

Sales Returns and Allowances.Sales Returns and Allowances. Goods sold and Goods sold and returned by the buyer are called returned by the buyer are called sales returnssales returns. . Reduction in the selling price of goods sold due to Reduction in the selling price of goods sold due to low quality or defect is called low quality or defect is called sales allowance.sales allowance.

Net Sales. Net Sales. This is the total selling price of the This is the total selling price of the goods sold less the amount of returns and goods sold less the amount of returns and allowances on such sale.allowances on such sale.

Purchases.Purchases. Goods bought for sale are called Goods bought for sale are called purchases. The cost of goods purchased purchases. The cost of goods purchased includes not only the invoice price but also the includes not only the invoice price but also the incidental costs incidental costs relating to merchandise relating to merchandise acquisition, preparation, and placement for sale. acquisition, preparation, and placement for sale. Example of incidental costs are transportation Example of incidental costs are transportation charges, taxes etc.charges, taxes etc.

Inventory.Inventory. Goods bought and remaining unsold Goods bought and remaining unsold are called inventory.are called inventory.

Cost of goods sold. Cost of goods sold. The cost of the goods The cost of the goods bought and sold is called bought and sold is called cost of goods sold. cost of goods sold. This is computed by adding inventory at the This is computed by adding inventory at the beginning plus net purchases less inventory beginning plus net purchases less inventory ending.ending.

Example 1. At the beginning of the year, the Example 1. At the beginning of the year, the inventory of the Erwin Gas Station was inventory of the Erwin Gas Station was P1,250,125.55. Purchases during the year P1,250,125.55. Purchases during the year amounted to P698.254.05. At the year end, the amounted to P698.254.05. At the year end, the manager verified that the station had P968,056.58 manager verified that the station had P968,056.58 remaining inventory. Determine cost of goods sold.remaining inventory. Determine cost of goods sold.

Cost of Goods Sold = Cost of Goods Sold = P1,250,125.55 + P698, P1,250,125.55 + P698,

254.05─ P968,056.58 254.05─ P968,056.58

7. 7. Gross Profit. Gross Profit. It is the excess of net sales over It is the excess of net sales over the cost of goods sold. It is also called gross the cost of goods sold. It is also called gross margin on sales. The formula to determine the margin on sales. The formula to determine the gross profit is as follows.gross profit is as follows.

Example 2. An Auto Supply bought spare parts for Example 2. An Auto Supply bought spare parts for P26,245. After a month of operation all the spare P26,245. After a month of operation all the spare parts were sold of P27,895.45 net. Determine the parts were sold of P27,895.45 net. Determine the gross profit.gross profit.

G.P.G.P. == Net Sales – Cost of Goods Net Sales – Cost of Goods SoldSold

G.P.G.P. == Php27,895.45- Php26,245Php27,895.45- Php26,245 == Php1,650.45Php1,650.45

Selling Expenses. Selling Expenses. Expenses in connection with Expenses in connection with the selling function of the business are calledthe selling function of the business are called Selling expenses. Selling expenses. Examples are advertising Examples are advertising expenses, traveling expenses of Salesmen and expenses, traveling expenses of Salesmen and the like.the like.

Administrative expenses. Administrative expenses. These are items of These are items of expenses incurred in the administrative expenses incurred in the administrative operations of the business which are not operations of the business which are not connected with the selling function. Examples are connected with the selling function. Examples are insurance expense, postage dues, property insurance expense, postage dues, property taxes, office supplies and the like.taxes, office supplies and the like.

Net Profit. Net Profit. The excess of the gross profit over The excess of the gross profit over the total operating expenses (selling and the total operating expenses (selling and administrative expenses is the net income (net administrative expenses is the net income (net profit) from operations before income taxes.profit) from operations before income taxes.

Example 5.Example 5. The gross profit of Nancy’s Grocery Store last year The gross profit of Nancy’s Grocery Store last year was P346,985.50. Below is the illustration to determine net profit.was P346,985.50. Below is the illustration to determine net profit.

Net profit= Gross Profit – Operating expensesNet profit= Gross Profit – Operating expenses

Gross ProfitGross Profit………………………… ……….P346,985.50………………………… ……….P346,985.50

Less: Less: Operating ExpensesOperating Expenses Selling ExpensesSelling Expenses P 56,925.00 P 56,925.00

Light and water 10,850.00Light and water 10,850.00

Salaries and wagesSalaries and wages 25,345.70 25,345.70 Repair and Maintenance Repair and Maintenance 61,250.4661,250.46 156,371.16156,371.16

Net ProfitNet Profit Php Php 192,614.34192,614.34

The life or existence of a business concern The life or existence of a business concern usually covers a long period of time involving many usually covers a long period of time involving many years. During this period, the businessman must be years. During this period, the businessman must be guided by some guided by some reliable business datareliable business data in order that in order that he may be informed of the trend of his business. In he may be informed of the trend of his business. In other words, even with estimates only, he must be of other words, even with estimates only, he must be of whether he is making a profit or suffering a loss. whether he is making a profit or suffering a loss.

Hence, the life of a business is divided into periods of Hence, the life of a business is divided into periods of equal length at the end of each of which summary equal length at the end of each of which summary reports are submitted to him. Usually, each of these reports are submitted to him. Usually, each of these periods cover one year or 12 months. These summary periods cover one year or 12 months. These summary reports are the following:reports are the following:

1. 1. Balance sheetBalance sheet or or statement of assetsstatement of assets and and liabilitiesliabilities - - This report or statement lists the assets This report or statement lists the assets (properly), liabilities (obligations), and the owner’s (properly), liabilities (obligations), and the owner’s equity as of a given date.equity as of a given date.

2. 2. Income statementIncome statement or or statement of operationsstatement of operations – – This statement shows the detail of the revenues, costs This statement shows the detail of the revenues, costs expenses and losses for the period which is usually a expenses and losses for the period which is usually a year.year.

This present chapter will deal with the income This present chapter will deal with the income statement. Below is an illustration of the income statement. Below is an illustration of the income statement of a trading concern.statement of a trading concern.

TRADING COMPANYTRADING COMPANYINCOME STATEMENTINCOME STATEMENT

YEAR ENDED DECEMBER 31,19_YEAR ENDED DECEMBER 31,19_ SalesSales PXXPXX Less: Sales Returns, discounts, and allowances Less: Sales Returns, discounts, and allowances XX XX Net SalesNet Sales PXXPXX Cost of goods soldCost of goods sold Merchandise, beginningMerchandise, beginning PXXPXX Add – Net Purchases Add – Net Purchases XXXX Available for saleAvailable for sale PXXPXX Merchandise, endingMerchandise, ending XXXX XXXX Gross Margin on SalesGross Margin on Sales PXX PXX

Less: Operating ExpensesLess: Operating Expenses PXXPXX Selling ExpensesSelling Expenses PXXPXX Administrative expensesAdministrative expenses XXXX Net Operating ProfitNet Operating Profit PXXPXX Less: Income TaxesLess: Income Taxes XXXX Net Income for the YearNet Income for the Year PXXPXX ==========

Thank Thank you…….you…….

Jasmin A. Jasmin A. Abiva & Abiva &

Marie Carl Marie Carl PasacsacPasacsac

BOA IV-1BOA IV-1