Math In Business boa

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Math In Business boa

  1. 1. Math in BusinessMath in Business Presents……….Presents……….
  2. 2. Contents……Contents……  Chapter I.Chapter I. Basic operationBasic operation  Chapter 2. Operation withChapter 2. Operation with DecimalsDecimals  Chapter 3. FractionsChapter 3. Fractions  Chapter 4 Percent in BusinessChapter 4 Percent in Business  Chapter 5: Employee’s CompensationChapter 5: Employee’s Compensation
  3. 3.  Chapter 6: BUYING AND SELLINGChapter 6: BUYING AND SELLING  Chapter 7:SIMPLE INTERESTChapter 7:SIMPLE INTEREST  Chapter 8: COMPOUND INTERESTChapter 8: COMPOUND INTEREST  Chapter 9: DEPRECIATIONChapter 9: DEPRECIATION  Chapter 10: INCOME STATEMENTChapter 10: INCOME STATEMENT
  4. 4. Chapter I.Chapter I. CheckingChecking answers ofanswers of the Basicthe Basic OperationsOperations
  5. 5.  Addition, subtraction, multiplication,Addition, subtraction, multiplication, and division are the basic operations inand division are the basic operations in mathematic. Users confronted withmathematic. Users confronted with different problems in business requiredifferent problems in business require thorough knowledge of these basicthorough knowledge of these basic operations. With the invention ofoperations. With the invention of calculators, a person can solve acalculators, a person can solve a mathematical problem with greatermathematical problem with greater speed. Since calculators are merespeed. Since calculators are mere instrument in performing theinstrument in performing the operations, accuracy is largelyoperations, accuracy is largely dependent on mastery of the operationsdependent on mastery of the operations by the user and on his skills in checkingby the user and on his skills in checking answersanswers..
  6. 6. Basic Math OperationsBasic Math Operations  AdditionAddition  Commutative Property of Addition ·Commutative Property of Addition · Associative Property of Addition ·Associative Property of Addition · Distributive Property · AdditiveDistributive Property · Additive Identity: ZeroIdentity: Zero  We'll assume the reader can add digits, so thatWe'll assume the reader can add digits, so that  2 + 2 = 42 + 2 = 4  is not a surprise. In addition, we'll also assume that theis not a surprise. In addition, we'll also assume that the fundamentals of "carrying" are not that big of a problem,fundamentals of "carrying" are not that big of a problem, and so most readers will immediately know how we didand so most readers will immediately know how we did  We're more interested here in the general properties ofWe're more interested here in the general properties of addition that impact algebra. With that in mind, let a, b andaddition that impact algebra. With that in mind, let a, b and c be three real numbers. Then the following properties ofc be three real numbers. Then the following properties of addition turn out to be usable and important:addition turn out to be usable and important:
  7. 7.  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 isone adds the numbers, i.e., forwards addition is the same as backwards addition).the same as backwards addition).  ExamplesExamples  Note that negative numbers are sometimesNote that negative numbers are sometimes enclosed in parentheses to avoid confusionenclosed in parentheses to avoid confusion between the sign of the number and the additionbetween 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 themany textbook writers use spacing to set off the difference instead.difference instead.
  8. 8.  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 doesof which are added together first, i.e., grouping does not matter)not matter)  ExamplesExamples
  9. 9.  MultiplicationMultiplication  Commutative Property of Multiplication ·Commutative Property of Multiplication · Associative Property of Multiplication ·Associative Property of Multiplication · Distributive Property · MultiplicativeDistributive Property · Multiplicative IdentityIdentity  Again, we’ll assume that the basics ofAgain, 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 different2=2*2=4. There are obviously several different notations in use, depending when one learns itnotations in use, depending when one learns it and what context one learns it. We will use all ofand what context one learns it. We will use all of these notations, as well as another: when usingthese notations, as well as another: when using variables, multiplication is assumed whenvariables, 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 justMultiplication follows the same two laws just described for addition.described for addition.
  10. 10.  Commutative Property of MultiplicationCommutative Property of Multiplication  ExamplesExamples  We are following the same convention as withWe are following the same convention as with addition. Note that just writing numbers next toaddition. 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 need32 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 – inThere are several correct ways to do this – in other articles, we’ll make a lot of use ofother articles, we’ll make a lot of use of parentheses, so that 3· 2 will be written 3(2) orparentheses, 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.
  11. 11.  Associative Property ofAssociative Property of MultiplicationMultiplication  ExamplesExamples  Please note that we do not have to limit ourselvesPlease note that we do not have to limit ourselves to parentheses; the last computation could haveto parentheses; the last computation could have been writtenbeen written  This looks a bit neater. Again, such things are aThis looks a bit neater. Again, such things are a matter of style, and the reader is encouraged to usematter of style, and the reader is encouraged to use whatever bracketing makes the most sense andwhatever bracketing makes the most sense and allows clear, proper ordering of calculations.allows clear, proper ordering of calculations.
  12. 12.  Distributive PropertyDistributive Property  When both addition and multiplication appear in aWhen both addition and multiplication appear in a single mathematical expression, this distributive lawsingle mathematical expression, this distributive law controls the operation. This is probably one of thecontrols the operation. This is probably one of the most important laws in mathematics; getting itmost important laws in mathematics; getting it wrong guarantees bad calculations! The "reverse" iswrong guarantees bad calculations! The "reverse" is NOT CORRECT:NOT CORRECT:  ExamplesExamples
  13. 13.  SubtractionSubtraction  We’re going to define subtraction in terms ofWe’re going to define subtraction in terms of addition of the negativeaddition of the negative::  This means that subtraction is a shortcut or anThis means that subtraction is a shortcut or an abbreviation of the above addition operation. Theabbreviation of the above addition operation. The "triple equals" sign used here means"triple equals" sign used here means definitiondefinition,, and is meant to signify anand is meant to signify an operation that is alwaysoperation 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 notSubtraction is not in generalin general commutative:commutative:
  14. 14.  ExampleExample  Nor is itNor is it in generalin general associative:associative:  ExampleExample
  15. 15.  DivisionDivision  We’re going to define division in terms ofWe’re going to define division in terms of multiplicationmultiplication 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 > 1inverse below for more info. For now, suppose that a > 1 and thatand that  The number b has a very important function – it is calledThe number b has a very important function – it is called the multiplicative inverse of a, also known as the inversethe 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 abut it should be pointed out that this is not really a definition. In order to properly define division, we woulddefinition. In order to properly define division, we would have to discusshave 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 thatInstead of doing that now, we’ll simply take it as given that the reader intuitively understands fractions. With thisthe reader intuitively understands fractions. With this definition of inverse, we can tackle division:definition of inverse, we can tackle division:
  16. 16.  ExampleExample  Suppose a = 2. ThenSuppose a = 2. Then  Note that the decimal expansion 0.5000… is theNote that the decimal expansion 0.5000… is the result of long division, a subject we are avoidingresult of long division, a subject we are avoiding here (for a discussion, see the division article). Wehere (for a discussion, see the division article). We can "prove" that 0.5000… is the inverse of 2 bycan "prove" that 0.5000… is the inverse of 2 by multiplying:multiplying:  To the extent that this is rather unsatisfying, weTo the extent that this is rather unsatisfying, we must ask the reader to suspend disbelief. Themust ask the reader to suspend disbelief. The mechanics of division are complicated, and deservemechanics 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.
  17. 17.  ExampleExample  Note that we expanded 6 into 2· 3 and canceled theNote that we expanded 6 into 2· 3 and canceled the 3’s. This is a bit sloppy, but again, our definition of3’s. This is a bit sloppy, but again, our definition of division is intuitive rather than precise.division is intuitive rather than precise.
  18. 18. Chapter 2.Chapter 2. OperationOperation withwith DecimalsDecimals
  19. 19.  Our money representation is based on a decimalOur 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 aSince we rarely deal with amounts less than a centavo, the lowest value we use is P.01. Thus, in anycentavo, the lowest value we use is P.01. Thus, in any mathematical computation, the final answer shouldmathematical computation, the final answer should be exact to the nearest centavo. An answer of Pbe exact to the nearest centavo. An answer of P 135.256 should be written as P 135.26. Usually, it135.256 should be written as P 135.26. Usually, it serves no purpose to express an answer id pesoserves no purpose to express an answer id peso value with fractional part of a centavo. For discussionvalue with fractional part of a centavo. For discussion purpose, the money value P 135.26 is read as “onepurpose, 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” toNote that the decimal point is read as “and” to separate peso value from the centavo. Sometimes, itseparate peso value from the centavo. Sometimes, it is necessary to indicate fractional parts of a centavois necessary to indicate fractional parts of a centavo in such cases as statistical figures, indices, realin such cases as statistical figures, indices, real estate tax, and surveys.estate tax, and surveys.  In performing addition and subtraction, the numbersIn performing addition and subtraction, the numbers containing decimal components are arrangedcontaining decimal components are arranged according to their place value. In multiplication andaccording to their place value. In multiplication and division, the numbers with decimal components aredivision, 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 willplacing the decimal points. The basic operations will be discussed in detail in the following sections.be discussed in detail in the following sections.
  20. 20. Addition with DecimalsAddition with Decimals  Addends with decimal components should be arrangedAddends with decimal components should be arranged according to their proper order (column). The decimalaccording to their proper order (column). The decimal points should fall in one vertical column in order to alignpoints should fall in one vertical column in order to align the figures according to their values. Hence, units of equalthe 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. 0853. 085  12. 31412. 314  94. 6594. 65  110. 0492110. 0492  Same columnSame column
  21. 21. Subtraction with DecimalsSubtraction with Decimals  The rule used in addition applies equally toThe rule used in addition applies equally to subtraction. Decimal points of the minuend andsubtraction. 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
  22. 22. Multiplication with DecimalsMultiplication with Decimals  The number of the decimal places of the product isThe number of the decimal places of the product is equal to the total number of decimal places (total digitequal to the total number of decimal places (total digit numbers to the right of the decimal points) of thenumbers 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 digitNote: 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)
  23. 23. Division with decimalsDivision with decimals  When the dividend contains decimal fractions and theWhen the dividend contains decimal fractions and the divisor is a whole number, the decimal point of thedivisor 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 remainder0 remainder
  24. 24. Rounding off NumbersRounding off Numbers  Generally, numbers are rounded offGenerally, numbers are rounded off in making estimates or forecasts. Inin making estimates or forecasts. In monetary computations, it is themonetary computations, it is the practice to round off the final answerpractice to round off the final answer to the nearest centavo because theto the nearest centavo because the third decimal place (thousandths) isthird decimal place (thousandths) is often immaterial.often immaterial.
  25. 25. Chapter 3.Chapter 3. FractionFraction
  26. 26. 1/41/4  While our exposure to the numeration system dealtWhile our exposure to the numeration system dealt mostly with whole numbe5rs, we also come acrossmostly 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” areSuch terms as “halves,” “fourths” or “quarters” are used with reference to measurement like “quarter toused 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 uniteach equal part is the fractional part of the unit which is usually expresses in a fraction. Forwhich is usually expresses in a fraction. For example, a parent bought a whole pizza pie andexample, 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 fourchildren. The parent would cut the pie into four equal parts and one part is given to each which isequal 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
  27. 27.  ¼ means¼ means oneone part of the whole which was divided into 4part of the whole which was divided into 4 equal parts.equal parts.  The number written above the line (numeratorThe number written above the line (numerator indicates the number of parts and the number writtenindicates the number of parts and the number written below the line (denominator) indicates the total units orbelow the line (denominator) indicates the total units or parts the cake was divided into. A fraction may be used toparts 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. Thesewhose 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 thewhose numerators are greater than or equal to the denominator. These fractions indicate values equal to ordenominator. 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 awritten as the sum of integer or whole number and a proper fraction. An improper fraction can also beproper fraction. An improper fraction can also be expressed as a mixed number or vice versa.expressed as a mixed number or vice versa.
  28. 28. Fundamental Rules in Dealing withFundamental Rules in Dealing with FractionsFractions  One should understand the fundamental rules inOne should understand the fundamental rules in dealing with fractions. These rules are very importantdealing 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 aremultiplication and division of fractions. The rules are the following:the following:  The value of a fraction is not changed when theThe value of a fraction is not changed when the numerator and the denominator are both multipliednumerator 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 we2. The value of a fraction is not changed if we divide both the numerator and the denominator bydivide both the numerator and the denominator by the same number other than zero. The result is likethe same number other than zero. The result is like dividing the fraction by 1.dividing the fraction by 1.
  29. 29. Changing Improper Fraction to aChanging Improper Fraction to a Whole or Mixed NumberWhole or Mixed Number  Mathematically, the line between the numerator andMathematically, the line between the numerator and denominator indicates division. In solving problems, it isdenominator indicates division. In solving problems, it is more convenient or easier to work with an impropermore 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 mixedTo change an improper fraction to a whole or mixed number, divide the numerator by the denominator.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 mixedChange 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
  30. 30. Changing Mixed Numbers toChanging Mixed Numbers to Improper FractionsImproper Fractions This conversion is very convenient when we perform mostThis conversion is very convenient when we perform most of the basic operations with fractions. Below are the waysof 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 fractionMake the result of (2) as the numerator of the new fraction and use the same denominator as that of the originaland use the same denominator as that of the original mixed number.mixed number.  Example 5Example 5. Change 4. Change 4 22 to improper fractionto improper fraction 33  Solution:Solution: follow the steps given abovefollow 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 the14/3 (improper fraction with same denominator as the original mixed number).original mixed number).
  31. 31. Reduction of FractionsReduction of Fractions  The process of converting fractions to otherThe process of converting fractions to other equivalent forms of either higher or lower termsequivalent forms of either higher or lower terms without changing the value of the fraction is calledwithout changing the value of the fraction is called reduction. As in division, multiplying or dividing bothreduction. As in division, multiplying or dividing both the numerator and denominator by the samethe 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 53 x 5 == 1515  4 4 x 3 124 4 x 3 12 4 4 x 5 204 4 x 5 20   Reduce 25/625 to lower terms.Reduce 25/625 to lower terms.  2525 == 2525 ÷÷ 55 == 55 oror 11 625625 625 ÷ 5 125 25625 ÷ 5 125 25
  32. 32.  When the numerator and the denominator have noWhen the numerator and the denominator have no more common divisor except 1 the fraction hasmore common divisor except 1 the fraction has been reduced to its lowest term. To reduce abeen reduced to its lowest term. To reduce a fraction to its lowest term, both the numerator andfraction to its lowest term, both the numerator and the denominator should be divided by their greatestthe denominator should be divided by their greatest possible common divisor.possible common divisor.  For example,For example,  in the proper fractionin the proper fraction 4646 23 is the greatest possible23 is the greatest possible 161161  common divisor of 46 and 161, hence,common divisor of 46 and 161, hence,  4646 == 4646 ÷÷ 2323 == 22  161 161161 161 23 723 7
  33. 33. Finding the greatest CommonFinding the greatest Common Divisor (g.c.d)Divisor (g.c.d)  The greatest common divisor is calledThe greatest common divisor is called highesthighest commoncommon factorfactor. If the greatest common divisor is not. If the greatest common divisor is not readily apparent, the following steps arereadily apparent, the following steps are recommended to determine the desirable divisor:recommended to determine the desirable divisor:  Divide the larger number of the numerator andDivide 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 smallerIf 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 theIf 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 succeedingContinue dividing each remainder by its succeeding remainder until the remainder is 0. The last divisor isremainder until the remainder is 0. The last divisor is the greater common divisor.the greater common divisor.
  34. 34.  Example: Find the g.c.d. ofExample: 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 11  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
  35. 35. Changing Fractions to DecimalChanging Fractions to Decimal  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 convertingMost of the problems in business require converting fractions into decimals. For example, if afractions into decimals. For example, if a mathematical operation with fractions is performedmathematical operation with fractions is performed on a pocket calculator, all fractions must beon a pocket calculator, all fractions must be converted to decimals before entering them in theconverted to decimals before entering them in the calculator.calculator.  Fraction indicates division, which means thatFraction indicates division, which means that the numerator is to be divided by the denominator.the numerator is to be divided by the denominator. An example is ¼ which is same as 1 divided by 4.An example is ¼ which is same as 1 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
  36. 36. Changing Decimal to FractionsChanging Decimal to Fractions  When a decimal fraction is written in a commonWhen a decimal fraction is written in a common fractions, the decimal fraction without the decimalfractions, the decimal fraction without the decimal point is used as the numerator, and thepoint is used as the numerator, and the denominator is 1 with as many zeros annexed asdenominator is 1 with as many zeros annexed as there are decimal places in the original decimalthere are decimal places in the original decimal fraction. Then the common fraction is reduced to itsfraction. 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
  37. 37. Addition of fractionsAddition of fractions  Two different things cannot be added. The same is true withTwo 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 aTo add fractions, all denominators must be converted to a common number without changing the values of thecommon number without changing the values of the fraction.fraction.  When two or more fractions with different denominators areWhen two or more fractions with different denominators are to be added, we must change the fractions to theirto be added, we must change the fractions to their equivalent fractions with the same denominator. Theequivalent fractions with the same denominator. The procedure involves the convertion of the fractions to itsprocedure 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 sameAfter converting 1/2 to ¼, lets add this to ¼ w/c has same denominatordenominator  22 ++ 11 == 33  4 4 44 4 4 
  38. 38.  Solution:Solution:  1st step – Determine the1st step – Determine the “lowest possible”“lowest possible” numbernumber that can be divided by the different denominatorsthat 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 each2nd 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 62 x 6 1212  3rd step – Add the equivalent fractions with the same3rd step – Add the equivalent fractions with the same denominators.denominators.  11 ++ 11 ++ 11 == 66 ++ 44 ++ 33  2 32 3 4 12 12 124 12 12 12  == 6 + 4 + 36 + 4 + 3  1212  = 13/12 or 1 ½= 13/12 or 1 ½
  39. 39. Addition of Mixed NumbersAddition of Mixed Numbers  When mixed numbers are added,When mixed numbers are added, it is unnecessary to convert theit is unnecessary to convert the numbers to their equivalentnumbers to their equivalent improper fractions. Add the wholeimproper fractions. Add the whole integers and the fractionsintegers and the fractions separately. The fractional part ofseparately. The fractional part of the answer should reduce to itsthe answer should reduce to its lowest term.lowest term.
  40. 40. Subtraction of FractionsSubtraction of Fractions  Fractions subtracted must firstFractions subtracted must first be converted to theirbe converted to their equivalents with theequivalents with the leastleast commoncommon denominatordenominator (l.c.d.).(l.c.d.). then the numerators of thethen the numerators of the converted fractions areconverted fractions are subtracted. The answer willsubtracted. The answer will have the numerator differencehave the numerator difference as the numerator and the leastas the numerator and the least common denominator as thecommon denominator as the denominator.denominator.
  41. 41. Subtracting a Mixed Number fromSubtracting a Mixed Number from a Whole Numbera Whole Number  By way of an example, theBy way of an example, the following are the steps infollowing are the steps in subtracting a mixed from asubtracting a mixed from a whole number:whole number:
  42. 42.  Step 1. Convert one unit of the minuend into anStep 1. Convert one unit of the minuend into an improper fraction with the same denominator as theimproper fraction with the same denominator as the fraction in the subtrahend. Reduce the wholefraction in the subtrahend. Reduce the whole number by one so as not to change the value of thenumber 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
  43. 43. Subtracting Mixed NumbersSubtracting Mixed Numbers  When necessary, convert the fractional parts ofWhen necessary, convert the fractional parts of both the minuend and the subtrahend so that theyboth the minuend and the subtrahend so that they have a common denominator. If the fraction in thehave a common denominator. If the fraction in the subtrahend is smaller than the fraction in thesubtrahend is smaller than the fraction in the minuend, you can subtract, but if the fraction isminuend, you can subtract, but if the fraction is greater than the fraction in the minuend, follow thegreater than the fraction in the minuend, follow the steps below.steps below.  Convert one unit of the minuend into an improperConvert one unit of the minuend into an improper fraction with “correct’ denominator, and add this unitfraction 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 oneReduce 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.
  44. 44. Multiplication of FractionsMultiplication of Fractions  There is no need to determine the commonThere 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 fractionThe basic principle in multiplication of fraction is to determine the product of both numeratorsis to determine the product of both numerators and both denominators and the productsand both denominators and the products become the numerator and denominator of thebecome the numerator and denominator of the answer.answer.  In multiplying a whole number by a mixedIn multiplying a whole number by a mixed number, change the mixed number to annumber, change the mixed number to an improper fraction and determine the product.improper fraction and determine the product.  If both factors are mixed numbers, convertIf both factors are mixed numbers, convert each mixed number to an improper fractioneach mixed number to an improper fraction before multiplying.before multiplying.
  45. 45.  Example: Multiply 1/2 by 2/3Example: Multiply 1/2 by 2/3 1/2 x 2/3 =1/2 x 2/3 = 1 x 21 x 2 == 22 oror 11 reduce to lowest termreduce to lowest term 2 x 3 = 6 32 x 3 = 6 3
  46. 46. Divisions of FractionsDivisions of Fractions  The simplest way to divide fractions is toThe simplest way to divide fractions is to multiply the dividend by the reciprocalmultiply the dividend by the reciprocal (inverted form) of the divisor. The(inverted form) of the divisor. The numbers 3 and 1/3 are called reciprocalnumbers 3 and 1/3 are called reciprocal to each other. Likewise, 4/5 is theto each other. Likewise, 4/5 is the reciprocal of 5/4.reciprocal of 5/4.  When dividing a mixed number byWhen dividing a mixed number by another mixed number, change eachanother mixed number, change each mixed number to an improper fractionmixed number to an improper fraction and then multiply after inverting theand then multiply after inverting the divisor.divisor.
  47. 47.  Example:Example:  Divide: 5/8 by 2/3Divide: 5/8 by 2/3  55 ÷÷ 22 == 55 xx 33  88 3 8 23 8 2 == 55 xx 33 == 1515 8 2 168 2 16
  48. 48. Chapter 4Chapter 4 Percent inPercent in BusinessBusiness
  49. 49.  The use of percent is gainingThe use of percent is gaining wide acceptance in all sphereswide acceptance in all spheres of activity. When relating theof activity. When relating the parts to a whole, relationship isparts to a whole, relationship is expressed in percent.expressed in percent.  Percent is also frequentlyPercent is also frequently used in presenting accountingused in presenting accounting and statistical data relationship.and statistical data relationship.
  50. 50. Percent to FractionPercent to Fraction  Percent numbers are not really “new” sincePercent numbers are not really “new” since they are in reality fractions.they are in reality fractions.  To express 80% as a fraction in its lowestTo express 80% as a fraction in its lowest term, convert 80% in its fractional form firstterm, convert 80% in its fractional form first and perform the operation of division ofand perform the operation of division of proper fractions as discussed in Chapterproper fractions as discussed in Chapter III.III.  80% =80% = 8080  100100   Then =Then = 80 ÷ 2080 ÷ 20  100÷20100÷20  == 44  55  thus, 80% is 4/5 in fractionthus, 80% is 4/5 in fraction
  51. 51. Percent to DecimalPercent to Decimal  In any mathematical computation, percent isIn any mathematical computation, percent is converted to decimal before multiplying of dividing itconverted to decimal before multiplying of dividing it with other quantifying numbers. There are severalwith other quantifying numbers. There are several percent values encountered in businesspercent values encountered in business transactions that do not convert into fractionstransactions that do not convert into fractions making such percent values as decimals rather thanmaking such percent values as decimals rather than as fractions.as fractions.  The basic principle in converting percent toThe basic principle in converting percent to decimal is to move the decimal point two places todecimal 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% == 1515 = .15= .15  100100
  52. 52. Representing Decimals as PercentsRepresenting Decimals as Percents  The fastest way to convert decimals to percent is toThe fastest way to convert decimals to percent is to the decimal point two places to the right. Thisthe decimal point two places to the right. This method is actually the reverse order of convertingmethod is actually the reverse order of converting percent to its decimal equivalent as discussed in thepercent 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 %
  53. 53. Representing Fractions asRepresenting Fractions as PercentsPercents  In converting a fraction to percent, changeIn converting a fraction to percent, change the fraction to its decimal equivalent asthe fraction to its decimal equivalent as discussed in the previous chapter. Thendiscussed in the previous chapter. Then multiply the decimal equivalent by 100 ormultiply the decimal equivalent by 100 or simply move the decimal point two placessimply 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 percentExample: Convert 1 7/8 to its equivalent percent form,form,  Solution:Solution:  11 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 %
  54. 54. Fraction-Decimal-PercentFraction-Decimal-Percent EquivalentsEquivalents  The equivalents discussed in the previous chaptersThe equivalents discussed in the previous chapters can be tabulated into what are known as aliquotcan be tabulated into what are known as aliquot parts of 100. An aliquot part is a portion of a numberparts of 100. An aliquot part is a portion of a number by which the number may be divided leaving noby which the number may be divided leaving no remainderremainder  Example:Example:  Multiply 60 by 33Multiply 60 by 33 11 %%  33   60 by 3360 by 33 11 %% = 60 x= 60 x  33 = 20= 20
  55. 55. Determining the PercentageDetermining the Percentage  In the statement, 40 is 40/700 0f 700 theIn the statement, 40 is 40/700 0f 700 the denominator of the fraction which is thedenominator of the fraction which is the basis of comparison is called the base. Thebasis of comparison is called the base. The rate is the percent indicating the number orrate is the percent indicating the number or quantity for every 100. The percentagequantity for every 100. The percentage refers to the number of items in the desiredrefers to the number of items in the desired situation or condition.situation or condition.  Based on the above concept, the principleBased 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 = percentageP = percentage  B = baseB = base  r = rate, usually expresses in %r = rate, usually expresses in %
  56. 56.  Stated as a principle, percentage is the product ofStated as a principle, percentage is the product of the base times the rate. In any mathematicalthe base times the rate. In any mathematical computation, before we add, subtract, multiply orcomputation, before we add, subtract, multiply or divide, we must convert the percent to its decimal ordivide, we must convert the percent to its decimal or fractional equivalent whichever is convenient to use.fractional equivalent whichever is convenient to use.  Example: An employee who earned P 1,500 spentExample: An employee who earned P 1,500 spent 30 % of his earnings to buy a wrist watch. How30 % 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
  57. 57. Finding the RateFinding the Rate  One of the most commonly used representations ofOne of the most commonly used representations of a number in business is the rate. Althougha number in business is the rate. Although representation of number based on rate involvesrepresentation of number based on rate involves different interpretations in business or surveys.different interpretations in business or surveys.  Based on the preceding discussion, a formulaBased on the preceding discussion, a formula can be derived from Equation 4.1 as follows:can be derived from Equation 4.1 as 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
  58. 58. Finding the BaseFinding the Base  Basically, the base quantity can be determined byBasically, the base quantity can be determined by applying the same concept in Equation 4.2. theapplying 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
  59. 59. Ratio and ProportionRatio and Proportion  As mentioned earlier, percent is another way ofAs mentioned earlier, percent is another way of comparing a number with the whole, while itscomparing a number with the whole, while its equivalent in decimal is comparing with one unit.equivalent in decimal is comparing with one unit.  RatioRatio is a relation between two numbers expressedis a relation between two numbers expressed in terms of a quotient.in terms of a quotient.  The result of reducing ratios to lowest terms isThe result of reducing ratios to lowest terms is called equivalent reduced ratio.called equivalent reduced ratio.  The equality of two ratios is calledThe equality of two ratios is called proportionproportion..
  60. 60. Chapter 5:Chapter 5: Employee’sEmployee’s CompensatioCompensatio nn
  61. 61.  Among other expenses, a business enterprise paysAmong other expenses, a business enterprise pays the salaries to its personnel. Salary is thethe salaries to its personnel. Salary is the renumeration received by an employees forrenumeration received by an employees for services rendered. The labor code of the Philippinesservices rendered. The labor code of the Philippines defines wages or salaries as “ the renumeration ordefines wages or salaries as “ the renumeration or earning, however designated, capable of beingearning, however designated, capable of being expressed in terms of money, whether fixed orexpressed in terms of money, whether fixed or ascertained on a time, task, piece or commissionsascertained on a time, task, piece or commissions basis or other method of calculating the same whichbasis or other method of calculating the same which is payable by an employer to an employee andis payable by an employer to an employee and includes the fair and reasonable value of board,includes the fair and reasonable value of board, lodging or other facilities customarily furnished bylodging or other facilities customarily furnished by the employer to the employee.”the employer to the employee.”  SalariesSalaries 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 theEmployees may be classified according to the bases of their salary payments. There are the fixedbases of their salary payments. There are the fixed wags earners, the piece-workers, the commissionwags earners, the piece-workers, the commission earners and the salary-plus-commission earners.earners and the salary-plus-commission earners.
  62. 62. Computing Wages Based on TimeComputing Wages Based on Time (period)(period)  Fixed wage earnersFixed wage earners are paid based on the numberare paid based on the number of hours worked. For work done beyond the regularof hours worked. For work done beyond the regular eight (8) working hours a day, the employee iseight (8) working hours a day, the employee is given additional pay calledgiven additional pay called overtime pay.overtime pay.  The Labor Code fixes the minimum rates ofThe Labor Code fixes the minimum rates of overtime pay, night shift differential, and work onovertime pay, night shift differential, and work on regular and special holidays. The rates range fromregular and special holidays. The rates range from 25 %, 30% to 100% over and above the regular pay25 %, 30% to 100% over and above the regular pay rate. For purposes of discussion, overtime pay asrate. For purposes of discussion, overtime pay as herein used will be regular pay plus 50% of said payherein used will be regular pay plus 50% of said pay which means that for every one hour overtime, thewhich means that for every one hour overtime, the pay will be equivalent to 1.5 of the regular hourlypay will be equivalent to 1.5 of the regular hourly rate.rate.
  63. 63. Computing Wages on PieceworkComputing Wages on Piecework BasisBasis  The minimum rates set by the Labor CodeThe minimum rates set by the Labor Code do not apply to employees whose salariesdo not apply to employees whose salaries are not based on the number of workingare not based on the number of working hours. Employees may be paid by resultshours. Employees may be paid by results likelike “pakyaw” (pakiao),“pakyaw” (pakiao), “takay”“takay” and byand by the pieces of product.the pieces of product.  Workers in selected industries such asWorkers in selected industries such as those engaged in manufacturing, packing,those engaged in manufacturing, packing, and handicrafts are paid onand handicrafts are paid on straightstraight pieceworkpiecework basis. The wage of a straightbasis. The wage of a straight pieceworker depends solely on number ofpieceworker depends solely on number of units completed each period of time.units completed each period of time.
  64. 64. Computing Wages on CommissionComputing Wages on Commission BasisBasis  Most of the commission wage earners are staff inMost of the commission wage earners are staff in the sales of a business. They are paid on the basisthe sales of a business. They are paid on the basis of number of units sold. Since sales is the bloodlineof number of units sold. Since sales is the bloodline of the company, the sales people are doublyof the company, the sales people are doubly motivated to increase sales output. The three mostmotivated to increase sales output. The three most commonly used compensation schemes in salescommonly used compensation schemes in sales are the straight commission, commission andare the straight commission, commission and bonus, and salary plus commission.bonus, and salary plus commission.  The regular sales force of a company isThe regular sales force of a company is usually paid a fixed monthly rate plus commissionusually paid a fixed monthly rate plus commission earned from sales.earned from sales.  To compute for the commission, multiply theTo compute for the commission, multiply the sales (in number of units or peso value) by thesales (in number of units or peso value) by the commission rate.commission rate.
  65. 65. Incentive or BonusesIncentive or Bonuses  Incentive pay and bonuses in money or inIncentive pay and bonuses in money or in kind are given to employees to motivatekind are given to employees to motivate them. The incentive pay may be based onthem. The incentive pay may be based on the number of units completed or sold inthe number of units completed or sold in excess of a required minimum.excess of a required minimum.
  66. 66. DeductionsDeductions  The earnings of an employee are subject toThe earnings of an employee are subject to deductions which are required by law of bydeductions which are required by law of by agreement between the employer and theagreement between the employer and the employee. Deductions required by law areemployee. Deductions required by law are income taxes, social security taxes,income taxes, social security taxes, medicare contributions, contributions tomedicare contributions, contributions to pension plans and contributions to pag-ibig.pension plans and contributions to pag-ibig.
  67. 67. Chapter 6:Chapter 6: Buying andBuying and SellingSelling
  68. 68.  Buying and selling are the essentialBuying and selling are the essential functions of a trading concern. Manufacturersfunctions of a trading concern. Manufacturers or producers buy raw materials or parts toor producers buy raw materials or parts to produce finished products. A trading concernproduce finished products. A trading concern tries to buy merchandise at the lowest possibletries to buy merchandise at the lowest possible cost in order to maximize its profits. Acost in order to maximize its profits. A manufacturer endeavors to acquire materialsmanufacturer endeavors to acquire materials and/or finished parts in order to minimize itsand/or finished parts in order to minimize its cost of production which is a way of increasingcost of production which is a way of increasing the income upon the sale of the finishedthe income upon the sale of the finished products.products.  In a trading business, the trader of theIn a trading business, the trader of the buyer should determine the goods needed bybuyer should determine the goods needed by the customers and the price the customers arethe customers and the price the customers are willing to pay for the goods. He should alsowilling to pay for the goods. He should also know where to obtain the goods at the lowestknow 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 productsIn general, a manufacturer sells its products in bulk to a wholesaler. In turn, the wholesalerin bulk to a wholesaler. In turn, the wholesaler sells in smaller lots to retailers who sell thesells in smaller lots to retailers who sell the product to the end users or consumers.product to the end users or consumers.
  69. 69. 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 ato induce trading. This is usually offered by a manufacturer or wholesaler. This type is usuallymanufacturer or wholesaler. This type is usually encountered in catalogs where a list price is printedencountered in catalogs where a list price is printed together with the trade discount thereon. The listtogether with the trade discount thereon. The list price less the trade discount is the suggestedprice 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 sellingIt is a reduction on the selling price offered to a buyer to induce him to payprice offered to a buyer to induce him to pay promptly.promptly.
  70. 70. Finding the Net PriceFinding the Net Price  There are two methods in determining the net priceThere are two methods in determining the net price to the buyer when trade discount is given. Theseto the buyer when trade discount is given. These are theare the Discount Rate MethodDiscount Rate Method and theand the Net PriceNet Price Rate Method.Rate Method.  Method A.Method A. Discount Rate Method.Discount Rate Method. The formulaThe formula for discount is the same for percentage where thefor discount is the same for percentage where the terms percentage (P), base (b) and rate (r) interms percentage (P), base (b) and rate (r) in Equation 4.1 on page 56 are substituted with theEquation 4.1 on page 56 are substituted with the terms discount (D), selling or list price (L.P) andterms 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 percent= Discount rate usually expressed in percent
  71. 71.  The net price then is the difference betweenThe net price then is the difference between the selling price and the discount which is shownthe 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 tapeExample 1. The list price of a cassette tape recorder at AVESCO is P1, 850. The Traderecorder at AVESCO is P1, 850. The Trade discount rate is 12 %. How much will the buyer paydiscount rate is 12 %. How much will the buyer pay for the recorder? Usefor the recorder? Use Discount Rate MethodDiscount Rate Method andand thethe Net Price Rate Method.Net Price Rate Method.  Given:Given: L.P. = P1,850L.P. = P1,850 d = 12%d = 12%
  72. 72.  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.)  ── 222222 (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
  73. 73. Finding the Discount RateFinding the Discount Rate  A supplier may list only the net price and list price ofA supplier may list only the net price and list price of the products being sold. The buyer should knowthe products being sold. The buyer should know how to determine the discount rate in order tohow to determine the discount rate in order to evaluate the reasonableness of the discountevaluate the reasonableness of the discount offered.offered.  Example 2. What is the discount rate if the listExample 2. What is the discount rate if the list price is P1,500 and the discount is P450.00?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= 1,500PhpPhp x dx d d =d = 450450 = .3 or 30%= .3 or 30%  1,5001,500
  74. 74.  Example 3.Example 3. What is the discount rate, if theWhat 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 = P83D = 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 6.1Substituting the list price in Equation 6.1::  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%
  75. 75. Discounts in SeriesDiscounts in Series  To dispose their goods quickly, wholesalers mayTo dispose their goods quickly, wholesalers may offer successive trade discount rates which we calloffer successive trade discount rates which we call discounts in series.discounts in series. If a manufacturing concernIf a manufacturing concern would give 15% and 10% trade discounts on itswould give 15% and 10% trade discounts on its product, this does not mean that the total discountproduct, this does not mean that the total discount rate is 25%. As a discount series, it simply meansrate is 25%. As a discount series, it simply means that the first discount of 15% is applied to the originalthat the first discount of 15% is applied to the original list price and the second discount of 10% to thelist price and the second discount of 10% to the balance of the original list price and the first discountbalance of the original list price and the first discount price.price.  Example 4.Example 4. A home appliance dealer was offeredA home appliance dealer was offered television set with a list price of P5,680 less 15% andtelevision set with a list price of P5,680 less 15% and 10%. What is the net price?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
  76. 76.  2nd Step. To get the balance (L.P2) on which the2nd Step. To get the balance (L.P2) on which the second discount rate (d2) will be applied, subtractsecond 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)  ── 852852 (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 d2D2 = 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.) by4rth 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.)  ==============
  77. 77. Method B.Method B. Net Price RateNet Price Rate MethodMethod  1rst Step1rst Step. The net price rate is the difference. The net price rate is the difference between 100% and the discount rate (See Equationbetween 100% and the discount rate (See Equation 6.3). The same formula is also applied when two or6.3). The same formula is also applied when two or more discounts are given. The final net price is themore discounts are given. The final net price is the product of all the net price rates found by usingproduct 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% ─ 15%) (100% -= (100% ─ 15%) (100% - 10%)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
  78. 78.  Obviously, the net price rate method isObviously, the net price rate method is relatively shorter and simpler than the discountrelatively shorter and simpler than the discount method, more so when there are more than twomethod, more so when there are more than two series of discounts. In such cases, the formula toseries 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 theIt is immaterial as to which order the discounts are taken.discounts are taken.
  79. 79. Single-Discount RateSingle-Discount Rate EquivalentEquivalent  There are two practical ways to evaluate discountThere are two practical ways to evaluate discount series offered by two or more sellers. Let usseries offered by two or more sellers. Let us assume that in terms of services, neither firm heldassume that in terms of services, neither firm held any advantage over the other. Hence, the buyer’sany advantage over the other. Hence, the buyer’s only concern is to determine which firm’s net sellingonly concern is to determine which firm’s net selling price was smaller. As earlier discussed, theprice was smaller. As earlier discussed, the discount series of, for example, 10% and 15% arediscount 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 toOne way of evaluating discount series is to determine the final net price rate which wasdetermine the final net price rate which was illustrated in Example 4, Method B. From theillustrated in Example 4, Method B. From the buyer’s point of view, the smaller the final net pricebuyer’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 rates inAn easier way to evaluate the discount rates in a discount series is thea discount series is the equivalent single discountequivalent single discount raterate.. From the buyer’s point of view, the biggerFrom the buyer’s point of view, the bigger the single discount rate equivalent, the lesser is thethe single discount rate equivalent, the lesser is the net price.net price.
  80. 80.  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 theproduct 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 each. Derive the net price rate (NPR) of each discount rate.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 rateDetermine the single discount rate equivalent (SDRE).equivalent (SDRE).  SDRE = 100% ─ (NPR1 x NPR2)SDRE = 100% ─ (NPR1 x 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%
  81. 81.  Example 6.Example 6. A product is being offered in the market byA product is being offered in the market by Supplier A and Supplier B. The trade discounts beingSupplier A and Supplier B. The trade discounts being offered by Supplier A are 20% and 10% while that ofoffered 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 equivalentSupplier A. Solving for the single discount rate equivalent of Supplier A, determine first the net price rate applicableof 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%
  82. 82.  Supplier B. Solve for the net price applicable to theSupplier 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, theBased on the net price rates being offered, the offer of a lower net price rate (72%) by Supplier A isoffer of a lower net price rate (72%) by Supplier A is the better offer. But, based on the single discountthe better offer. But, based on the single discount rate equivalent, the offer of a higher discount (28%)rate equivalent, the offer of a higher discount (28%) by Supplier A is also the better offer.by Supplier A is also the better offer.
  83. 83. Cash DiscountsCash Discounts  To encourage wholesalers or retailers to buy is one thingTo encourage wholesalers or retailers to buy is one thing and to induce the buyers to pay their purchases is anotherand to induce the buyers to pay their purchases is another thing. While sellers give buyers credit lines in that thething. While sellers give buyers credit lines in that the buyer are given several days after delivery within which tobuyer are given several days after delivery within which to pay their purchases, sellers encourage buyers to paypay 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 tradeshould not be confused with a trade discount. The former is a reduction in thediscount. The former is a reduction in the net pricenet price oror invoice priceinvoice price upon payment on a specified period of timeupon 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 writtenThe cash discount rates offered are clearly written on the invoice. An example of a cash discount is”on the invoice. An example of a cash discount is”5,5, n ”n ”  10 3010 30  which means that a cash discount of 5% will be deductedwhich means that a cash discount of 5% will be deducted  from the net price if the invoice paid within 10 daysfrom the net price if the invoice paid within 10 days from the date of the invoice. The "n/30" means that afterfrom the date of the invoice. The "n/30" means that after 10 days, no discount is given and that the invoice price10 days, no discount is given and that the invoice price should be paid not later than 30 days of the invoice date.should be paid not later than 30 days of the invoice date.  If the retailer or buyer would avail of the cashIf the retailer or buyer would avail of the cash discount offered, the price to be paid is the invoice pricediscount offered, the price to be paid is the invoice price less the amount of cash discount.less the amount of cash discount.
  84. 84.  Example 7. An invoice with a list price of P18,200Example 7. An invoice with a list price of P18,200 dated October 12, has trade discounts of 15% anddated 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. Hown/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 = 44 ,, 22 , 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
  85. 85.  2nd Step.2nd Step. Having solve for the N.P., which isHaving solve for the N.P., which is P13,923.00, we can now proceed to solved the cashP13,923.00, we can now proceed to solved the cash discount (Dc). If the purchaser pays on October 22 whichdiscount (Dc). If the purchaser pays on October 22 which is 10 days from the invoice date, the cash discount term isis 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. - DcNIP = 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
  86. 86.  2nd Step.2nd Step. If the Purchaser pays on November 11If the Purchaser pays on November 11 which is 30 days from the invoice date, the cashwhich 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 asAlternatively, 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
  87. 87. Mathematics of PricingMathematics of Pricing  The price of a commodity is determined by manyThe price of a commodity is determined by many factors obtaining in the market. It is greatly affectedfactors obtaining in the market. It is greatly affected by the need of the consumers for the commodity, itsby the need of the consumers for the commodity, its availability, its quality level and the number ofavailability, its quality level and the number of different lines with similar products. If there is adifferent lines with similar products. If there is a great demand for the commodity and the stock isgreat 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 thatBut if the price demanded is prohibitive such that the average consumer could not afford it, thethe average consumer could not afford it, the consumers may shift to other available similarconsumers 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 termsThe business should understand pricing terms such as cost, mark-up, margin and the like tosuch as cost, mark-up, margin and the like to enable himself to price his commodities reasonably.enable himself to price his commodities reasonably. CostCost refers to the amount the purchaser acquiredrefers 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, theIf an original selling price is adjusted upward, the addition is calledaddition is called Mark-upMark-up.. The reverse is calledThe reverse is called Mark-down.Mark-down.
  88. 88.  To illustrate clearly the above terminologies, letTo illustrate clearly the above terminologies, let us take an example. A car dealer bought a carus 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, assumingP14,500 is the cost of the article, assuming he did not spend any amount for reconditionhe did not spend any amount for recondition the car before selling it.the car before selling it.  P 17,200 is the selling price, tag price, orP 17,200 is the selling price, tag price, or retail price of the article. (P17,200 – P14,500) =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 P2,700In the above illustration, the margin P2,700 can be expressed in percent based in either thecan be expressed in percent based in either the selling price or cost. This is calledselling price or cost. This is called percentpercent margin.margin. If the selling price is used as the base,If the selling price is used as the base, thethe percent margin on selling pricepercent margin on selling price isis P2,700P2,700 = .15697 = 15.7%.= .15697 = 15.7%.
  89. 89.  If the cost is P17,200 used as the base, theIf the cost is P17,200 used as the base, the percentpercent marginmargin on cost ison cost is P2,700P2,700 = .1862 =1 =18.62%= .1862 =1 =18.62%  P14,500P14,500   Formula-wise, the equations for the above are theFormula-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(%) = MarginMargin x 100, based onx 100, based on selling priceselling price  Selling priceSelling price  Percent Margin(%) =Percent Margin(%) = MarginMargin x 100, based on costx 100, based on cost  CostCost
  90. 90.  Example 8.Example 8. The merchandise that cost a dealerThe merchandise that cost a dealer P4,150 was sold for P5,200. Determine the percentP4,150 was sold for P5,200. Determine the percent margin (a) based on selling price, (b) based on cost.margin (a) based on selling price, (b) based on cost.  Given:Given:  Selling price = P5,200Selling price = P5,200  Cost = P4,150Cost = P4,150  Solution:Solution: Immediately subtract P4,150 fromImmediately subtract P4,150 from P5,200 to get the margin.P5,200 to get the margin.  MarginMargin = 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 sellingIf 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 100x 100  P5,200P5,200  = .202 x 100= .202 x 100  = 20.2%= 20.2%
  91. 91.  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 100x 100  P4,150P4,150  = .253 x 100= .253 x 100  = 25.3%= 25.3%  NoteNote:: Another term used for gross margin isAnother term used for gross margin is mark on.mark on.
  92. 92. Chapter 7:Chapter 7: SimpleSimple InterestInterest
  93. 93.  Money borrowed usually bears a cost calledMoney borrowed usually bears a cost called interest.interest. The amount borrowed is calledThe amount borrowed is called principal.principal.  Sources of loans are the banks, investmentSources 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 inThe interest or interest charge is usually expressed in percent, such as 6%,which means a charge of P6.00percent, such as 6%,which means a charge of P6.00 for every P100.00 for a definite period of time. Unlessfor 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 chargeThe 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 value.I = interest expressed in monetary value.  P = principal or the amount borrowed.P = principal or the amount borrowed.  R = interest rate on the principal, usuallyR = interest rate on the principal, usually stated in percent.stated in percent.  T = time or duration of the loan, expressedT = time or duration of the loan, expressed in number of years, months,in number of years, months,  days, etc.days, etc.
  94. 94.  To illustrate, if Mr. X borrowed P2,600.00 from theTo illustrate, if Mr. X borrowed P2,600.00 from the bank at an interest of 5%, what is the interest in 3bank 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
  95. 95.  The amount to be paid by the borrower upon maturity of the loanThe 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 anExample 1. Pedro borrowed from Jose P5,200 with an interest of 6%. How much should Pedro pay after 16interest 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
  96. 96.  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
  97. 97. Determining the Rate ofDetermining the Rate of Interest (r)Interest (r) This particular problem is encountered in evaluating two orThis particular problem is encountered in evaluating two or more alternative choices. A lender wants a higher rate ofmore alternative choices. A lender wants a higher rate of interest for his money, while a borrower prefers a lesser rate ofinterest 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 afterAssume that Mr. Y would like to borrow P5,000 payable after one (1) year from a bank whose interest charge is 18% perone (1) year from a bank whose interest charge is 18% per year. On the other hand, Mr. Z is willing to lend him the sameyear. On the other hand, Mr. Z is willing to lend him the same principal and interest charge is P950 per year. Since the rateprincipal and interest charge is P950 per year. Since the rate of interest on the bank loan is already know, Mr. Y must knowof 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 whichthe 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 = II Equation 7.4Equation 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% interestThus, as compared to Mr. Y’s proposal of a 19% interest rate per year, the bank loan with 18% interest rate perrate per year, the bank loan with 18% interest rate per year is definitely the better offer.year is definitely the better offer.
  98. 98. Determining the Time (t)Determining the Time (t)  As derived from Equation 7.1As derived from Equation 7.1  t =t = II Equation 7.5Equation 7.5  PrPr  Example 3. How long will it take for a deposit ofExample 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 7Php1,500 x 7 11%%  22  == Php186.00Php186.00  Php1,500 x .075Php1,500 x .075  == 1.65 years1.65 years
  99. 99.  The time may be converted to its appropriate unitsThe 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 121.65 years x 12 months per yearmonths per year  == 19.8 months19.8 months  OrOr  (Days) t(Days) t == 1.65 years x 360 days1.65 years x 360 days per yearper year  == 594 days594 days  If it is required to express the time (t) inIf 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 = 1.65In 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) yearThe period of 1.65 indicates one(1) year and a fraction of year expressed in decimal part asand 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 permultiplied by the conversion factor of 12 months per year.year.  Thus, number of months is 7.Thus, number of months is 7.
  100. 100.  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.
  101. 101. Determining the Principal,Determining the Principal, Given (A) (r) (t)Given (A) (r) (t)  This particular problem involves Equations 7.1 andThis particular problem involves Equations 7.1 and 7.2. In any mathematical computation, solving two7.2. In any mathematical computation, solving two unknown variables using one equation or formula isunknown variables using one equation or formula is not possible. However, two equations with at leastnot possible. However, two equations with at least one common variable can be combined to derive aone common variable can be combined to derive a new equation.new equation.  If A = P + I (Equations 7.2) and I = PrtIf 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 followingequivalent (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 7.6A = P (I + rt) Equation 7.6 P =P = AA Equation 7.7Equation 7.7 I + rtI + rt 
  102. 102.  Example 4.Example 4. If a bank offers 8 ½% interest rate andIf 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 = 8r = 8 11%%  22  A = P3,878.20A = P3,878.20  t = 5 yearst = 5 years  Solution:Solution: Using EquationUsing Equation P =P = A___A___  1 + r x t1 + r x t  PP == P3,878.20P3,878.20  1 + 81 + 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
  103. 103. Determining the Sum (A)Determining the Sum (A)  Example 5. The bank offers an annualExample 5. The bank offers an annual rate of 10rate of 10 11 % interest, for 3 ½ years,% interest, for 3 ½ years, 22  how much will the accumulated sum be ifhow 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 11 %%  22  t = 3 ½ yearst = 3 ½ years   In this particular problem, two methodsIn this particular problem, two methods may be used to determine the amount (A).may be used to determine the amount (A).
  104. 104.  1rst Method.1rst Method. Determine the interest by usingDetermine the interest by using  Equation 7.1, I = Prt and then, add the interestEquation 7.1, I = Prt and then, add the interest and the principal to determine the amount A, usingand 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 11 % x 3% x 3 11))  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
  105. 105. Promissory NotesPromissory Notes  A promissory note may be used for raising money. TheA promissory note may be used for raising money. The borrower , in exchange for a loan, may issue his ownborrower , in exchange for a loan, may issue his own promissory note or an immature note received by himpromissory note or an immature note received by him form another person. Aform another person. A promissorypromissory notenote is a writtenis a written promise to pay to a person a certain sum of moneypromise to pay to a person a certain sum of money on a specified date.on a specified date. The person who signs theThe person who signs the promissory note is thepromissory note is the maker.maker. The person to whom hisThe person to whom his payment is promised is thepayment is promised is the payee.payee. TheThe period,period, oror termterm ofof the note is the duration or length of time from the date ofthe 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 alsois 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. TheThe maturity valuematurity value is the total amount to be paid on theis the total amount to be paid on the maturity or due date. Thematurity or due date. The face valueface value of a promissoryof a promissory note is the amount specifically mentioned in the note. Ifnote is the amount specifically mentioned in the note. If the interest is mentioned in the note, it is anthe interest is mentioned in the note, it is an interest-interest- bearing notebearing note.. If no interest is mentioned, then it is aIf 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 onMr. Darwin borrowed P10,000 from his friend Carlos on November 8, Mr. Darwin issued a promissory noteNovember 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:
  106. 106.   P 10,000P 10,000 November 8November 8  Ninety daysNinety days afterafter II promise to pay to the order ofpromise to pay to the order of Mr. CarlosMr. Carlos the sum ofthe sum of Ten Thousand Pesos andTen Thousand Pesos and no/100no/100For value received with interest at 10%.DueFor value received with interest at 10%.Due Date:Date: February 8February 8 Mr. DarwinMr. Darwin 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 theTo compute for the Maturity Value (a) of the promissory note in the above example –promissory note in the above example –
  107. 107.  1rst Step. Determine the interest by using the1rst 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% xP10,000 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 value2nd 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 beAlternatively, 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
  108. 108.  On the other hand, if no interest is mentioned in theOn the other hand, if no interest is mentioned in the note, the face value is the maturity value. Tonote, the face value is the maturity value. To illustrate, convert above interest-bearing note to aillustrate, 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 8November 8  Ninety daysNinety days afterafter II promise to pay to the order ofpromise to pay to the order of Mr. CarlosMr. Carlos the sum ofthe sum of Ten Thousand Two hundredTen Thousand Two hundred fifty Pesos and no/100fifty Pesos and no/100For value received with interestFor value received with interest at 10%.at 10%.  Due Date:Due Date: February 8February 8 Mr. DarwinMr. Darwin SignedSigned   The omission of a stated interest rate does notThe omission of a stated interest rate does not necessarily mean that the original debt bears nonecessarily mean that the original debt bears no interest. The interest, if any, might have beeninterest. The interest, if any, might have been added to the original debt when the face value wasadded to the original debt when the face value was determined. As illustrated, the original debt was Pdetermined. As illustrated, the original debt was P 10,000 but the face value indicated in the note was10,000 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.
  109. 109. Discounting of NotesDiscounting of Notes  Discounting a noteDiscounting a note is based on the maturity value whileis 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 anythe basic principles of discounting a note or any unmatured drafts or negotiable instruments at a bank areunmatured drafts or negotiable instruments at a bank are the same as those for obtaining a loan from a bank fromthe 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 banka businessman was granted a one (1) year loan by a bank worth P10,000. If the bank charges 10% interest theworth P10,000. If the bank charges 10% interest the amount the businessman will receive is P10,000 lessamount 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 theP1,000. Therefore, the businessman will receive only the discounted amount of P9,000 but the maturity value afterdiscounted amount of P9,000 but the maturity value after one year is still P10,000.one year is still P10,000.  TheThe discount rate (d)discount rate (d) is usually expressed inis usually expressed in percent or its equivalent decimal and is generally quotedpercent or its equivalent decimal and is generally quoted on a yearly basis. Its is the ration of the discount for theon a yearly basis. Its is the ration of the discount for the period to the maturity value. If the discount rate is notperiod to the maturity value. If the discount rate is not mentioned in any given problem, it is assumed that thementioned in any given problem, it is assumed that the interest rate is also the discount rate. Theinterest 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. Thematurity value. The proceedsproceeds (Pd)(Pd) is the amount a creditor is willing to pay ais the amount a creditor is willing to pay a note before its maturity or due date.note before its maturity or due date.

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