Business-Math-Module-1-Fraction-Ratio-and-Proportion.pdf

 Business Mathematics Governor Pack Road, Baguio City, Philippines 2600 Tel. Nos.: (+6374) 442-3316, 442-8220; 444-2786; 442-2564; 442-8219; 442-8256; Fax No.: 442-6268...

 Business Mathematics Governor Pack Road, Baguio City, Philippines 2600 Tel. Nos.: (+6374) 442-3316, 442-8220; 444-2786; 442-2564; 442-8219; 442-8256; Fax No.: 442-6268 Grade Level/Section: ABM-11 Email: [email protected]; Website: www.uc-bcf.edu.ph MODULE 1 – Business Math Subject Teacher: JOAN A. MALA FUNDAMENTAL OPERATIONS ON FRACTIONS, DECIMALS, AND PERCENTAGE Learning Objectives: At the end of the modules, the student should be able to: a. Identify the fundamental operations needed on fractions, decimals and percentages b. Solve word problems on fractions, decimals, and percentages INTRODUCTION TO BUSINESS MATHEMATICS BUSINESS MATH A branch of business where mathematical tools and/or techniques are applied to aid the effective and critical analysis of problems in the business. AS APPLIED IN VARIOUS AREAS IN BUSINESS 1. Financial Accounting 2. Management Accounting Services 3. Economics 4. Marketing 5. Financial Management 6. Taxation A. FRACTIONS - A numerical figure that is not a whole number - It represents a part of a whole Parts of a fraction: A fraction has four parts namely: a) Fraction bar- the horizontal line that separates the numerator and denominator of fractions. b) Whole number-a number written before the fraction and is only present in a mixed number. It represents a whole part. c) Numerator- is the number placed on the top of the fraction bar representing the number of parts taken from the whole number. d) Denominator- the number below the fraction bar to represent the total number of equal parts the whole number is subdivided into. To further understand fraction, let us make use of the sample fraction below: 3 2 cakes 4 In this fraction, the parts are as follows: 2 =whole number 3 =numerator 4 =denominator - =fraction bar This fraction has a whole number which means that the fraction represents a number greater than one. In this example, the whole number is two which states that there are two whole cakes. In addition to the two cakes, there is another cake equally divided into four (the denominator). However, only three (the numerator) equal parts of the said cake is left. Hence, there are two whole cakes and a 3/4 parts cake. 2 5 More Examples: ½;3; ¼; 3; 1 ¼ BUSINESS MATHEMATICS Page 1 of 18  Business Mathematics Governor Pack Road, Baguio City, Philippines 2600 Tel. Nos.: (+6374) 442-3316, 442-8220; 444-2786; 442-2564; 442-8219; 442-8256; Fax No.: 442-6268 Grade Level/Section: ABM-11 Email: [email protected]; Website: www.uc-bcf.edu.ph MODULE 1 – Business Math Subject Teacher: JOAN A. MALA Kinds of Fractions Fractions are subdivided into the following kinds: 1. Proper Fractions (e.g., ½)- This kind of fraction represents a number lower than one. The numerator of a proper fraction is always less than the denominator. Proper fractions do not have whole numbers. 2. Improper Fractions (e.g., 5/4)- Improper fractions represent a number greater than 1 though it does not have a whole number. Improper fractions always have higher numerator than the denominator, which is the opposite of proper fractions. Improper fractions become mixed number should they be converted into whole numbers since it is greater than 1. Same with the proper fractions, improper fractions do not have whole numbers. 3. Mixed Numbers/ Fraction (e.g., 1 ¼)-Mixed fractions are fractions with whole numbers. If converted, mixed numbers become improper fractions. Conversion of fractions In performing the fundamental operations on fractions, there is a need to understand how to convert improper and mixed numbers.  Improper Fractions to Mixed Numbers To convert an improper fraction to a mixed number, simply divide the numerator with the denominator. 5 Example: 4 =4√5 Therefore, the mixed number derived is 1 ¼. 1 whole number =4√5 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 -4 1 numerator  Mixed Numbers to Proper Fractions To convert mixed number to improper fractions, multiply the denominator to the whole number then add the product to the numerator. The sum derived is the new numerator for the improper fraction. The new denominator is the same with the old denominator of the mixed number. Example: 2 ¼ =2 ¼ 4 (denominator) x2 (whole number) =2 ¼ 8 (product of 4x2) + 1 (numerator) =2 ¼ 9 (product +numerator) =9 (new numerator) 4 (copied denominator) ADDITION and SUBTRACTION OF FRACTIONS  Similar Fractions- similar fractions are fractions with same denominator. - To add similar fractions, simply add the numerators then copy the denominator - To subtract similar fractions, simply deduct the latter numerator from the first numerator then copy the denominator BUSINESS MATHEMATICS Page 2 of 18  Business Mathematics Governor Pack Road, Baguio City, Philippines 2600 Tel. Nos.: (+6374) 442-3316, 442-8220; 444-2786; 442-2564; 442-8219; 442-8256; Fax No.: 442-6268 Grade Level/Section: ABM-11 Email: [email protected]; Website: www.uc-bcf.edu.ph MODULE 1 – Business Math Subject Teacher: JOAN A. MALA 𝑎 𝑐 𝑎+𝑐 Rule: + = 𝑏 𝑏 𝑏 𝑎 𝑐 𝑎−𝑐 - = 𝑏 𝑏 𝑏 *Note: Always write your answers in its simplest form. Examples: 1. 1/4 + 2/4 = 3/4 2. 1/2 + 1/2 =2/2 =1 3. 3/4 – 2/4 =1/4 4. 7/8 – 6/8 + 1/8 =2/8 =1/4  Dissimilar Fractions - To add or deduct dissimilar fractions, follow the following steps:  Find the lowest multiple of the two denominators called least common denominator (LCD). LCD is the lowest whole number that you may use in dividing the denominators of the dissimilar fractions. The LCD computed will be the new denominator for all the fractions.  Rewrite the original equation into similar fractions by dividing the LCD by the original denominator of each fraction. Then, the answer is to be multiplied with the respective old numerators. The product of this multiplication is the new numerator for each respective fraction.  Add or deduct the similar fractions. 𝑎 𝑐 𝑎(𝑛1)+𝑐(𝑛2) Rule: 𝑏 +𝑑= 𝐿𝐶𝐷 Where: n1= LCD/b; n2=LCD/d 𝟏 𝟑 Example #1: 𝟏𝟐 + 𝟖 *Look for the LCD of 12 and 8. In this problem, it is 96. *= 1 + 3 * 8 + 36 *44 simplify = *44÷4 = 11 12 8 96 96 96 96÷4 24 Example #2: 2/3 – 2/7 *Look for the LCD of 3 and 7. In this case, it is 21. *=2 - 2 * 14 - 6 *8 3 7 21 21 21 Example #3: 3/4 – 1/3 + 1/5 *= 3 - 1 + 1 * 45 - 20 +12 *37 4 3 5 60 60 60 60  Mixed Numbers In adding mixed numbers, convert the mixed number into improper fraction and perform the addition/subtraction using the rules in adding/deducting improper fractions. Another possible method is to add/subtract the whole numbers and add/subtract the fractions separately before putting them together. BUSINESS MATHEMATICS Page 3 of 18  Business Mathematics Governor Pack Road, Baguio City, Philippines 2600 Tel. Nos.: (+6374) 442-3316, 442-8220; 444-2786; 442-2564; 442-8219; 442-8256; Fax No.: 442-6268 Grade Level/Section: ABM-11 Email: [email protected]; Website: www.uc-bcf.edu.ph MODULE 1 – Business Math Subject Teacher: JOAN A. MALA Example #1: 1 ½ + 1 ½ Method 1: Convert first the mixed numbers into improper fraction before performing the addition/subtraction. *1 ½ + 1 ½ = 3/2 + 3/2 = 6/2 Simplify= 3 Method 2: Add/subtract the whole number and fractions separately. *1 ½ + 1 ½ = 1 +1 (whole numbers) ; ½ + ½ (fractions) * = 2 (whole number) ; 2/2 fractions * = 2 2/2; then simplify * =3 Note: Whichever method you use, the answers should be the Example #2: 1 6/8 – 5/8 + 1/4 Method 1: Convert first the mixed numbers into improper fraction before performing the addition/subtraction. *=1 6/8 – 5/8 + ¼ *=14/8 - 5/8 + ¼ *after converting the mixed number to improper numbers, perform the operations required =14/8 -5/8+2/8 *=11/8 Should you use method two, the answer should still be the same. MULTIPLICATION and DIVISION OF FRACTIONS Rule in multiplying: Numerator by numerator; denominator by denominator 𝑎 𝑐 𝑎𝑐 = x = 𝑏 𝑑 𝑏𝑑 TIP: Convert all mixed numbers to improper fractions before multiplying. Examples: 1 2 2 𝟏 1. 4 x 3 = 12 = 𝟔 2 3 7 9 63 𝟐𝟏 2. 1 5 x 1 6 = 5 x 6 = 30 = 𝟏𝟎 6 3 14 3 42 2 𝟐𝟏 3. 1 8 x 4 = 8 x 4 = 32 ÷2 =𝟏𝟔 𝑎 𝑐 𝑎𝑑 Rule in dividing: ÷ = 𝑏 𝑑 𝑏𝑐 TIP: get the reciprocal of the divisor, then proceed to multiplication. In the event that there is a mixed number, convert it first to improper fraction. 1 𝟐 1 𝟑 𝟑 1. 4 ÷ 𝟑 = 4 x 𝟐= 𝟖 1 1 5 𝟏 𝟓 1 𝟓 2. 2 2÷8 =2 ÷ 𝟖 = 𝟐 x 8 =𝟏𝟔 B. PERCENTAGE - A number expressed as a fraction of 100. - Derived from the Latin phrase per centum- figured or expressed on the basis of a rate or proportion per hundred. BUSINESS MATHEMATICS Page 4 of 18  Business Mathematics Governor Pack Road, Baguio City, Philippines 2600 Tel. Nos.: (+6374) 442-3316, 442-8220; 444-2786; 442-2564; 442-8219; 442-8256; Fax No.: 442-6268 Grade Level/Section: ABM-11 Email: [email protected]; Website: www.uc-bcf.edu.ph MODULE 1 – Business Math Subject Teacher: JOAN A. MALA Examples: 1. 1/100 = 1% 2. 20/100 = 20% 3. 8/10 = 80% Convert Fractions to Percentage TIP: Perform the basic mathematical operation of division, then multiply the quotient by 100 to get the rate. 1. 2/5 =.40 x 100 = 40% 2. 6/7 =.86 x 100 = 86% Rate - Facilitates assignment of a measurement or quantity against some other quantity or measure. Rate of increase/ decrease 𝑞2−𝑞1 % increase/ decrease = 𝑞1 Where: q2= new value; q1= original value C. DECIMALS - These are fractions where denominators are expressed as positive powers of ten. (decimal fractions) - Numbers which uses the base of ten to extend the number system to include digits with place values that are negative powers of ten. (decimal digits) - Whole numbers and decimal digits shall be separated by a decimal point. IMPORTANCE: In the actual conduct of trade or business, monetary values are not expressed in terms of pesos. Here in the Philippines, we use pesos and centavos in their decimal forms for convenience sake. Thus, instead of writing 800 pesos and ¼ centavos, we write: P 800.25. Application of fractions, percentages, decimals, and rates in word problems 1. A 12-meter log will be divided into 8 equal pieces of woods. How long shall a piece of wood be measured? Answer: 12 𝑚 4 𝟑 𝟑 =12 meter log ÷ 8; = 8 ÷ 4 = 𝟐 𝒎; Therefore, there will be 𝟐 𝑚 of each piece of wood. 2. In the testamentary will of Don Potenciano, the 100 square meter of land shall be divided among his three children: Anna= ¼ ; Bitoy= 3/6; Chinita= 2/8. Determine the share of each from the total piece of land. In this word problem, the 100 square meters of land is to be distributed to the three children. Hence, the equation is to be written as: 100𝑚2 For Anna : 100 m2 x ¼ = 4 = 𝟐𝟓 𝒔𝒒𝒖𝒂𝒓𝒆 𝒎𝒆𝒕𝒆𝒓𝒔 𝒐𝒇 𝒍𝒂𝒏𝒅 3 300𝑚2 6 For Bitoy: 100m2 x = ÷ = 𝟓𝟎 𝒔𝒒𝒖𝒂𝒓𝒆 𝒎𝒆𝒕𝒆𝒓𝒔 𝒐𝒇 𝒍𝒂𝒏𝒅 6 6 6 2 200𝑚2 4 For Chinita: 100m2 x 8 = 8 ÷4 = 𝟐𝟓 𝒔𝒒𝒖𝒂𝒓𝒆 𝒎𝒆𝒕𝒆𝒓𝒔 𝒐𝒇 𝒍𝒂𝒏𝒅 Anna, Bitoy, and Chinita will inherit 25, 50, 25 square meters of land respectively. BUSINESS MATHEMATICS Page 5 of 18  Business Mathematics Governor Pack Road, Baguio City, Philippines 2600 Tel. Nos.: (+6374) 442-3316, 442-8220; 444-2786; 442-2564; 442-8219; 442-8256; Fax No.: 442-6268 Grade Level/Section: ABM-11 Email: [email protected]; Website: www.uc-bcf.edu.ph MODULE 1 – Business Math Subject Teacher: JOAN A. MALA 3. After an hour, the inlet pipe can fill 1/6 of the tank, and the outlet pipe can drain 1/8 of the water in the tank. Suppose it takes x hours to fill the tank if both pipes are open, how many hours shall it take to fill the tank if both outlet and inlet pipes are open? The equation is to be written as: 1 1 X number of hours = - 6 8 8 6 2 X=48 - 48 X=48 X= 24 hours It takes 24 hours to fill the tank. 4. According to the available data gathered by the Philippine Statics Authority, the population in the Philippines a decade ago is 78 Million. For the current year, the population has reached 120 Million. Determine the growth rate of the population in the Philippines. 𝑞2−𝑞1 % of increase/ decrease =( 𝑞1 ) x 100 Where: q2= new value; q1= original value Solution: % of increase/decrease= [(120 million-78 million) ÷ 78 million] x 100 % of increase/decrease=.5384 x 100 % of increase/decrease= 53.84 % increase There is an increase of 53.84% in the Philippine population compared to a decade ago. *positive value is an increase and a negative value represents decrease Note that the required is rate of change. Hence, the answer should be stated in percentage. 5. A house was sold for 250% of what the original owners paid for it. If the house was sold for P 2.4 Million, what was the original price paid for the house? Given: Original/ Initial Price X Mark-up rate 250% New selling price ₱2,400,000 Solution: X=new selling price÷ new selling price rate X= ₱2,400,000 /250% X= ₱960,000 The original selling price is ₱960,000. 3. 25% of a man’s monthly income goes to income tax. If the tax on his monthly income was ₱9,250, how much is the man’s monthly income? X=tax amount÷ tax rate X= ₱9,250 ÷25% X= ₱37,000 The man’s monthly income is ₱37,000. BUSINESS MATHEMATICS Page 6 of 18  Business Mathematics Governor Pack Road, Baguio City, Philippines 2600 Tel. Nos.: (+6374) 442-3316, 442-8220; 444-2786; 442-2564; 442-8219; 442-8256; Fax No.: 442-6268 Grade Level/Section: ABM-11 Email: [email protected]; Website: www.uc-bcf.edu.ph MODULE 1 – Business Math Subject Teacher: JOAN A. MALA RATIO AND PROPORTION Learning Objectives: At the end of this module, student must be able to: a) Identify the different kinds of proportion. b) Compute for direct, inverse, and partitive proportion Have you ever checked the back of your favorite food product? What do you often see? Well, in all products, whether it is food or non-food, ingredients and all necessary information about the product is written there, we call it the label (you will learn more of it in your marketing subject). Below is an example of what you might see at the back of almost every product. This is one of my favorite food products and at the back of its packaging is the label where I can see the nutritional facts which includes the ingredients used. What is the use of this? What’s the point? You see, every product we create, there is an appropriate percentage or amount of every ingredient for it to become what it must become. For example, for a rubbing alcohol to be an alcohol, there is a certain amount of its ingredients needed. Put too much or too little of any of its ingredients, it becomes something else. For it to be a rubbing alcohol, the producer must make sure to strictly follow the appropriate amount of all ingredients otherwise he may fail to create what is expected. If you are fond of watching news, you will often hear terms like “…the car is running 70km per hour” or “…the typhoon has a speed of 80km per hour” or even things like “…every 10 Filipino household have at least 2 to 3 members who are jobless”. Every one of these has something in common. All of them applied ratio and proportion. Thus, learning the concept of ratio and proportion is critical not only in making products but also in other aspects of life. It may also apply in other aspects of a business like allocation of funds, how much should go to the marketing department for the advertising of our product? How much should we pay each employee? How much do we need to spend on product development? In many part of our lives, knowledge on ratio and proportion gives us the advantage to make proper decision on allocating, grouping, distributing, and others. BUSINESS MATHEMATICS Page 7 of 18  Business Mathematics Governor Pack Road, Baguio City, Philippines 2600 Tel. Nos.: (+6374) 442-3316, 442-8220; 444-2786; 442-2564; 442-8219; 442-8256; Fax No.: 442-6268 Grade Level/Section: ABM-11 Email: [email protected]; Website: www.uc-bcf.edu.ph MODULE 1 – Business Math Subject Teacher: JOAN A. MALA RATIO A ratio is a relationship between two quantities. It is also defined as the quantitative relation between two amounts showing the number of times one value contains or is contained within the other. We use the following notations in writing the ratio of two quantities a and b: 𝑎  read as “a over b”; or 𝑏  𝑎: 𝑏 read as “a is to b” In this module, we will use these two notations interchangeably. Note that fractions and ratios have similarities. The first notation of a ratio given above is just like the notation for a fraction. We also call a and b as terms of the ratio. Moreover, a ratio may be reduced in its lowest terms using the same process as in simplifying fractions. To illustrate the concept of ratio, consider this situation. Suppose a vendor sells two baskets of fruits. One basket contains 50 mangoes while the other contains 60 avocados. The ratio of mangoes to avocados is 50 50: 60 𝑜𝑟 60 This ratio can be reduced in lowest terms by dividing both terms of the ratio by their Greatest Common Factor (GCF), as shown below. 50 ÷ 10 5 = 𝑜𝑟 5: 6 60 ÷ 10 6 Here is another example. Six cups of flour are mixed with 2 cups of sugar. 1. What is the ratio of the number of cups of sugar to that of flour? 2. What part of the mixture is flour? SOLUTIONS AND ANSWERS 2 1. The ratio of the number of cups of sugar to that of flour is 2:6 or. In lowest terms, the ratio is 6 2 ÷2 1 = 𝑜𝑟 1: 3 6 ÷2 3 2. The total amount of mixture is the sum of the number of cups of flour and the number of cups of sugar; that is, 6 + 2= 8cups. This means that the amount of flour is 6 cups out of the 8 6 cups, or of the mixture. In lowest terms, 8 6 ÷2 3 = 𝑜𝑟 3: 4 8 ÷2 4 3 Therefore, the amount of flour is of the mixture. 4 In the example above, notice that we used the concept of a ratio to compare two quantities with the same unit; that is, in terms of number of cups. In certain cases, however, we need to express a relationship between quantities with different units. To do that, we can use a special type of ratio called a rate. A rate is a comparison of two quantities with different units - for example, number of kilometers to number of hours so we have kilometers per hour (km/hr), number of feet to number of seconds then we will have feet per second (ft/s), or Philippine peso to US dollar (PhP/USD). These are BUSINESS MATHEMATICS Page 8 of 18  Business Mathematics Governor Pack Road, Baguio City, Philippines 2600 Tel. Nos.: (+6374) 442-3316, 442-8220; 444-2786; 442-2564; 442-8219; 442-8256; Fax No.: 442-6268 Grade Level/Section: ABM-11 Email: [email protected]; Website: www.uc-bcf.edu.ph MODULE 1 – Business Math Subject Teacher: JOAN A. MALA not new to us as we often hear them in the news when they are reporting about an accident due to reckless driving or when they are reporting about the weather. Here is an example of how to compute for the rate. Steff Cory spends 32 minutes to make 8 pancakes for her children. What is her rate in cooking pancakes? Express your answer in simplest form. SOLUTIONS AND ANSWERS Steff Cory can cook 8 pancakes in 32 minutes. So her rate is: 8 𝑝𝑎𝑛𝑐𝑎𝑘𝑒𝑠 32 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 In simplest form, this rate can be written as follows: 8 8 ÷ 8 1 𝑝𝑎𝑛𝑐𝑎𝑘𝑒 1 = ; 𝑜𝑟 𝑝𝑎𝑛𝑐𝑎𝑘𝑒 𝑝𝑒𝑟 𝑚𝑖𝑛𝑢𝑡𝑒 32 32 ÷ 8 4 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 4 This means that is takes her 4 minutes to make 1 pancake because she is able to make ¼ of it in a minute. Here is another example. Braun Lee drives 6 km to bring his daughter to school. Their travel tie usually takes 0.5 hour. What is Braun Lee’s driving rate? Express your answer in simplest form. SOLUTIONS AND ANSWERS Braun Lee drives 6 km for 0.5 hour or 30 minutes. So his rate is: 6 𝑘𝑖𝑙𝑜𝑚𝑒𝑡𝑒𝑟𝑠 6 𝑘𝑖𝑙𝑜𝑚𝑒𝑡𝑒𝑟𝑠 𝑜𝑟 0.5 ℎ𝑜𝑢𝑟 30 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 In simplest form, this rate can be written as follows: 6 6 𝑥 10 60 60 ÷ 5 12 𝑘𝑖𝑙𝑜𝑚𝑒𝑡𝑒𝑟𝑠 = ; = ; 𝑜𝑟 12𝑘𝑚 𝑝𝑒𝑟 ℎ𝑜𝑢𝑟 0.5 0.5 𝑥 10 5 5 ÷5 1 ℎ𝑜𝑢𝑟 Notice that before we divided the numerator and denominator to its GCF, we took the time to convert first the decimal number into a whole number. We do this to make solving easier but you may not do it if you can find the GCF between the numerator and denominator PROPORTION When two ratios are equal, we may write the equality in either of the following notations: 𝑎 𝑐 = 𝑜𝑟 𝑎: 𝑏 = 𝑐: 𝑑 𝑏 𝑑 Such equality is called a proportion. The outer terms a and d are called the extremes, while the inner terms b and c are called means. An important property of any proportion is the cross- product property, which is explained on the next page. BUSINESS MATHEMATICS Page 9 of 18  Business Mathematics Governor Pack Road, Baguio City, Philippines 2600 Tel. Nos.: (+6374) 442-3316, 442-8220; 444-2786; 442-2564; 442-8219; 442-8256; Fax No.: 442-6268 Grade Level/Section: ABM-11 Email: [email protected]; Website: www.uc-bcf.edu.ph MODULE 1 – Business Math Subject Teacher: JOAN A. MALA FORMULA in CROSS-PRODUCT PROPERTY Given the following proportion: The product of the extremes is equal to the product of the means; that is, a x d = b x c or ad = bc To illustrate this property, consider the following proportion: 3 9 3 : 5 = 9 : 15 or = 5 15 Observe that if we apply the cross-product property on this proportion, the resulting products are indeed equal; that is, 3 x 15 = 5 x 9 45 = 45 The cross-product property is useful when one of the terms in a proportion in unknown and we need to solve for the value of that term. Study how this is done in the next examples. Solve for the value of x in the given proportions. 1 𝑥 3 15 𝑥 15 90 8 = = = = 2 10 4 𝑥 8 20 𝑥 12 SOLUTIONS AND ANSWERS To solve for x in each proportion, get the cross-products of the terms. Then use the appropriate properties of equality to isolate x on one side of the equation. 1 𝑥 3 15 𝑥 15 90 8 = = = = 2 10 4 𝑥 8 20 𝑥 12 2x = 1(10) 3x = 4(15) 20x = 8(15) 8x = 12(90) 2x = 10 3x = 60 20x = 120 8x = 1,080 2x/2 = 10/2 3x/3 = 60/3 20x/20 = 120/20 8x/8 = 1,080/8 x=5 x = 20 x=6 x = 135 BUSINESS MATHEMATICS Page 10 of 18  Business Mathematics Governor Pack Road, Baguio City, Philippines 2600 Tel. Nos.: (+6374) 442-3316, 442-8220; 444-2786; 442-2564; 442-8219; 442-8256; Fax No.: 442-6268 Grade Level/Section: ABM-11 Email: [email protected]; Website: www.uc-bcf.edu.ph MODULE 1 – Business Math Subject Teacher: JOAN A. MALA Here is another example. Lue Kha is a shoemaker who finishes 3 pairs of men’s shoes for every 5 pairs of ladies’ shoes that his son finishes. At the same rate, if the shoemaker finishes 45 pairs of men’s shoes, how many pairs of ladies’ shoes can his son finish? SOLUTIONS AND ANSWERS The ratio of the number of men’s shoes that Lue Kha finishes to the number of pairs of ladies’ 3 shoes that his son finishes is 3:5 or. 5 Suppose x is the number of pairs of ladies’ shoes that Lue Kha’s son can finish when Lue Kha finishes 45 pairs of men’s shoes. We can form the following proportion: 3 45 = 5 𝑥 Notice that both the numerators are Lue Kha’s and the denominators are both his son’s. we must consider proper placement of values otherwise the answer will be wrong. We can now solve for the value of x using the cross-product property, as shown below. 3x = 5(45) 3x = 225 3x/3 = 225/3 x = 75 Therefore, when Lue Kha finishes 45 pairs of men’s shoes, his son can finish 75 pairs of ladies’ shoes. Let’s have one more example. For every 2,000 jars that AD company ships, an average of 3 jars break during transit. If the company ships 10,000 jars, how many jars can be expected to break during transit? SOLUTIONS AND ANSWERS The ratio of the total number of jars to the total number of jars that break during transit is 2,000 2,000:3 or. 3 Suppose x is the number of jars that can be expected to break during transit when AD company ships 10,000 jars. We can form the following proportion: 2,000 10,000 = 3 𝑥 Notice that both the numerators are the number of jars shipped and the denominators are both the number of jars that break during transit. we must consider proper placement of values otherwise the answer will be wrong. We can now solve for the value of x using the cross-product property, as shown below. 2,000x = 3(10,000) 2,000x = 30,000 2,000x/2,000 = 30,000/2,000 x = 15 BUSINESS MATHEMATICS Page 11 of 18 Therefore, when AD company ships 10,000 jars, 15 jars can be expected to break during transit.  Business Mathematics Governor Pack Road, Baguio City, Philippines 2600 Tel. Nos.: (+6374) 442-3316, 442-8220; 444-2786; 442-2564; 442-8219; 442-8256; Fax No.: 442-6268 Grade Level/Section: ABM-11 Email: [email protected]; Website: www.uc-bcf.edu.ph MODULE 1 – Business Math Subject Teacher: JOAN A. MALA In the two previous examples, you will see that solving or computing for proportion is helpful in making decisions which in our problems, in relation to production. KINDS OF PROPORTION a. Direct Proportion In the previous example, notice that the number of jars that break increases as the number of jars shipped increases. When two or more quantities increase or decrease in the same ratio, then their relationship is called a direct proportion. It is also defined as a proportion of two variable quantities when the ratio of the two quantities is constant. The two previous examples show how a direct proportion is computed. You can know that a proportion is direct by taking into consideration what will happen when one value changes. For example, Jhong Key can write about 120 words in 3 minutes. If Jhong Key will write for about 5 minutes, how many words will he be able to write? SOLUTIONS AND ANSWERS 120 The ratio of words Jhong Key can write in 3 minutes is 120:3 or. 3 Suppose x is the number of words that Jhong Key can write within 5 minutes. We can form the following proportion: 120 𝑥 = 3 5 Notice that both the numerators are the number of jars shipped and the denominators are both the number of jars that break during transit. we must consider proper placement of values otherwise the answer will be wrong. We can now solve for the value of x using the cross-product property, as shown below. 3x = 5(120) 3x = 600 3x/3 = 600/3 x = 200 Therefore, when Jhong Key writes for 5 minutes straight, he is able to write 200 words. In this example, you will know that this is a direct proportion because of the fact that as the relationship between production and time is direct. When Jhong Key is given only 2 minutes, he is only able to write about 120 words. This meant that if he is given more time, then he will be able to write more; If he will be given lesser time, he will write lesser. Hence, we can say that this proportion is direct because of that relationship. Having an understanding of the relationship between two variables is critical in identifying what kind of proportion we are dealing with. Different kind of proportion will have its own way of solving as you will observe in the next kind of proportion. b. Inverse or Indirect Proportion In the previous kind of proportion, when two or more quantities either increase or decrease in the same ratio, then their relationship is a direct proportion. Now, we will discuss another kind of proportion called inverse or indirect proportion. Before we define this type of proportion, let us consider the situation below. BUSINESS MATHEMATICS Page 12 of 18  Business Mathematics Governor Pack Road, Baguio City, Philippines 2600 Tel. Nos.: (+6374) 442-3316, 442-8220; 444-2786; 442-2564; 442-8219; 442-8256; Fax No.: 442-6268 Grade Level/Section: ABM-11 Email: [email protected]; Website: www.uc-bcf.edu.ph MODULE 1 – Business Math Subject Teacher: JOAN A. MALA Suppose you are putting up a tailoring business. Using your budget of P6,000, you need to buy cloth for your business and you have the following kinds of cloth to choose from:  Cloth A: P120 per yard; and  Cloth B: P100 per yard If you choose Cloth A, the number of yards that you can buy can be computed as follows: 6,000 ÷ 120 = 50 yards From this equation, you can also form the following equation: (120)(50) = 6,000 On the other hand, if you choose cloth B, the number of yards that you can buy can be computed as follows: 6,000 ÷ 100 = 60 yards From this equation, you can also form the following equation: (100)(60) = 6,000 Notice that as the price of the cloth decreases, the number of yards of cloth you can buy increases. This illustrates the concept of indirect proportion, which we now define as follows: An indirect proportion exists between two quantities if, as one quantity decreases, the other quantity increases proportionately. Let us analyze this type of proportion further. From the given situation, we can form this equality: (120)(50) = (100)(60) Both the left-hand and right hand sides of equality are equal to 6,000, which is called the constant of proportionality of the indirect proportion between the price of cloth and the number of yards of cloth. In general, if the following equality represents an indirect proportion: a:b=c:d then ab = cd Take note that in an indirect proportion, we do not apply the cross-product property that we used for a direct proportion. The difference between the two types of proportion is summarized below: DIRECT PROPORTION If a : b = c : d, then ad = bc INDIRECT or INVERSE PROPORTION If a : b = c : d, then ab = cd Here is a sample problem. The table shows the number of carpenters working together and their corresponding number of days to finish a certain task. Solve for x, which is the number of days that the 12 carpenters can finish the same task. Number of Carpenters Number of Days 8 18 12 x BUSINESS MATHEMATICS Page 13 of 18  Business Mathematics Governor Pack Road, Baguio City, Philippines 2600 Tel. Nos.: (+6374) 442-3316, 442-8220; 444-2786; 442-2564; 442-8219; 442-8256; Fax No.: 442-6268 Grade Level/Section: ABM-11 Email: [email protected]; Website: www.uc-bcf.edu.ph MODULE 1 – Business Math Subject Teacher: JOAN A. MALA SOLUTIONS AND ANSWERS The situation illustrates an indirect proportion because as the number of carpenters increases, it is expected that the number of days to finish the task decreases. So if the proportion is 8:18 = 12:x, we have the following: (8)(18) = 12x 144 = 12x 144/12 = 12x/12 x = 12 Therefore, 12 carpenters can finish the task in 12 days. Here is another example. If it takes 3 days for 8 factory workers to finish 40 pairs of shoes, how many days will it take 6 factory workers to finish the same job? Assume that all of them work at the same rate. SOLUTIONS AND ANSWERS Since the number of factory workers decreases from 8 to 6 workers, we can expect that the number of days to finish 40 pairs of shoes will increase. Suppose x is the number of days that 6 factory workers can complete the task. Then we have the proportion 8 : 3 = 6 : x. solving for x, we have: (8)(3) = 6x 6x = 24 6x/6 = 24/6 x=4 Therefore, factory workers can finish 40 pairs of shoes in 4 days. Here are more examples. 1. If 2 pencils cost P10.50, how many pencils can you buy with P36.75? 2. If it takes 7 days for 6 weeks to tile the wall of a building, how many workers are needed to tile the same wall in only 2 days? Assume that all of them work at the same rate. 3. A farmer has enough corn to feed 300 hens for 20 days. If he buys 100 more hens, how long will the same amount of corn be consumed? BUSINESS MATHEMATICS Page 14 of 18  Business Mathematics Governor Pack Road, Baguio City, Philippines 2600 Tel. Nos.: (+6374) 442-3316, 442-8220; 444-2786; 442-2564; 442-8219; 442-8256; Fax No.: 442-6268 Grade Level/Section: ABM-11 Email: [email protected]; Website: www.uc-bcf.edu.ph MODULE 1 – Business Math Subject Teacher: JOAN A. MALA SOLUTIONS AND ANSWERS 1. The problem involves a direct proportion. So if the proportion is 2 : 10.50 = x : 36.75, we have the following: 10.5x = (2)(36.75) 10.5x = 73.5 10.5x/10.5 = 73.5/10.5 x=7 Therefore, you can buy 7 pencils with P36.75. 2. The problem involves an indirect proportion. So if the proportion is 6 : 7 = x : 2, we have the following: 2x = (6)(7) 2x = 42 2x/2 = 42/2 x = 21 Thus, it takes 21 workers to tile the wall in 2 days. 3. The problem involves an indirect proportion. So if the proportion is 300 : 20 = 400 : x, we have the following: 400x = (300)(20) 400x = 6,000 400x/400 = 6,000/400 x = 15 Therefore, the corn feed will last 15 days if 400 hens will be fed. c. Partitive Proportion There are cases when we need to divide a number or a quantity based on a given ratio. In such instances, we need to apply another concept in ratio and proportion called partitive proportion. Consider the situation below. Suppose a man owns 15 hectares (ha) of land. He wants to divide the land into two unequal areas based on a ratio of 2 : 3. The smaller area will be used for the shelter of his animals and the other area for planting crops. Let us analyze the given conditions using the following illustration: 15 hectares |----2 Parts----|---------3 Parts---------| The given ratio 2 : 3 means that 2 parts of the land ( which is the smaller area ) will be for the animals, and 3 parts will be for the crops. As we can also see in the illustration above, in order to divide the land in a ratio of 2 : 3, we first needed to “partition” the whole land into 5 parts ( that is, 2 parts + 3 parts ). So we can also say that:  2 out of the 5 total parts of the land ( or 2/5 ) are for the animals; and  3 out of the 5 total parts of the land ( or 3/5 ) are for the crops. To calculate the number of hectares of land for the animals, we just need to find 2/5 of 15 ha; that is, BUSINESS MATHEMATICS Page 15 of 18  Business Mathematics Governor Pack Road, Baguio City, Philippines 2600 Tel. Nos.: (+6374) 442-3316, 442-8220; 444-2786; 442-2564; 442-8219; 442-8256; Fax No.: 442-6268 Grade Level/Section: ABM-11 Email: [email protected]; Website: www.uc-bcf.edu.ph MODULE 1 – Business Math Subject Teacher: JOAN A. MALA 2 𝑥 15 = 6 ℎ𝑎 5 Similarly, the number of hectares of land for the crops is 3/5 of 15 ha; that is, 3 𝑥 15 = 9 ℎ𝑎 5 In this situation, we applied the concept of partitive proportion, which we can now define as a proportion that requires dividing a number or a quantity into two or more parts that are not necessarily equal, according to a given ratio. Here is another example. Gohan and his father Goku own a total of 35 ha of land. The ratio of the areas of Gohan’s land to Goku’s land is 3 : 4. How many hectares of land does each of them own? SOLUTIONS AND ANSWERS To solve the problem, we need to partition the 35 ha of land into 7 parts ( that is 3 + 4 = 7 ). Gohan owns 3 out of the 7 parts, or 3/7 of 35 ha; that is, 3 𝑥 35 = 15 ℎ𝑎 7 Goku on the other hand owns 4 out pf the 7 parts, or 4/7 of 35 ha; that is, 4 𝑥 35 = 20 ℎ𝑎 7 Therefore, Goku owns 20 ha and Gohan owns 15 ha of the 35 ha of land they own. To know if you are correct or not, you add up the answers and it should match the original value being divided, which in this problem is 35 ha ( that is 20 ha + 15 ha ). What if there are more than 2 partitions? Then we apply the same process. Here is an example. Meruem is a rich man who would like to divide his P2,000,000 among his three children, Neferpitou (a.k.a. Pitou), Shaiapouf (a.k.a. Pouf), and Menthuthuyoupi (a.k.a. Youpi)in the ratio of 2 : 3 : 5. How much will each receive? SOLUTIONS AND ANSWERS To solve the problem, we need to apply the same process as the previous example. We need to partition the P2,000,000 into 10 parts ( that is 2 + 3 + 5 ). Pitou will get 2 out of the 10 parts, or 2/10 of P2,000,000; that is, 2 𝑥 2,000,000 = 𝑃400,000 10 Pouf will get 3 out of the 10 parts, or 3/10 of P2,000,000; that is, 3 𝑥 2,000,000 = 𝑃600,000 10 Youpi will get 5 out of the 10 parts, or 5/10 of P2,000,000; that is, 5 𝑥 2,000,000 = 𝑃1,000,000 10 Thus, Pitou, Pouf, and Youpi will receive P400,000; P600,000; and P1,000,000 respectivelyPage from16 of 18 BUSINESS MATHEMATICS their father Meruem.  Business Mathematics Governor Pack Road, Baguio City, Philippines 2600 Tel. Nos.: (+6374) 442-3316, 442-8220; 444-2786; 442-2564; 442-8219; 442-8256; Fax No.: 442-6268 Grade Level/Section: ABM-11 Email: [email protected]; Website: www.uc-bcf.edu.ph MODULE 1 – Business Math Subject Teacher: JOAN A. MALA Here’s another problem that requires you to think further. Lollipop Machine A works twice as fast as Machine B. Machine B works twice as fast as Machine C. If the 3 machines will function at the same time to produce 1,400,000 lollipops, how many lollipops will each machine produce? To answer this problem, you must take into consideration the given conditions for each machine. You must also remember that the solution for this problem is not necessarily the same with other problems. Solution will differ depending on the information provided and the conditions given. The solution for this particular problem is presented below. SOLUTIONS AND ANSWERS Based on the given information in the problem, we can conclude that whenever Machine C makes 1 lollipop, Machine B makes 2 lollipops ( which is twice as much as Machine C), and Machine A makes 4 lollipops ( which is twice as much as Machine B). this means that we can use the ratio 4 : 2 : 1 or 1 : 2 : 4 to partition the number of lollipops that each machine will produce respectively. From this ratio, the total number of partitions is 1 + 2 + 4 = 7 parts. So Machine C will produce 1/7 of 1,400,000 lollipops; that is, 1 𝑥 1,400,000 = 200,000 𝑙𝑜𝑙𝑙𝑖𝑝𝑜𝑝𝑠 7... … Machine B will produce 2/7 of 1,400,000 lollipops; that is, 2 𝑥 1,400,000 = 400,000 𝑙𝑜𝑙𝑙𝑖𝑝𝑜𝑝𝑠 7 Machine A will produce 4/7 of 1,400,000 lollipops; that is, 4 𝑥 1,400,000 = 800,000 𝑙𝑜𝑙𝑙𝑖𝑝𝑜𝑝𝑠 7 Thus, Machine A,B, and C will produce 800,000; 400,000; and 200,000 lollipops respectively. Again, remember that to check if your answer is correct, simply add the results and it should match the original value partitioned. Here’s another example. Three employees, namely Biz, Ness, and Matt work together to produce a certain number of baskets. Biz can make 5 baskets every time Ness and Matt make 3 and 4 baskets respectively. 1. What is the ratio of the number of baskets that Biz, Ness and Matt can make? 2. What part of the total number of baskets can each employee make? BUSINESS MATHEMATICS Page 17 of 18  Business Mathematics Governor Pack Road, Baguio City, Philippines 2600 Tel. Nos.: (+6374) 442-3316, 442-8220; 444-2786; 442-2564; 442-8219; 442-8256; Fax No.: 442-6268 Grade Level/Section: ABM-11 Email: [email protected]; Website: www.uc-bcf.edu.ph MODULE 1 – Business Math Subject Teacher: JOAN A. MALA SOLUTIONS AND ANSWERS 1. Based on the given information in the problem, the ratio of the number of baskets that Biz, Ness, and Matt can make is 5 : 3 : 4. 2. From the ratio of 5 : 3 : 4, the total number of partitions is 5 + 3 + 4 = 12 parts 5 3 1 Therefore, Biz can make of the total number of baskets; Ness can make or of the 12 12 4 4 1 total number of baskets; and Matt can make or of the total number of baskets. 12 3 Hence, if the total number of baskets produced is 24 baskets, then Biz, Ness, and Matt produced 10, 6, and 8 baskets respectively. In summary, there are three kinds of proportion, direct, indirect or inverse, and partitive proportion. You can know what type of proportion it is by the relationship of the values. If a value increases in relation to another value which also increased, then it is a direct proportion. When one value increases and another related value decreases, that is an indirect or inverse proportion. And when a value is being partitioned or divided into two or more parts, then it is a partitive proportion. Knowing what kind of proportion, a problem is critical because it will affect the way we deal with the problem. In a direct proportion, we use the cross-product property but in an indirect proportion we do not use it. In a partitive proportion, we simply add the ratios and then create a fraction out of the result of the addition of the ratios then multiplying it to the value being divided or partitioned. References: 1. Banggawan, R., Asuncion, D.(2017).Fundamentals of Accountancy, Business and Management 1. Aurora Hill, Baguio City: Real Excellence Publishing. 2. BAL 650.0151 Si79 2016 Sirug, Winston S. (2016), Business mathematics for senior high school - ABM specialized subject: a comprehensive approach, Mindshaper Co., Inc., Intramuros, Manila BUSINESS MATHEMATICS Page 18 of 18