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- 1. Solving Direct Variations 9 Mathematics Quarter 2 Self-Learning Module 3
- 2. Mathematics – Grade 9 Quarter 2 – Self-Learning Module 3: Solving Direct Variations First Edition, 2020 Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them. Published by the Department of Education Division of Pasig City Printed in the Philippines by Department of Education – Schools Division of Pasig City Development Team of the Self-Learning Module Writer: Rochelle B. Laranang Editor: Cristina DC. Prado Reviewer (Language): Ma. Cynthia P. Badana, Ma. Victoria Peñalosa (Technical): Glady O. Dela Cruz Illustrator: Edison P. Clet Layout Artist: Anthony G. Fijo , Clifchard D. Valente Management Team: Ma. Evalou Concepcion A. Agustin OIC-Schools Division Superintendent Carolina T. Rivera OIC-Assistant Schools Division Superintendent Victor M. Javeña EdD Chief, School Governance and Operations Division and OIC-Chief, Curriculum Implementation Division Education Program Supervisors Librada L. Agon EdD (EPP/TLE/TVL/TVE) Liza A. Alvarez (Science/STEM/SSP) Bernard R. Balitao (AP/HUMSS) Joselito E. Calios (English/SPFL/GAS) Norlyn D. Conde EdD (MAPEH/SPA/SPS/HOPE/A&D/Sports) Wilma Q. Del Rosario (LRMS /ADM) Ma. Teresita E. Herrera EdD (Filipino/GAS/Piling Larang) Perlita M. Ignacio PhD (EsP) Dulce O. Santos PhD (Kindergarten/MTB-MLE) Teresita P. Tagulao EdD (Mathematics/ABM)
- 3. 9 Mathematics Quarter 2 Self-Learning Module 3 Solving Direct Variations
- 4. Introductory Message For the facilitator: Welcome to the Mathematics 9 Self-Learning Module on Solving Direct Variations! This Self-Learning Module was collaboratively designed, developed and reviewed by educators from the Schools Division Office of Pasig City headed by its Officer-in-Charge Schools Division Superintendent, Ma. Evalou Concepcion A. Agustin, in partnership with the City Government of Pasig through its mayor, Honorable Victor Ma. Regis N. Sotto. The writers utilized the standards set by the K to 12 Curriculum using the Most Essential Learning Competencies (MELC) in developing this instructional resource. This learning material hopes to engage the learners in guided and independent learning activities at their own pace and time. Further, this also aims to help learners acquire the needed 21st century skills especially the 5 Cs, namely: Communication, Collaboration, Creativity, Critical Thinking, and Character while taking into consideration their needs and circumstances. In addition to the material in the main text, you will also see this box in the body of the module: As a facilitator you are expected to orient the learners on how to use this module. You also need to keep track of the learners' progress while allowing them to manage their own learning. Moreover, you are expected to encourage and assist the learners as they do the tasks included in this module. Notes to the Teacher This contains helpful tips or strategies that will help you in guiding the learners.
- 5. For the Learner: Welcome to the Mathematics 9 Self-Learning Module on Solving Direct Variations! This module was designed to provide you with fun and meaningful opportunities for guided and independent learning at your own pace and time. You will be enabled to process the contents of the learning material while being an active learner. This module has the following parts and corresponding icons: Expectations - This points to the set of knowledge and skills that you will learn after completing the module. Pretest - This measures your prior knowledge about the lesson at hand. Recap - This part of the module provides a review of concepts and skills that you already know about a previous lesson. Lesson - This section discusses the topic in the module. Activities - This is a set of activities that you need to perform. Wrap-Up - This section summarizes the concepts and application of the lesson. Valuing - This part integrates a desirable moral value in the lesson. Posttest - This measures how much you have learned from the entire module.
- 6. 1. Translate statements involving direct variations to mathematical equations 2. Solve problems involving direct variations Directions: Read each question carefully and choose the letter that corresponds to the correct answer. 1) Which of the following situations represents direct variation? A. The number of hours to do a job as to the number of people doing the job B. The gas consumed as to the length of cooking time. C. The time traveled by a car as to its speed. D. The atmospheric pressure as to the altitude. 2) If y varies directly as x and y is 15 when x is 3, what is the value of the constant of variation? A. k = 3 C. k = 5 B. k = 4 D. k = 6 3) Which of the following equations illustrates direct variation? A. y = 0.25x C. y = xz + 5 B. y = 3 𝑥 D. y = √6𝑥𝑧 4) If a varies directly as b and b is 108 when a is 12, find b when a is 7. A. 1296 C. 63 B. 84 D. 9 5) A recipe for salad requires 250 mL of cream for every 300 mL of condensed milk. How many liters of cream would be needed for 1.5 liters of milk? A. 7.5 liters C. 2.25 liters B. 5 liters D. 1.25 liters EXPECTATIONS PRETEST
- 7. Directions: Find the value of the unknown of each of the following proportions: 1) 6 : a = 4 : 10 2) b : 6 = 8 : 24 3) 4 : 20 = c : 15 4) 2 3 = 8 𝑑 5) 𝑒 20 = 3 7 Direct Variation Direct variation exists whenever the ratio between two quantities is a nonzero constant. The statement “y varies directly as x”, “y is directly proportional to x” and “y is proportional to x” may be written as y = kx where k is the constant of variation. This means, as one quantity increases, the other quantity also increases. Similarly, as one quantity decreases, the other quantity also decreases. Example 1: Write the equation for the statement “the circumference (C) of a circle varies directly with its radius r”. Solution: Using the given variables, the corresponding equation will be C = kr where k is the constant of variation. Example 2: Write in symbols: “r varies directly with the square root of s” with k as the constant of variation. Solution: Using the given variables, the corresponding equation will be 𝑟 = 𝑘√𝑠. LESSON RECAP (References: Wikimedia Commons. Accessed June 13, 2020. https://commons.wikimedia.org/wiki/File:HK_Central_結志 街_Gage_Street_market_雞蛋_Chicken_n_鴨蛋 _Duck_Eggs_on_sale_March-2012.jpg; Free Images & Free Stock Photos. Accessed June 13, 2020. https://pxhere.com/en/photo/661869.
- 8. Example 3: If y varies directly as x and y is 32 when x is 4, find the variation constant and the equation of the variation. Solution: y = kx Translate “y varies directly as x” into an equation 32 = k(4) Substitute the given values in the equation 32 4 = k Solve for k 8 = k Therefore, the constant of variation is 8 and the equation is y = 8x. Example 4: If a varies directly as b and a is 54 when b is 9, what is the value of a when b is 12? Solution: a = kb Translate “a varies directly as b” into an equation 54 = k(9) Substitute the first given set of values in the equation 54 9 = k Solve for k 6 = k Therefore, the constant of variation is 6 and the equation is a = 6b. a = 6b a = 6(12) Solve for a when b is 12 a = 72 Hence, a = 72 when b is 12. Example 5: If p varies directly as the square of r and p is 324 when r is 9, what is the value of r when p is 100? Solution: p = k𝑟2 Translate “p varies directly as the square of r” into an equation 324 = k(9)2 Substitute the first given set of values in the equation 324 = k(81) Solve for k 324 81 = k 4 = k Thus, the constant of variation is 4 and the equation is p = 4𝑟2 p = 4𝑟2 100 = 4(𝑟)2 Solve for r when p is 100
- 9. 100 4 = 𝑟2 √25 = √𝑟2 Get the square root of both sides ± 5 = r Then, r = ± 5 when p is 100. Example 6: Anna is going to make a leche flan for dessert. She knows that 8 egg yolks are needed for 300 mL of condensed milk, but she is planning to use all the 450 mL condensed milk that she has. How many egg yolks does she need to maintain the proportion of the recipe? Method 1 Solution: Let e be the number of egg yolks needed m be the milliliter for the condensed milk e = km working equation for the variation 8 = k(300) substitute the first given set of values to the equation 𝑘 = 8 300 solve for k 𝑘 = 2 75 Therefore, the constant of variation is 2 75 and the equation is e = 2 75 𝑚 e = 2 75 𝑚 e = 2 75 (450) solve for e when m is 450 e = 12 Hence, there are 12 egg yolks needed for 450 ml of milk. Method 2 Solution: Since the constant of variation is k = 𝑦 𝑥 , we can establish the proportion that 𝑦1 𝑥1 = 𝑦2 𝑥2 where x will be the number of egg yolks and y will be the milliliters of milk needed. Let 𝑥1 = 8 egg yolks 𝑦1 = 300 ml condensed milk 𝑦2 = 450 ml condensed milk 𝑥2 = number of egg yolks needed 𝑦1 𝑥1 = 𝑦2 𝑥2 establish the proportion needed 300 8 = 450 𝑥2 substitute the given information
- 10. 300𝑥2 = 450(8) apply cross product property of proportions 𝑥2 = 450 (8) 300 solve for 𝑥2 𝑥2 = 12 Thus, there are 12 egg yolks needed for 450 ml of milk. ACTIVITY 1: LET’S PRACTICE! Directions: Translate the following direct variation statements into equations. 1) m varies directly as n. 2) d varies directly as the square of c. 3) j varies directly as the cube root of h 4) The fare (f) cost is directly proportional to the distance (d) traveled. 5) The pressure (p) at the bottom of the sea is directly proportional to the depth (d) reached. ACTIVITY 2: KEEP PRACTICING! Directions: Determine the constant of variation of the following statements: 1) y varies directly as x. If y is 36 then x is 9. 2) R varies directly as S. If R is 56 then S is 8. 3) Q is directly proportional to P. If Q is 150 then P is 10. 4) c is directly proportional to the square of b. If c is 16 then b is 3. 5) m is proportional to the square root of n. If m is 125 then n is 25. ACTIVITY 3: TEST YOURSELF! Directions: Write each of the following statements into direct variation equation and then solve for the unknown. 1) If a varies directly as b and a is 15 when b is 5, find a when b is 8. 2) If e is directly proportional to f and e is 40 when f is 16, find e when f is 30. 3) If r varies directly as the square of t and r is 20 when t is 2, find r when t is 7. ACTIVITIES
- 11. 4) If g is proportional to the cube root of h and g is 24 when h is 27, find h when g is 16. 5) If w varies directly as v + 4 and w is 42 when v is 2, find v when w is 63. 6) A recipe for cake requires 3 teaspoonfuls of yeast for 5 cupfuls of flour. How much yeast would be needed for 18 cupfuls of flour? 7) Liza, a professional typist, can type 70 words per minute. How many minutes will it take her to finish a manuscript of 1610 words? 8) At a local market, a kilo of pork costs 240 pesos. If you plan to buy 2 1 2 kilos, how much will you have to pay? 9) The weight on the moon varies directly with the weight on Earth. A person that weighs 70 kg on Earth weighs 12 kg on the moon. How much will a person who weigh 120 kg on Earth weighs on the moon? 10) The distance that a body falls from rest varies directly as the square of the time it falls. If a ball falls 180 feet in two seconds, how far will the ball fall in five seconds? How are you going to identify whether the given equation or situation illustrates direct variation? How do we solve problems involving direct variation? In a family, the amount of money that can be spent varies directly with the household income. With the recent problem in the COVID-19 pandemic such as the implementation of lockdown, a lot of families lose their source of income. Many families do not have enough savings to be used during emergency needs. How will you describe your family’s economic situation during the lockdown? What valuable experiences have you learned from this situation? As a young person, how can you help in managing your family expenses? Write your answer in your notebook. WRAP-UP VALUING
- 12. Directions: Read each question carefully and choose the letter that corresponds to the correct answer. 1) Which of the following situations DOES NOT represent direct variation? A. The area of a circle as to the length of its radius. B. The salary as to the number of hours worked. C. The time to reach the destination as to the speed of the car. D. The cost of fare as to the distance traveled. 2) If y varies directly as x and y is 36 when x is 9, what is the value of the constant of variation? A. k = 3 C. k = 5 B. k = 4 D. k = 6 3) Which of the following equations illustrates direct variation? A. z = √6𝑥𝑦 C. f = 3gh B. p = 7 𝑟 D. a = 3𝑏 4 4) If m varies directly as n and m is 60 when n is 5, find n when m is 108. A. 15 C. 9 B. 12 D. 5 5) Phoebe deposits P 5,000 in her savings account every 3 months. How many years will it take her to have a savings of P 100,000? A. 5 years C. 25 years B. 15 years D. 60 years POSTTEST
- 13. KEY TO CORRECTION PRETEST 1) B 2) C 3) A 4) C 5) D POSTTEST 1) C 2) B 3) D 4) C 5) A RECAP 1) a = 15 2) b = 2 3) c = 3 4) d = 12 5) e = 60 7 ACTIVITY 1: LET’S PRACTICE 1) m = kn 2) d = k𝑐 2 3) 𝑗 = 𝑘 √ℎ 3 4) f = kd 5) p = kd ACTIVITY 2: KEEP PRACTICING 1) k = 4 2) k = 7 3) k = 15 4) k = 16 9 5) k = 25 ACTIVITY 3: TEST YOURSELF! 1) a = 24 6) 54 5 or 10.8 teaspoonfuls of yeast 2) e = 75 7) 23 minutes 3) r = 245 8) 600 pesos 4) h = 8 9) 144 7 or 20.57 kilos 5) v = 5 10) 1125 feet
- 14. References BOOKS: Bryant, Merden L., Bulalayao, Leonides E., Callanta, Melvin M., Cruz, Jerry D., De Vera, Richard F., Garcia, Gilda T. and Javier, Sonia E., et. al. Mathematics Grade 9 Learner’s Material. First Edition. Pasig City: Department of Education, 2014. Diaz, Zenaida B., Mojica, Maharlika P., Suzara, Josephine L., Mercado, Jesus P., Esparrago, Mirla S. and Reyes, Nestor Jr. V. Next Century Mathematics 9. Quezon City: Phoenix Publishing House, Inc., 2014. Dilao, Soledad Jose and Bernabe, Julieta G. Intermediate Algebra Textbook for Second Year. Pilot Edition. Quezon City: JTW Corporation, 2002. Oronce, Orlando A. and Mendoza, Marilyn O. E-Math 9. Revised Edition. Manila: Rex Book Store Inc., 2015. ONLINE RESOURCES: Free Images & Free Stock Photos - PxHere. 2020, https://pxhere.com/en/photo/661869. (Accessed June 13, 2020). Pike, Scott. Welcome to MAT 120/121/122 Intermediate Algebra. Mesa Community College, 2020. https://www.mesacc.edu/~scotz47781/mat120/notes/ variation/direct/ direct.html. (Accessed June 17, 2020). Practical Algebra Lessons │ Purplemath, 2020, https://www.purplemath.com/ modules/variatn.html. (Accessed June 17, 2020). Wikimedia Commons. 2020, https://commons.wikimedia.org/wiki/File:HK_Central _結志_Gage_Street_market_雞蛋_Chicken_n_鴨蛋 _Duck_Eggs_on_sale_March-2012.jpg. (Accessed June 13, 2020).