e-Module 3 in 2 GED MAT 01 (MMW) Problems and Reasoning.pdf

Philippine Normal University North Luzon The National Center for Teacher Education The Indigenous Peoples Education Hub Kaway-Aralan sa Bagong Kadawyan MATHEMATICS IN THE MODERN WORLD e-MODULE 3 in 2 GED MAT 01 Problems and Reasoning...

Philippine Normal University North Luzon The National Center for Teacher Education The Indigenous Peoples Education Hub Kaway-Aralan sa Bagong Kadawyan MATHEMATICS IN THE MODERN WORLD e-MODULE 3 in 2 GED MAT 01 Problems and Reasoning JERRY B. TOLENTINO Nurturing Innovative Teachers and Education Leaders  VISION PNU North Luzon is an internationally recognized and nationally responsive teacher education university specializing in living traditions and indigenous education. Ang Pamantasang Normal ng Pilipinas Hilagang Luzon ay kinikilala sa sandaigdigan bilang pambansang pamantasan na tumutugon sa edukasyong pangguro na dalubhasa sa mga nakagawiang tradisyon at katutubong edukasyon. Ti Philippine Normal University North Luzon ket mabigbigbig iti sangalubungan kas pagadalan ti pagilian a mangisayangkat para kadagiti manursuro nga eksperto iti kannawidan nga edukasion ken tradision. MISSION PNU North Luzon is dedicated to nurturing innovative teachers and education leaders. Ang Pamantasang Normal ng Pilipinas Hilagang Luzon ay nakatuon sa paghubog ng mga makabagong guro at namumuno sa edukasyon. Ti Philippine Normal University North Luzon ket napudno a mangpatanor kadagiti nalalaing a manursuro ken mangidadaulo ti edukasion. PNU North Luzon Quality Policy As the National Center for Teacher Education and the Hub for Indigenous Peoples Education, the Philippine Normal University North Luzon commits to provide responsive teacher education programs focused on indigenous peoples education. The Campus further commits to the continual improvement of its Quality Management System through compliance with International Quality Standards and statutory and regulatory requirements. PNU North Luzon shall achieve these commitments through its core functions of instruction, research, extension, and production.  A MODULE 3 Pacauna (Introduction) ✓ In this module, you will learn problem solving and reasoning technique when you take the following lessons: Lesson 1 Reasoning Lesson 2 Problem Solving Dagiti Tarigagayen nga Pagbanagan Kalpasan ti Leksyon (Target Learning Outcomes) ✓ Employ critical thinking and logical reasoning in examining social issues as experienced in one’s life ✓ Apply Polya’s 4-steps in Problem Solving Nasisita nga Saludsod (Essential Questions) What is the difference between inductive and deductive reasoning? How do we apply inductive and deductive reasoning to solve problems? How do we apply Polya’s problem solving strategy? Pannubok Sakbay ti Leksyon (Pretest) ✓ Pre-test will be done asynchronously. The web link will be posted in our chatbox. Lecture and Discussion (Panangipalawag, Panangsingaysingay ken Pannakabinsa-binsa kadagiti Kapanunutan Maigapu iti Leksyon) Umuna nga Leksyon Lesson 1: Reasoning What You’ll Learn ✓ Differentiate inductive and deductive reasoning ✓ Apply inductive and deductive reasoning to solve problems Panangipakaammo ti Leksyon (Lesson Introduction) THIS COURSEPACK IS FOR PNU NL USE ONLY 55 NO PART OF THIS MATERIAL MAY BE REPRODUCED  Logic is the study of the principles of correct reasoning. It helps to differentiate correct reasoning from poor reasoning. Logic allows us to determine the validity of arguments in and out of Mathematics. It illustrates the importance of precision and conciseness of the language of Mathematics. The importance of logic to Mathematics cannot be overstated. No conjecture in Mathematics is considered fact until it has been logically proven and truly valid Mathematical analysis is done only within the rigors of logic (https://faculty.math.illinois.edu/-vddries/main.pdf). Reasoning skills are essential to good judgment. And without good judgment your creativity will lead to projects that make no sense. The point of little dialogue is that good reasoning skills are essential to doing anything right. The business person uses reasoning skills in writing a report or preparing a presentation; the scientist uses them in designing an experiment or clinical trial, the department manager uses them in maximizing worker efficiency, the lawyer uses them in composing an argument to a judge or jury. And that’s where logic comes in. The chief purpose of logic is to develop good reasoning skills. Panangipalawag, Panangsingaysingay ken Pannakabinsa- binsa kadagiti Kapanunutan Maigapu iti Leksyon (Discussion and Learning Activities) Statement or Proposition A statement or proposition is a declarative sentence that is true or false but not both. Examples: Determine which of the following statements are proposition 1. Ilagan City is the Capital of Isabela Answer: Proposition because it has a true value 2. Mark is handsome and tall Answer: NOT a proposition because it neither true or false 3. Where are you going? Answer: NOT a proposition because the sentence is interrogative 4. Please give face mask and face shield. Answer: NOT a proposition because the sentence is imperative 5. 1+1=2 Answer: Proposition because it has a truth value 6. x > 7 if x = 4 Answer: Proposition because the sentence has a truth value 7. x + 1 = 2 8. x + y = z Answer to 7 & 8: NOT a proposition because they are neither true nor false since the variables have no assigned values. ✓ If a proposition is true we say its truth value is true, and if a proposition is false, we say its truth value is false. THIS COURSEPACK IS FOR PNU NL USE ONLY 56 NO PART OF THIS MATERIAL MAY BE REPRODUCED  Mathematical reasoning refers to the ability of a person to analyse problem situations and to construct logical arguments to create both conceptual foundations and connections to be able to process the available information and solve the problems. The NCTM (2000) says people who can reason and think analytically “tend to note patterns, structure, or regularities in both real – world situations and symbolic objects; ask if those patterns are accidental or if they occur for a reason; and conjecture and prove.” Mathematical reasoning requires more than simply memorizing facts but thinking beyond rules and procedures to form one’s own questions and conjectures (Nocon & Nocon, 2018). When is a Mathematical reasoning correct? What methods can be used to construct effective Mathematical reasoning? The drawing of inferences or conclusions from known or assumed facts is called Reasoning Types of Reasoning 1. Analogy- is a type of reasoning that uses the relationship involved in one situation and applies it to another. Comparison between the two situations is the key in using this type of reasoning. 2. Inductive- is an specific information, usually through observation and measurement, regarding a conjecture and then proceeds to make a general conclusion based on the gathered information. 3. Deductive-is a general facts or statements and applies them to specific situations 4. Intuition- is a drawing conclusions through senses 5. Abstract Analogy- Example: APPLE : FRUIT :: A. Salad : Mix B. Orange: Banana C. Juice : Can D. Slice : Core E. Butterfly: Insect PEN : WRITE :: A. Pencil : Sharpen B. Ink : Blue C. Letter : Compose D. Knife : Cut E. Mistake : Erase Try to come up with your examples of reasoning by analogy. Inductive and Deductive Reasoning ✓ INDUCTIVE REASONING- the process of reasoning that arrives at a general conclusion based on the observation of specific examples. It involves: Looking for patterns A conclusion is formed is called a Conjecture (an educated guess) THIS COURSEPACK IS FOR PNU NL USE ONLY 57 NO PART OF THIS MATERIAL MAY BE REPRODUCED  Use inductive reasoning to find a pattern, and then find the next three numbers by using that pattern. Example 1: Using Inductive Reasoning to find for a pattern Use inductive reasoning to find a pattern, and then find the next three numbers by using that pattern. 1, 2, 4, 5, 7, 8, 10, 11, 13, __, __, __ SOLUTION : To find the pattern, look at the first number and see how to obtain the second number. Then look at the second number and see how to obtain the third number, etc. 1, 2, 4, 5, 7, 8, 10, 11, 13, ___, ___, __ The pattern seems to be: add 1, then add 2, then add 1, then add 2, etc. So a reasonable conjecture for the next three numbers is 14, 16, and 17. Example 2: Using Inductive Reasoning to find a pattern Make a reasonable conjecture for the next figure in the sequence. SOLUTION: The flat part of the figure is up, right, down, and then left. There is a solid circle in each figure. The sequence then repeats with an open circle in each figure. So we could reasonably expect the next figure to be Example 3: Using Inductive Reasoning to make conjecture When two odd numbers are added, will the result always be an even number? Use inductive reasoning to determine your answer. SOLUTION :We will try several specific examples: 3 +7 = 10 25 + 5 = 30 5+ 9 = 14 1 + 27 = 28 19 + 9 = 28 21 + 33 = 54 Since all the answers are even, it seems reasonable to conclude that the sum of two odd numbers will be an even number. Note: Since the sum of every pair of odd numbers hasn’t been tried, we can’t be 100% sure that the answer will always be an even number by using inductive reasoning. COUNTEREXAMPLE: One specific example that proves the conjecture false. To find a counterexample… THIS COURSEPACK IS FOR PNU NL USE ONLY 58 NO PART OF THIS MATERIAL MAY BE REPRODUCED  Start with a number(s) or object(s) that matches the premise of the conjecture. Pick them in a way that will give you an incorrect response. Example 4: Finding a counterexample Find a counterexample that proves the conjecture below is false. Conjecture: A number is divisible by 3 if the last two digits are divisible by 3. SOLUTION : We’ll pick a few numbers at random whose last two digits are divisible by 3, then divide them by 3, and see if there’s a remainder. Start with 1,527: 1,527 divided by 3 = 509 Next 11,745: 11, 745 divided by 3 = 3,915 At this point, you might start to suspect that the conjecture is true, but you shouldn’t! We’ve only checked two cases. Now try 1,136: 1,136 divided by 3 = 378 3/8 This counterexample shows that the conjecture is false. DEDUCTIVE REASONING- the process of reasoning that arrives at a conclusion based on previously accepted general statements. It does not rely on specific examples. EXAMPLE 1: Using Deductive Reasoning to Prove a Conjecture Consider the following problem: Think of any number. Multiply that number by 2, then add 6, and divide the result by 2. Next subtract the original number. What is the result? (a) Use inductive reasoning to make a conjecture for the answer. (b) Use deductive reasoning to prove your conjecture. SOLUTION: (a) Inductive reasoning will be helpful in forming a conjecture. We’ll choose a couple specific numbers at random and perform the given operations to see what the result is. Number: 12 5 Multiply by 2: 12 x 2 = 24 5 x 2 = 10 Add 6: 24 + 6 = 30 10 + 6 = 16 Divide by 2: 30 / 2 = 15 16 / 2 = 8 Subtract the original number: 15 – 12 = 3 8–5=3 Result: 3 3 THIS COURSEPACK IS FOR PNU NL USE ONLY 59 NO PART OF THIS MATERIAL MAY BE REPRODUCED  So we might form a conjecture that the result will always be number 3. But this doesn’t prove the conjecture, as we’ve tried only two of infinitely many possibilities. SOLUTION: (b) The problem with the inductive approach is that we can’t check every possible number. Instead, we’ll choose an arbitrary number and call it x. If we can show that the result is 3 in this case, that will tell us that this is the result for every number. Remember, we’ll be doing the exact same operations, just on an arbitrary number x. Number: x Multiply by 2: x X 2 = 2x Add 6: 2x + 6 = 2x + 6 Divide by 2: ( 2x + 6) / 2 = x + 3 Subtract the original number: x+3–x =3 Result: 3 Now we proved our conjecture for all numbers. EXAMPLE 6 :Using Deductive Reasoning to Prove a Conjecture Consider the following problem: Select a number. Add 50 to the number. Multiply the sum by 2. Subtract the original number from the product. What is the result? (a) Use inductive reasoning to arrive at a general conclusion. (b) Use deductive reasoning to prove your conclusion is true. SOLUTION (a) Inductive reasoning will be helpful in forming a conjecture. We’ll choose a couple specific numbers at random and perform the given operations to see what the result is. Number: 12 50 Add 50: 12 + 50 = 62 50 + 50 = 100 Multiply by 2: 62 x 2 = 124 100 x 2 = 200 Subtract the original number: 124 – 12 = 112 200 – 50 = 150 Result: 112 150 The conjecture is that the final answer is 100 more than the original number. But this doesn’t prove the conjecture, as we’ve tried only two of infinitely many possibilities. (b) Now we’ll try using deduction. Remember, we’ll be doing the exact same operations, just on an arbitrary number x. Number: x Add 50: x + 50 Multiply by 2: 2( x + 50) = 2x + 100 Subtract the original number: 2x + 100 - x Result: x + 100 Our conjecture was right: the final answer is always 100 more than the original number. THIS COURSEPACK IS FOR PNU NL USE ONLY 60 NO PART OF THIS MATERIAL MAY BE REPRODUCED EXAMPLE 7 :Comparing Inductive and Deductive Reasoning Determine whether the type of reasoning used is inductive or deductive. The last six times we played our archrival in football, we won, so I know we’re going to win on Saturday. SOLUTION: Inductive Reasoning! This conclusion is based on six specific occurrences, not a general rule that we know to be true. EXAMPLE 8: Comparing Inductive and Deductive Reasoning Determine whether the type of reasoning used is inductive or deductive. The syllabus states that any final average between 80% and 90% will result in a B. If I get a 78% on my final, my overall average will be 80.1%, so I’ll get a B. SOLUTION: Deductive Reasoning! Although we’re talking about a specific person’s grade, the conclusion that I’ll get a B is based on a general rule: all scores in the 80s earn a B. Applikasyon wenno aktibidades Kalpasan ti Leksyon (Practice Exercises or Activities after the Lesson) Identify the type of reasoning used in each of the following statements. 1. Kristin is a law student. Most law students own laptops. So, probably Kristin owns a laptop. 2. No Texans are architects. No architects are Democrats. So, no Texans are Democrats. 3. Either Kurt voted in the last election, or he didn't. Only citizens can vote. Kurt is not, and has never been, a citizen. So, Kurt didn't vote in the last election. 4. Arnie: Harry told me his grandmother recently climbed Mt. Everest. Sam: Well, Harry must be pulling your leg. Harry's grandmother is over 90 years old and walks with a cane. 5. Every quiz has been easy. Therefore, the test will be easy. 6. The teacher used PowerPoint in the last few classes. Therefore, the teacher will use PowerPoint tomorrow. 7. Every fall there have been hurricanes in the tropics. Therefore, there will be hurricanes in the tropics this coming fall. 8. 90% of humans are right handed. Joe is human, therefore Joe is right handed 9. Mikhail hails from Russia and Russians are tall, therefore Mikhail is tall 10. Tess: Are there any good Italian restaurants in town? Don: Yeah, Luigi's is pretty good. I've had their Neapolitan rigatoni, their lasagne col pesto, and their mushroom ravioli. I don't think you can go wrong with any of their pasta dishes. THIS COURSEPACK IS FOR PNU NL USE ONLY 61 NO PART OF THIS MATERIAL MAY BE REPRODUCED  Pannubok Kalpasan ti Leksyon (Assessment) NAME _____________________________________________ Score: _________ Yr/Section:___________________________________________ Date: _________ Answer the following and send scanned copy to your teacher. Instructions will be given before you upload your answers. A) Use inductive reasoning to predict the next letter in the following list O, T, T, F, F, S, S, E, ? B) Use deductive reasoning to show that the following procedure produces a perfect cube. Pick 3 CONSECUTIVE INTEGERS. Multiply these three consecutive integers and then add the middle integer to the product C) Gina, Shane, and Mona are Basketball players. Their positions are guard, forward, and center. Given the conditions (a) Gina and the center bought an avocado shake for Mona, (b) Gina is not a guard. Find who plays each position in the team. D) Prove or disprove the following statements using counterexamples. 1. If x is a whole number, then x 2 is also a whole number. 2. If two numbers are both even and both odd, their sum and difference are even number. 3. The only odd product is when you multiply odd times odd number. 4. All even numbers greater than 2 can be expressed as the sum of two primes. THIS COURSEPACK IS FOR PNU NL USE ONLY 62 NO PART OF THIS MATERIAL MAY BE REPRODUCED  Maikadua nga Leksyon Lesson 2: Problems Solving What You’ll Learn ✓ Organize one’s Polya’s method involving problems ✓ Apply the 4 steps of Polya’s method in solving problems Panangipakaammo ti Leksyon (Lesson Introduction) This lesson will discuss the basic of problem solving. We will have the opportunity to solve number and age problems, work and geometry problems. Each type of problems has an illustrative examples for better understanding. Panangipalawag, Panangsingaysingay ken Pannakabinsa- binsa kadagiti Kapanunutan Maigapu iti Leksyon (Discussion and Learning Activities) Problem Solving Problem Solving is the activity of completing mathematical tasks for which one has not yet been shown a procedure. It does not necessarily refer to difficult or more complex routine exercises nor to routine tasks which are presented in words rather than in symbols or numbers. But rather, it refers to problem solving activities where you are expected to make decisions about how they find solutions themselves and not to depend solely on the process presented by the teacher in the solution. One of the recent mathematicians who outlined a strategy for solving problems from virtually any discipline is George Polya (1887-1985). He was the father of problem-solving among mathematicians. Focus on Problem Solving The Problem-Solving Plan To be a good problem solver you need a good problem-solving plan. Using a problem-solving plan along with a problem-solving strategy helps you organize your work and correctly solve the problem. The plan is outlined below. The following is named as Polya’s four-step problem solving strategy A. UNDERSTAND the Problem What are you asked to find? Make sure you understand exactly what the problem is asking. Restate the problem in your own words. THIS COURSEPACK IS FOR PNU NL USE ONLY 63 NO PART OF THIS MATERIAL MAY BE REPRODUCED  What information is given in the problem? List every piece of information the problem gives you. Is all the information relevant? Sometimes problems have extra information that is not needed to solve the problem. Try to determine what is and is not needed. This helps you stay organized when you are making a plan. Were you given enough information to solve the problem? Sometimes there simply is not enough information to solve the problem. List what else you need to know to solve the problem. B. MAKE a PLAN What problem-solving strategy or strategies can you use to help you solve the problem? Think about strategies you have used in the past to solve problems. Would any of them be helpful in solving this problem? Create a step-by-step plan of how you will solve the problem. Write out your plan in words to help you get a clearer idea of how to solve the problem mathematically. C. SOLVE Use your plan to solve the problem. Translate your plan from words to Math. Show each step in your solution and write your answer completely. D. LOOK BACK Did you completely answer the question that was asked? Be sure you answered the question that was asked and that your answer is complete. Is your answer reasonable? Your answer should make sense. Could you have used a different strategy to solve the problem? Solving the problem again with a different strategy is a good way to check your answer. Did you learn anything that could help you solve similar problems in the future? You may want to take notes about this kind of problem and the strategy you used to solve it. A. Number Problems Word problems that involve real numbers, integers or fractions are considered as number problems. Let us consider the following problems Illustrative Example 1: The sum of two positive integers is 64 and their difference is 28. What is the value of the positive difference of the squares of the integers? THIS COURSEPACK IS FOR PNU NL USE ONLY 64 NO PART OF THIS MATERIAL MAY BE REPRODUCED  Solution: Note that: a2 – b2 = (a + b)(a – b) The sum (a + b) = 64 The difference (a – b) = 28 a2 – b2 = (a + b)(a – b) = (64) x (28) a2 – b2 = 1792 Illustrative Example 2: How many three-digit numbers can be formed such that the sum of the digits is 6? Solution: There are several triples that have a sum of 6. These are 006, 150, 240, 330, 123, 222, 411. The numbers are: 600, 150, 105, 501, 510, 240, 204, 420, 402, 330, 303, 123, 132, 321, 312, 213, 231, 222, 411, 141, and 114. There are 21 numbers. B. Age Problems Problems involving people’s ages are another example of word problems that can be translated into equations. A problem of this type may refer to ages at some time in the past or in the future, as well as to present ages. A person age’s some number of years in the past is simply the present age minus that number of years; the age some number of years in the future is found by adding that number of years to the present age. Illustrative Example 3: Angela is now four times as old as Beth. Five years ago the sum of their ages was 45. how old will Angela be in five years? Solution: Present Age Ages 5 years ago Angela X X–5 Beth 4x 4x -5 (x – 5) + (4x – 5) = 45 5x – 10 = 45 5x = 55 x = 11 4x = 44 Angela’s age now 4x + 5 = 49 Angela’s age in five years Illustrative Example 4: Kathy’s age is half her mother’s age. In 2 years the sum of their ages will be 64. how old is her mother now? THIS COURSEPACK IS FOR PNU NL USE ONLY 65 NO PART OF THIS MATERIAL MAY BE REPRODUCED  Solution: Let: x – age of Kathy now x + 2 – age of Kathy in two years 2x – age of Kathy’s mother 2x + 2 – age of Kathy’s mother in two years (x + 2) + (2x + 2) = 64 3x + 4 = 64 3x = 60 x = 20 years old (age of Kathy now) 2x = 40 years old ( Kathy’s Mother age) C. Work Problems Problems involving people or machine doing a job are called work problems. If a piece of work is to be done in t units of time, then in a unit of time, 1/t of the work can be completed. The rate of work is the amount of work done over the actual time to finish the work. Mostly, a work problem involves two persons (M and N) each having different rates of work to combine their efforts to finish a job. The working equation in such cases is Work done by M + work done by N = work done together Illustrative Example 5: Joseph can build a garage in 3 days, and Sam can build the same garage in 6 days. How many days would it take them to build the garage working together? Solution: Let x be the number of days for both Joseph and Sam to build the garage. The rate of work for Joseph to finish the entire job is (1/3)x or x/3 while that of Sam is (1/6)x or x/6. The work is 1 whole job. Thus, the working equation is (1/3)x + (1/6)x = 1 Multiplying both side by 6, the LCD, and continuing the process, we will get the value of x 6[(1/3)x + (1/6)x = 1] 2x + x = 6 3x = 6 x=2 days THIS COURSEPACK IS FOR PNU NL USE ONLY 66 NO PART OF THIS MATERIAL MAY BE REPRODUCED Illustrative Example 6: A father and son working together can finish painting a room in 6 hours. Working alone, the son takes 9 hours to do the painting. How many painting hours will it take the father, working alone, to finish painting the room? Solution: Let x be the number of hours for the father to finish painting the room. Since the job can be done in 6 hours, the rte of work for the father to finish the whole job is (1/x)6 or 6/x while that of his son is (1/9)6 or 6/9. The work is 1 entire job. Thus, the working equation is 9x(6/x + 6/9 = 1) 54 + 6x = 9x 54 =3x 18 =x working alone, the father can finish the job in 18 hours. D. Geometry Problems Problems involving geometric figures such as rectangles and triangles are considered as geometry problems. To solve any problem of this type, one technique is to sketch a picture of the figure involved. Another would be to recall the formulas associated with the figure. Review also some terminologies and characteristics of simple types of geometric figures. Illustrative Example 7: The length of a rectangle is 7 feet less than four times its width. The perimeter is 96 feet. Find the length and width of the rectangle. Solution: Let x = width of the rectangle 4x – 7 = the length of the rectangle Twice the width + twice the length = perimeter 2W + 2L = P 2x + 2(4x – 7) = 96 2x + 8x – 14 = 96 10x = 110 X = 11 Width = 11 feet length -4x – 7 = 4(11) – 7 = 37 feet THIS COURSEPACK IS FOR PNU NL USE ONLY 67 NO PART OF THIS MATERIAL MAY BE REPRODUCED Illustrative Example 8: A rectangle and a square have the same width. The rectangle is 6 cm longer than the square. One perimeter is twice the other. Find the dimensions of the rectangle. Solution: Let x = width of the square and rectangle X + 6 = length of rectangle Rectangle: 2W + 2L = perimeter 2x + 2(x + 6) = perimeter Square : 4s = perimeter 4x = perimeter The problem states that one perimeter is twice the other perimeter. Which perimeter will be larger? The rectangle would clearly have the larger perimeter. Since the perimeter of the rectangle is twice the perimeter of the square, the equation would be 2x + 2(x + 6) = 2(4x) Solving for x, we have 2x + 2x + 12 = 8x 4x + 12 = 8x 12= 4x X= 3 Width of the rectangle and square: x = 3 centimeters Length of the rectangle: x + 6 =3 = 6 = 9 centimeters THIS COURSEPACK IS FOR PNU NL USE ONLY 68 NO PART OF THIS MATERIAL MAY BE REPRODUCED  Applikasyon wenno aktibidades Kalpasan ti Leksyon (Practice Exercises or Activities after the Lesson) Solve the following problems. Show your solution following the 4 steps by George Polya 1. A number is two more than four times another number. Thrice the smaller number and half the larger number is sixty-one. Find the numbers. 2. The sum of three consecutive integers is thirty-eight more than twice the first integer. Find the three integers. 3. One-half the sum of three consecutive multiples of 10 is 75. Find the three multiples of 10. 4. Rex is twice as old as his brother Ben and the sum of their ages is 24 years. How old is each? 5. A narra tree is twenty years younger than a pine tree. In eight years, the narra will be three times as old as the pine tree will be. How old is each tree now? 6. An old machine can do a certain unit of work in 12 hours. To increase production, a more modern machine is put into operation, and both machines now take four hours to do the work. If the old machine breaks down, how long will it take the new machine to do the work alone? 7. The length of a rectangle is 25 cm more than its width. If the perimeter is 98 cm, what is its width? 8. The second angle of a triangle is ten degrees more than half the measure of the first angle. The third angle is ten degrees less than four times the second angle. What are the measures of the three angles of the triangle? THIS COURSEPACK IS FOR PNU NL USE ONLY 69 NO PART OF THIS MATERIAL MAY BE REPRODUCED  Pannubok Kalpasan ti Leksyon (Assessment) NAME _____________________________________________ Score: _________ Yr/Section:___________________________________________ Date: _________ Solve the following problems. Show your solution and upload an scanned copy of your solution. 1. The sum of two numbers is 50. Three times the first is five more than twice the second. What are the numbers? 2. The second of three numbers is six more than the first, and the third is seven times the second. If the third number is twelve more than ten times the first number, what are the three numbers? 3. Ray is twelve years younger than Liza. Four years ago, she was four times as old as he was. Find their present ages. 4. Dennis estimates that he could paint Mrs. Abaya’s house in 6 days. Adam, her assistant, calculated that he could do it in 10 days. How long would it take both of them to finish the job if they worked together? 5. A pipe can fill a tank alone in 5 hours, while a drain can do the contrary in 15 hours. How long would it take the pipe to fill the tank if the drain was simultaneously opened during the duration of the filling job and if the tank was entirely emptied at the start? 6. The first side of a triangle is 5 cm less than the second side, the third side is 3 cm more than the first and the perimeter is 17 cm. how long is each side? 7. When the side of a square is increased by four cm, its area is increased by 80 square cm. what is the length of the side? 8. Peter asked Jane how old she was. Joy answered (a) my age is between 18 and 35, (b) my age is a multiple of 3, (c) my age is an odd number and (d) my age is divisible by 11. How old is Jane? 9. Each of the four friends Liza, Sarah, Bea, and Claudine has a different pet (fish, cat , dog, and snake). From the following clues, determine the pet of each individual. (a) Sarah is alder than her friend who owns the cat and younger than her friend who owns the dog. (b) Bea and her friend who owns the snake are both of the same age and are the youngest members of their group. (c) Liza is older than her friend who owns the fish. THIS COURSEPACK IS FOR PNU NL USE ONLY 70 NO PART OF THIS MATERIAL MAY BE REPRODUCED Nakaadawan kadagiti Kapanunutan nga Nadakamat ken Nayon a Babasaen/Panagparang-ay (References and other Related Readings) Baltazar, E. C. et. Al. (2018). Mathematics in the Modern World. Quezon City: C & E. Publishing, Inc. Nocon, R,C, & Nocon E.G. (2018). Essential Mathematics for the Modern World. Quezon City: C & E. Publishing, Inc. THIS COURSEPACK IS FOR PNU NL USE ONLY 71 NO PART OF THIS MATERIAL MAY BE REPRODUCED