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480611738 Gen Math11 Q1 Mod10 solving real life problems involving rational 08082020 pdf
Civil Engineering (BSCE 01)
Ateneo de Davao University
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General Mathematics
Quarter 1 – Module 10:
Solving Real-Life Problems
Involving Rational Functions,
Equations, and Inequalities
General Mathematics Alternative Delivery Mode Quarter 1 – Module 10: Solving Real-Life Problems Involving Rational Functions, Equations, and Inequalities First Edition, 2020
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Development Team of the Module Writer: Dennis S. Vergara Editors: Elizabeth B. Dizon, Anicia J. Villaruel, and Roy O. Natividad Reviewers : Fritz A. Caturay, Necitas F. Constante, Celestina M. Alba, and Jerome A. Chavez Illustrator: Dianne C. Jupiter Layout Artist: Noel Rey T. Estuita Management Team: Wilfredo E. Cabral, Job S. Zape Jr., Eugenio S. Adrao, Elaine T. Balaogan, Catherine P. Talavera, Gerlie M. Ilagan, Buddy Chester M. Repia, Herbert D. Perez, Lorena S. Walangsumbat, Jee-ann O. Borines, Asuncion C. Ilao
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Introductory Message
For the facilitator:
Welcome to General Mathematics Grade 11 Alternative Delivery Mode (ADM) Module on Solving Real-Life Problems Involving Rational Functions, Equations and Inequalities.
This module was collaboratively designed, developed and reviewed by educators from public institutions to assist you, the teacher or facilitator in helping the learners meet the standards set by the K to 12 Curriculum while overcoming their personal, social, and economic constraints in schooling.
This learning resource hopes to engage the learners into guided and independent learning activities at their own pace and time. Furthermore, this also aims to help learners acquire the needed 21st century skills 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. Furthermore, you are expected to encourage and assist the learners as they do the tasks included in the module.
For the learner: Welcome to General Mathematics Grade 11 Alternative Delivery Mode (ADM) Module on Solving Real-Life Problems Involving Rational Functions, Equations and Inequalities. The hand is one of the most symbolized parts of the human body. It is often used to depict skill, action and purpose. Through our hands, we may learn, create and accomplish. Hence, the hand in this learning resource signifies that you as a learner is capable and empowered to successfully achieve the relevant competencies and skills at your own pace and time. Your academic success lies in your own hands!
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 resource while being an active learner.
This module has the following parts and corresponding icons:
Notes to the Teacher This contains helpful tips or strategies that will help you in guiding the learners.
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The following are some reminders in using this module:
- Use the module with care. Do not put unnecessary mark/s on any part of the module. Use a separate sheet of paper in answering the exercises.
- Don’t forget to answer What I Know before moving on to the other activities included in the module.
- Read the instruction carefully before doing each task.
- Observe honesty and integrity in doing the tasks and checking your answers.
- Finish the task at hand before proceeding to the next.
- Return this module to your teacher/facilitator once you are through with it. If you encounter any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator. Always bear in mind that you are not alone. We hope that through this material, you will experience meaningful learning and gain deep understanding of the relevant competencies. You can do it!
References This is a list of all sources used in developing this module.
What I Know
Choose the letter of the best answer. Write the chosen letter on a separate sheet of paper.
Mayor Rodriguez received 5000 sacks of rice to be distributed among the families in his municipality during the lockdown. If the municipality has x families, write the function which represents the relationship of the allotted sack of rice per family(y-variable) versus the total number of families. a. ÿ = 5000 𝕥 b. ÿ = 5000 𝕥 c. ÿ = 5000 𝕥𝕥 d. ÿ = 5000 𝕥+𝕥
To beat the heat of summer, Mang Berto built a rectangular swimming pool that has a perimeter of 200 meters. Write the function which represents the width( y ) of the swimming pool as a function of the length(x).
a. ÿ = 200 𝕥 b. ÿ = 200 𝕥 c. ÿ =𝕥+1 200 d. ÿ = 100 2 þ
It takes Brad 2 hours to mow his rice field. It takes Kris 3 hours to mow the same rice field. At the same pace, how long would it take them to mow the rice field if they do the job together? a. 2 Ω hours b. 1 1/5 hours c. 1 1/6 hours d. 5/6 hours
Anne and Maria play tennis almost every weekend. So far, Anne has won Āā out of āÿ matches. a. How many matches will Anne have to win in a row to improve her winning percentage to ĆĄ%? a. 15 b. 12 c. 9 d. 6
In a basket, there are 12 apples and 32 oranges. A buyer requires having a basket of apples and oranges with the ratio greater than or equal to 3:4 respectively. How many apples must be added to the basket to satisfy the buyer’s request? a. 10 apples b. 15 or more apples c. 12 or more apples d. 8 apples
Mario was given 3 hours to practice driving his motorcycle. He plans to travel 100 kilometers at an average speed of 40 kilometers per hour. He wants to maximize his time in driving his motorcycle. How many kilometers more does he need to travel to spend at most 3 hours? a. less than or equal 20 kilometers b. greater than or equal 20 kilometers c. exactly 30 kilometers d. less than or equal 30 kilometers
Jessie works as a salesman. He earns a daily wage of 250 pesos and an additional 10 pesos for every 3 pieces of cell phone sold. If x represents the number of cell phones sold, write the function for his daily earning (y) as a function of the number of cell phones sold (x).
a. ÿ = 250 10𝕥 b. ÿ = 250 + 10(𝕥 3 ) c. ÿ = 250 + 3 𝕥+ 10 d. ÿ = 2500 3𝕥
Using the problem in number 7, if Jessie sold 48 cell phones in a day, how much money did he earn for that day? a. 410 pesos b. 250 pesos c. 500 pesos d. 480 pesos
Melissa walks Ă miles to the house of a friend and returns home on a bike. She averages ă miles per hour faster when cycling than when walking, and the total time for both trips is two hours. Find her walking speed. a. 1 mph b. 2 mph c. 3 mph d. 4 mph
You have Āÿ liters of a juice blend that is ąÿ% juice. How many liters of pure juice needs to be added to make a blend that is ĆĄ% juice? a. 10 liters b. 8 liters c. 6 liters d. 4 liters
If the sum of a number (x) and 3 is divided by 5, the result is greater than 2. What are the possible values for the given number (x)? a. x > 5 b. x > 7 c. x < 5 d. x < 7
During a pandemic, Brgy. Captain Gerry was given 1,000,000 pesos to support 500 households in his barangay. He plans to give at least 3,000 pesos for every household. How much money does he need to solicit to realize his plan? a. at least 300,000 b. at least 400,000 c. at least 500,000 d. at least 100,
Coronavirus infection is spreading fast worldwide. The number of people infected
by the virus each day is given by the function 𝕃(þ)=100𝕥𝕥+3, 0 f þ f 10 where x is the
number of days, and 𝕃(þ) is the number of people infected (in thousands). How many people are infected on the first day? a. 25 b. 25,000 c. 50,000 d. 75,
- Sir Paco is thrice as old as his son Javy. 10 years from now, the ratio of their ages will be 2:1 respectively. How old is Javy?
a. 5 b. 15 c. 12 d. 10
- As part of his exercise routine, Jerson runs 20 kilometers at an average speed of 3 kilometers per hour. If he decided to run at most 2 hours on a specific day, how may kilometers less does he need to run?
a. at least 14 km b. at most 14 km c. exactly 14 km d. less than 14 km
Rational Equation
Solution to Rational Equations
Rational Functions
Rational Inequalities
Solution to Rational Inequalities
Recall your skill in solving a rational equation and rational inequalities to match the correct data in the appropriate column. This skill is a prerequisite in this module because you cannot solve real-life problems involving rational functions, equation, and rational inequalities if you do not master your previous skill. In that case, let me help you.
On the given, you observed that 𝖇(ý) =ý
ā2āý+ă ý and þ =
Āÿÿÿ+ý āÿ are written in the form Ą(þ)=ý(𝕥)þ(𝕥) where ā(þ) and Ă(þ) are both polynomial functions, therefore these are
examples of rational functionals provided that Ă(þ) is not equal to zero. While ăý=ý+ĂĀÿ
and ăý 2 ý2ĀĄ =ĀăĀĄ are both rational equations because they involve rational
expressions. Intuitively, you may think that 3 and 5 are the solutions but you need
to solve it for you to see the result. On the other hand, ý+Āý2Ąf ÿ and ý2āă > ā are rational
inequalities because they are inequalities that involve rational expressions. If you master the skills in solving them, I am sure you got the correct data on the appropriate column.
If you think you are not confident that you are correct, review first your previous lesson before you proceed to take this module, But I am sure, you will do your part because you are willing to learn.
What’s New
Speed Me Up!
Read and analyze each situation below and answer the questions that follow.
Mario rides his motorcycle in going to school. He drives at an average speed of 30 kilometers per hour. The distance between his house and the school is 15 kilometers. Every time he sees his best friend Jessica walking on the road, he invites her for a ride and lowers his speed. On the other hand, he increases his speed when he wakes up late for school.
15 kilometers
Questions:
a. How long does it take Mario to reach school considering his average speed?
b. If x represents the time it takes Mario to drive to school with the given distance of 15 kilometers, how will you represent the relationship of his speed (y) versus the time (x)? c. Mario’s average speed as 30 kilometers per hour. Suppose Mario lowers his speed by 10 kilometers per hour, how long will he reach the school given the same distance?
d. Suppose Mario’s speed is unknown and represented by (x), he lowers his speed by
10 kilometers per hour at a distance of 15 kilometers and reaches school at 34 hours,
how will you write the equation to find his average speed (x)? e. Mario’s average speed was 30 kilometers per hour. He plans to drive for another 30 kilometers from school, how long will it take him to cover the whole distance (house to school to 30 kilometers from school)?
f. If Mario drives another (x) kilometers from his school at an average speed of 30 kilometers per hour and he plans to drive in at most 2 hours, how will you write the inequality to find the additional distance?
What is It
The Speed Me Up Activity is an example of the real-situation involving rational equation and inequality, and to be able to answer the questions given above, it is very important to know the distance-speed-time relationship. The following illustrates these relationships.
Question number 6 requires you to write rational inequality to be able to find the additional distance. Additional distance will be represented by x and the total distance will be <15 + x=. Since his speed remains at 30 kilometers per hour and the time that will require him to cover the distance is at most 2 hours (less than or equal to 2), we write the inequality:
ÿ Ąf ą 15 + þ 30 f 2
Solving this inequality will give þ f 45. Mario needs to travel an additional distance of not more than 45 kilometers to spend at most 2 hours.
The idea of riding a motorcycle seems very enjoyable. But, always bear in mind that accidents may happen. So, be cautious and consider safe driving by following street rules. Just like analyzing Math problems, little by little, we would arrive at answers if we only know how to follow rules.
Another skill that you will learn in this module is solving real-life problems involving rational function. Consider the examples below:
Example 1
Bamban National High School is preparing for its 25th founding anniversary. The chairperson of the activity allocated ₱ 90 ,000 from different stakeholders to be divided among various committees of the celebration. Construct a function þ(ÿ) which would give the amount of money each of the ÿ numbers of committees would receive. If there are six committees, how much would each committee have?
Solution:
The function þ(ÿ)= 90000 𝕛 would give the amount of money each of the ÿ numbers of
committees since the allocated budget is ₱ 9 0,000 and it will be divided equally to the ÿ number of committees.
If there are six committees, then you need to solve for þ( 6 ), thus
þ( 6 )=
90000
6 = 15000
Therefore, each committee will receive ₱15,000.
Example 2
Barangay Masaya allocated a budget amounting to ₱100,000 to provide relief goods for each family in the barangay due to the Covid-19 pandemic situation. The amount is to be allotted equally among all the families in the barangay. At the same time a philanthropist wants to supplement this budget and he allotted an additional ₱500 to be received by each family. Write an equation representing the relationship of the allotted amount per family (y-variable) versus the total number of families (x-variable). How much will be the amount of each relief packs if there are 200 families in the barangay?
Solution:
The amount to be received by each family is equal to the allotted (₱100,000), divided by the number of families plus the amount to be given by the philanthropist.
Thus the rational function is described as ÿ = 100000 𝕥 + 500. The amount of each relief
packs can be computed by finding the value of ÿ when þ = 50, since there are 50 families in the barangay. Thus,
ÿ =
100000
200 + 500 = 1000
Therefore, the amount of each relief packs to be distributed to each family worth ₱1,000.
Notes to the Teacher
Remind students that they must: (a) read and analyze the problem carefully, (b) paraphrase and summarize the problem in their own words, (c) find an equation that models the situation, and (d) say how it represents the quantities involved and (e) check to make sure that they understand the problem before they begin trying to solve it.
Complete the following to solve the problem. a. The part of the job accomplished by Rodalyn on the first day is 1 5., So, the part of the job accomplished by Apple on the first day is _____. b. If x represents the time it will take them to do the job together, the part of the job accomplished on the first day of working together is ________. c. Looking at the relationship, we arrive at the equation: 1 5
+
1
3
=
1
þ d. Solving the rational equation, the value of x is ________ together, they can finish the job in ____ day and_____ hours.
Independent Assessment 2
Practice Activity 3
a. Complete table to understand the relationship.
Original Added Result
Concentration 20% = 10020 100% = 1 25% = 10025
Amount 30 liters x 30 + x
Multiply 20 100
(30) 1(x)?
Paint my Wall
Analiza can paint a room in 3 hours. Leoben can do it in 2 hours. Walter can do the painting job in 5 hours. If all of them worked together, how long will it take them to paint the room?
Mix mix mix! How many liters of pure alcohol must be added to 30 liters of 20 % alcohol solution to make a 25% alcohol solution.
Note: We use 100% or 1 because pure alcohol was added.
b. Use the relationship to make an equation. 20 100
( 30 )+ 1(þ)=
25
100 (30 + þ) c. Solve the equation by finding the value of x. Multiply the whole equation by LCM which is 100. 600 + _______ = _______(30 + þ) 600 + 100þ = 750 + 25þ 75þ = 150 þ = _______. Independent Assessment 3
Practice Activity 4
Complete the following to solve the problem. a. The formula to find the volume of the box is _________________. b. The equation relating to find the value of / is ___________________. Since the height is greater than the length of the edge, the inequality can be described as 27 þ 2
2 þ > 0
c. The possible value of þ should be _____________________.
(Hint: Solve for x in the inequality 27 𝕥 2 2 þ > 0.)
Independent Assessment 4
Volume of a Box
A box with a square base has a volume of 27 cubic inches. If þ is the length of its edge and / is the height of the box. What are the possible measurement of its edge if the height should be longer than the edge?
Who am I? I am thinking of a number, the sum of twice a number and 8 divided by 12 is greater than or equal to 4. Find the number/numbers.
Salt solution Joey has 40 liters of 10% salt solution. How much salt should be added to make it a 20% salt solution?
480611738 Gen Math11 Q1 Mod10 solving real life problems involving rational 08082020 pdf
Course: Civil Engineering (BSCE 01)
University: Ateneo de Davao University
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