Mathematics 8 Quarter 1 Module 5C: Solving Problems Involving Rational Algebraic Expressions

Mathematics Quarter 1 – Module 5C: “Solving Problems Involving Rational Algebraic Expressions”

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CO_Q1_Mathematics8_M5c

Mathematics – Grade 8 Alternative Delivery Mode Quarter 1 – Module 5: Solving Problems Involving Rational Algebraic Expressions First Edition, 2020

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Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio

Development Team of the Module
Writer:	Alicia E. Gonzales
Language Editor:	Merjorie G. Dalagan
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Layout Evaluator:	Jake D. Fraga
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8

Mathematics

Quarter 1 – Module 5C: “Solving Problems Involving Rational Algebraic Expressions”

Introductory Message

This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson.

Each SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you.

Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these.

In addition to the material in the main text, Notes to the Teacher are also provided to our facilitators and parents for strategies and reminders on how they can best help you on your home-based learning.

Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task.

If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator.

Thank you.

What I Need to Know

[illustration: hand pointing with a finger, icon beside the part title]

This module was designed and written with you in mind. It is here to help you master the skills of solving problems involving rational algebraic expressions. You are provided with varied activities to process the knowledge and skills learned and to deepen and transfer your understanding of the lesson. The scope of this module enables you to use it in many different learning situations. The lessons are arranged to follow the standard sequence of the course. But the order in which you read them can be changed to correspond with the textbook you are now using.

This module contains:

Lesson 1- Solving Problems Involving Rational Algebraic Expressions

After going through this module, you are expected to solve problems involving rational algebraic expressions.

What I Know

[illustration: hand holding a pencil, icon beside the part title]

Directions: Choose the letter of the correct answer. Write the chosen letter on a separate sheet of paper. Exclude values of the variable/s that will make the denominator/s zero.

1. What is the correct rational expression of the phrase “the quotient of $2x$ and $3y$?
   A. $\frac{3y}{2x}$
   B. $\frac{2x}{3y}$
   C. $(2x)(3y)$
   D. $(2x – 3y)$

   
   

2. Which of the following is the correct rational expression of “the fraction whose denominator is $6$ more than the numerator $z$”?
   A. $\frac{z}{z + 6}$
   B. $\frac{z}{6}$
   C. $\frac{z + 6}{z}$
   D. $\frac{1}{z} + 6$

   
   

3. Write the verbal phrase into rational algebraic expression: “the average speed in $x^2 + 7$ kilometers traveled in $x – 2$ hours.”
   A. $(x^2 + 7)(x – 2)$
   B. $\frac{x – 2}{x^2 + 7}$
   C. $\frac{x^2 + 7}{x – 2}$
   D. $\left(\frac{1}{x^2 + 7}\right)(x – 2)$

   
   

4. There are two numbers whose sum is 48. One number is one-third of the other. What are the two numbers?
   A. 10 and 38
   B. 12 and 36
   C. 14 and 34
   D. 20 and 28

   
   

For items 5 & 6: Think of a number. Double a number. Subtract 6 from the result and divide the answer by 2, the quotient will be 20.

5. What is the correct rational equation to represent the problem?
   A. $2x – \frac{6}{2} = 20$
   B. $\frac{2x}{2} = \frac{x – 6}{20}$
   C. $\frac{2x – 6}{2} = 20$
   D. $\frac{1}{2}x = \frac{2x – 6}{20}$

   
   

6. What is the number?
   A. 15
   B. 18
   C. 20
   D. 23

   
   

For items 7 & 8: Dianne and Agnes are siblings. Dianne can complete her household chores in $x$ hours while Agnes can do the same chores in 4 hours.

7. What part of the job can be completed by Dianne after 5 hours?
   A. $x + 5$
   B. $x – 5$
   C. $\frac{x}{5}$
   D. $\frac{5}{x}$

   
   

8. Which expression below represents the rate of Dianne and Agnes working together?
   A. $4 + x$
   B. $x – 4$
   C. $\frac{1}{4} – \frac{1}{x}$
   D. $\frac{1}{4} + \frac{1}{x}$

   
   

For items 9-12: Joe and John are planning to paint a house together. John thinks that if he worked alone, it would take him 3 times as long as it would take Joe to paint the entire house. Working together, they can complete the job in 24 hours.

9. What is the time it takes for John to complete the job?
   A. $x$
   B. $3x$
   C. $\frac{1}{x}$
   D. $\frac{1}{3x}$

   
   

10. What is the work rate of Joe?
    A. $3x$
    B. $\frac{1}{3x}$
    C. $\frac{1}{x}$
    D. $\frac{1}{x} + \frac{1}{3x}$

    
    

11. What is the correct rational algebraic equation that can represent both Joe and John working together?
    A. $\frac{1}{x} + \frac{1}{24} = \frac{1}{3x}$
    B. $\frac{1}{3x} – \frac{1}{x} = \frac{1}{24}$
    C. $\frac{1}{x} + \frac{1}{3x} = \frac{x}{24}$
    D. $\frac{1}{x} + \frac{1}{3x} = \frac{1}{24}$

    
    

12. How long would it take Joe to complete the job alone?
    A. 32 hours
    B. 96 hours
    C. 72 hours
    D. 24 hours

    
    

13. Three years ago, Aiko’s age is one-half of the age she will be in 6 years. How old is she now?
    A. 12 years old
    B. 15 years old
    C. 18 years old
    D. 21 years old

    
    

14. If the same number is added to both numerator and denominator of the fraction $\frac{7}{9}$, the result is $\frac{8}{9}$. What is the number?
    A. 1
    B. 2
    C. 8
    D. 9

    
    

15. It took six hours for Danny to clean the attic while Piolo took twelve hours to do the same job. If Danny and Piolo worked together, how many hours had it took them to complete the job?
    A. 6 hours
    B. 11 hours
    C. 3 hours
    D. 4 hours

    
    

Lesson 1: Solving Problems Involving Rational Algebraic Expressions

What’s In

Verbal Phrases and Algebraic Expressions

In translating a verbal phrase to an algebraic expression, assign a variable to an unknown quantity. Then, write an expression for any other unknown quantities involved in terms of that variable.

Examples:

A. Translate each verbal phrase into an algebraic expression

1. 10 added to twice a number n – 10 + 2n
2. A number x decreased by 3 – x −3
3. Twice a number z divided by 4 – 2z/4

B. Formulas are equations that state relationships between quantities. These formulas can be translated into verbal sentences.

1. P = 2(l + w) → The perimeter of a rectangle is the sum of twice its length l and twice its width w.
2. d = rt → The distance travelled by a moving body is the product of its rate r and the time t elapsed.

Try this out

A. Match the verbal phrase in column A with the corresponding algebraic expression in column B. Write only the letter.

A	B
1. A number x subtracted from 5	a. n/6
2. Twice the product of 4 and y	b. 5x
3. A number n divided by 6	c. 5 − x
4. Five times the number x	d. x − 5
5. A number x increased by 7	e. 2(4y)
	f. x + 7

B. Translate each formula into a verbal statement.

1. P = 4s
2. A = s²
3. A = ½bh

Questions:

1. Were you able to correctly identify the correct algebraic expressions in A?
2. Were you able to correctly translate each formula in B into its equivalent verbal statement?
3. Did you have any difficulty in performing the activity? If so, what did you do to overcome this difficulty?
4. Find s in formula P = 4s.
5. Find s in formula A = s².
6. Find h in the formula A = ½bh.

What’s New

Will the Pool Be Full?

Read the situation below and answer the questions that follow.

Suppose you are tasked to organize a pool party for your best friend’s birthday. Your other friends suggested that it has to be held in the lone inland resort in the city. You decided not to book the event in advance and planned to just come to the resort early to arrange for the pool party on that same day. However, when you came to the resort, you had been informed that the pool was drained and is scheduled to be refilled within the day. One pipe can fill the empty pool in 12 hours, and another can fill the empty pool in 18 hours. Suppose both pipes are opened at 8:00A.M. and you have scheduled the pool party for your best friend’s birthday at 2:00 P.M. on the same day.

Questions:

1. Will the pool be filled by 2:00 P.M.? Elaborate your answer.

   
   

2. At what exact time will the pool be filled?

   
   

3. Would the situation be different if you have booked for the event ahead of time?

   
   

4. What do you think should be the best decision to make?
   A. Cancel the event.
   B. Move the schedule later within the day.
   C. Move the schedule at an earlier time within the day.
   D. Wait for the pool to be filled before inviting your friends to come over.

   
   

What is It

The situation presented in the previous section is one of the applications of the concepts on rational algebraic expressions.

To determine the number of hours it will take for both pipes to fill the pool given that one of the pipes can fill the pool in 12 hours while the other in 18 hours, we need to do the following steps:

Step 1. Read and understand the problem. Identify what is the given and what is being unknown. Choose a variable to represent the unknown number.

Let x = the number of hours it will take to fill the pool if both pipes are used

Step 2. Express the other unknowns, if there are any, in terms of the variable chosen in step 1.

Let 1/x what both pipes can do together to fill the pool

Step 3. Write an equation to represent the relationship among the given and the unknown/s.

[chart: diagram; equation for the two pipes, unnumbered]

• What the first pipe can do to fill the pool | plus | What the second pipe can do to fill the pool | equals | what both pipes can do together to fill the pool
• Equation: 1/12 + 1/18 = 1/x

Step 4. Solve the equation for the unknown and use the solution to find the quantities being asked.

Work	Note
1/12 + 1/18 = 1/x	Equation to represent the situation
36x(1/12 + 1/18) = 36x(1/x)	Multiply both sides by the LCD 36x, to clear the fractions
36x(1/12) + 36x(1/18) = 36x(1/x)	Distribute
3x + 2x = 36	Simplify
5x = 36	Combine similar terms
5x/5 = 36/5	Divide both sides by 5
x = 7 1/5 or 7.2

Hence, it will take 7.2 hours or 7 hours and 12 minutes to fill the pool.

Step 5. Check.

In 7.2 hours or 7 hours and 12 minutes, the first pipe fills 1/12 of 36/5 and the second pipe can fill 1/18 of 36/5. If we get the sum of these, we have:

1/12 (36/5) + 1/18 (36/5) = 3/5 + 2/5 = 5/5 = 1, which is equal to one completely filled swimming pool.

Knowing that it will take 7 hours and 12 minutes to completely fill the pool, will the pool be filled by 2:00 P.M. if both pipes are opened at 8:00 A.M.? The answer is no since there are only six hours in between 8:00 A.M. to 2:00 P.M and that the pool be completely filled by 3:12 P.M. This means, that the best decision that you could make is to move the schedule after 3:12 P.M.

Let us now explore some of the other examples of problems involving rational algebraic expressions.

A. Number Problem

Example: If the same number is added to both numerator and denominator of the fraction 1/2, the result is 3/4. Find the number.

Solution:

Step 1: Let x = the number Step 2: 1 + x = numerator 2 + x = denominator

Step 3: (1 + x)/(2 + x) = 3/4

Step 4:

Work	Note
(1 + x)/(2 + x) = 3/4	Equation
4(1 + x) = 3(2 + x)	Cross multiply
4 + 4x = 6 + 3x	Distributive Property
4 + (−4) + 4x + (−3x) = 6 + (−4) + 3x + (−3x)	Addition Property of Equality
x = 2	Combine like terms

Hence, the number is 2.

Step 5. Check

(1 + x)/(2 + x) = 3/4 → (1 + 2)/(2 + 2) = 3/4 → 3/4 = 3/4

B. Age Problem

Age problems are algebraic problems that deal with the ages of people currently, in the past or in the future.

Example: Five years ago, John’s age was half of the age he will be in 8 years. How old is he now?

Solution:

Step 1: Let x = John’s age Step 2: x − 5 is John’s age five years ago ½(x + 8) is half of the age he will be in 8 years Step 3: x − 5 = ½(x + 8)

Step 4: Solve.

Work	Note
x − 5 = ½(x + 8)	Equation
2(x − 5) = x + 8	Cross multiply
2x − 10 = x + 8	Distributive Property
2x + (−x) − 10 + 10 = x + (−x) + 8 + 10	Addition Property of Equality
x = 18	Combine like terms

Hence, John’s age now is 18 years old.

Step 5. Check

(1 + x)/(2 + x) = 3/4 → (1 + 2)/(2 + 2) = 3/4 → 3/4 = 3/4

C. Work Problem

The formula for work problem that involves two persons is

1/t₁ + 1/t₂ = 1/t₃

where t₁ is time taken by the first person t₂ is the time taken by the second person t₃ is the time taken by both

Example: Joey can mow the lawn in 40 minutes and Pet can mow the lawn in 60 minutes. How long will it take for them to mow the lawn together?

Solution:

Step 1: Assign variables Let x = time to mow lawn together

Step 2: Joey = 1/40, Pet = 1/60, Joey and Pet will mow together = 1/x

Step 3: Write the equation

1/40 + 1/60 = 1/x

Step 4: Solve.

Work	Note
1/40 + 1/60 = 1/x	Equation
120x(1/40 + 1/60) = 120x(1/x)	Multiply both sides by 120x, the LCM of 40, 60, and x
120x/40 + 120x/60 = 120x/x	Distributive Property
3x + 2x = 120	Simplify
5x = 120	Combine like terms
5x/5 = 120/5	Divide both sides by 5
x = 24	Simplify

Hence, it will take 24 minutes for both of them to mow the lawn together.

Step 5. Check

1/40(24) + 1/60(24) = 24/40 + 24/60 → 3/5 + 2/5 → (3 + 2)/5 → 5/5 = 1

D. Speed/Travel Problem

An object is said to be in uniform motion when it moves without changing its speed, or rate.

distance = rate x time d = rt

Example: Macky won a two-day bicycle race. He travelled 60 km each day and his average speed on the second day was doubled that of the first day. If Macky rode for a total of 6 hours, what was his average speed each day?

Solution:

Step 1: Assign variables Let x = speed on the first day

Step 2:

(blank)	distance	Speed (rate)	time
Day 1	60	x	60/x
Day 2	60	2x	60/2x
Total			6 hours

Step 3: Write the equation

60/x + 60/2x = 6

Step 4: Solve.

Work	Note
60/x + 60/2x = 6	Equation
2x(60/x + 60/2x) = 2x(6)	Multiply both sides by 2x, the LCM of x and 2x
120x/x + 120x/2x = 12x	Distributive Property
120 + 60 = 12x	Simplify
180 = 12x	Combine like terms
180/12 = 12x/12	Divide both sides by 12

Work	Note
15 = x	Simplify

Hence, the speed on Day 1 is 15 km/h and on Day 2 is 2x = 2(15) = 30 km/h.

Step 5.

60/15 + 60/2(15) = 4 + 2 = 6

What’s More

Activity 1: Find a Number (Number Problem)

Directions: Analyze the problem and fill in with the correct answers in every step. Write your answer in a clean sheet of paper.

Problem: The denominator of a fraction is one more than the numerator. If 3 is subtracted to the numerator and to the denominator, the resulting fraction is equivalent to 1/2. What is the original fraction?

Solution:

Step 1: _______________________________

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Step 2: _______________________________

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Step 3: _______________________________

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Step 4: _______________________________

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Step 5: _______________________________

Activity 2: What is my Age? (Age Problem)

To add more skills on solving problems involving rational algebraic expressions, let us answer this activity. Fill in with the correct answer by choosing the answer form the box below.

Problem: One-half of Alvin’s age two years from now plus one-third of his age three years ago is twenty years. How old is he now?

Solution:

Step 1: Let x = Alvin’s age now

Step 2: Create an expression using the problem: One-half of Alvin’s age two years from now plus one-third of his age three years ago is twenty years.

(1.)_____ → Alvin’s age two years from now (2.)_____ → Alvin’s age three years ago (3.)_____ → One-half of age 2 years from now (4.)_____ → One-third of age 3 years ago

Step 3: Write out the equation:

(5.)_____ Equation based on the problem.

Step 4: Solution:

$\frac{1}{2}x+1+\frac{1}{3}x-1=20$ Distributive Property

(6.)_____ Combine like terms and simplify

$6\left(\frac{1}{2}x+\frac{1}{3}x=20\right)6$ Multiply both sides by the LCM

(7.)_____ Distributive Property

$3x+2x=120$ Simplify

(8.)_____ Combine similar terms

$\frac{5x}{5}=\frac{120}{5}$ Divide both sides by 5

(9.)_____ Final Answer

Choices:

$\frac{1}{2}(x+2)+\frac{1}{3}(x-3)=20$	$5x=120$	$\frac{1}{3}(x-3)$

[transcriber’s note: table continued from the previous page]

$x+2$	$\frac{1}{2}(x+2)$	$\frac{6}{2}x+\frac{6}{3}x=120$
$x=24$	$\frac{1}{2}x+\frac{1}{3}x=20$	$x-3$

Activity 3: Working in Pair (Work Problem)

Directions: Read the problem carefully. Arrange the solutions based on the steps given. Write only the letter.

Problem: Bob and Pat are asked to paint a house. Bob can paint the house by himself in 12 hours and Pat can paint by herself in 16 hours. How long would it take to paint the house if they worked together?

Solution:

Step 1: Assign variables Let x = time of Bob and Pat are working together

Step 2: Bob = $\frac{1}{12}$, Pat = $\frac{1}{16}$, Bob and Pat work together = $\frac{1}{x}$

Step 3:

1. Write the equation

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Step 4:

2. The LCM of 12 and 16

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3. Multiply both sides with the LCM.

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4. Distribute the LCM to the formula

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5. Simplify

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6. Add

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7. Multiply both sides by x

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8. Divide both sides by the numerical coefficient

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9. Final Answer

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Choices:

A. $48\left(\frac{1}{12}+\frac{1}{16}=\frac{1}{x}\right)48$ B. $\frac{1}{12}+\frac{1}{16}=\frac{1}{x}$ C. $\frac{7x}{7}=\frac{48}{7}$ D. $\frac{48}{12}+\frac{48}{16}=\frac{48}{x}$ E. $x\left(7=\frac{48}{x}\right)x$ F. $4+3=\frac{48}{x}$ G. $x=6.9$ H. $7x=48$ I. $48$ J. $7=\frac{48}{x}$

Activity 4: How Fast Am I? (Speed/Travel Problem)

A passenger train can travel, on average, 20 miles per hour faster than a freight train. If the passenger train covers 390 miles in the same time it takes the freight train to cover 270 miles, then how fast is each train?

Solution:

Step 1: _____ Step 2: _____ Step 3: _____ Step 4: _____ Step 5. _____

[illustration: lightbulb in a cupped hand (part-title icon for “What I Have Learned”)]

What I Have Learned

Directions: Fill in the blanks with the correct word to make the statement true. Write your answer on a separate sheet of paper.

To solve word problems on rational algebraic expressions, we will know how to write equations. There are steps to follow in writing the equation and finding the solution.

A. Read and understand the (1) _____. Identify what is the given and what is being unknown. Choose a (2) _____ to represent the unknown number.

B. Express the other unknowns, if there are any, in terms of the variable chosen in step 1.

C. Write an (3) _____ to represent the relationship among the given and the unknown/s.

D. (4) _____ the equation for the unknown and use the solution to find the quantities being asked.

E. Check.

[illustration: hand holding flower stems (part-title icon for “What I Can Do”)]

What I Can Do

Let’s Cook!

In your TLE class, you are asked by your teacher to cook a recipe with the following ingredients:

Ingredients:

• 1/2 tsp baking powder
• 2/3 cup brown sugar
• 1/4 cup Cocoa powder
• 2 1/3 cups Flour
• 1/4 tsp nutmeg
• 1/2 tsp salt
• 2 cups semi-sweet chocolate chips
• 3/4 cup butter
• 10 1/2 oz condensed milk, sweetened

If the same number is added to both numerator and denominator of the amount of cocoa powder used in the recipe, the result is the amount of butter to be used. Find the number.

[illustration: hand holding a pen (part-title icon for “Assessment”)]

Assessment

Directions: Choose the letter of the correct answer. Write the chosen letter on a separate sheet of paper.

1. Write the expression “the ratio of five less than thrice a number x and four” in symbols.

A. $\frac{5-3x}{4}$ B. $\frac{5<3x}{4}$ C. $\frac{3x-5}{4}$ D. $\frac{3x<5}{4}$

2. Which of the following is the correct translation for the phrase, “ten less than twice the sum of two number x and y divided by five”?

A. $\frac{10-2(x+y)}{5}$ B. $\frac{-2(x+y)10}{5}$ C. $\frac{2(x+y)-10}{5}$ D. $\frac{10<2(x+y)}{5}$

3. Translate this phrase into algebraic expression: “The ratio of two and x less by the quotient of 2 and square of x”.

A. $\frac{2}{x}-\frac{2}{x^2}$ B. $\frac{2}{x}<\frac{2}{x^2}$ C. $\frac{x^2}{2}-\frac{x}{2}$ D. $\frac{x^2}{2}<\frac{2}{x}$

4. The numerator of a fraction is three less than the denominator. If 4 is added to the numerator and to the denominator, the resulting fraction is equivalent to $\frac{3}{4}$. What is the final equation of this problem?

A. $\frac{d-3+4}{d+4}=\frac{3}{4}$ B. $\frac{d-4+3}{d+4}=\frac{3}{4}$ C. $\frac{d+4}{d-3+4}=\frac{3}{4}$ D. $\frac{d+4-3}{d-4}=\frac{3}{4}$

5. Find the denominator of the given problem in number 4.

A. 5 B. 6 C. 7 D. 8

6. What is the original fraction in the problem #4?

A. $\frac{5}{6}$ B. $\frac{5}{8}$ C. $\frac{4}{7}$ D. $\frac{7}{8}$

7. Dividing 20 by a number gives the same result as dividing 12 by 2 less than the same number. What is a number?

A. 5 B. 6 C. 7 D. 8

8. Jay’s father is twice as old as Jay. In 20 years, Jay will be two-thirds as old as his father. How old is Jay now?

A. 10 years old B. 15 years old C. 20 years old D. 25 years old

9. What is the age of Jay’s father now in problem number 8?

A. 20 B. 40 C. 50 D. 60

10. Bob can do his project in x hours, what part of the job can be completed by Bob after 2 hours?

A. $x-2$ B. $x+2$ C. $\frac{2}{x}$ D. $\frac{x}{2}$

11. Shane can clean the house in 3 hours and Ana can clean the house in 5 hours, what is the correct equation below represents if they are working together?

A. $\frac{1}{3}+\frac{x}{5}=1$ B. $\frac{1}{3}-\frac{x}{5}=1$ C. $\frac{1}{3}+\frac{1}{5}=\frac{1}{x}$ D. $\frac{1}{3}-\frac{1}{5}=\frac{1}{x}$

12. Leah can clean the store in five hours. If Angelo helps, it takes them four hours. Without help, how many hours would it take Angelo to complete this job?

A. 10 B. 25 C. 20 D. 15

13. It takes Jordan 36 minutes to mow the lawn while it takes James 45 minutes to mow the same lawn. What is the completed part of James?

A. $\frac{1}{36}$ B. $\frac{1}{45}$ C. $\frac{x}{36}$ D. $\frac{x}{45}$

14. What is the correct equation will be used in problem number 13?

A. $\frac{1}{36}+\frac{1}{45}=x$ B. $\frac{x}{36}+\frac{x}{45}=1$ C. $\frac{x}{36}+\frac{x}{45}=x$ D. $\frac{1}{36}+\frac{x}{45}=1$

15. Using problem number 13, If Jordan and James work together, using two lawn mowers, how long would it take them to mow the lawn?

A. 10 minutes B. 15 minutes C. 20 minutes D. 25 minutes

Additional Activities

In the problem presented in the What’s New section of this module, one pipe could fill the pool in 12 hours while the other pipe could fill the same pool in 18 hours. You were asked to find how long would it take to completely fill the pool if both pipes were used. Explain why each of the following approaches is INCORRECT.

1. The time it would take to fill the pool is the sum of the lengths of time it takes each pipe to fill the pool:

$12\ hours+18\ hours=30\ hours$

[transcriber’s note: the printed line has an orange strike-through line across the equation]

2. The time it would take to fill the pool is the difference in the lengths of time it takes each pipe to fill the pool:

$18\ hours-12\ hours=6\ hours$

[transcriber’s note: the printed line has an orange strike-through line across the equation]

3. The time it would take to fill the pool is the average of the lengths of time it takes each pipe to fill the pool:

$\frac{12\ hours+18\ hours}{2}=\frac{30\ hours}{2}=15\ hours$

[transcriber’s note: the printed line has an orange strike-through line across the equation]

Answer Key

[transcriber’s note: this page is printed rotated 180 degrees; it has been read in its upright reading order, and some small entries in the rotated layout are hard to read and are marked with [?]]

What I Know

1. B
2. A
3. C
4. B
5. C
6. D
7. D
8. D
9. B
10. C
11. D
12. A
13. A
14. C
15. D

What’s New

1. No. It will take 7 hours and 12 minutes to completely fill the pool, hence, it will not yet be completed at 2 P.M.
2. 3:12 P.M.
3. Yes. Reasons may vary.
4. B

What’s In

A.

1. c
2. e
3. a
4. b
5. f

B.

1. The perimeter of a square is 4 times its side
2. The area of a square is the square of its side

What’s More

Activity 1

Step 1 n = numerator d = denominator

Step 2 [?]

Step 3 [?]

Step 4 the numerator the denominator the original fraction

Activity 2

1. $x+2$
2. $x-3$
3. $\frac{1}{2}(x+2)$
4. $\frac{1}{3}(x-3)$
5. $\frac{1}{2}(x+2)+\frac{1}{3}(x-3)=20$
6. $\frac{1}{2}x+\frac{1}{3}x=20$
7. $\frac{6}{2}x+\frac{6}{3}x=120$
8. $5x=120$
9. $x=24$

Activity 3

1. B
2. I
3. A
4. D
5. F
6. H
7. E
8. C
9. G

Activity 4

Step 1 Let x = speed of the passenger train

Step 2

distance	rate	time
Passenger train: 390	$x+20$	$\frac{390}{x+20}$
Freight train: 270	$x$	$\frac{270}{x}$
Total

[transcriber’s note: the total row and the exact layout of the table are hard to read in the rotated image]

Step 3 $390x=270(x+20)$

Step 4 $390x=270x+5400$ $120x=5400$ $x=45$

Final answer: 45 miles/hour speed of freight train; 65 miles/hour, speed of passenger train

What I Can Do

The number is 8

$\frac{x+1}{x+4}=\frac{3}{4}$ $4(x+1)=3(x+4)$ $4x+4=3x+12$ $4x-3x=12-4$ $x=8$

Assessment

1. C
2. C
3. A
4. A
5. D
6. B
7. A
8. C
9. B
10. C
11. C
12. C
13. D
14. B
15. C

What I Have Learned

1. Problem
2. Variable
3. Equation
4. Solve

References

Abuzo, E.P., Bryant, M.L., Cabrella, J.B., Caldez, B.P., Callanta, M.M., Castro, A.I., Halabaso, A.R., et. al (2013). Mathematics 8 – Learner’s Module. Department of Education, Pasig City, Philippines, pp. 81-104

Jose-Dilao, S., Bernabe, J.G. Intermediate Algebra, Revised Edition, SD, Publications, Inc. Quezon City, 2009.

Bernabe, Julieta .G. Elementary Algebra, Revised Edition, SD, Publications, Inc. Quezon City, 2009.

Orines, F.B., Mercado, J.P., Suzara, J.L., Manalo, C.B. Next Century Mathematics 7. Phoenix Publishing house, Inc. Quezon City. 2012

For inquiries or feedback, please write or call:

Department of Education – Bureau of Learning Resource Ground Floor, Bonifacio Building, Deped Complex Meralco Avenue, Pasig City, Philippines 1600

Telefax. Nos.: (632) 8634-1072; 8634-1054; 8631-4985

Email Address: bld.lrqad@deped.gov.ph * blr.lrpd@deped.gov.ph