RATIONAL ALGEBRAIC EXPRESSIONS

MODULE IN ALGEBRA

1

OBJECTIVES:

Describe and illustrate rational algebraic expression .

Interpret zero and negative exponents.

Evaluate algebraic expressions involving integral exponents.

Simplify rational algebraic expression

RATIONAL ALGEBRAIC EXPRESSIONS

Zero & Negative Exponents

Evaluation of Simplification of

LESSON 1 1

MODULE IN ALGEBRA

I. Matching Type:Match each item in column A with its simplest form in column B. Write the letter of the correct answer on the space provided for before each number. Assume that the variables are nonzero.Column A Column B1. 4 x−2 a. xy2. x0+ y0+z0 b.813. xy z0 c. x44. −34 d. 4 x35. ¿ e. 4

x26. ¿ f. 1x87. (x 72 y

45 )(x

12 y

−45 ) g. −6

x8. 4x−3

h. 39. (−30 x4a+5 ) ÷(5 x4a+6) i. y6

x210. ¿ j. -81k. x3II. Choose the letter that corresponds to each given statement.1. For a=b2, ______ is the square of _____A. a; b B. aC. b; a D. b; d2. When no index is indicated in a radical, then it is understood that the index is ____A. 3 B. 2C. 1 D. 0

2

Pre-requisite:

Hello guys!!!Let’s have a great time learning Mathematics. This

quarter will feed your minds with Rational Algebraic Expressions. Enjoy the activities while you explore, firm-

up what you will discover, deepen your learning and transfer through real-life applications.

MODULE IN ALGEBRA

3. The product of (3√2 + 4) (3 √2 – 4) is ____A. 2 B. 3√4C. 12 D. 184. To simplify√3

15, multiply it by ____

A. √3 B. 5C. √25 D. √55. The value of x in equation 3√ x+1 - 2 = 3 is ____A. 124 B. 26C. 7 D. 63

3

Let’s begin the lesson by reviewing some of the previous lessons and focusing your thoughts on the lesson. This may help you in understanding Rational Algebraic Expressions in Simplest Form. Perform each given activity.

DEFINITION OF ALGEBRAIC EXPRESSION

MODULE IN ALGEBRA

There are verbal phrases below. Look for the mathematical expression of each verbal phrase from the box. Knowing how to translate verbal phrase into mathematical expression is important in solving word problem of the lesson.

4

2x− 2

x2 pq3

b2

(b+2)

h2

√3 yy 9− 1

w2

10x + 6

√3 y

a2+2a

z3−9b2−(b+2)

x2−1x2−2x+1

c2

3

x4+2

Activity 1: Match Me

w - 3√910y

+4

1. The ratio of a number x and four added to two2. The product of the square root of three and the number y3. The square of a added to twice the a 4. The sum of b and two less than the square of b5. The product of p and q divided by three6. One-third of the square of c7. Ten times a number y increased by six8. The cube of the number z decreased by nine9. The cube root of nine less than a number w10. A number h raised to the fourth power

3

c2

3

z3 3

z3

LESSON 1.1

In a weather bulletin, it was announced that a very strong typhoon would hit the country in 3 days. Mr. Par is worried because his house still needs some repairs. If he will not do something, his house might be completely destroyed by the coming typhoon? Mr. Par knows that he needs 7 days to repair his house.

MODULE IN ALGEBRA

a. What should Mr. Par do?b. If he asks somebody to help him, how fast must that person work so that they can finish repairing the house in 2 days?c. Do you think the house can be repaired in 2 days even if somebody helps him?

Write your ideas on rational algebraic expressions and algebraic expressions with integral exponents. Answer the unshaded portion of the table.What I Know What I Want to Find Out What I Learned How I Can Learn More

5

1. The ratio of a number x and four added to two2. The product of the square root of three and the number y3. The square of a added to twice the a 4. The sum of b and two less than the square of b5. The product of p and q divided by three6. One-third of the square of c7. Ten times a number y increased by six8. The cube of the number z decreased by nine9. The cube root of nine less than a number w10. A number h raised to the fourth power

Activity 2: Let’s Explore!

Activity 3: KWLH

Questions:

1. Can you answer the first question? If yes, how will you answer it? If no, what must you do to answer the question?

2. How will you describe the second question?3. How will you model the above problem?

These problems are related to our lesson “Rational Algebraic Expressions”. You will be able to answer these problems after taking the next activities.

Your goal in this section is to learn and understand the key concepts on rational algebraic expressions and algebraic expressions with integral exponents.

As the concepts on rational algebraic expressions and algebraic expressions with integral exponents become clear to you through the succeeding activities, do not forget to apply these concepts in real-life problems especially to rate-related problems.

MODULE IN ALGEBRA

1. What are the polynomials in the activity “Match Me”? 2. Describe these polynomials. Which are not polynomials? 3. List these non-polynomials under set R.4. How do these non-polynomials differ from the polynomials?5. Describe these non-polynomials.P R

6

Activity 4: Match Me -

You were engaged in some of the concepts in the lesson but there are questions in your mind. The next section will answer your queries and clarify your thoughts regarding the lesson.

MODULE IN ALGEBRA

Use your answers in the activity “Match Me-Revisited” to complete the graphic organizer. Compare and contrast. Write the similarities and differences between polynomials and non-polynomials in the first activity.

Write your initial definition of rational algebraic expressions in the appropriate box. Your final definition will be written after some activities. 7

Activity 6: My Definition Chart

In the activity “Match Me”, the non-polynomials are called rational algebraic expressions. Your observations regarding the difference between polynomials and non-polynomials in activities are the descriptions of rational expressions. Now, can you define rational algebraic expressions? Write your own definition about rational algebraic expressions in the chart on the next page.

POLYNOMIALS NON-POLYNOMIALS

Activity 5: Compare and Contrast

How Alike?

In terms of . . . _______________________________________________________________________________________________

______________________________________________________________________________________________________________________________________________________________________________________

______________________________________________________________________________________________________________________________________________________________________________________

How Alike?

_________________________________________________________________________________________________________________________________________________________________________________________

_______________________________________________________________________________

_______________________________________________________________________________

MODULE IN ALGEBRA

Try to firm up our own definition regarding the rational algebraic expressions by doing the next activity.

8

ca−2

1−m

m31

a6

m+20

y+2y−2

My Initial Definition

My Final Definition

12

c4

m−m

k

3k2−6k

Activity 7: Classify Me

Note: Rational Algebraic Expression is a ratio of two polynomials, however 12 is not a polynomial. It is simply a fraction because considering the laws of exponent.

x0=120=x

30=0So, 12 can be written in the form 1

2x0 that will satisfy the definition of RAE.

12

x0=12(1)=1

2

Any number raised to zero is always 1.

Rational Algebraic Expressions Not Rational Algebraic Expressions

_______________________________________________________________________________

_______________________________________________________________________________

MODULE IN ALGEBRA

Write your initial definition of rational algebraic expressions in the appropriate box. Your final definition will be written after some activities.

9

In the first few activities, you might have some confusions regarding rational algebraic expressions. However, this section firmed up your idea regarding rational algebraic expressions. Now, put into words your final definition of a rational algebraic expression.Activity 8: My

Definition Chart

My Initial Definition

My Final Definition

MATH DETECTIVERational algebraic expression is a ratio of two polynomials where the denominator is not equal to zero. What will happen when the denominator is not equal to zero? What will happen when the denominator of a fraction becomes zero? Clue: Start investigating in 42=2

4=(2)(2)14

4=(1)(4)

Compare your initial definition with your final definition of rational algebraic expressions. Are you clarified with your conclusion by the final definition? How? Give at least three

rational algebraic expressions different from those given by your classmate.

Note:A rational algebraic expression is a ratio of two polynomials provided that the denominator is not equal to zero. An expression is a rational expression if it can be written in

the form

pq

, where p and q are polynomials and q ≠ 0.All terms or expressions are Rational Algebraic Expression as long as it is a rational number be it in a fraction form or in polynomial form (ax2+bx+1¿ .However any terms or expressions divided by zero is not Rational Algebraic Expression.

ZERO AND NEGATIVE EXPONENTS

MODULE IN ALGEBRA

Complete the table below and observe the pattern.

10

RECALLLAWS OF EXPONENTS

I-Product of PowersFor any real number

x, and any positive integers n and b:

xn . xn+b

II-Power of a PowerFor any real number

x, and any positive integers n and b:

¿

III-Power of a ProductFor any real

numbers x and y, and any positive integer n:

¿

IV-Power of a QuotientFor all integers n

and b, and any nonzero number x.Case I.

xn

xb=xn−b ,where n>b

Case II. xn

xb=¿

Activity 9: Let the Pattern Answer it!

You observed in the table that instead of writing the whole operation of multiplying a number by itself, it can be simplified by writing the number x and raised it to the number of times n it will multiplied. In symbols,xn, where x is the base and n is the exponent, index, or power.

In the first example, you have learned in your previous years that 25 is multiplying 2 five times by itself or that is 2x2x2x2x2 that results to 32.

In the activities above, you had encountered rational algebraic expressions. You might encounter some algebraic expressions with negative or zero exponents. In the next activities, you will define the meaning of algebraic expressions with integral exponents including negative and zero

LESSON 1.2

A B C A B C A B C A B2.2.2.2.2 25532 3.3.3.3.3 35 243 4.4.4.4.4 45 1024 x.x.x.x.x x5

2.2.2.2 3.3.3.3 4.4.4.4 x.x.x.x2.2.2 3.3.3 4.4.4 x.x.x2.2 3.3 4.4 x.x2 3 4 x

MODULE IN ALGEBRA

Use your observations in the activity above to complete the table below.

A B A B A B A B25 32 35 243 45 1024 x5 x.x.x.x.x24 34 44 s4

23 33 43 x3

22 32 42 x2

2 3 4 x

20 30 40 x0

2−1 3−1 4−1 x−1

2−2 3−2 4−2 x−2

2−3 3−3 4−3 x−3

Exercises:11

The exponent of a number says how many times to use a number in multiplication. For example, 82=8×8=64, the exponent 2 indicates that 8 will be multiplied twice to itself. In words, 82 can be read as 8 to the power of 2, 8 to second power or simply 8 squared. On the other hand, a negative exponent means how many times to divide by the number. Let’s take a look at 8−2 . When simplified8−2=1÷8÷8= 1

64=0.016. It could

also be done this way, 8−2=1×8×8= 1

82= 164

=0.016. The last example could be an easier way in handling negative exponents. First, disregard the negative sign of the exponent and calculate it, then get the reciprocal of the result. That is, x−n= 1

xn .

Oftentimes we mistook x0=0 instead of 1. Remember the rule in dividing variables raised to a power, xa

xb=xa−b. In same manner, we can investigate whyx0=1. For example, x3

x3=x3−3=x0. If we cancel, x3

x3= x . x . x

x . x . x=11=1. That’s why x0=0 butx0=1.

MODULE IN ALGEBRA Rewrite each item to expressions with positive exponents.1. b−4 6. de5 f

2. c3

d−8 7. ¿3. w−3 z−2 8. 14 t 0

4. n2m−20 9. p0

5. x+ y¿¿ 10. 2

(a−b+c)0

Simplify the following. Express your answer in fraction form.1. 2-4 2. 4-2 3. x-6

4. 3z-2 5. 1 3-2

6. 50

7. 2-5 2∙ 3 8. x3 x∙ -7 9. 3 3 35

10. x 4 x-6

11. x0 12. 1001-1

Match each of the expressions in the squares of the grid below with an equivalent simplified expression from the top. If an equivalent expression is not found among the choices A through D, then choose E (none of these).

12

RECALLLAWS OF NEGATIVE

EXPONENTS

I-Exponent of 0 or 1:

x0=1 or x1=x

II-Negative exponent:

x−1=1x

III-Negative exponent in the denominator:

1−x

=x

Activity 12: MATCH UP

Activity 11: SIMPLY--FY

MODULE IN ALGEBRA A. 1 B. 1x2

C. 9 x2 y3

D. −9 x4

y3E. none of the these

¿ ¿ 4 x−2 4 x0

¿ ¿ ¿¿ −¿¿

¿ ( 100 x27 y35

a4b4 ) ¿ y7 ¿

¿ ( − y

a4b4)¿ 3−2 y−3

x−4

4 ¿

8 x2( x−2

8 ) 3(x2 y2)¿ ¿¿ −12x4

5 ( 5−12 x4 )

Complete the chart below.

13 3 things you found

out2

interestin

Activity 13: 3-2-1 CHART

RECALLLAWS OF NEGATIVE

EXPONENTS

I-Exponent of 0 or 1:

x0=1 or x1=x

II-Negative exponent:

x−1=1x

III-Negative exponent in the denominator:

1−x

=x

MODULE IN ALGEBRA

Allan and Gina were asked to simplify n3

n−4 . Their solutions are shown below together with their explanation. Which of them has the right solution.Allan’s Solution Gina’s Solution

n3

n−4=n3−(−4)=n7

I used the quotient law of exponents in my solution since they have the same base and subtracted the exponents.

n3

n−4=n3

1n−4

=n3n4

1=n7

I expressed the exponent of the denominator as positive integer, then I followed the rules in dividing polynomials.

14

3 things you found

out2

interestin

1 question you still

have

You have learned some concepts of rational algebraic expressions as you performed the previous activities. Now, let us try to use these concepts in a different context.

Allan’s solution is wrong because in the law of negative exponent, variables with negative exponents should be expressed in its reciprocal to become positive before it can be simplified. In Gina’s solution, it is wrong that she expressed n−4 as 1n−4

. Remember that the law of negative exponent takes the form,x−n= 1

xn.

Activity 14: Who is right?

MODULE IN ALGEBRA

Par finished the 15-meter dash within three seconds. Answer the questions below.1. How fast did Par run?2. At this rate, how far can Par run after four seconds? five seconds? six seconds? 3. How many minutes can Par run for 50 meters? 55 meters? 60 meters?

Find the value of each expression below by evaluation.My Expression Value of a Value of b My Solution My Value

a2+b3 2 3 Example:a2+b3=22+33 ¿4+27 ¿31

31

3 4

2 415

Activity 15: SPEEDY PAR

RECALLSpeed is the rate of moving object as it transfers from one point to another. The speed is the ratio between the distance and time traveled by the object. In symbol,

v=dt

Activity 16 My value

What you just did was evaluating the speed that Par run, substituting the value of the time to your speed, you come up

with distance. When you substitute your distance to the formula of the speed, you get the time. This concept of evaluation is the same with evaluating algebraic expressions. Try to evaluate the

following algebraic expressions in the next activity.

MODULE IN ALGEBRA

a−2

b−3

-2 3 Example:a−2

b−3=(−2 )−2

3−3 ¿ 33

(−2 )2 ¿ 274

274

a−2

b−3

3 2

a−1b0 2 3

Match column A to its equivalent simplest fraction in column B.A B520

13

16

Activity 17: CONNECT TO MY MATCH EQUIVALENT

Questions:

1. What have you observed in the solution of the examples?2. How did these examples help you find the value of the expression?3. How did you find the value of the expression?

SIMPLIFYING RATIONAL ALGEBRAIC EXPRESSION

MODULE IN ALGEBRA 812

48

515

68

14

34

12

23

Let’s analyze and examine the following examples.Example 1: Simplify 90

240 by removing a factor equal to 1.Solution:

90240

= 30 ∙330∙8

= 3030 ∙ 38 17

Factoring the rational expression

You might wonder how to answer the last question but the key concept of simplifying rational algebraic expressions is the concept of reducing a fraction to its simplest form.Examine and analyze the following examples.

3030

=1

Questions:

1. How did you find the equivalent fractions in column A?2. Do you think you can apply the same concept in simplifying a rational algebraic

expression?

Factoring the numerator and the denominator

LESSON 1.3

MODULE IN ALGEBRA = 1 ∙ 38 = 38

Example 2:Simplify 3 x2 y4

21x3 y3

Solution 1: 3 x2 y4

21x3 y3= 3 x2 y3∙ y3 x2 y3 ∙7 x

= 3x2 y3

3x2 y3 ∙ y7 x

= y7 x (x ≠ 0)( y ≠ 0)

Solution 2:3 x2 y4

21x3 y3= 3x2 y4

7 x3 y3 ∙3

= y4−3

7 x3−2

= y7 x (x ≠ 0)(y ≠ 0)

Example 3:Simplify x− y

y−x . Solution:x− yy−x

= x− y−(x− y )

= -1 , if y ≠ x18

Factoring the numerator and the denominator

Factoring the rational expression

Removing a factor of 1

Factoring

Applying xm

xn = xm−n or 1

xn−m

Removing a factor of (Multiplicative Property)

The quotient of opposites is -1

Read and understand important notes about Rational Algebraic Expressions in Simplest Forms. Use the mathematical ideas and the examples that will be presented in answering the activities provided.

MODULE IN ALGEBRA

Rational expression that can be written as a quotient of two polynomials p and q, where q ≠ 0, is a rational algebraic expression. The following are examples of rational expressions which are not rational algebraic expression. Examples: 1. The expression 5x

0 is undefined or has no meaning because its denominator is zero. 2. In the expression x+7x−7 the value of x that would make the denominator equal to zero is 7. Thus, if 7 is substituted to the given expression, the result is 7+7

7−7=140

.3. The expression 5

3abbecomes undefined or has no meaning if any or both of the variables a and b are equal to zero. If 0 is substituted to any variable in the denominator, the product of the factors in the denominator becomes zero.

19

NOTE

The quotient of rational expression becomes undefined when the denominator is equal to zero.

RECALL

Property

The quotient of two polynomials that have opposite signs and are additive inverses is -1.

Web- based Booster

Watch the videos for more examples. http://www.youtube.com/watch?v=Mxm-eUBr5eI

http://www.youtube.com/results?search_query=simplifying+rational+algebraic+expressi

on

MODULE IN ALGEBRA If a = 0, then 3ab= 3

(0 ) b=30 .

If b = 0, then 3ab= 3

a (0 )=30 .

If both a and b are equal to zero, then = 3(0)(0)

=30

s.

If the numerator and denominator of a rational algebraic expression have no common factor other than 1, then it is in its simplest form or it is in lowest terms.The Fundamental Property of Rational Expressions states that if p

q, where q ≠ 0,

is a rational expression and k is any polynomial not equal to zero, then pkqk

= pq

. Thus, you can cancel common factors from a rational expression to simplify. This is true becausekk=1.

The fundamental property of rational expressions permits us to write a rational algebraic expression in lowest terms, in which the numerator and denominator have no common factor other than 1.

Match each rational algebraic expression to its equivalent simplified expression from choices A to E. Write the rational expression in the appropriate column. If the equivalent is not among the choices, write it in column 1.A. -1 B. 1 C. a+5 D. 3a E. a

3

a2+6a+5a+1

a3+2a2+a3a2+6a+3

3a2−6 aa−2

a−11−a

(3a+2)(a+1)3a2+5 a+2

3a3−27a(a+3)(a−3)

a3+125a2−25

a−8−a+8

18a2−3a−a+6a

3a−11−3a

3a+11+3a

a2+10a+25a+5

20

Activity 18: Match it Down

MODULE IN ALGEBRA

A B C D E F

In each circle write the steps in simplifying rational algebraic expressions. You can add or delete circles if necessary.

A. Below are different fractions. Use these in answering the questions that follow.

21

Activity 20: Simplify Me!

Activity 19: The Circle Process

Web- based Booster

For more activities, you may check http://www.onlinemathlearning.com/grade7-8-math-worksheets.html

412

15

−525

7−35

315

MODULE IN ALGEBRA

1. Which of the fractions are expressed in simplest forms? Which are not? Why? 2. Simplify the fractions that are not written in simplest forms.

3. Which of the fractions are equivalent? Justify your answer.

4. When do you say that a fraction is in its simplest form?

5. Give five fractions that are written in simplest forms.

B. Identify the rational algebraic expressions that are expressed in simplest forms. Answer the questions that follow.

1.3x−3 y

x− y

2. 5x2

5x+2

3.3m3+m

4. x2−36x−6

5. x2+8 x+16x6−16

6.t+10t−10

22

Your goal in this section is to relate the operations of rational expressions to real life problems, especially rate problems.

MODULE IN ALGEBRA

7. a2+6 a+9a2−6 a+9

8.x−2 y2 y−x

9. 15a6b7

5x4 y5

10.t2−4 t−5

t2−1

Jul-ann can paint the wall in five hours. What part of the wall is painted in three hours?23

Let’s Examine

Work problems are one of the rate-related problems and usually deal with persons or machines working at different rates or speed. The first step in solving these problems involves determining how much of the work an individual or machine can do in a given unit of time called the rate.

In this section, the discussions are introduction to rational algebraic expressions. How much of your initial ideas are found in the discussion? Which ideas are different and need revision? Try to move a little further in this topic through the next activities.

MODULE IN ALGEBRA

Figure 1. The shaded cell represents an hour of Jul-ann’s work.SOLUTION:

The rate of work is the part of the task that is completed in one unit of time. In Jul-ann’s work, there are five units (refer to the figure) and out of that five units, she was able to complete only one unit which is written as 15 basing it from the formula for

rate v= td

.Since Jul-ann can paint in five hours, then in one hour, she can paint 1

5 of the wall.

Her rate of work is 15 of the wall each hour. Therefore, in three hours, she will be able

to paint (3) ( 15)=35of the wall.

You can also solve the problem by using a table. Examine the table below.Rate of work (wall painted per hour) Time worked Work done (wall painted)

15

1 hour 15

15

Next 1 hour 25

15

Another next 1 hour 35

In a weather bulletin, it was announced that a very strong typhoon would hit the country in 3 days. Mr. Par is worried because his house still needs some repairs. If he will not do something, his house might be completely destroyed by the coming typhoon? Mr. Par knows that he needs 7 days to repair his house.

24

Let’s explore 2

Analyzing the situation of Mr. Par, it is only right to conclude that he needed help to repair his house. He needs 7 days to repair his house but within 3 days the typhoon would hit his place that indicates that he will not be able to finish the repair before or on 3 days if he works alone. If Mr. Par needs to finish his house repair within two days, we can suggest that he should get a co-worker who is thrice faster than him.

MODULE IN ALGEBRA

You printed your 40-page reaction paper. You observed that printer A in the internet shop finished printing in two minutes. How long will it take printer A to print 150 pages? How long will it take printer A to print p pages? If printer B can print x pages per minute, how long will it take to print p pages> the rate of each printer is constant.

Let’s take another example that may help you out in answering the succeeding activities.Jay can construct a toy car made of wood in 6 hours while Jose can do the same in 8 hours. How long would it take Jay and Jose to construct a toy car if they will both work together?

First, let’s assign variables.Let x=the number of hours for Jay and Jose to construct the toy.

Next, we answer the question: “How much work is completed by each person per 1 unit of time?”In 1 hour,

25

Questions:1. How did you solve the rate of each printer?2. How did you compute the time of each printer?3. What will happen if the rate of the printer increases?4. How do time and number of pages affect the rate of the printer?

Activity 21: HOW FAST

Printer Pages Time RatePrinter A 40 pages 2 minutes45 pages150 pagesp pagesPrinter B p pages x pages per minute30 pages35 pages40 pages

MODULE IN ALGEBRA Jay completed 16 of the work.

Jose completed 18 of the work.

Then, we combine the number of hours of their work multiplied to the number of hours of working together which is represented by x that will result to 1 toy constructed.( 16 + 1

8 ) x=1

Working further on the equation created,( 16 + 1

8 ) x=1

16

x+ 18

x=1

Get the LCD of 6∧8 : 4 x24

+ 3 x24

=1

Add similar terms :( 7 x24

=1)24Multiply both sidesby24 ¿eliminate24 :7 x=24

Divide both sidesby7¿ get the value of x :7 x7

=247

x=247

∨3.429

Therefore, working together, it takes about 3.4 hours for Jay and Jose to construct a toy car made of wood.

1. Suppose a government office have a hundred employees. Every employee should receive a minimum salary of 18, 000 pesos a month and a maximum salary of 30, 000 pesos amount. Find out the least amount of budget allocation for the employees and the maximum budget allocation.26

Activity 22: TRY THIS

MODULE IN ALGEBRA 2. Elaine and Sam sells banana que for a living. Elaine can sell 20 sticks of banana que a day. If the two was able to sell 65 sticks of banana que on Tuesday, how many sticks was Samuel able to sell? 3. The students of Par School Academy goes to a tree planting project in Mt. Hanku-amo. During the first hour, the students were able to plant 50 trees. How fast should the students take to plant all 500 trees before one o’clock when they started at eight in the morning?4. Glesie noticed that Cathy was having a hard time answering the activity given by their teacher about the movie they have watched. Cathy, who cannot hear, did not totally understand the movie and still in number two question out of the 10 questions. If Glesie could answer a question 6 minutes each, would she be able to help Cathy 10 minutes before their one hour class ends? Justify your answer.5. Mother would like to choose the right fever remover for her son. Brand A can cool down a fever two hours after it’s taken while Brand B works twice the first brand but will cool down fever totally. Mother chooses Brand B. Is her decision correct? Why or why not?

1. Ryan walks 2 km in going to school while Susan rides a jeepney in travelling 6 km. Linda’s rate of travel while riding in a tricycle is four times than that of Ryan’s rate of walking.a. How would you represent Ryan’s rate of walking?How about Susan’s rate of travel while riding in a jeepney?b. How would you represent the time Ryan spends in walking?How about the time Susan spends in riding a jeepney?c. How much time does Ryan spend in walking if he walks at a rate of 5 kph?How about the time that Susan spends in riding a jeepney?d. Suppose they leave their respective house at the same time, who would reach the school first? Explain your answer.2. Working together, Yan and Ry can construct a cabinet in 6 days. If Dante works alone, he can finish constructing the cabinet in 8 days.a. What can you say about the given situation?b. If Yan and Ry work together in constructing the cabinet, what part of the job can they finish in 1 day?c. If Yan works alone, what part of the work can he finish in 1 day?27

Activity 23: RATES

MODULE IN ALGEBRA d. What expression represents the part of the work Ry can finish in 1 day?e. What equation would represent the part of the work Ry can finish in 1 day?f. How would you determine the number of days Ry would take in constructing the cabinet if he works alone? How about Yan?g. If you were them, would you rather work alone or work together? Justify your answer.3. A company that manufactures t-shirts has fixed cost of Php 50, 000 and variable cost of Php 205 per shirt.a. How many shirts must be manufactured to have an average cost of Php 500 per shirt?b. In what instance would the average cost per shirt increase to more than Php 300?c. If you were the manufacturer, what would you do to lower the average cost of each shirt?

28