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Mathematics I Quarter 3: RATIONAL ALGEBRAIC EXPRESSIONS Module 3.3: Applications of Rational Algebraic Expressions EXPLORE Your Understanding Activity 1

Directions: Perform each indicated operation, then express your answer in simplest form. Answer the questions that follow.

1. 33

4

a

c

cb

ba 6.

1n

9

3

5n5

2. 55

93

7

3

xn25

xn12

x6

n10 7.

a2c7

36

6

c7a2

3.

a16

b5

b

a2 33

8.

ba

16

4

ba 22

4.

3

2

4 m2

n

n

m8 9.

6

yx

yx

32

5. 27

5

2

53

yx21

z44

z11

yx7

10.

2x

4x

10x7x

5x 2

2

Assess your understanding of the different mathematics concepts previously studied. This may help you in understanding the different Applications of Rational Algebraic Expressions. Perform each given activity. If you find any difficulty in answering the exercises, seek the assistance of your teacher or peers or refer to modules you have previously gone over.

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11. p

2

p

3 21.

y

2

y2

1

12. 5

4

3

2

r6

r8

r9

r3 22.

2x5

3

x10

2

13. 4

6

5

10

m20

m15

m9

m25 23.

a

1

8a

1

14.

s4s2

s6

2s

s32

2

24.

y2x2

x3

yx

x2

15.

12

30m6

10

25m5

25.

6mm

2m

4m4m

3m22

16.

72

rpr

8

p2r2 2

26.

1m

1

1m

m

17.

y12

2y

8

y2

27.

7

ba

7

ba

18.

20yy

9y

10y2

12y42

2

28.

a4a2

a5

a4a2

a322

19.

2bb

12

6b5b

322

29.

4p2

3

4p

62

20.

16r12

12r4

8r2r3

18r62

30.

1t

1t2

1t

1t2

How did you perform each operation? Which operation did you find difficult to perform? Why? Were you able to express you answers in simplest forms? How?

Activity 2 Directions: Translate each of the following English phrases into algebraic expressions.

1. Twice the difference of two numbers

2. Six times a number increased by three

3. The difference of two numbers divided by twice their sum

4. Five times the sum of a number and 4 divided by their product

5. A fraction whose numerator is 5 less than the denominator

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6. The total cost of a number of folders that cost Php2 each

7. A distance that is 10 m shorter than x meters

8. The cost of one banana if a number of bananas costs Php350

9. The distance (d) travelled by a car divided by its rate (r) of travel when

d = 120 and r = x + 10.

10. A train that travelled 100 km farther than the distance a bus had

travelled

11. The part of the work Edna could finish in 5 hr, if she could finish it in x hours.

12. The cost of a motorcycle that is ten times the cost of a mountain bike

13. The ratio of the length of a rectangle to its width when its length is 7 cm

more than its width.

14. The ratio of the amount of water in a cleaning solution when the amount of solution is 10 cups less than the amount of water.

15. The time it took a bus to travel a distance of 90 km when its rate of

travel is (x + 15) kph.

FIRM UP Your Understanding

Learn About These

The concept of rational algebraic expressions finds its applications in different fields such as agriculture, physics, biology, business, economics, industry, and many others. Many real-life situations in these fields could be modeled and solved by applying the different operations on rational algebraic expressions.

Read and understand important notes about the different Applications of Rational Algebraic Expressions. Use the mathematical ideas and the examples that will be presented here in answering the activity provided.

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Examples: 1. Mang Pedro planted 53

of his piece of land with rice and 21

of the

remaining was planted with corn. What part of his land is planted with rice and corn?

Solution:

Let x = total area of Mang Pedro’s land

x53

= part of the land that is planted with rice

x52

= remaining part of Mang Pedro’s land not planted with

rice

x51

x52

21

= part of the land that is planted with corn

x51

x53

= part of the land that is planted with rice and corn

x51

x53

x54

Answer: The part of the land that is planted with rice and corn is

.x

54

Problem Extension:

Suppose x = 15,000 m2, what is the area of the land that is

planted with rice? How about corn?

Solution:

The part of the land that is planted with rice is .x53

Substituting the value of x gives 15,00053

.2m9,000

Answer: 9,000 m2 is planted with rice.

The part of the land that is planted with corn is .x51

Again,

substituting the value of x gives .2m3,00015,00051

Answer: 3,000 m2 is planted with tomatoes.

2. Alexa left her home at 6:30 AM and took a bus going to the university where she teaches. After travelling for about 20 km, the bus stopped due to engine malfunction. No other buses passed by

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that time so she decided to ride in a jeepney and travelled 12 km more. The bus travelled twice as fast as the jeepney. What expression represents Alexa’s total time of travel from her home to the university?

Solution:

Let x = speed of the jeepney

2x = speed of the bus

tb = time of travel riding in the bus

tj = time of travel riding in a jeepney

The time of travel can be determined by getting the ratio of the distance travelled to the rate of travel. Since Alexa travelled

20 km riding in the bus, her time of travel is given by 2x20

tb .

Likewise, since she travelled 12 km riding a jeepney, her time of

travel is given by x12

t j .

Alexa’s total time of travel then is x12

2x20

.

Activity 3 Directions: Read each situation then answer the questions that follow.

1. A boat travels 15 km upstream and 15 km back. The speed of the current is 10 kph. a. How would you represent the speed of the boat in still water?

How about its speed upstream? downstream?

b. What expression represents the boat’s time of travel upstream?

How about downstream?

c. What expression represents the boat’s total time of travel?

2. Melecio and Brian are asked to paint a room. If Melecio works alone, he can do the job in 6 days. If they work together, they can paint the room in 4 days.

Check Learned Processes or Skills

Apply the processes or skills learned related to the Applications of Rational Algebraic Expressions by performing Activity 3.

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a. What part of the job can Melecio finish in 1 day?

How about the part of the job that Brian can finish in 1 day?

b. If they work together, what expression would represent the part of the job they can finish in 1 day?

3. Grace spent three-fourths of her money for a blouse. Then she spent half of the remaining for a handkerchief. a. How would you represent the amount of money Grace originally had?

b. What expression represents the amount of money Grace spent for a blouse?

How about for the handkerchief? c. If Php25 is left from her money, what expression represents the

amount of money she originally had?

d. If the handkerchief costs Php25, how much did Grace spend for the blouse?

4. Ariel drove a distance of 290 km, part at 70 kph and part at 50 kph.

a. If x represents the distance traveled at 70 kph, how would you represent the distance traveled at 50 kph?

b. The distance (d) traveled is equal to the rate (r) of travel multiplied by the time (t) of travel or d = rt. How would you represent the time of travel at 70 kph?

How about the time of travel at 50 kph?

c. What expression represents the total time spent in traveling 290 km?

5. The time it took a faster runner to run a distance of 80 m is the same as

the time it took a slower runner to run a distance of 60 m. The rate of the faster runner was 1.5 meters per sec (m/s) more than the rate of the slower runner. a. What expression represents the time it took the faster runner to run a

distance of 80 m?

How about the expression that represents the time it took the slower runner to run a distance of 60 m?

b. Suppose the speed of the slower runner is 1.5 m/s, how long did the

faster runner cover the distance of 80 m?

6. A train left the terminal with some passengers. At the first station, one-

fourth of the passengers got off, and fifteen new ones got on. a. How would you represent the number of passengers when the train

left the terminal?

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How about the number of passengers who got off at the first station?

b. Suppose there were 120 passengers in the train when it left the terminal, how many passengers were aboard after leaving the first station? Justify your answer

7. A furniture shop can produce a table for Php1,200 in addition to an initial

investment of Php35,000. a. If a number of tables is to be produced, how would you represent the

total cost of manufacturing the product? b. What expression represents the average cost of each table?

c. If 200 tables are to be manufactured, what would be the cost of

each?

8. The graduating class of Mangaldan National High School went on an educational tour. A portion of their total expenses was shouldered by their municipal mayor and other sponsors. The remaining part amounting to Php42,000 was divided equally by the students who joined the tour. The day before the trip, 100 students decided not to join the tour. This increased the cost by Php10 per student. a. What expression represents the amount each student is supposed to

pay if all of them joined the tour? b. What expression represents the amount each student paid after 100

of them decided not to join? c. Suppose each student paid Php70, how many of them joined the

tour? 9. Andres goes to work by walking 1 km and traveling 8 km more by riding

a tricycle. He observes that the tricycle’s rate of travel is eight times his rate in walking. a. How would you represent Andres’ rate in walking?

How about the rate of travel of the tricycle?

b. How would you represent Andres’ time spent in walking?

How about his time of travel in riding a tricycle?

c. Suppose Andres walks at a rate of 4 kph, how much time does he

spend in walking?

d. How about the time that he spends in riding a tricycle? 10. The time it takes for a bus to travel a distance of 300 km is the same as

the time a car takes to travel a distance of 200 km. The bus travels 25 kph faster than the car. a. How would you represent the rate of travel of the bus?

How about the car’s rate of travel?

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b. How would you represent the bus’ time of travel in terms of the

distance travelled and its rate of travel? How about the car’s time of travel?

DEEPEN Your Understanding

Activity 4

Directions: Answer the following items. Show your complete solutions or explanations.

1. It take 6 hr for Angelo to install an air conditioning unit. If Allan helps him, it would take them 4 hr. a. How would you represent the part of the work that Angelo could

finish in 1 hr?

b. If Allan works alone, how would you represent the part of the work he can finish in 1 hr?

c. How would you represent the part of the work they will finish in 1 hr

if they work together?

d. Suppose Allan can do the same job in 6 hr, do you think they will take more than 4 hr in doing the job if they work together? Justify your answer.

2. Mr. Fernandez, a rice retailer, purchased a number of sacks of rice

over 3 months. In the first month, he bought one-fourth of the total number of sacks of rice. In the second month, he purchased two-thirds of the total number of sacks of rice. In the third month, he bought 10 sacks of rice. a. How would you represent the number of sacks of rice Mr.

Fernandez bought in the first month? How about in the second month?

b. What expression represents the total number of sacks Mr. Fernandez bought?

Think deeper and check your understanding of the Applications of Rational Algebraic Expressions by doing the following activity.

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c. Suppose the total number of sacks of rice bought by Mr. Fernandez is 120. How many sacks of rice did the retailer purchased in the first month? How about in the second month?

d. Suppose you are a rice retailer or somebody who is engaged in

business. Would you purchase a big number of goods and keep it in your storage then sell these when the right time comes? Explain your answer.

3. A shoe factory can produce a pair of shoes for Php700 in addition to an

initial investment of Php50,000. a. If x is the number of pairs of shoes to be produced, what expression

represents the average cost of each pair?

b. Suppose the production cost increases from Php700 to Php800 and the number of pairs to be produced remains the same. How would it affect the average cost of each pair of shoes?

How about if the production cost decreases from Php700 to Php600?

c. Suppose you are the shoe factory owner and you have noticed that

the production cost of each pair of shoes is increasing. What would you do if you wanted the average cost to remain the same?

TRANSFER Your Understanding

Activity 5

1. Visit any factory in your community and take note of the number of workers, number of hours each worker does a particular job, and the number of particular product each worker makes. If possible, ask also the owner of the factory the amount of his investment and the production cost for each product. Out of the information that you could obtain, formulate and solve problems involving rational algebraic expressions. If given the chance, discuss with the factory owner how he could increase his profit. You may use the problems formulated and solved.

Apply your understanding on Rational Algebraic Expressions through the following culminating activities that reflect meaningful and relevant problems/situations.

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2. Find at least 3 situations in real life where rational algebraic expressions are applied. Model each situation by rational algebraic expression and formulate problems out of these situations.

Answers Key Module 3.3: Applications of Rational Algebraic Expressions Activity 1

1. 2b

a 11.

2

3 21.

y2

5

2. 3x5

n4 12.

4

1 22.

2x10

6x2

3. 2

a5 2

13. 27

m100 3

23. a8a

8a22

4. 2

2

m

n 14. 1 24.

)yx(2

x7

5. 4

33

x

zy4 15. 1 25.

)3m()2m(

13m2m22

2

6. 15 16. r

18 26. -1

7. 6 17. 2

y3 27.

7

b2

8. 4(a + b) 18. 3y

)4y(2

28.

2a

1

9. 2

yx 19.

)3b(4

1b

29.

8p2

6p32

10. 1 20. 2r

6

30.

1t

t62

Activity 2 (Any variable could be used.)

1. 2(x – y) 6. 2x 11. x

5

2. 6t + 3 7. x – 10 12. 10x

3. )yx(2

yx

8.

x

350 13.

w

7w

4.

x4

4x5 9.

10x

120

14.

x

10x

5. x

5x 10. x + 100 15.

15x

90

Activity 3 1. a. speed of boat in still water: x

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speed of boat upstream: x – 10 speed of boat downstream: x + 10

b. time of travel upstream: 10x

15

time of travel downstream: 10x

15

c. boat’s total time of travel: 10x

15

+

10x

15

2. a. part of the job Melecio can finish in 1 day: 6

1

part of the job Bryan can finish in 1 day: x

1

b. part of the job Melecio and Bryan can finish in 1 day working together:

x

1

6

1

or 4

1

3. a. amount of money Grace originally had: x

b. amount of money Grace spent for a blouse: x4

3

amount of money Grace spent for the handkerchief: x8

1orx

4

1

2

1

c. 25x8

1x

4

3 d. Php150

4. a. 290 – x

b. time of travel at 70 kph: 70

x time of travel at 50 kph:

50

x290

c. 50

x290

70

x

5. a. time it took the faster runner to run a distance of 80 m: 5.1x

80

time it took the slower runner to run a distance of 60 m: x

60

b. 3

226 sec

6. a. number of passengers when the train left the terminal: x

number of passengers who got off at the first station: x4

1

b. 105

7. a. total cost of manufacturing the product: 1,200x + 35,000

b. average cost of each table: x

000,35x200,1

c. Php1,375

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8. a. x

000,42 b.

100x

000,42

c. 600

9. a. Andres rate of walking: x c. 4

1 hr or 15 min

rate of travel of the tricycle: 8x

b. Andres’ time spent in walking: x

1 d. 15 min

time of travel in riding a tricycle: x8

8

10. a. rate of travel of the bus: x + 25 b. bus’ time of travel: 25x

300

car’s rate of travel: x car’s time of travel: x

200

Activity 4

1. a. 6

1 b.

x

1 c.

6

1 +

x

1 or

4

1 d. No

2. a. 1st month: x4

1 2nd month: x

3

2

b. x4

1+ x

3

2 + 10

c. 1st month: 30 2nd month: 80

3. a. x

000,50x700

b. The average cost would increase. The average cost would decrease. c. Increase the number of shoes to be manufactured