1. Mathematics I
Quarter 3: RATIONAL ALGEBRAIC EXPRESSIONS
Module 3.3: Applications of Rational Algebraic Expressions
EXPLORE Your Understanding
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.
Activity 1
Directions: Perform each indicated operation, then express your answer in simplest
form. Answer the questions that follow.
a 4b c 5n  5 9
1.   6.  
b3c a3 3 n 1
10n3 12n3 x 9  2a  7c  36
2.   7.  
6x 7 25n5 x 5 6  7c  2a
 5b 3 
3
 2a  a 2  b 2 16
3.   
 16a  
 8.  
 b    4 ab
x  y  
3
 8m  n 
2 2
3
4.  4   
 9. 
 n  2m  xy 6
7x 3 y 5 44z 5 x5 x2  4
5.   10. 2  
11z 2 21x 7 y 2 x  7x  10 x  2
1

2. 3 2 1 2
11.   21.  
p p 2y y
3r 2 8r 4 2 3
12. 3  5  22.  2 
9r 6r 10x 5x
25m10 15m6 1 1
13.   23.  
9m5 20m 4 a8 a
3s 6s 2 2x 3x
14.  2  24.  
s  2 2s  4s x  y 2x  2y
5m  25 6m  30 m3 m2
15.    2 
25. m  4m  4 m  m  6
2
10 12
2r  2p r 2  rp m 1
16.   26.  
8 72 m 1 m 1
2y y2 ab ab
17.   27.  
8 12y 7 7
4y  12 y2  9 3a 5a
18.   28.  2 
2y  10 y 2  y  20 2a  4a 2a  4a
2
3  12 6 3
19.  2  29.  
b  5b  6 b  b  2
2
p  4 2p  4
2
6r  18 4r  12 2t  1 2t  1
20. 2   30.  
3r  2r  8 12r  16 t 1 t 1
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
2

3. 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
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.
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.
3

4. 3 1
Examples: 1. Mang Pedro planted of his piece of land with rice and of the
5 2
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
3
x = part of the land that is planted with rice
5
2
x = remaining part of Mang Pedro’s land not planted with
5
rice
1 2  1
 x   x = part of the land that is planted with corn
2 5  5
3 1
x  x = part of the land that is planted with rice and corn
5 5
3 1 4
x x x
5 5 5
Answer: The part of the land that is planted with rice and corn is
4
x.
5
Problem Extension:
Suppose x = 15,000 m2, what is the area of the land that is
planted with rice? How about corn?
Solution:
3
The part of the land that is planted with rice is x.
5
3
Substituting the value of x gives 15,000   9,000 m 2 .
5
Answer: 9,000 m2 is planted with rice.
1
The part of the land that is planted with corn is x. Again,
5
1
substituting the value of x gives 15,000   3,000 m 2 .
5
2
Answer: 3,000 m 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
4

5. 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
20 km riding in the bus, her time of travel is given by tb  .
2x
Likewise, since she travelled 12 km riding a jeepney, her time of
12
travel is given by t j  .
x
20 12
Alexa’s total time of travel then is  .
2x x
Check Learned Processes or Skills
Apply the processes or skills learned
related to the Applications of Rational
Algebraic Expressions by performing
Activity 3.
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.
5

6. 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?
6

7. 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?
7

8. 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
Think deeper and check your
understanding of the Applications of
Rational Algebraic Expressions by doing
the following activity.
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?
8

9. 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
Apply your understanding on Rational
Algebraic Expressions through the following
culminating activities that reflect meaningful
and relevant problems/situations.
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.
9

10. 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
a 3 5
1. 2 11. 21.
b 2 2y
4n 1 2x  6
2. 12. 22.
5x 3 4 10x 2
5a 2 100m3 2a  8
3. 13. 23. 2
2 27 a  8a
n2 7x
4. 14. 1 24.
m2 2( x  y )
4y 3 z3 2m2  2m  13
5. 15. 1 25.
x4 (m  2)2 (m  3)
18
6. 15 16. 26. -1
r
 3y  2b
7. 6 17. 27.
2 7
2( y  4) 1
8. 4(a + b) 18. 28. 
y3 a2
xy b 1  3p  6
9. 19.  29.
2 4(b  3) 2p 2  8
6  6t
10. 1 20. 30. 2
r2 t 1
Activity 2 (Any variable could be used.)
5
1. 2(x – y) 6. 2x 11.
x
2. 6t + 3 7. x – 10 12. 10x
xy 350 w 7
3. 8. 13.
2( x  y ) x w
 x  4 120 x  10
4. 5  9. 14.
 4x  x  10 x
x 5 90
5. 10. x + 100 15.
x x  15
Activity 3
1. a. speed of boat in still water: x
10

11. speed of boat upstream: x – 10
speed of boat downstream: x + 10
15
b. time of travel upstream:
x  10
15
time of travel downstream:
x  10
15 15
c. boat’s total time of travel: +
x  10 x  10
1
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
b. part of the job Melecio and Bryan can finish in 1 day working together:
1 1

6 x
1
or
4
3. a. amount of money Grace originally had: x
3
b. amount of money Grace spent for a blouse: x
4
1 1  1
amount of money Grace spent for the handkerchief:  x  or x
24  8
3 1
c. x  x  25 d. Php150
4 8
4. a. 290 – x
x 290  x
b. time of travel at 70 kph: time of travel at 50 kph:
70 50
x 290  x
c. 
70 50
80
5. a. time it took the faster runner to run a distance of 80 m:
x  1 .5
60
time it took the slower runner to run a distance of 60 m:
x
2
b. 26 sec
3
6. a. number of passengers when the train left the terminal: x
1
number of passengers who got off at the first station: x
4
b. 105
7. a. total cost of manufacturing the product: 1,200x + 35,000
1200x  35,000
,
b. average cost of each table:
x
c. Php1,375
11

12. 42,000 42,000
8. a. b. c. 600
x x  100
1
9. a. Andres rate of walking: x c. hr or 15 min
4
rate of travel of the tricycle: 8x
1
b. Andres’ time spent in walking: d. 15 min
x
8
time of travel in riding a tricycle:
8x
300
10. a. rate of travel of the bus: x + 25 b. bus’ time of travel:
x  25
200
car’s rate of travel: x car’s time of travel:
x
Activity 4
1 1 1 1 1
1. a. b. c. + or d. No
6 x 6 x 4
1 2
2. a. 1st month: x 2nd month: x
4 3
1 2
b. x + x + 10
4 3
c. 1st month: 30 2nd month: 80
700x  50,000
3. a.
x
b. The average cost would increase.
The average cost would decrease.
c. Increase the number of shoes to be manufactured
12