Mathematics 7 Quarter 1 Module 2: Problems Involving Sets

Mathematics Quarter 1 – Module 2: Problems Involving Sets

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CO_Q1_MATHEMATICS 7_Module 2

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Mathematics – Grade 7 Alternative Delivery Mode Quarter 1 – Module 2: Problems Involving Sets First Edition, 2020

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Development Team of the Module

Writers: Joy D. Padernal , Isabelita D. Tenorio, Joven V. Felongco, Josephine G. Tibay Editors: Randy L. Pendilla, Raul Pojas, Mary Jean Nequinto Reviewers: Evelyn C. Frusa PhD, Noemi E. Parcon, Rolex H. Lotilla, Arvin M. Tejada Illustrator: None Layout Artist: Iza May S. Agrazamendez Management Team: Dr. Allan G. Farnazo, CESO IV – Regional Director Gilbert B. Barrera – Chief, CLMD Arturo D. Tingson, Jr. – REPS, LRMS Peter Van C. Ang-ug – REPS, ADM Jade T. Palomar – REPS, Mathematics Belen L. Fajemolin , PhD -CID Chief Evelyn C. Frusa, PhD – EPS – LRMS Bernardita M. Villano – ADM Coordinator

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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-test 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 Teachers are also provided to the 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. 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

This module was designed and written with you in mind. It is here to help you master your skills in solving mathematical problems involving sets. The scope of this module permits it to be used in many different learning situations. The language used recognizes the diverse vocabulary level of students. 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.

The module is all about Solving Problems Involving Sets.

After going through this module, you are expected to:

1. solve problems involving sets using Venn diagram;
2. apply set operations to solve a variety of word problems.

What I Know

Read each question below. You may draw a Venn diagram to help you find the answer. Select your answer from the choices lettered A to D and write the letter of your choice on a separate sheet of paper.

For items 1 – 2.

Let A = { 1, 2, 3, 4, 5, 6 } , B = { 2, 4, 5, 6 }

1. Find A U B.

   • A. { 1, 2, 3, 4, 5, 6 }
   • B. { 2, 4, 5, 6 }
   • C. { 2, 4, 6}
   • D. { 1, 3, 5 }

2. Find A ∩ B.

   • A. { 1, 2, 3, 4, 5, 6 }
   • B. { 2, 4, 5, 6 }
   • C. { 2, 4, 6}
   • D. { 1, 3, 5 }

For items 3 – 4. Given the Venn Diagram below.

[chart: Venn diagram; given for items 3 – 4] Labels: U (universal set), S (left circle), M (right circle). Values: 10 (in S only), 15 (in the overlap of S and M), 5 (in M only).

3. Find M U S .

   • A. { 5, 10, 15 }
   • B. { 10, 15 }
   • C. { 5, 10 }
   • D. { 15 }

4. Find M ∩ S.

   • A. { 5, 10, 15 }
   • B. { 10, 15 }
   • C. { 5, 10 }
   • D. { 15 }

5. In the Venn diagram below, which of the following is represented by the shaded region?

   
   

[chart: Venn diagram; item 5] Labels: U, X (left circle), Y (right circle). Values are not labeled on the chart. [transcriber’s note: the shaded region covers both circles X and Y]

• A. X U Y
• B. X ∩ Y
• C. X ‘
• D. all of the above

6. Which of the following is represented by the Venn diagram below?

[chart: Venn diagram; item 6] Labels: U, A (left circle), B (right circle). Values are not labeled on the chart. [transcriber’s note: the part of circle A that does not overlap circle B is shaded black]

Uu [transcriber’s note: this line is the leftover label “U” and “u” from the item 6 diagram, printed at the top of this page]

• A. A – B
• B. B – A
• C. A U B
• D. A ∩ B

For items 7 – 10. Refer to the diagram below.

[chart: Venn diagram; items 7 – 10] Labels: U, A (left circle), B (right circle). Values: 2, 4, 1 (top row, A only, both, B only); 3, 7, 6 (middle row, A only, both, B only); 5 (in A only, lower part); 8 (in U, outside both circles).

7. What is A ∩ B?

   • A. { 4,7 }
   • B. { 1, 6 }
   • C. { 2, 3, 5 }
   • D. { 1, 4, 6, 7, 8}

8. What is A – B?

   • A. { 4,7 }
   • B. { 1, 6 }
   • C. { 2, 3, 5 }
   • D. { 1, 4, 6, 7}

9. What is B – A?

   • A. { 4,7 }
   • B. { 1, 6 }
   • C. { 2, 3, 5 }
   • D. { 1, 4, 6, 7}

10. What is A U B?

    • A. { 1,2,3,4,5,6,8 }
    • B. { 1,2,3,4,5 ,6,7 }
    • C. { 2, 3,4, 5,7 }
    • D. { 1, 4, 6, 7,8}

For items 11 – 15. Refer to the problem below.

A survey was conducted among 50 people on the ice cream flavor they liked. It was found out that 28 liked chocolate, 15 liked mango, and 20 liked strawberry. Furthermore, 8 liked chocolate and strawberry, and 7 liked chocolate and mango, 11 liked mango and strawberry and 4 liked all the 3 flavors.

11. How many people liked Chocolate only?

    • A. 28
    • B. 17
    • C. 8
    • D. 7

12. How many people liked mango only?

    • A. 15
    • B. 14
    • C. 7
    • D. 1

13. How many people liked strawberry only?

    • A. 20
    • B. 8
    • C. 5
    • D. 4

14. How many people liked all the three flavors?

    • A. 3
    • B. 4
    • C. 7
    • D. 9

15. How many people did not like the three flavors at all?

    • A. 3
    • B. 4
    • C. 7
    • D. 9

Lesson 1 Problems Involving Sets

What’s In

Try solving the following problem:

In a group of 120 students, 68 had ridden a bus, 78 had ridden the LRT, 33 had ridden a jeep, while 40 had ridden both the bus and the LRT , 20 had ridden the bus and the jeep,19 had ridden the LRT and the jeep and 15 had ridden the bus, the LRT and the jeep.

• a. How many had ridden the bus only?
• b. How many had ridden the LRT only?
• c. How many had ridden the jeep only?
• d. How many did not ride on any of the three modes of transportation?

Notes to the Learner

Find the solution using any method.

What’s New

Venn diagram is a principal way of showing sets diagrammatically. This method consists primarily of entering the elements of a set into a circle or circles. It can be used to solve word problems involving union and intersection of sets.

In solving set operations using the Venn diagram, the following are the steps to be followed:

Step 1. Determine what is given and what are being asked. Step 2. Illustrate using the Venn diagram. Step 3. Determine what operations to be used. Step 4. Use the operations. Step 5. Answer the questions being asked.

Here are some worked out examples:

Example 1.

Let A and B be two finite sets such that n(A) = 20, n(B) = 28 and n(AUB)=36. Find n(A∩B).

Solution:

Step 1. Determine what is given and what are being asked.

Given :

• n(A) = 20,
• n(B) = 28
• n( A U B ) = 36.

Asked: Find n( A∩ B).

Step 2. Illustrate using the Venn diagram if possible. The Venn diagram is shown below

[chart: Venn diagram; Example 1] Labels: U (universal set), A (left circle), A∩B (overlap), B (right circle). Values are not labeled on the chart.

Step 3. Determine what operations to be used.

Using the formula n( A U B ) = n(A) + n(B) – n( A ∩ B). then ,

n( A ∩ B ) = n(A) + n( B) – n( A U B )

n( A ∩ B ) = 20 + 28 – 36

n( A ∩ B ) = 48 – 36

n( A ∩ B) = 12

Step 4. Use the operations.

[chart: Venn diagram; unnumbered. Labels: U (universal set). Values: 8, 12, 16. The 8 is in the left circle, the 12 in the overlap, and the 16 in the right circle.]

Step 5. Answer the questions being asked.

Find n( A∩ B) = 12

Example.

In a Junior High School, 200 students were randomly selected. 140 liked tea, 120 liked coffee and 80 liked both tea and coffee.

a. How many students liked only tea? b. How many students liked only coffee? c. How many students liked neither tea or coffee?

Solution:

Step 1. Determine what is given and what are being asked.

Given:

200 Junior High School students who were randomly selected

140 students who liked tea

120 students who liked coffee

80 students who liked both tea and coffee

Asked:

a. How many students liked only tea? b. How many students liked only coffee? c. How many students liked neither tea nor coffee?

Step 2. Illustrate using the Venn diagram.

Let T = set of students who like only tea C = set of students who liked only coffee X = set of students who liked neither tea nor coffee

The Venn diagram is shown below

[chart: Venn diagram; unnumbered. Labels: U (universal set), T, C, X. Value: 80 (overlap of T and C).]

Step 3. Determine what operations to be used.

(1) To obtain T,

T = 140 – 80 students who liked tea minus students who liked both tea and coffee

T = 60 set of students who liked only tea

(2) To obtain C,

C = 120 – 80 students who liked coffee minus students who liked both tea and coffee

C = 40 set of students who liked coffee only

(3) To obtain X,

X = 200 – ( T + C + 80) total number of students who were randomly selected minus the sums of T , C and 80

X = 200 – ( 60 + 40 + 80) by substitution

X = 200 – 180 by simplifying

X = 20 set of students who liked neither tea nor coffee

Step 4. Use the operations.

The number of elements in each region is shown below:

[chart: Venn diagram; unnumbered. Labels: U (universal set). Values: 60 (left circle only), 80 (overlap), 40 (right circle only), 20 (outside both circles).]

Step 5. Answer the questions being asked.

a. How many students liked only tea? 60 students b. How many students liked only coffee? 40 students e. How many students liked neither tea nor coffee? 20 students

Example 3.

A group of 25 high school students were asked whether they used either Facebook or Twitter or both. Fifteen (15) of these students used Facebook, and twelve (12) used Twitter.

a. How many students used Facebook only? b. How many students used Twitter only? c. How many students used both Social networking sites?

Solution:

Step 1. Determine what is given and what are being asked.

Given:

25 high school students who were asked whether they use either Facebook or Twitter or both

15 students who used Facebook

12 students who used Twitter

Asked:

a. How many used Facebook only? b. How many used Twitter only? c. How many used both Social networking sites?

Step 2. Illustrate using the Venn diagram.

Let F = set of students who used Facebook only T = set of students who used Twitter only B = set of students who used both social networking sites

The Venn diagram is shown below

[chart: Venn diagram; unnumbered. Labels: U (universal set), F, B, T.]

Step 3. Determine what operations to be used.

(1) To obtain B,

B = (F + T) – 25 sums of F and T minus 25 (total number of high school students)

B = (15 + 12) – 25 by substitution

B = 27 – 25 by simplifying

B = 2 set of students who used both social networking sites

(2) To obtain F,

F = 15 – B students who used Facebook minus B

F = 15 – 2 by substitution

F = 13 set of students who used Facebook only

(3) To obtain T,

T = 12 – B set of students who used Twitter only minus B

T = 12 – 2 by substitution

T = 10 set of students who used Facebook only

Step 4. Use the operations.

The number of elements in each region is shown below:

[chart: Venn diagram; unnumbered. Labels: U (universal set). Values: 13 (left circle only), 2 (overlap), 10 (right circle only).]

Step 5. Answer the questions being asked.

a. How many used Facebook only? 13 students b. How many used Twitter only? 10 students c. How many used both Social networking sites? 2 students

Example 2.

A group of 50 students went for a tour in South Cotabato province. Out of 50 students, 24 joined the trip in Lake Sebu for a zipline experience, 18 went to the flower farm in Tupi, 20 went to Si – ok falls in Koronadal City, 12 joined the trip to Lake Sebu and Tupi, 15 went to Tupi and Si-ok falls and 11 made a trip to Lake Sebu and Si-ok falls and 10 visited the three tourists spots.

a. How many of the students went to Lake Sebu only? b. How many of the students went to Tupi only? c. How many joined the Si-ok trip in Koronadal City only? d. How many did not go to any of the tourist spots?

Solution:

Step 1. Determine what is given and what are being asked.

Given:

50 students went for a tour 24 students who visited Lake Sebu 18 students who went to Tupi 20 students who went to Koronadal City 12 students who joined the trip to Lake Sebu and Tupi 15 students who went to Tupi and Koronadal City 11 students who went to Lake Sebu and Koronadal City 10 students who visited the three tourist spots

Asked:

a. How many of the students went to Lake Sebu only?

b. How many of the students went to Tupi only? c. How many joined the Si-ok trip in Koronadal City only? d. How many did not go to any of the tourist spots?

Step 2. Illustrate using the Venn diagram.

We will let,

LKT = number of students who visited the THREE tourist spots L = number of students who visited Lake Sebu only T = number of students who went to Tupi only K = number of students who went to Koronadal City only LT = number of students who joined the trip to Lake Sebu and Tupi only TK = number of students who went to Tupi and Koronadal City only LK = number of students who went to Lake Sebu and Koronadal City only N = number of students who DID NOT see any of the THREE tourist spots

[chart: Venn diagram; unnumbered. Three overlapping circles. Labels: U (universal set), K, LK, L, LT, LKT, TK, LT, T, N.]

Step 3: Determine what operations to be used.

(1) To obtain LKT,

LKT = 10 students who visited the THREE tourist spots

(2) To obtain TK,

TK = 15 – LKT students who went to Tupi and Koronadal Cityminus LKT

TK = 15 – 10 by substitution

TK = 5 students who went to Tupi and Koronadal City only

(3) To obtain LT ,

LT = 12 – LKT students who went to Lake Sebu and Tupi minus LKT

LT = 12 – 10 by substitution

LT = 2 students who joined the trip to Lake Sebu and Tupi only

(4) To obtain LK,

LK = 11 – LKT students who went to Lake Sebu and Koronadal City minus LKT

LK = 11 – 10 by substitution

LK = 1 students who went to Lake Sebu and Koronadal City only

( 5) To obtain L ,

L = 24 – (LKT + LT + LK) students who joined the trip to Lake Sebu minus the sums of LKT, LT and LK)

L = 24 – (10 + 2 + 1) by substitution

L = 24 – 13 by simplying

L = 11 students who visited Lake Sebu only

(6) To obtain K,

K = 20–(LKT+LK+TK) students who went to Koronadal City minus the sums of LKT,LK and TK

K = 20 – (10 + 1 + 5) by substitution

K = 20 – 16 by smplifying

K = 4 students who went to Koronadal City only

(7) To obtain T,

T = 18 – (LKT + TK + LT) students who went to Tupi minus the sums of LKT + TK + LT

T = 18 – (10 + 5 + 2) by substitution

T = 18 – 17 by simplifying

T = 1 students who went Tupi only

(8) To obtain N,

N = 50 – (LKT + TK + LT + LK + L + K + T) total number of students went for a tour minus the sums of LKT, TK,LT,LK,L,K and T

N = 50 – (10 + 5 + 2 + 1 + 11+ 4 + 1) by substitution

N = 50 – 34 by simplifying

N = 16 students who DID NOT visited any of the three spots

Step 4. Use the operations.

Step 5. Answer the questions being asked.

a. How many of the students went to Lake Sebu only? 11 students b. How many of the students went to Tupi only? 1 student c. How many joined the Si-ok trip in Koronadal City only? 4 students d. How many did not go to any of the tourist spots? 16 students

What is It

In a group of 120 students, 68 had ridden a bus, 78 had ridden the LRT, 33 had ridden a jeep, while 40 had ridden both the bus and the LRT, 20 had ridden the bus and the jeep,19 had ridden the LRT and the jeep and 15 had ridden the bus, the LRT and the jeep.

a. How many had ridden the bus only? b. How many had ridden the LRT only? c. How many had ridden the jeep only? d. How many did not ride on any of the three modes of transportation?

[chart: Venn diagram; unnumbered. Three overlapping circles. Labels: U (universal set), K, L, T. Values: 4, 1, 11, 10, 5, 2, 1, 16. The 4 is in the K circle only, the 11 in the L circle only, the 1 at the bottom of the T circle only, and the 16 outside all circles. The 1 at the top is in the K and L overlap, the 10 at the center is in all three, the 5 is in the K and T overlap, and the 2 is in the L and T overlap.]

Solution:

[chart: Venn diagram; Solution (no number printed), with a universal set labeled U]

• U (universal set label, top left)
• LRT (circle label)
• Bus (circle label)
• Jeep (circle label)
• 28 (LRT only)
• 25 (LRT and Bus)
• 23 (Bus only)
• 15 (LRT, Bus and Jeep)
• 4 (LRT and Jeep)
• 5 (Bus and Jeep)
• 9 (Jeep only)
• 11 (outside the three circles)

Can you explain the numbers?

What’s More

Do the following. Represent the sets and draw a Venn Diagram.

1. Among the 70 residents in Barangay General P. Santos, 53 liked eating in Restaurant A while 42 liked eating in Restaurant B. How many liked eating both in Restaurant A and Restaurant B? In Restaurant A only? In Restaurant B only?

   
   

2. The following diagram shows how all the Grade Seven students of Koronadal National Comprehensive High School go to school.

   
   

[chart: Venn diagram; the transport diagram of Grade Seven students (no number printed)]

• Walking: 100 (outside the circles)
• Tricycle (circle label): 76
• Car (circle label)
• Motorcycle (circle label)
• 19 (Car and Tricycle)
• 15 (Car, Tricycle and Motorcycle)
• 17 (Tricycle and Motorcycle)
• 55 (Car only)
• 20 (Car and Motorcycle)
• 67 (Motorcycle only)

a. How many students ride in a car, tricycle and the motorcycle in going to school? b. How many students ride both in a car and a tricycle? c. How many students ride both in a car and the motorcycle? d. How many students ride both in a tricycle and the motorcycle? e. How many students go to school in a car only? Tricycle only? in the motorcycle only? walking? f. How many Grade Seven students of Koronadal National Comprehensive High School are there in all?

What I Have Learned

In solving problems involving sets, application of the knowledge in sets and the use of Venn diagram and sets operations are necessary.

Here are the steps to follow in solving problems involving sets.

1. Determine what is given and what is being asked.
2. Illustrate using the Venn diagram.
3. Determine what operations to be used.
4. Use the operations.
5. Answer the questions being asked.

What I Can Do

Think of a scenario in real life situation where problems involving sets is shown.

Here is an example:

This is a diagram which shows the relationship between solids and liquids.

[chart: Venn diagram; example diagram showing the relationship between solids and liquids (no number printed)]

• SOLID (circle label)
• LIQUID (circle label)
• Definite Shape (solid only)
• Does NOT take shape of its container (solid only)
• Not a definite shape (liquid only)
• take the shape of its container (liquid only)
• takes up space (overlap)
• Made of molecules (overlap)
• definite size (overlap)

Assessment

Solve the following problems using Venn diagram.

1. The 150 Grade 7 students participated in school’s Math Olympics. 64 students registered in Math Trail, 78 students registered in Amazing Race and 28 students registered in both Math Trail and Amazing Race. How many students registered in Math Trail only and in Amazing Race only? Are there students who did not registered in both events, how many?

   
   

2. The Hair Company offers two brands of shampoo: brand H and brand S. 100 people were surveyed and their brand preferences are as follows: 50 people liked brand H and 35 people liked brand S.
   What is the maximum number of people who liked both H and S?
   What is the minimum number of people who liked both H and S?
   What is the minimum number of people who do not like any of the brand?

   
   

3. In a group of students, 30 played chess, 19 played volleyball, 25 played basketball, 14 played both volleyball and chess, 8 played both basketball and volleyball, 15 played both basketball and chess and 5 played both three events. How many played chess only, basketball only and volleyball only? How many students are there in all?

   
   

4. A group of 300 incoming Grade 7 students were given questionnaires to find out who like Online classes as belonged to Group A, Offline classes as belonged to Group B and physical but no-contact policy as belonged to Group C. It was found out that 138 students belonged to A, 150 students belonged to B, 95 students belonged to C, 58 students belonged to both A and B, 47 students belonged to both A and C, 42 students belonged to both B and C and 25 students belonged to the three options. How many students belonged to A only, to B only, to C only and how many students are still undecided?

   
   

5. There are 35 students in art class and 57 students in dance class. Find The number of students
   a. who are either in art class or in dance class.
   b. when two classes meet at the same hour.
   c. when two classes meet at different hours.

   
   

Additional Activities

Answer the question based on the following Venn Diagram.

For items 1 – 4, Refer to the diagram below.

[chart: Venn diagram; likes indoor and outdoor games (no number printed)]

• Likes indoor games (circle label): 28 (indoor only)
• 11 (indoor and outdoor)
• Likes outdoor games (circle label): 56 (outdoor only)

1. How many students like to play indoor games only?
2. How many students like to play outdoor games only and don’t like to play indoor?
3. How many students don’t like both indoor or outdoor games?
4. Do more students like to play indoor or outdoor games?

For items 5 – 6, refer to the diagram below.

[chart: Venn diagram; Math, English and Science survey (no number printed)]

• Math (circle label): 24 (Math only)
• 12 (Math and English)
• English (circle label): 60 (English only)
• 14 (Math and Science)
• 14 (all three subjects, center)
• 8 (English and Science)
• Science (circle label): 22 (Science only)

5. Find how may had taken
   a. Math only?
   b. English only?
   c. Science only?
   d. all three subjects?
6. How many students had been considered in a survey?

[transcriber’s note: this page is printed rotated 180 degrees; it has been read upright and its three columns are given in reading order]

Answer Key

What I Know

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

What’s More

1. 25 liked eating in Both Restaurant A and Restaurant B
   28 liked eating in Restaurant A only
   17 liked eating in Restaurant B only

   
   

2. a. 15 students ride in a car, tricycle and motorcycle
   b. 34 students ride in both car and tricycle
   c. 35 students ride both in car and in motorcycle
   d. 32 students ride both in tricycle and in motorcycle
   e. 55 students ride in car only
   67 students ride in motorcycle only
   76 students ride in tricycle only
   100 students preferred walking
   f. there are 369 students

   
   

Assessment

1. 36 students registered in Math Trail only; 50 students registered in Amazing Race only; 36 students did not register for an event.

   
   

2. 85 is the maximum number of people who liked H & S; None or 0 for the minimum number; 15 is the minimum number of people who did not like H & S.

   
   

3. 6 students played chess only; 7 students played basketball only; 2 students played volleyball only; There are 42 students in all.

   
   

4. 58 students who belonged to A only; 75 students who belonged to B only; 31 students belonged to C only; 39 students who are still undecided.

   
   

5. a. 22 students
   b. 70 students
   c. 92 students

   
   

References

Amid, D.M., Tibulan, M.V. ( 2017), Mathematics for Grade 7 – A Spiral Approach Explanations, Examples, Exercises, 16-18.

Crisostomo, R.M.. de Sagun, P. C., Padua, A.L.( 2013 ) , K-12 Developmental Math, Our World of MATH 7, 11-12.

Diagram showing the relationship between Solids and Liquids, pinterest.ph/pin/50876670760320688/.

Mathematics 7 Learners’ Material ( 1st ed.). (2013), 13 – 17.

Oronce, O. A., Mendoza, M.O., ( 2015) , K-12 Curriculum Aligned Seamless, E-MATH 7, Worktext in Mathematics, 14

Torio, V. A. G.,Paz, J.M., Vargas, E.A., Angcos, A.J., Matalog, E. ,( 2010) , K-12 Curriculum Based SMART IN MATH 7, 13-17.

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