math for smart kids gr.6

Math for Smart Kids6

Math for Smart Kids Grade 6Textbook

Philippine Copyright 2010 by DIWA LEARNING SYSTEMS INCAll rights reserved. Printed in the Philippines

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ISBN 978-971-46-0125-3

The Editorial BoardAuthorsDr. Estrella P. Mercado finished her PhD in Educational Management (with honor) and MA in Education at Manuel L. Quezon University. She also holds an MEd in Special Education degree and a BS in Elementary Education degree from the Philippine Normal University (PNU). She has been a classroom teacher, an Education supervisor, and an assistant chief of the Elementary Division of the Department of Education, Culture and Sports (DECS-NCR). She was awarded as Outstanding Female Educator in 1998 by the Filipino Chinese Women Federation. She presently heads the Special Education Department at PNU.

Angelo D. Uy is currently pursuing his master’s degree in Mathematics Education at PNU, where he also obtained his BS in Mathematics for Teachers. He was also a trainer in different Math competitions and a participant in various seminar-workshops sponsored by the Mathematics Teachers Association of the Philippines. He taught at Hotchkiss Learning Center in Surigao del Sur and at the De La Salle Santiago Zobel School in Ayala Alabang. He is presently a member of the faculty of Jacobo Z. Gonzales Memorial National High School in Biñan, Laguna.

ConsultantMa. Portia Y. Dimabuyu holds an MA in Education degree, major in Mathematics, and a BS Education degree, major in Mathematics, from the University of the Philippines-Diliman. She was a recipient of the Excellence in Mathematics Teaching Award in 2007 from the UP College of Education. She is presently an Assistant Professor at the Mathematics Department of UP Integrated School and a lecturer of undergraduate and graduate courses in teaching Mathematics at the UP College of Education.

ReviewerReina M. Rama has a bachelor’s degree in Mathematics from Silliman University and is currently pursuing her master’s degree in Mathematics from Ateneo de Manila University. Before teaching full time, she was a researcher/teacher-trainer at the University of the Philippines-National Institute of Science and Mathematics Education Development (UP-NISMED). She taught Mathematics at Colegio de San Lorenzo and Miriam College. She is the Subject Coordinator for Mathematics Area of Miriam College-High School Unit.

Preface

Math for Smart Kids is a series of textbooks in Mathematics for grade school,

which is designed to help pupils develop appreciation and love for mathematics.

This series also aims to help the learners acquire the skills they need to become

computationally literate.

The lessons in each textbook present mathematics concepts and principles that

are anchored on the competencies prescribed by the Department of Education. Each

lesson starts with Let’s Do Math, where mathematics concepts and principles are

introduced through problems, stories, games, or puzzles. This section is followed by

Let’s Look Back, which lists questions that will help the pupils to think critically on

what has been introduced in the lesson and allow them to discover things on their

own. For easy recall of important points or concepts taken up in a lesson, the section

Let’s Remember Our Learning has been included. Multilevel exercises are provided in

Let’s Practice and Let’s Test Our Learning that will assess how much the pupils have

learned from the lesson. The exercises will also determine if the pupils are ready to

learn new mathematics skills. The development of the multiple intelligences of an

individual is reflected in the different activities that the pupils will perform—from

concrete to semi-concrete, and from semi-abstract to abstract kind of learning.

Situations and real-life problems are provided in Let’s Look Forward to give the pupils

opportunities to apply what they have learned to their daily life experiences.

This series of textbooks gives the learners the opportunity to explore and enjoy

Mathematics. Let’s have fun learning together!

The Authors

Table of Contents

Unit 1 Whole Numbers, Number Theory, and Fractions

Chapter 1 Review of Whole NumbersLesson 1 Properties of Operations………………………………………… ......................2 2 Exponents ...........................................................................................6 3 Order of Operations ............................................................................9 4 Problem Solving Involving Series of Operations ................................. 12

Chapter 2 Number TheoryLesson 1 Divisibility ........................................................................................ 19 2 Prime Factorization ............................................................................ 25 3 Greatest Common Factor ................................................................... 31 4 Least Common Multiple .................................................................... 36

Chapter 3 Fractions Lesson 1 Equivalent Fractions .......................................................................... 41 2 Reducing Fractions to Lowest Terms ................................................. 47 3 Changing Mixed Numbers to Improper Fractions, and Vice Versa ...... 51 4 Fractions Close to 0, 1

2 , and 1 ........................................................... 55 5 Comparing Fractions ......................................................................... 59 6 Ordering Fractions ............................................................................ 65

Chapter 4 Operations on Fractions Lesson 1 Addition and Subtraction of Similar Fractions and Mixed Numbers .......................................................................... 69 2 Addition and Subtraction of Dissimilar Fractions and Mixed Numbers .......................................................................... 75 3 Mental Addition and Subtraction of Fractions .................................... 82 4 Multiplication of Fractions ................................................................ 86 5 Division of Fractions ......................................................................... 93

Unit 2 Decimals, Ratio, Proportion, and Percent

Chapter 5 Decimals Lesson 1 Place Values of Decimals ................................................................. 102 2 Changing Fractions to Decimals, and Vice Versa .............................. 107 3 Comparing and Ordering Decimals .................................................. 111 4 Rounding off Decimals ................................................................... 117

Chapter 6 Operations on Decimals Lesson 1 Addition and Subtraction of Decimals .............................................. 121 2 Mental Addition of Decimals ........................................................... 127 3 Problem Solving Involving Addition and Subtraction of Decimals ..................................................................................... 131 4 Multiplication of Decimals............................................................... 136 5 Estimation of Decimal Products ....................................................... 141 6 Mental Multiplication of Decimals ................................................... 145 7 Division of Decimals ....................................................................... 148 8 Estimation of Decimal Quotients ..................................................... 154 9 Mental Division of Decimals............................................................ 157 10 Problem Solving Involving Two or More Operations on Decimals .................................................................................... 162

Chapter 7 Ratio and ProportionLesson 1 Ratio and Proportion ....................................................................... 167 2 Direct, Inverse, and Partitive Proportions ........................................ 173

Chapter 8 Percent Lesson 1 Percents, Fractions, and Decimals .................................................... 181 2 Finding the Percentage, Rate, and Base ............................................ 186 3 Applications of Percent .................................................................... 194

Unit 3 Geometry and Measurement

Chapter 9 Geometry Lesson 1 Subsets of a Line ............................................................................. 206 2 Angles ............................................................................................. 210 3 Polygons and Circles ....................................................................... 215 4 Space Figures .................................................................................. 224

Chapter 10 MeasurementLesson 1 Surface Area of Basic Solids ............................................................. 228 2 Volume of Solids ............................................................................. 237 3 Meter Reading ................................................................................ 245 4 Metric Conversion ........................................................................... 252

Unit 4 Statistics, Probability, and Introduction to Algebra

Chapter 11 Statistics and ProbabilityLesson 1 Circle Graph ................................................................................... 256 2 Mean, Median, and Mode ................................................................ 266 3 Possible Outcomes .......................................................................... 270 4 Simple Probability ........................................................................... 274

Chapter 12 Introduction to AlgebraLesson 1 Integers ........................................................................................... 278 2 Comparing Integers ......................................................................... 282 3 Ordering Integers ............................................................................ 286 4 Algebraic Expressions and Equations ............................................... 290

Glossary …………………………………………………………………………... ...................... 293Bibliography ........................................................................................................ 296Index ................................................................................................................... 297

Whole Numbers, Number Theory, and Fractions 1

Unit

� MathforSmartKids6

Review of Whole Numbers

Lesson 1 Properties of Operations

Mr. Cruz planted 20 tomatoes and 35 ampalaya plants in one of his garden plots. Then, he planted 35 tomatoes and 20 ampalaya plants in another plot. Compare the combined number of tomatoes and ampalaya plants in the two plots. What do you observe? plot A plot B 20 + 35 = 35 + 20 55 = 55

Mr. Cruz has the same number of plants in both garden plots, that is, 55.The above equation illustrates the commutative property of addition. This

property states that the order of the addends does not change the sum.

Look at this other example.

Mr. Cruz planted 5 rows of eggplants with 6 eggplants in each row of his garden. In another garden, he had 6 rows of okra plants with 5 okra plants in each row.

Chapter 1

WholeNumbers,NumberTheory,andFractions �

Compare the number of eggplants with the number of okra plants. What do you observe?

1. What are the properties of operations?2. What is the identity element for addition? for multiplication?

eggplants okra plants 5 6 = 6 5 30 = 30

The number of eggplants and the number of okra plants are the same, that is, 30.In the above equation, the symbol used for multiplication is the multiplication

dot (⋅). The equation shows the commutative property of multiplication. This property states that the order of the factors does not affect the product.

Aside from the commutative property, there are other properties of operations that you should know about.

Other Properties of Operations

PropertyOperation

Addition Multiplication

Associative property 5 + (6 + 9) = (5 + 6) + 9 5 + (15) = (11) + 9 20 = 20

6 (2 3) = (6 2) 3 6 (6) = (12) 3

36 = 36

Identity property7 + 0 = 78 + 0 = 8

7 1 = 78 1 = 8

Zero property of multiplication

7 0 = 025 0 = 0

Distributive property of multiplication over addition

9 (2 + 3) = (9 2) + (9 3) 9 (5) = (18) + (27) 45 = 45

Inverse property of multiplication

5 15

= 1


These are the properties of operations:

1. Commutative property – The order of the addends or factors does not change the sum or the product.

2. Associative property – The grouping of the addends or factors does not change the sum or the product.

3. Identity property of addition – Any number added to 0 will give the number itself. Zero (0) is the identity element for addition.

4. Identity property of multiplication – Any number multiplied by 1 will give the number itself. One (1) is the identity element for multiplication.

5. Zero property of multiplication – Any number multiplied by 0 will give a product of 0.

6. Distributive property of multiplication over addition – The product of a number and the sum of two or more addends is equal to the sum of the products of that number multiplied to each of the addends.

7. Inverse property of multiplication – Any number multiplied by its reciprocal will give a product of 1.

A. Supply the missing number and identify the property used. The first one has been done for you.

1. 25 + 35 = 35 + 25 Commutative property of addition

2. 85 25 = 25 __________________________________

3. 85 = 0 __________________________________

4. 75 = 75 __________________________________

5. 100 + = 100 __________________________________

6. 12 + (13 + 26) = (12 + 13) + __________________________________

7. (89 2) 14 = 89 (2 ) __________________________________

8. 32 (5 + 8) = (32 ) + (32 ) __________________________________

9. 45 145

= __________________________________

10. 89 + (24 + 25) = ( + 24) + 25 __________________________________


B. Identify the property of operation shown in each equation.

_________________________ 1. 7 (32 + 4) = (7 32) + (7 4) _________________________ 2. 89 + (45 + 32) = (89 + 45) + 32 _________________________ 3. 360 + 0 = 360 _________________________ 4. 39 3 = 3 39 _________________________ 5. 85 (3 40) = (85 3) 40 _________________________ 6. 1 720 = 720 _________________________ 7. 45 + 89 = 89 + 45 _________________________ 8. 39 (40 + 32) = (39 40) + (39 32) _________________________ 9. (36 + 34) + 45 = 36 + (34 + 45)

_________________________10. 37 137

= 1

Read and solve the following problems carefully.

1. Mother gave Rosa P45 while Father gave her P39. Mother gave Sonny P39 while Father gave him P45. Who had more money, Rosa or Sonny? Explain your answer.

2. Eloisa gathered 20 chicos on Sunday, 32 chicos on Monday, and 45 chicos on Tuesday. Renato gathered 32 chicos on Sunday, 45 chicos on Monday, and 20 chicos on Tuesday. Who had more chicos, Eloisa or Renato? Explain your answer.

Explain why both sides of the equations are equal.

1. 41 + (24 + 69) = (41 + 24) + 69

2. 89 (3 + 8) = (89 3) + (89 8)

3. 179 + 184 = 184 + 179

4. 85 79 = 79 85

5. (46 + 39) 62 = (46 62) + (39 62)


Scientists and engineers who often use very large or very small numbers find it useful to write these numbers this way:

1. 9.46 × 1015 meters (m) – the distance light travels in one year 2. 1.9891 × 1030 kilograms (kg) – the mass of the sun

Observe how the two numbers are written. The numbers 1015 and 1030 are written in exponential form. The number 10 is called base. The base is the number multiplied by itself. The numbers written at the upper right-hand side of the base (15 and 30) are called exponents. The exponent indicates how many times the base is to be multiplied by itself. The value of an exponential expression after multiplying the base by itself as many times as indicated by the exponent is called power.

Here are some examples of numbers written in exponential form and expanded form.

Exponential Form Base Exponent Expanded Form Power

30 3 0 1

52 5 2 5 5 25

43 4 3 4 4 4 64

65 6 5 6 6 6 6 6 7 776

79 7 9 7 7 7 7 7 7 7 7 7 40 353 607

1002 100 2 100 100 10 000

Lesson 2 Exponents

1. What do you call the number written at the upper right-hand side of the base? 2. What does this number tell you?3. How do you find the value of a number written in exponential form?


In ax = y , a is the base, x is the exponent, and y is the power or value of the expression.

The power or value (y) of a number with exponent is computed by multiplying the base (a) by itself as many times as indicated by the exponent (x).

The value of any nonzero number raised to the zero power is equal to 1.

A. Complete the table. Identify the base, exponent, and power of each number written in exponential form.

Exponential Form Base Exponent Power

1. 107

2. 394

3. 873

4. 2502

5. 3403

B. Write the following in expanded form.

1. 532 _______________________________

2. 105 _______________________________

3. 273 _______________________________

4. 854 _______________________________

5. 767 _______________________________

6. 956 _______________________________

7. 398 _______________________________

8. 4010 _______________________________

9. 459 _______________________________

10. 2604 _______________________________


A certain bacterium reproduces exponentially. If 10n is the number of bacteria in n days, how many of these bacteria will be present after 15 days? Write the answer in expanded and exponential forms.

Perform the indicated operations on the following exponential expressions. An example has been provided for you.

Example: 23 + 32 = (2 2 2) + (3 3) = 8 + 9 = 17

1. 35 + 43 ___________________________________________________________________________________________________________

2. 29 − 27 ___________________________________________________________________________________________________________

3. 19 56 ___________________________________________________________________________________________________________

4. 105 54 ___________________________________________________________________________________________________________

5. 163 − 54 ___________________________________________________________________________________________________________

6. 117 116 ___________________________________________________________________________________________________________

7. 20 + 50 + 4341 ___________________________________________________________________________________________________________

8. 73 + 84 + 25 ___________________________________________________________________________________________________________

9. (93 − 28) + 75 ___________________________________________________________________________________________________________

10. (95 73) 92 ___________________________________________________________________________________________________________


Letty’s transportation fare to school is P15. She spends the same amount for her fare in going home. Letty also spends P30 for snacks and P50 for lunch. If her daily allowance is P150, how much money would she have left at the end of the day?

To find the answer, you have to perform a series of operations.

1. How much does Letty spend for her fare, snacks, and lunch in a day?

n=(15 2) + 30 + 50

2. How much money would she have left at the end of the day if her daily allowance is P150?

n = 150 − [(15 2) + 30 + 50] = 150 − [(30) + 30 + 50] = 150 − [110] = 40

Therefore, Letty has P40 left at the end of the day.

To solve an equation that involves a series of operations, you must follow the PEMDAS rule:

P − Do the operation inside the parentheses first. E − Evaluate the expressions with exponents. M

−

Multiply or divide from left to right, whichever comes first. D A

−

Add or subtract from left to right. S

Lesson 3 Order of Operations

10 MathforSmartKids6

Study these other examples.

1. n = 35² + (32 5) ÷ 5 = 1 225 + 160 ÷ 5 = 1 225 + 32 = 1 257

2. n = (2 5) − (6 4 ÷ 8) = (10) − (3) = 10 − 3 = 7

3. n = (6 − 4 + 2)² + 8 6 = (2 + 2)² + 8 6 = 16 + 8 6 = 16 + 48 = 64

4. n = (5 + 2) 10² − 10 = 7 10² − 10 = 7 100 − 10 = 700 − 10 = 690

1. What rule must you follow to perform a series of operations?2. Which operation must you do first? Why? 3. Which operation must you do last? Why?

To solve an equation that involves a series of operations, follow the PEMDAS rule.

P – Do the operation inside the parentheses first. E – Evaluate the expressions with exponents.M

–

Multiply and divide from left to right, whichever comes first.DA

–

Add and subtract from left to right. S

Do the operation inside the parentheses. Evaluate the expression with exponents.

Divide, then add.

Do the operation(s) inside the parentheses.Multiply, then divide. Subtract.

Do the operation(s) inside the parentheses. Evaluate the expression with exponents. Multiply, then add.

Do the operation inside the parentheses.Evaluate the expression with exponents.Multiply, then subtract.

WholeNumbers,NumberTheory,andFractions 11

A. Write the order of operations to be performed.

1. 103 + (8 4) 2 4 = n ______________________________________________

2. 44 20 − (85 ⋅ 2) =n ______________________________________________

3. (92 45) 10 − 400 = n ______________________________________________

4. 122 + (86 5) 10 = n ______________________________________________

5. 104 − (45 87) + 999 = n ______________________________________________

6. (145 3) + (97 2) – 229 =n ______________________________________________

7. (985 + 5) 10 − 98 = n ______________________________________________

8. 396 + 492 7 − 586 = n ______________________________________________

9. 625 5 + 701 − 289 = n ______________________________________________

10. 798 + 42 3 − 375 = n ______________________________________________

B. Find the value of n in Exercise A. Show every step of your solution on a separate sheet of paper.

Andrea plans to spend her summer vacation in Boracay. She needs P5 000 for the one-way airfare, P3 000 for the one-day hotel accommodation, and P2 000 for one-day meals. How much does Andrea need, including the two-way air fare, if she will spend her summer vacation in Boracay for 7 days?

Find the value of n. Show every step of your solution. Do this on a separate sheet of paper.

1. 375 + 3 878 − 245 = n 2. 104 + (92 4) − 304 =n

3. (87 9) + 63 8 = n

4. 105 + (810 9) − 45 = n

5. (38 5) + 10 + 169 =n

6. (140 − 101) + 10 2 = n 7. (784 4) 2 + 1 589 − 379 = n 8. (475 5) (45 – 85) = n 9. 37 804 − 439 8 + 6 000 = n 10. 105 − 82 81 + 37 809 = n

1� MathforSmartKids6

Lesson 4 Problem Solving Involving a Series of Operations

Rita got the following grades in her subjects.

93, 88, 84, 91, 86, and 86 93, 88, 84, 91, 86, and 86 93, 88, 84, 91, 86, and 86

If the required average for the entrance examination to a science high school is 85 or higher, would she be qualified to take the exam? By how much less or more is her average grade compared with the required average?

To solve the problem, follow these steps:

Understand and analyze the problem.

What facts are given in the problem?

• Rita’s grades: 93, 88, 84, 91, 86, 86

• Average required is 85 or higher.

What is asked in the problem?

• Is Rita qualified or not to take the entrance exam?

• By how much less or more is her average grade compared with the required average?

What are the hidden questions?

• What is Rita’s total grade in all her subjects?

• What is Rita’s average grade?

What operations will you use?

Addition, division, subtraction

WholeNumbers,NumberTheory,andFractions 1�

93 + 88 + 84 + 91 + 86 + 86

6 85 n

Visualize the problem.

How will you illustrate the problem?

Plan. What are the number phrases for the hidden questions?

Total grade: 93 + 88 + 84 + 91 + 86 + 86

Average grade:(93 + 88 + 84 + 91 + 86 + 86) 6

What is the equation for the problem?

(93 + 88 + 84 + 91 + 86 + 86) 6 − 85 = n

Carry out the plan.

Solve the equation. Apply the PEMDAS rule.

Total grade: 93 + 88 + 84 + 91 + 86 + 86 = 528

Average grade:(93 + 88 + 84 + 91 + 86 + 86) 6 = (528) 6 = 88

The difference between Rita’s actual average grade and the required average for the exam is given by: 88 − 85 = 3

State the complete answer.

What is the complete answer?

Rita is qualified to take the exam. Her average grade is higher than the required average by 3 points.

1. What steps did you take to solve the word problem?2. Why is it important to follow the order of operations in solving word

problems? 3. Will you get a different answer if you will not follow the order of operations?

−


To solve word problems involving a series of operations, follow these steps:

1. Understand and analyze the problem. Find out what facts are given, what is asked in the problem, what the hidden questions are, and what operations will be used.

2. Visualize the problem. Draw a diagram to illustrate the problem.3. Plan to solve the problem. Write the number phrases for the hidden questions

and the equation for the problem.4. Carry out the plan. Solve the equation by applying the PEMDAS rule.5. State the complete answer.

A. Read and solve each problem. Follow the steps in problem solving and the rule on the order of operations.

Problem1

A laundrywoman earns P350 in one day. If she spends P15 on transportation and P100 on meals daily, how much money will she have left after 3 days?

Problem2

There are 8 classrooms in a building. If the desks in each classroom are arranged in 4 columns and 10 rows, how many desks are there in the building?

Understand and analyze the problem

• What facts are given in the problem?

• What is asked in the problem?

• What is/are the hidden question(s)?

• What operation(s) will you use?


Visualize the problem.

• How will you illustrate the problem?

Plan.

• What is/are the number phrase(s) for the hidden question(s)?

• What is the equation for the problem?

Carry out the plan. • Solve the equation.

State the complete answer.

• What is the complete answer?

B. Read and solve each problem. Show your solutions on a separate sheet of paper.

1. A landowner owns 6 hectares (ha) of farmland. He harvested 340 sacks of palay from each hectare of farmland. He set aside 90 sacks to be divided among his helpers and stored the rest equally in his 5 kamaligs. How many sacks of palay were stored in each kamalig?

2. A supermarket has a delivery of P370 450 worth of canned goods and P232 370 worth of kitchenware. The manager gives P200 000 as initial payment and will pay the remaining amounts in 5 installments. How much will each installment be?

3. Nena gathered 7 350 eggs from the poultry house. Her husband gathered 3 850 eggs more. They delivered the eggs equally to 50 vendors. How many eggs did each vendor get?

4. A school choir held a concert for three days to raise funds. The choir members were able to raise P6 900 on the first day, P2 500 on the second day, and P10 550 on the third day. If they spend P500 on transportation and P900 for miscellaneous expenses for a day, how much money do they have left after three days?

5. A golf club had 148 members last month. Eighty-six new members joined the club this month. Thirty-five club members from last month will no longer renew their membership. How many members would the golf club have this month?


Study the pictures below. Write five word problems using the given information and solve them on a separate sheet of paper.

School Supplies

P25

P30

P12

P600

P500

Mr. Fernando harvested 8 500 coconuts last week and 9 800 coconuts this week. He gave his helpers 300 coconuts. If a truck can load 600 coconuts in one trip, how many trips does the truck need to make to transport the remaining harvested coconuts?

A. Read and analyze the problem. Answer the questions that follow by encircling the letter of your answer.

1. What facts are given in the problem? a. 8 500 coconuts, 9 800 coconuts, 300 helpers, 600 trips b. 8 500 coconuts, 9 800 coconuts, 300 coconuts, 600 coconuts per trip c. 8 500 coconuts, 9 800 coconuts, 300 coconuts given away, 600 coconuts

per trip


2. What is asked in the problem? a. number of trips b. number of coconuts per trip c. number of coconuts per hectare

3. What are the hidden questions? a. How many trips need to be made? How many coconuts are to be

transported? b. How many coconuts are transported per trip? How many coconuts are

left? c. How many coconuts were harvested in all? How many coconuts are to

be transported?

4. What operations will be used? a. addition, subtraction, and division b. addition, division, and multiplication c. addition, subtraction, and multiplication

5. What is the equation for the problem? a. (8 500 + 9 800 – 300) 600 = n b. [(8 500 − 9 800) + 300] 600 = n c. [(9 800 + 8 500) 300] 600 = n

6. What is the complete answer? a. There will be 3 trips. b. There will be 30 trips. c. There will be 300 trips.

B. Solve these problems.

1. There were 36 male volleyball players and 40 female volleyball players during an athletic meet. Fifty-two of them participated in the meet last year. How many volleyball players were not part of last year’s athletic meet?

2. During the orientation for the new students, 289 students were present. Of

these, 109 were not first year students. If 23

of the first year students were

female, how many male first year students were there?


3. Gina can decorate 25 baskets in the morning and 20 baskets in the afternoon. If she continues to work at this rate, how many baskets can she decorate in 4 weeks?

4. Harry sold 460 tickets for their school’s benefit concert. For every 5 tickets he sold, Harry earned P8. How much did he earn from selling the tickets?

5. Roy earned P575 from selling barbecue. Jim earned P195 more than Roy. Dan earned twice as much as Jim. How much is the total earning of the three boys?

6. Mr. Lacson takes public transportation in going to work. He rides a tricycle and pays P14 for his fare. He also rides a jeepney and pays P12 for his fare. How much is his daily fare if he also rides a jeepney and a tricycle in going home?

7. Two members of an organization are selling tickets for a benefit concert. Member A sold 62 tickets, while member B sold 76 tickets. How much did the two members earn from selling the tickets if each ticket costs P15?

8. Mother gave me P35. Father gave me P40. How many notebooks can I buy if a notebook costs P15 each?

9. Mariel sold 3 blouses and 2 pairs of pants for P2,100. If the blouses cost P350 each, how much did each pair of pants cost?

10. Carmen bought 12 notebooks at P25 each and 10 pencils at P7 each. She sells the notebooks for P26.50 each and the pencils for P7.75 each. How much profit would she make if she will be able to sell all the items?


Number Theory

Chapter �

Lesson 1 Divisibility

Angelo has 24 oranges. He wants to divide the oranges equally among his 3 sisters. Is it possible to divide the oranges equally among his 3 sisters? If yes, how many oranges will each one get?

To answer this problem, you need to divide 24 by 3.

)3 2424

0

8

−))3 23 2))3 2)) 44

242400

88

−−

Based on the solution, there is no remainder after dividing 24 by 3. Therefore, Angelo’s sisters will have the same number of oranges and each of them will get 8 oranges.

The idea of having no remainder after dividing two numbers is called divisibility. A whole number is divisible by another whole number if, after dividing, the remainder is zero. However, you can determine if a whole number is divisible by another whole number even without performing division. This can be done by applying the divisibility rules.