Student Researchers/ Authors:

PAMN FAYE HAZEL M. VALINRON ANGELO A. DRONA

ASST. PROF. BEATRIZ P. RAYMUNDOModule Consultant

MR. FOR – IAN V. SANDOVALModule Adviser

A premier university in CALABARZON, offering academic programs and related services designed to respond to the requirements of the Philippines and the global economy, particularly, Asian countries.

The University shall primarily provide advanced education, professional, technological and vocational instruction in agriculture, fisheries, forestry, science, engineering, industrial technologies, teacher education, medicine, law, arts and sciences, information technology and other related fields. It shall also undertake research and extension services, and provide a progressive leadership in its areas of specialization.

In pursuit of college mission/vision the college of education is committed to develop the full potential of the individuals and equip them with knowledge, skills and attitudes in teacher education allied fields effectively responds to the increasing demands, challenge and opportunities of changing time for global competitiveness.

1. Produce graduates who can demonstrate and practice the professional and ethical requirements for the Bachelor of Elementary Education such as: 2. Acquire basic and major trainings in Bachelor of Elementary Education focusing on General Education and Pre - School Education. 3. Produce mentors who are knowledgeable and skilled in teaching pre - school learners and elementary grades and with desirable values and attitudes or efficiency and effectiveness. 4. Conduct research and development in teacher education and other related fields. 5. Extend services and other related activities for the advancement of community life.

This Teacher’s Guide Module entitled “Learning to Solve Decimal Numbers (Modular Workbook for Grade VI)” is part of the requirements in Educational Technology 2 under the revised curriculum for Bachelor in Secondary Education based on CHED Memorandum Order (CMO)-30, Series of 2004. Educational Technology 2 is a three (3)-unit course designed to introduce both traditional and innovative technologies to facilitate and foster meaningful and effective learning where students are expected to demonstrate a sound understanding of the nature, application and production of the various types of educational technologies.

The students are provided with guidance and assistance of selected faculty members of the university through the selection, production and utilization of appropriate technology tools in developing technology-based teacher support materials. Through the role and functions of computers especially the Internet, the student researchers and the advisers are able to design and develop various types of alternative delivery systems. These kind of activities offer a remarkable learning experience for the education students as future mentors especially in the preparation of instructional materials.

The output of the group’s effort may serve as an educational research of the institution in providing effective and quality education. The lessons and evaluations presented in this module may also function as a supplementary reference for secondary teachers and students.

FOR-IAN V. SANDOVAL Computer Instructor / Adviser Educational Technology 2

BEATRIZ P. RAYMUNDO Assistant Professor II /

Consultant

LYDIA R. CHAVEZ Dean College of Education

This instructional modular workbook entitled “Learning to Solve Decimal Numbers (Modular Workbook for Grade VI)”, which geared toward the objective of making quality education available to all and offers you a very interesting and helpful friend in your journey to the world of decimal numbers.

This modular workbook offers you many experiences in learning decimal numbers. This time, you will study how to read, write, and name decimal numbers and how to compare order and round off decimal numbers. Of course you will also express the equivalent fractions and decimals.

You will also experience the four (4) fundamental operations (Addition, Subtraction, Multiplication and Division) dealing with decimal numbers. Lastly, you will also solve more difficult problems involving the four (4) fundamental operations using decimal numbers. Learning decimal content is much more skillful in drilling with the application of FUN WITH MATH which designed to achieve with outmost skill and convenience.

The authors feel that you can benefit much from this modular workbook if you follow the direction carefully. Be mindful as you lead yourselves to challenge the situation and circumstances and as you faces in every living as well as for the near future. If you do these, you will realize that indeed this modular workbook can be a very interesting and helpful companion.

We would like to express our sincerest gratitude for the following whom in are ways or another help us making this modular workbook become possible:

To Prof. Corazon N. San Agustin, for her kindness and understanding to this modular workbook.

To Mr. For – Ian V. Sandoval, our instructor and adviser in Educational Technology 2, for giving sufficient technical trainings, suggestions, constructive criticism and unending support in our every needs.

To Assistant Professor Beatriz P. Raymundo, our Module Consultant, for making her available most of the time for comments, suggestions and revision of the modular workbook.

To Professor Lydia R. Chavez, our Dean, College of Education, for inspiring advises and encouragement.

To our classmates and friends for their never ending support.

To our beloved families, for unconditional love, emotional, spiritual and financial support all the way to used and for the filling up our duties in our home.

And most importantly to Almighty God, for rendering abilities, wisdom, good health, strength, courage, source of enlightenment and inspiration to pursue doing this piece of material.

VMGO’s

FOREWORD

PREFACE

ACKNOWLEDGEMENT

TABLE OF CONTENTS

UNIT I Decimal NumbersLesson 1 What is Decimal?Lesson 2 Reading and Writing Decimal NumbersLesson 3 Reading and Writing Mixed Decimal

NumbersLesson 4 Reading and Writing Decimal Numbers Used

in Technical and Science WorkLesson 5 Place ValueLesson 6 Comparing Decimal NumbersLesson 7 Ordering Decimal NumbersLesson 8 How to Round Decimal Numbers?Lesson 9 The Self-Replicating Gene

UNIT II Equivalent Fractions and DecimalsLesson 10 Expressing Fractions to DecimalsLesson 11 Expressing Mixed Fractional

Numbers to Mixed DecimalsLesson 12 Expressing Decimals to FractionsLesson 13 Expressing Mixed Decimals Numbers

to Mixed Numbers (Fractions)

UNIT III Addition and Subtraction of Decimal Numbers

Lesson 14 Meaning of Addition and Subtraction of Decimal Numbers

Lesson 15 Addition and Subtraction of Decimal Numbers without Regrouping

Lesson 16 Addition and Subtraction of Decimal Numbers with Regrouping

Lesson 17 Adding and Subtracting Mixed DecimalsLesson 18 Estimating Sum and Difference of Whole

Numbers and DecimalsLesson 19 Minuend with Two ZerosLesson 20 Problem Solving Involving Addition and

Subtraction of Decimal Numbers

UNIT IV Multiplication of Decimals

Lesson 21 Meaning of Multiplication of Decimals

Lesson 22 Multiplying Decimals

Lesson 23 Multiplying Mixed Decimals by Whole Numbers

Lesson 24 Multiplication of Mixed Decimals by Whole Numbers

Lesson 25 Multiplying Decimals by 10, 100 and 1000

Lesson 26 Estimating Products of Decimal Numbers

Lesson 27 Problem Solving Involving Multiplication of Decimal Numbers

UNIT V Division of Decimal Numbers

Lesson 28 Meaning of Division of Decimals

Lesson 29 Dividing Decimals by Whole Numbers

Lesson 30 Dividing Mixed Decimals by Whole Numbers

Lesson 31 Dividing Whole Numbers by Decimals

Lesson 32 Dividing Whole Numbers by Mixed Decimals

Lesson 33 Dividing Decimals by Decimals

Lesson 34 Dividing Mixed Decimals by Mixed Decimals

CURRICULUM VITAE

REFERENCES

OVERVIEW OF THE MODULAR WORKBOOK

In this modular workbook, you will understand the concept of the language of decimal numbers. This modular workbook, will help you to read, write, and name decimal numbers for a given models, standard, mixed and technical and science work form. It provides the knowledge about place value, with the aid of a place - value chart. It also provides information on how to compare and order decimal numbers and also how to round off decimal numbers. This module will provide you a more difficult work in mathematics. Exercises will help the learners evaluate themselves to understand decimal numbers.

OBJECTIVES OF THE MODULAR WORKBOOK

After completing this modular workbook, you expected to:

1. Know the language of decimal numbers.2. Read, write, and name decimal numbers in

different forms.3. Read and write decimal numbers with the aids of place - value chart.4. Compare and order decimal numbers.5. Rounding off decimal numbers by following

its rule.

Lesson 1 WHAT IS DECIMAL?

Lesson ObjectivesAfter accomplishing the lesson, the students are expected to:

1. Define decimals.2. Identify the terms in decimal numbers.3. Familiarize the language of decimal numbers.

Lesson ObjectivesAfter accomplishing the lesson, the students are expected to:

1. Define decimals.2. Identify the terms in decimal numbers.3. Familiarize the language of decimal numbers.

One important feature of our number system is the decimal. It involved many computational operations. It is very useful in the measurement of very thin sheets and in the computation involving in exact amount.

But what is decimal? Look at the following examples:

a. .3 = 3 .03 = 3 10 100

.003 = 3 .0003 = 3 1000 10000

b. .5 = 5 .05 = 5 10 100.005 = 5 .0005 = 5 1000 10000

From the example given above, a “decimal” may be defined as a fraction whose denominator is in the power of 10.

Numbers in the power of 10 are 10, 100, 1000, 1000, etc. The dot before a digit in a decimal is called “decimal point” which is an indicator that the number is a decimal. The place on the position occupied by a digit at the right of the decimal point is called a “decimal place”.

I. Give the meaning and explain the use of the following.

1. What are

decimals?

1. What are

decimals?

2. What is decimal point?

2. What is decimal point?

3. What is decimal place?

3. What is decimal place?

4. Give some

examples of decimal numbers.

4. Give some

examples of decimal numbers.

1. Decimals ____________________________________________________________________________________

2. Decimal Point ____________________________________________________________________________________

3. Decimal Place ____________________________________________________________________________________

4. Examples

____________________________________________________________________________________

II. Change the decimal numbers to fractional form.

Example: 0.8 = 8 10

1. 0.9 =_______________2. 0.1 =_______________3. 0.04 =_______________4. 0.06 =_______________5. 0.09 =_______________6. 0.001 =_______________7. 0.009 =_______________8. 0.0071 =_______________9. 0.0009 =_______________10. 0.0003 =_______________

11. 0.0004 =________________12. 0.0005 =________________13. 0.00008 =________________14. 0.00009 =________________15. 0.148 =________________16. 0.79 =________________17. 0.1459 =________________18. 0.6 =________________19. 0.01 =________________20. 0.051 =________________

Lesson 2READING AND WRITING DECIMAL

NUMBERS

Lesson ObjectivesAfter accomplishing the lesson, you are expected to:

1. Read and write decimal numbers.2. Follow the rules in reading and writing decimal numbers.3. Use the place value chart in order to read and write decimal numbers.

Lesson ObjectivesAfter accomplishing the lesson, you are expected to:

1. Read and write decimal numbers.2. Follow the rules in reading and writing decimal numbers.3. Use the place value chart in order to read and write decimal numbers.

How to read and write decimals or decimal numbers? A decimal is read and write according to the number of decimal place it has.

Here are the rules in reading and writing decimal numbers.

RULE I. A decimal of one decimal place is to be read and to be written as tenth.

.4 is read as “4 tenths” and is to be written as “four tenths”; 4/10.2 is read as “2 tenths” and is to be written as “two tenths”. 2/10

RULE II. A decimal of two decimal places is to be read and to be written as hundredth.

.35 is read as “35 hundredths” and is to be written as “thirty – five hundredths”; 35/100.43 is read as “43 hundredths” and is to be written as “forty – three hundredths”.43/100

RULE III. A decimal of three decimal places is to be read and written as thousandth.

.261 is read as “261 thousandths” and is to be written as “two hundred sixty – one thousandths”; 261/1000.578 is read as “578 thousandths” and is to be written as “five hundred seventy – eight thousandths”.578/1000

RULE IV. A decimal of four decimal places is to be read and to be written as ten thousandth.

.4917 is read as “4917 ten thousandths” and is to be written as “four thousand, nine hundred seventeen ten thousandths”; 4917/10,000.5087 is read as “5087 ten thousandths” and is to be written as “five thousand eighty - seven ten thousandths”.5078/10,000

A decimal is read and written like an integer with the name of the order of the right most digits added.

tenths

hundredths

thousandths

ten thousandths

hundred thousandths

Millionths

ten millionths

hundred millionths

billionths

ten billionths

hundred billionths

trillionths

0 . 4 3 5 7 8 9 6 1 2 5 3 4

Note: the names of the order of the different

decimal places.

QuadrillionthsPentillionthsHexillionthsHeptillionthsOctillionthsNonillionthsDecillionths

UndecillionthsDodecillionthsTridecillionths

…

Examples: 0.4 Read as four tenths.

0.43 Read as forty-three hundredths.

0.435Read as four hundred thirty-five thousandths.

0.4357 Read as four thousand, three hundred fifty-seven ten thousandths.

0.43578 Read as forty-three thousand, five hundred seventy-eight hundred thousandths.

0.435789 Read as four hundred thirty-five thousand, seven hundred eighty nine millionths.

0.4357896 Read as four million, three hundred fifty-seven thousand, eight hundred ninety-six ten millionths.

0.43578961 Read as forty three million, five hundred seventy-eight thousand, nine hundred sixty-one hundred millionths.

0.435789612Read as four hundred thirty-five million, seven hundred eighty nine thousand, six hundred twelve billionths.

0.4357896125 Read as four billion, three hundred fifty seven million, eight hundred ninety six thousand, one hundred twenty five ten billionths.

0.43578961253 Read as forty-three billion, five hundred seventy eight million, nine hundred sixty-one thousand, two hundred fifty three hundred billionths.

0.435789612534 Read as four hundred thirty-five billion, seven hundred eighty-nine million, six hundred twelve thousand, five hundred thirty-four trillionths.

I. Write each decimal numbers in words on the space provided.

1. 0.167213143__________________________________________________________________

2. 0.52541876_____________________________ ______________________________________3. 0.263411859____________________________

______________________________________4. 0.984562910____________________________ ______________________________________5. 0.439621512____________________________ _______________________________________

II. Write the decimal number in standard form.

1. Nine tenths______________________________________________2. Four hundredths______________________________________________

3. Two thousand, two hundred and two hundred thousandths____________________________________________4. Four hundred seventy – six millionths________________________________________________

5. Forty thousand, one hundred forty – one millionths________________________________________________

Lesson 3READING AND WRITING MIXED

DECIMAL NUMBERS

Lesson ObjectivesAt the end of the lesson, the students were expected to:

1. Read mixed decimal numbers.2. Follow the rules in reading and writing mixed decimal numbers.3. Write mixed decimal numbers.

Lesson ObjectivesAt the end of the lesson, the students were expected to:

1. Read mixed decimal numbers.2. Follow the rules in reading and writing mixed decimal numbers.3. Write mixed decimal numbers.

Look at the following examples:

a. 5.8 is read as “5 and 8 tenths” and is to be written as “five and eight tenths”

b. 26.38 is read as “26 and 38 hundredths” and is to be written as “twenty – six and thirty – eight hundredths”

c. 49.246 is read as “49 and 246 thousandths” and is to be written as “forty – nine and two hundred forty – six thousandths”

d. 348.578 is read as “348 and 578 thousandths” and is to be written as “three hundred forty – eight and five hundred seventy – eight thousandths”

It is seen that the following rule has been followed in the above examples.

RULE:

In reading a mixed decimal numbers, read the integral part as usual “and” in place of the decimal point, the decimal point is read as usual also.

1.246.819_____________________________________________________________________________________

2.65.42387____________________________________________________________________________________

3.9023.145867_________________________________________________________________________________

4.87.5843_____________________________________________________________________________________

5.48.0089_____________________________________________________________________________________

I. Write the words of decimal number for each of the following:

II. Write decimal numbers for each of the following sentences:

1. Sixteen and sixteen hundredths _____________________________________________

2.Two and one ten – thousandths _____________________________________________

3.Ten thousand four and fourteen ten – thousandths _____________________________________________

4. Ninety – nine billion and eight tenths _____________________________________________

5. Twelve hundred two and seven millionths _____________________________________________

6. Ninety – nine and nine hundred nine thousand, nine millionths_________________________________7. Five billion and sixty – five hundredths ______________________________________________8. Three billion, six thousand and three thousand six millionths _____________________________________9. Seventy – one million, one hundred and fifty – five hundred thousandths ______________________________________________10. Two hundred two million, two thousand, two and two hundred two thousand two millionths ______________________________________________

Lesson 4READING AND WRITING DECIMALS USED

INTECHNICAL AND SCIENCE WORK

Lesson ObjectivesAt the end of the lesson, the pupil should be able to:

1. Read and write decimals used in technical and science work.2. Follow the rules in reading and writing decimals

used in technical and science work.3. Know the simple way of reading and writing decimals that can be used in technical and science

work.

Lesson ObjectivesAt the end of the lesson, the pupil should be able to:

1. Read and write decimals used in technical and science work.2. Follow the rules in reading and writing decimals

used in technical and science work.3. Know the simple way of reading and writing decimals that can be used in technical and science

work.

This method of reading decimals and mixed decimals is often used by people engaged in technical and science work.

But this can be used by lay people especially if the part of the number has many digits.Observe the following examples:

a. 5.8 is read as “5 point 8” and is to be written as “five point eight”

b. .9 is read as “point 9” and is to be written as “point nine”

c. 6.893 is read as “6 point 893” and is to be written as “six point eight nine three”

d. 348.09536 is read as “348 point 09536” and is to be written as “three four eight point zero nine five three six“

e. 8945.874205 is read as “8945 point 874205” and is to be written as “eight nine four five point eight seven four two zero five”

The rule followed in the above examples is as follows:

To read decimals or mixed decimal numbers used in technical and science work or when the numbers of digits in the decimal is too many, just mention the values of the digits and separate the integral part by saying “point” instead of “and”.

RULE:

1. 0.009Read:___________________________________________Write:___________________________________________2. 45.78Read:__________________________________________Write:__________________________________________3. 3148Read: ___________________________________________Write:___________________________________________4. 3.456Read:___________________________________________Write:__________________________________________

I. Read and write the following in technical or science way.

4. 3.456Read:______________________________________Write:______________________________________5. 47.629Read: ___________________________________________Write:___________________________________________6. 5.78456Read: ___________________________________________Write:___________________________________________7. 0.491Read:___________________________________________Write:__________________________________________

8. 28.652Read:__________________________________________Write:_________________________________________9. 4928.95Read:__________________________________________Write:_________________________________________9. 4928.95Read:__________________________________________Write:_________________________________________10. 376.732Read:__________________________________________Write:_________________________________________

11. 841.50Read:__________________________________________Write:_________________________________________12. 3.62Read:__________________________________________Write:_________________________________________13. 0.03Read:__________________________________________Write:_________________________________________14. 97.5Read:__________________________________________Write:________________________________________15. 2.3148Read:_________________________________________Write:________________________________________

II. Write the following using decimal numbers.

1. one seven point three ___________________________________________

2. point five four two nine ___________________________________________

3. one two point zero nine ___________________________________________

4. four three point one eight nine ___________________________________________

5. two four point seven three two __________________________________________

6. three point seven six nine ______________________________________________7. two one seven point one five ____________________________________________8. point zero eight zero zero zero ___________________________________________9. nine point zero four zero ______________________________________________10. two point six seven two five ____________________________________________11. zero point nine eight nine ______________________________________________

12. zero point five two six eight two nine ____________________________________________13. five six zero point four zero one eight ____________________________________________14. one point one nine one eight ____________________________________________15. eight point five four three ____________________________________________

Lesson 5 PLACE VALUE

Lesson ObjectivesIn this lesson, the pupils are expected to:

1. Distinguish the relationship of place value in its place.

2. Write common fractions in decimal forms.3. Give the place value for every digit.

Lesson ObjectivesIn this lesson, the pupils are expected to:

1. Distinguish the relationship of place value in its place.

2. Write common fractions in decimal forms.3. Give the place value for every digit.

PLACE VALUE CHART

PlaceValueNames

MILLIONS

H TU H N OD UR S E AD N D S

T TE HN O U S A ND

S

THOUS ANDS

HUNDREDS

TENS

ONES

TENTHS

HUNDREDTHS

THOUS ANTHS

T T E H N O

U S A N T H S

H TU H N OD UR S E AD N T H S

MILLIONTHS

Numerals

1 9 4 6 3 4 1 . 1 3 4 5 8 7

× × × × × × × . × × × × × ×

106 105 104 103 102

101 1/10

0 1/101 1/102 1/103 1/104 1/105 1/106

What relationship exists in the diagram? What does the 1 in the tenths place mean? What does the 3 in the hundreds place represent? How about the 3 in the hundredths place?

Notice that:0.1 = 1 × 1/10 = 1/10 (one tenth)0.13 = 13 × 1/102 = 13/100 (thirteen hundredths)0.134 = 134 × 1/103 = 134/1000 (one hundred thirty –

four thousandths) 0.1345 = 1345 × 1/104 = 1345/10000 (one thousand

three hundred forty – five ten thousandths)

0.13458 = 13458 × 1/105 = 13458/100000 (thirteen thousand four hundred fifty – eight

hundred thousandths)0.134587 = 134587 × 1/106 = 134587/1000000 (one

hundred thirty – four thousand five hundred eighty – seven millionths)

I. Complete the equivalent decimals to fractions.

Decimal Fraction

1. 0.23  

2. 4.165  

3. 0.937  

4. 1.52  

5. 0.041  

6. 2.003  

7. 0.1527

8. 16.775  

9. 0.000658  

10. 685.95  

II. Answer the following.

1. In 246.819, what number is in each of the following place value?Example: __6__a. ones _246_c. hundreds

_46__b. tens __.8__d. tenths _.81__e. hundredths __.819_f. thousandths

2. In 65.42387, tell what number is in each of the following places._____a. tenths _____d. ten–thousandths _____b. hundredths _____e. hundred – thousandths_____c. thousandths _____f. ones

_____g. tens

3. In 9023.45867, tell what number is in each of the following places._____a. ones _____e. hundredths_____b. tens _____f. thousandths_____c. tenths _____g. thousands_____d. hundreds _____h. ten – thousandths

_____i. hundred–thousandths_____j. millionths

Lesson 6COMPARING DECIMAL NUMBERS

Lesson ObjectivesAt the end of the lesson, the pupils are expected to:

1. Compare decimal numbers.2. Use fractional number to compare decimals.3. Know the sign in comparing decimal numbers.

Lesson ObjectivesAt the end of the lesson, the pupils are expected to:

1. Compare decimal numbers.2. Use fractional number to compare decimals.3. Know the sign in comparing decimal numbers.

If there are two decimal numbers we can compare them. One number is either greater than (>), less than (<) or equal to (=) the other number.

A decimal number is just a fractional number. Comparing 0.7 and 0.07 is clearer if we compared 7/10 to 7/100. The fraction 7/10 is equivalent to 70/100 which is clearly larger than 7/1000.

Therefore, when decimals are compared start with tenths place and then hundredths place, etc. If one decimal has a higher number in the tenths place then it is larger than a decimal with fewer tenths. If the tenths are equal, compare the hundredths, then the thousandths, etc. Until one decimal is larger or there are no more places to compare. If each decimal place value is the same then the decimals are equal.

I. Fill the frame with the correct sign (>) “less than”, (=) “equal to”, or (>) “greater than” between two given numbers.

Example:0.9 = 9/10 0.90 = 10/100 =

b. 9.004 0.040 f. 51.6 51.59

c. 20.80533 20.06 g. 50.470 50.469

d. 0.070 0.07 h. 0.90 0.9

e. 0.540 0.054 i. 0.003 0.03

j. 0.8000 0.080

Lesson 7ORDERING DECIMAL NUMBERS

Lesson ObjectivesAfter accomplishing the lesson, the pupils are

expected to:1.Order decimal numbers.2.Know the terms in arranging decimal

numbers.3.Understand how to arrange decimal

numbers.

Lesson ObjectivesAfter accomplishing the lesson, the pupils are

expected to:1.Order decimal numbers.2.Know the terms in arranging decimal

numbers.3.Understand how to arrange decimal

numbers.

Numbers have an order or arrangement. The number two is between one and three. Three or more numbers can be placed in order. A number may come before the other numbers or it may come between them or after them.

Examples: If we start with numbers 4.3 and 8.78, the number 5.2764 would come between them, the number 9.1 would come after them and the number 2 would come before them.(Descending- 9.1; 8.87; 5.2764; 4.3) 9.1> 8.87>5.2764>4.3

If we start with the numbers 4.3 and 4.78, the number 4.2764 would come before both of them; the number 4.5232 would come between them.(Ascending- 4.2764; 4.3; 4.5232; 4.78) 4.2764< 4.3<4.5232<4.78

REMEMBER:The order may be

ascending (getting larger in value) or descending (becoming smaller in value).

I. Write in order from ascending order and descending order by completing the table.

Ascending Order

Descending Order

1. 2.0342; 2.3042; 2.3104

Example:2.03422.30422.3104

2.31042.30422.0342

2. 5; 5.012; 5.1; .502

3. 0.6; 0.6065; 0.6059;0.6061

5. 6.3942; 6.3924; 6.9342; 6.4269

6. 0.0990; 0.0099; 0.999; 0.90

7. 3.01; 3.001; 3.1; 3.001

8. 0.123; 0.112; 0.12; 0.121

9. 7.635; 7.628; 7.63; 7.625

4. 12.9; 12.09; 12.9100; 12.9150; 12

10. 4.349; 4.34; 4. 3600; 4.3560

Arrange the given decimal numbers from the least to greatest and you will find a famous quotation by Shakespeare.

Shakespeare(least)7.301All

8.043climb

7.8except

7.310ambitious

8.88or

7.84those

9.100of

7.911which

10.5mankind

7.33are

8.43up

8.513upward

7.352lawful

8.901the

9.003miseries

All___ _______ ________ ________ ________ ________ 7.301 _______ ________ ________ ________ ________

_______ ________ ________ ________ ________ _______________ ________ ________ ________ ________ ________ _______ ________ ________

_______ ________ ________ . - Shakespeare II. Answer the following.

a. The list below is the memory recall time of 5 personal computers. Which model has the fastest memory recall?

Model Recall Time

Sterling PC 0.0195 sec.

XQR 2000 0.01936 sec.

Redi-mate 0.02045 sec.

Vision 0.1897 sec.

Sal 970 0.019 sec.

Answer: ______________________________________________________________________________________

b.Arrange the memory recall time of computers in number 1 in ascending order.

Answer: __________________________________________________________________________________

c. A carpenter uses different sizes of drill bits in boring holes. The sizes are in fractional form and their equivalent decimals. Arrange the decimal equivalent in descending order.

1/16 = 0.0625 ¼ = 0.251/8 = 0.125 5/6 = 0.3125

Answer: ______________________________________________________________________________________

d.Which has the smallest decimal equivalent among the drill bits in item C?

Answer: ________________________________________

________________________________________

e. Which has the greatest decimal equivalent the drill bits in item C?

Answer: ________________________________________________________________________________

Lesson 8ROUNDING OFF DECIMALS

Lesson Objectives After accomplishing the lesson, the pupils are expected to:

1. Round decimals. 2. Tabulate data in the chart. 3. Show rules in rounding decimal numbers.

Lesson Objectives After accomplishing the lesson, the pupils are expected to:

1. Round decimals. 2. Tabulate data in the chart. 3. Show rules in rounding decimal numbers.

To round decimal numbers means to drop off the digits to the right of the place-value indicated and replace them by zeros.

The accuracy of the place-value needed must be stated and it depends on the purpose for which rounding is done. We give rounded decimal numbers when we do not need the exact value or number. Instead, we are after an estimated value or measure that will serve our purpose. These are many instances in daily life when rounded numbers are what we need to use.

How well do you remember in rounding whole numbers? Study the example below.

Round to the nearest4935 ten 4940

hundred 4900thousand 5000

See how the following decimals are rounded.

Rounded to the nearest0.31659 tenths 0.3

hundredths 0.32thousandths 0.317ten thousandths 0.3166

To round decimals, follow these rules:

1. Look at the digit immediately to the right of the digit in the rounding place.

2. All digits to the right of the place to which the number is rounded are dropped.

3. If the first of the digits to be dropped is 0,1,2,3 or 4, the last kept digit is not changed.

4. Increase the last kept digit by 1, when the first digit dropped is:

a. 6,7,8 or 9;orb. 5 followed by non-zero digit(s); orc. 5 (alone or followed by zero or zeros) and the

last kept digit is odd.

Example:

Round off 78.4651 to the nearest hundredths.

7 8 . 4 6 5 1 = 78.47

Dropping digit Decimal number to be rounded off

Examples: Round the following.

a. 5.767 to the nearest tenths = 5.8Since the digit to the right of 7 is 6.

b. 65.499 to the nearest hundredths = 65.50Since the digit to the right of 9 is 9.

c. 896.4321 to the nearest thousandths= 896.432Since the digit to the right of 2 is 1.

d. 32.28 to the nearest tenths = 32.3Since the digit to the right of 2 is 8

e. 1000.756 to the nearest hundredths = 1000.80Since the digit to the right of 5 is 6

f. 56.58691 to the nearest thousandths = 56.5870Since the digit to the right of 6 is 9

1. 29.8492 to the nearest:a. tenths

___________________b. ones

___________________c. hundredths

___________________d. thousandths

___________________e. tens

___________________

I. Round off the following decimal numbers.

2. 3.097591 to the nearest: a. ones _______________________ b. tenths _______________________ c. hundredths _______________________ d. thousandths _______________________ e. ten-thousandths _______________________

3. 6.152292 to the nearest: a. ones ______________________ b. tenths ______________________ c. hundredths ______________________ d. thousandths ______________________ e. ten-thousandths ______________________

4. 10.01856 to the nearest: a. ones ____________________ b. tenths ____________________ c. hundredths ____________________ d. thousandths ____________________ e. ten-thousandths ____________________

5. 123.831408 to the nearest: a. ones ____________________ b. tenths ____________________ c. hundredths ____________________ d. thousandths ____________________ e. ten-thousandths ____________________

III. Select the best answer.

A. 0.4278 rounded to the nearest thousandthsa. 0.462b. 0.46c. 0.430d. 0.464

B. 0.0042 rounded to the nearest thousandthsa. 0.003b. 0.0031c. 0.004d. 0.04

C. 0.6354 rounded to the nearest thousandthsa. 0.635b. 0.6c. 0.630d. 0.64

D. 0.635`4 rounded to the nearest hundredthsa. 0.635b. 0.6c. 0.630d. 0.64

E. 0.6354 rounded to the nearest tenthsa. 0.635b. 0.6c. 0.630d. 0.64

IV. Answer the following with TRUE or FALSE.

________________ 1. 0.32 rounded to the nearest tenths is 0.3.

________________ 2. 0.084 rounded to the nearest hundredths is 0.09.

________________3. 0.483 rounded to the nearest thousandths is 0.048.

________________4. 0.075 rounded to the nearest hundredths is 0.06.

________________5. 0.375 rounded to the nearest tenths is 0.4.

V. Round each of the following by completing the tables. Number 1 serves as an example.

DecimalsRound to the nearest

Tenths Hundredths ThousandthsTen

Thousandths

Example:1. 0.89432

0.9 0.89 0.8940.8943

2. 5.09998        

3. 2.96425        

4. 5.2358        

5. 5.39485        

6. 0.86302        

7. 28154      

8. 42356        

9. 2.38425        

10. 0.56893        

11. 2.9625        

12. 62.84213        

13. 29.04347        

14. 85.42998        

15. 1539485        

I. Find the answer by rounding off to the nearest place value indicated. Draw a line to the correct rounded number. Each line will pass through a letter. Write the letter next to the rounded number.

ONES1.6 ● ● 1.63 __________5.38 ● ● 3.4 __________52.52 ● ● 2 __________TENTHS0.45 ● ● 3.433 __________3.421 ● ● 53 __________12.76 ● ● 0.35 __________88.55 ● ● 5 __________HUNDREDTHS0.345 ● ● 12.8 __________1.634 ● ● 0.044 __________13.479 ● ● 0.5 __________201.045 ● ● 11.68 __________11.677 ● ● 16.778 __________THOUSANDTHS0.0437 ● ● 88.6 __________3.4325 ● ● 105.312 __________16.7777 ● ● 13.48 __________23.40092 ● ● 23.401 __________105.31238 ● ● 201.05 __________

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What happened to the man who

stole the calendar?

What happened to the man who

stole the calendar?

Lesson 9FACTS AND FIGURES

(The self-replicating Gene)

or centuries, generations of scientists in Numerica had been working relentlessly on what had been dubbed as The Genetic Enterprise. It was founded for the purpose of controlling a runaway gene that had beleaguered the Decimal citizens of Numerica for millennia: the repeating decimal gene.

Their history revealed that it all began when a woman named Four (4) united with a man named Forty-Four (44) positioned as 4/44. Their first offspring, a fraction, came out all right, and they named her Three-Thirty-thirds (3/33). Such a beautiful fraction she was. But their second child came out with the first problematic replicating gene--- the boy looked different and came with a long tail: 0.0909090909…

Arbitrarily multiplying both sides of the equation by any power of ten does not change the value of the decimal nor does it destroy the balance of the equation. This is because of the Multiplication Property of Equality of the Real Numbers.

That wasn’t the end of it. Every week, the boy’s tail added a new segment of 09, and it just never ended! It wasn’t so much the unusual appearance of the boy that worried every one, as the difficulty of naming him. Point-Zero-Nine-Zero-Nine-Zero-Nine- Zero -Nine-Zero-Nine, going on forever, was just too long a name!

There were many others in the community whose tails also continuously and regularly increased. Despite their peculiar form, though, those trouble with the repeating gene were never shunned, were treated equally with love, respect, and total acceptance. Still, the continually growing tail proved cumbersome for the Decimals, and they prayed to be changed to regular forms.

Arbitrarily multiplying both sides of the equation by any power of ten does not change the value of the decimal nor does it destroy the balance of the equation. This is because of the Multiplication Property of Equality of the Real Numbers.

One fateful day, the newspaper headlines screamed: “The Genetic Enterprise: Finally Success!” All of Numerica was thrilled!

At the state conference the next day, every citizen was present, especially those with continuously growing tails.

“And now, I must ask for a volunteer,” said Doctor One Half (½), head scientist of the project, over the microphone.

Immediately, 0.33333…, friend of Point-Zero-Nine-Zero-Nine…, was up the stage.

“O-Point-Three-Three-Three…” One Half began, “do not be afraid. Go into that glass capsule to your left, for we must first clone you.”

The crowd was aghast. One Half reassured every one immediately. “Do not worry, it is only temporary.”

When 0.33333… came out, his clone came out from the other capsule.

The doctor spoke again. “I will run you all through the process as we proceed. First, we shall designate the clone as Ex = 0.33333… “Now, 0.33333…, go back into the capsule---we will introduce a new gene into you. This gene is called Tenn (10), and this will change your appearance, so that we will temporarily call you Tennexx (10x). Do not be alarmed!” One Half added quickly at every one’s reaction.

When 0.33333… stepped out of the capsule, he had become 3.33333…

10x = 10 x 0.3333… = 3.3333…10x = 3.3333…

The crowd was mesmerized. “Tennexx, go back into the capsule. Exx, go into the other capsule. This time, we shall remove the repeating gene from Tennexx---by taking out Exx!”10x – x = 3.3333… - 0.333… 9x = 3

Now, Tennex come out! Let us all see what you have become…” One Half said dramatically.

The purpose of assigning a variable and multiplying both sides of the equation by 10 is to come up with whole numbers on both sides of the equation (on one side, with the variable, and on the other side of the equation, with just an integer). From this form we obtain a fraction equal to the original decimal. Note when finally making a subtraction, the digits in the decimal parts MUST be the same in order for the difference to be an integer.

The purpose of assigning a variable and multiplying both sides of the equation by 10 is to come up with whole numbers on both sides of the equation (on one side, with the variable, and on the other side of the equation, with just an integer). From this form we obtain a fraction equal to the original decimal. Note when finally making a subtraction, the digits in the decimal parts MUST be the same in order for the difference to be an integer.

When Tennex came out, he had become 1/3.9x = 3x = 3/9 or 1/3

The applause was thunderous! 1/3 spoke on the microphone, tears of joy poring down his cheeks. “Once! Was Zero-point-Three- -Three-Three-Three-Three… now I am One-Third. Thank you, Doctor One Half!” he said.

LESSON LEARNEDIn the article, we see that there is still

hope for repeating Decimal genes like Point O Nine O Nine O Nine and O Point Three-Three-Three. It is all about representing them using any variable, say, x, and taking away their never-ending tail.

Any repeating decimal represents a geometric series 0.3333… is 0.3 + 0.03+ 0.03+… The common ratio is 0.1, that is, the next term is obtained by multiplyingthe previous term by 0.1. The formula S= a1/1-rmay be used if (r) < 1.

S = a1/1-rS = 0.3/1-0.1 = 0.3/0.9 or 1/3

Any repeating decimal represents a geometric series 0.3333… is 0.3 + 0.03+ 0.03+… The common ratio is 0.1, that is, the next term is obtained by multiplyingthe previous term by 0.1. The formula S= a1/1-rmay be used if (r) < 1.

S = a1/1-rS = 0.3/1-0.1 = 0.3/0.9 or 1/3

PROBLEM BUSTER

PROBLEM BUSTERA DIFFERENT GENE

May I have another Volunteer? Ah, yes.

May I have another Volunteer? Ah, yes.

Do you think we can change 0.833333...in exactly the same way as what we did to 0.33333…? Notice that in this case, the numeral 8 does not repeat. If we introduced 10 like we did to0.33333…, by multip- lying 0.833333…by 10, 0.833333…will become 8.33333… Following the process, we have, 10x –x = 8.33333… - 0.833333… which is the same as 9x =7.5 where the

right side of the equation is not an integer! What are we to do?

What we need to do is keep multiplying by 10, until we get two numbers whose digits or numerals in the decimal parts are exactly the same. Thus,

x = 0.833333 – 10x = 10 x 0.833333… -- 10x = 8.33333… -- 10x (10) = 8.33333… x 10 -- 100x = 83.3333…

What we need to do is keep multiplying by 10, until we get two numbers whose digits or numerals in the decimal parts are exactly the same. Thus,

x = 0.833333 – 10x = 10 x 0.833333… -- 10x = 8.33333… -- 10x (10) = 8.33333… x 10 -- 100x = 83.3333…

With 0.33333…, it was enough to subtract x from 10x because the digits or numerals in their decimal parts are already exactly the same. Recall that when we subtracted them, we arrived at an integer on either side of the equation. This time, we subtract 10x from 100x, because the decimal parts of these two numbers have exactly the same digits or numerals. So that,100x – 10x =83.3333… - 8.3333…

-- 90x =75 -- x = 75/90 -- x = 5/6 -- 0.833333… 5/6 Therefore, 0.833333… is 5/6!

With 0.33333…, it was enough to subtract x from 10x because the digits or numerals in their decimal parts are already exactly the same. Recall that when we subtracted them, we arrived at an integer on either side of the equation. This time, we subtract 10x from 100x, because the decimal parts of these two numbers have exactly the same digits or numerals. So that,100x – 10x =83.3333… - 8.3333…

-- 90x =75 -- x = 75/90 -- x = 5/6 -- 0.833333… 5/6 Therefore, 0.833333… is 5/6!

1. 0.88888…as a fraction is:

a. 5/11 b. 7/8 c. 8/9 d. 10/11

2. 0.22222… in fraction form is:

a. 3/7 b. 2/11 c. 3/10 d. 2/9

I. Choose the correct answer:

II. Change the following to fraction in simplest form.

3. 0.77777…

4. 0.9166666…

5. 0.9545454…

6. 0.891891891…

7. 0.153846153846153846…

8. 0.9692307692307692307…

This modular workbook provides knowledge about different form or ways of computing fractions to decimals and decimals to fraction. This will help you to understand better what equivalent fraction and decimal is and you can use it in your everyday life.

OVERVIEW OF THE MODULAR WORKBOOK

After completing this Unit, you are expected to:1. Transform fraction/mixed fractional numbers to decimals/mixed decimal.2. Change decimal/mixed decimal to fraction /mixed numbers (fractions).3. Follow the rules in expressing equivalent fractions and decimals.

OBJECTIVES OF THE MODULAR WORKBOOK

Lesson 10EXPRESSING FRACTIONS TO DECIMALS

Lesson ObjectivesAfter accomplishing the lesson, you are expected to:

1. Change fractions to decimals.2. Know the rules in changing fractions to decimals.3. Understand the equivalent fractions and decimals.

Lesson ObjectivesAfter accomplishing the lesson, you are expected to:

1. Change fractions to decimals.2. Know the rules in changing fractions to decimals.3. Understand the equivalent fractions and decimals.

Decimals are a type of fractional number.

Let us now study how to write fractions to decimal form.

We will apply the principle of equality of fractions that is,if a/b =c/d then ad = bc.

Example 1:

Write the fraction 2/5 as a tenth decimal. In this case we are interested to find the value of x such that 2/5=x/10.

Since the two fractions name the same rational number, we can proceed:

5x = 2(10) – applying equality principle5x = 20 x = 20/5 or 4

Hence, 2/5 = 4/10 = 0.4

Example 2:

Write the fraction 3 as a hundredth decimal. We are 4interested to find the value of x such 3 that = x . 4 100 Applying the principle of equality we have

4x = 3(100)4x = 300 x = 75

Hence, ¾ = 75/100 = 0.75