Math for Smart Kids Gr.5

Math for Smart Kids
Grade 5
Textbook

Philippine Copyright 2010 by DIWA LEARNING SYSTEMS INC
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The Editorial Board
Authors
Dr. 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. She has been a classroom teacher, an Education supervisor, and 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.
Bernardo R. Daquiz Jr. holds a Bachelor of Arts in Social Science degree, major in Behavioral Studies, from the
University of the Philippines-Manila . He took units in BS Education at PNU. He was a former grade school teacher at
PAREF-Northfield Private School for Boys. He is currently teaching grade school Mathematics at Xavier School.
Consultant
Ma. 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.
Reviewer
Reina 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 computa-
tionally 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 will 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 Whole Numbers
Lesson 1 Place Values of Numbers ...................................................................2
2 Comparing and Ordering Numbers ....................................................8
3 Rounding off Numbers ..................................................................... 12
4 Adding Whole Numbers ................................................................... 15
5 Subtracting Whole Numbers ............................................................ 19
6 Multiplying Whole Numbers ............................................................ 22
7 Dividing Whole Numbers................................................................. 28
8 Solving Word Problems Involving More than One Operation ........... 32
9 Roman Numerals ............................................................................. 38

Chapter 2 Number Theory
Lesson 1 Divisibility Rules ............................................................................. 41
2 Prime and Composite Numbers ........................................................ 46
3 Prime Factorization .......................................................................... 49
4 Greatest Common Factor.................................................................. 52
5 Least Common Multiple ................................................................... 57

Chapter 3 Fractions
Lesson 1 Reducing Fractions to Lowest Terms ................................................ 61
2 Changing Dissimilar Fractions to Similar Fractions ........................... 65
3 Equivalent Fractions ........................................................................ 70
4 Comparing and Ordering Fractions .................................................. 73
1
5 Estimating Fractions Close to 0, , or 1 ........................................... 79
2

Unit 2 Operations on Fractions and Decimals
Chapter 4 Addition and Subtraction of Fractions
Lesson 1 Adding Similar Fractions ................................................................. 84
2 Adding Dissimilar Fractions ............................................................ 89
3 Adding Mixed Numbers ................................................................... 96
4 Mental Addition of Fractions .......................................................... 102
5 Subtracting Similar Fractions.......................................................... 105
6 Subtracting Dissimilar Fractions ..................................................... 115

Chapter 5 Multiplication and Division of Fractions
Lesson 1 Multiplying Fractions and Whole Numbers .................................... 124
2 Multiplying Mixed Numbers and Fractions..................................... 128
3 Dividing Whole Numbers and Fractions ......................................... 135
4 Dividing Fractions.......................................................................... 140
5 Solving Multistep Word Problems ................................................. 144

Chapter 6 Decimals
Lesson 1 Place Values of Decimals ................................................................ 149
2 Reading and Writing Decimals ....................................................... 153
3 Comparing and Ordering Decimals ................................................ 156
4 Rounding off Decimals ................................................................... 160

Chapter 7 Operations on Decimals
Lesson 1 Adding Decimals ............................................................................ 164
2 Subtracting Decimals ..................................................................... 170
3 Multiplying Decimals ..................................................................... 176
4 Multiplying Mixed Decimals and Whole Numbers.......................... 179
5 Multiplying Mixed Decimals .......................................................... 182
6 Mental Multiplication Involving Decimals ...................................... 186
7 Dividing Decimals.......................................................................... 189
8 Dividing Decimals by Whole Numbers ........................................... 195

Unit 3 Ratio, Proportion, and Percent

Chapter 8 Ratio and Proportion
Lesson 1 Ratio .............................................................................................. 200
2 Proportion...................................................................................... 208
3 Direct and Indirect Proportions ...................................................... 211
4 Problem Solving on Proportion ....................................................... 215

Chapter 9 Percent
Lesson 1 Meaning of Percent ........................................................................ 219
2 Percents, Decimals, Fractions, and Ratios ....................................... 223
3 Finding the Percentage ................................................................... 228
4 Finding the Rate ............................................................................. 233
5 Finding the Base ............................................................................ 236

Unit 4 Geometry, Measurement, and Graphs

Chapter 10 Geometry
Lesson 1 Polygons: Quadrilaterals and Triangles ........................................... 240
2 Similar and Congruent Polygons..................................................... 246
3 Circles ........................................................................................... 251
4 Space Figures ................................................................................. 254

Chapter 11 Measurement and Graph
Lesson 1 Perimeter and Circumference ......................................................... 258
2 Area ............................................................................................... 265
3 Volume of Space Figures ................................................................ 273
4 Temperature .................................................................................. 280
5 Line Graph .................................................................................... 284
6 Finding the Average ....................................................................... 290

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

Unit
1
Whole Numbers, Number
Theory, and Fractions

Chapter 1

Whole Numbers

Lesson 1 Place Values of Numbers

Pluto, once known as the coldest, smallest, and
most distant planet from the sun, was discovered by an
American astronomer named Clyde Tombaugh in 1930.
Its distance from the sun is 5 906 380 000 kilometers
(km). Its equatorial radius is 1 151 km, and its volume
is 6 390 000 000 cubic kilometers (km³). It has a mass of
13 000 000 000 000 000 000 000 kilograms (kg).
On 24 August 2006, the International Astronomical
Union (IAU) reclassified Pluto from one of the nine major
planets of the solar system to a minor planet. The IAU said
that Pluto does not meet the three criteria to be classified
as a planet. One criterion is that a planet should have “cleared the neighborhood”
around its orbit. Astronomers have observed that Pluto orbits among the icy debris in
the asteroid belt.
What is Pluto’s distance from the sun? 5 906 380 000 km
What is its equatorial radius? 1 151 km
To help you read these numbers, study the place-value chart below.
Place-value Chart

Billions Millions Thousands Units
Hundreds Tens Ones Hundreds Tens Ones Hundreds Tens Ones Hundreds Tens Ones
(H) (T) (O) (H) (T) (O) (H) (T) (O) (H) (T) (O)

1 1 5 1
5 9 0 6 3 8 0 0 0 0

You read 1 151 as “one thousand, one hundred fifty-one.”
You read 5 906 380 000 as “five billion, nine hundred six million, three hundred
eighty thousand.”
Math for Smart Kids 5

Look at the place-value chart again. Each group of three digits in a number is a period.
In numbers with four or more digits, the periods are separated by a single space.
Consider the number 1 426 725 400. This number can be written in three ways:
• Standard form: 1 426 725 400
• Expanded form: 1 000 000 000 + 400 000 000 + 20 000 000 + 6 000 000 +
700 000 + 20 000 + 5 000 + 400
• Word form: one billion, four hundred twenty-six million, seven hundred
twenty-five thousand, four hundred

Here are more examples:
1. Standard form: 57 909 175
Expanded form: 50 000 000 + 7 000 000 + 900 000 + 9 000 + 100 + 70 + 5
Word form: fifty-seven million, nine hundred nine thousand, one hundred
seventy-five
Expanded form: 200 000 000 + 20 000 000 + 7 000 000 + 900 000 + 30 000 +
6 000 + 600 + 40
Word form: two hundred twenty-seven million, nine hundred thirty-six
thousand, six hundred forty
Expanded form: 700 000 000 + 70 000 000 + 8 000 000 + 400 000 + 10 000 +
2 000 + 20
Word form: seven hundred seventy-eight million, four hundred twelve
thousand, twenty
4. Standard form: 1 426 725 400
Expanded form: 1 000 000 000 + 400 000 000 + 20 000 000 + 6 000 000 +
700 000 + 20 000 + 5 000 + 400
Word form: one billion, four hundred twenty-six million, seven hundred
twenty-five thousand, four hundred
5. Standard form: 2 870 972 201
Expanded form: 2 000 000 000 + 800 000 000 + 70 000 000 + 900 000 +
70 000 + 2 000 + 200 + 1
Word form: two billion, eight hundred seventy million, nine hundred
seventy-two thousand, two hundred one
6. Standard form: 4 498 252 900
Expanded form: 4 000 000 000 + 400 000 000 + 90 000 000 + 8 000 000 +
200 000 + 50 000 + 2 000 + 900
Word form: four billion, four hundred ninety-eight million, two hundred
fifty-two thousand, nine hundred

Whole Numbers, Number Theory, and Fractions

Examine the numbers in the place-value chart below.

Place-value Chart
period period period period

Billions Millions Thousands Units
H T O H T O H T O H T O

6 2 0 0 1 7 9 5 0 8 0 0

7 0 1 0 3 0 8 7

1 5 9 6 0 5 6 3 8

4 1 2 0 7 3 1 5 0 0 4

You read: six hundred twenty billion, seventeen million, nine hundred fifty
thousand, eight hundred
You write: 620 017 950 800

You read: seventy million, one hundred three thousand, eighty-seven
You write: 70 103 087

You read: one hundred fifty-nine million, six hundred five thousand, six hundred
thirty-eight
You write: 159 605 638

You read: forty-one billion, two hundred seven million, three hundred fifteen
thousand, four
You write: 41 207 315 004

Look at the numbers in the given place-value chart.
1. How are the digits in each number grouped?
2. How many digits are there in each period?
3. How are the digits in each period read?
4. How are the digits in each period written?


The place value of a digit depends on its place in the number. A place-value
chart can be used to easily determine the place value of a digit. Zero is used as a
place holder.
In reading a number, read first the leftmost digit, then the remaining digits going
to the right.
Starting from the rightmost digit in a number, each group of three digits in a
number is a period.
In writing the standard form of a number, leave a single space to separate each
period.

A. Write the value of each underlined digit.
1. 4 396 _________________ 6. 945 672 389 _________________
2. 59 485 _________________ 7. 8 835 693 000 _________________
3. 104 629 _________________ 8. 73 729 486 925 _________________
4. 3 495 829 _________________ 9. 182 920 825 496 _________________
5. 46 829 405 _________________ 10. 393 003 934 286 _________________

B. Write the numbers in standard form.
1. three million, two hundred twenty-four thousand, one hundred thirty-five
_________________
2. four billion, four hundred million, nine hundred seventy-five thousand, three
_________________
3. twenty-five billion, three million, one hundred twelve thousand, two hundred
twenty-six _________________
4. one hundred three billion, thirty-four million, six hundred fifty-four thousand,
seven hundred thirteen _________________
5. eight hundred billion, nine hundred forty-five million, five thousand, three
hundred seventy-nine _________________

C. Write these numbers in words.
1. 6 922 436 ______________________________________________________
______________________________________________________


2. 89 345 782 ______________________________________________________
______________________________________________________
______________________________________________________

3. 120 407 315 ______________________________________________________
______________________________________________________
______________________________________________________

4. 10 985 375 112 ______________________________________________________
______________________________________________________
______________________________________________________

5. 465 123 400 327 ______________________________________________________
______________________________________________________
______________________________________________________

You were asked to help in canvassing the number of votes during a presidential
election. How would you read the following votes that each candidate received?
Presidential candidate A – 2 305 439
Presidential candidate B – 6 692 307

A. Write the place value of each underlined digit.
1. 825 _________________
2. 9 262 _________________
3. 34 827 _________________
4. 505 948 _________________
5. 6 828 725 _________________
6. 73 263 827 _________________
7. 825 684 936 _________________
8. 9 836 825 000 _________________
9. 63 527 043 924 _________________
10. 425 682 426 135 _________________


B. Write the following numbers in standard form.
1. three million, five hundred twenty-three thousand, thirty-nine

2. seven billion, ninety-five million

3. eighty billion, three hundred twenty-seven million, three hundred fifteen
thousand, one hundred twenty-one

4. one hundred five billion, three hundred two million, nine hundred thirty
thousand, two hundred two

5. nine hundred two billion, three million, twenty-five thousand, seven

C. Write the following numbers in words.
1. 4 326 175

2. 35 002 872

3. 8 350 432 175

4. 92 001 426 127

5. 105 000 345 278


Lesson 2 Comparing and Ordering Numbers

Tally Board
Candidate Number of Votes
A 85 361 425 126
B 85 136 395 427

Rita and Steve were
discussing the results of the
presidential elections. Find
out how they were able to
compare two numbers.

Steve: Who has the most number of votes?
Rita: Let’s look at the figures.
Steve: How do we find out who got the most number of votes just by looking at the
figures?
Rita: Look at the digit in the highest place value of each number. In this case, the
highest place value of the numbers is ten billions. Are the digits the same? If
they are, compare the digits in the next highest place value of the two numbers.
Do this until you find different digits in the same place value of the numbers.
Steve: Since both numbers are more than 85 billion, we can compare the digits in the
hundred millions place.
Rita: Right, Steve! One hundred million is less than 3 hundred million. So,
85 361 425 126 is greater than 85 136 395 427.

Here is another example. Compare 5 395 432 375 with 5 395 324 896.
First, compare the digits in the highest place value which is the billions place.
5 395 432 375 and 5 395 324 896
5 billion = 5 billion
Since the billions digits are the same, compare the digits in the next highest place
value, which is the hundred millions place. Continue comparing until you find different
digits in the same place value.
395 million = 395 million


Since all the digits in the millions period are the same, compare the digits in the
hundred thousands period.
400 thousand 300 thousand
Therefore, 5 395 432 375 is greater than 5 395 324 896.
You write 5 395 432 375 5 395 324 896.

Study this example.
Arrange these numbers in descending order.
17 432 187 19 807 453 18 476 389
Since the digits in the highest place value are the same, compare the digits in the
next place value.
17 432 187 19 807 453 19 807 453 18 476 389 17 432 187 18 476 389
Arranged in descending order: 19 807 453, 18 476 389, 17 432 187

1. How do you compare numbers with the same number of digits?
2. How do you order numbers in a given set?

To compare two numbers, start comparing the digits in the highest place
value. If the digits in the highest place value are the same, compare the digits in the
next highest place value. Continue comparing until you find different digits in the
same place value. After comparing numbers, order the numbers according to the
indicated arrangement.
To order numbers means to arrange the numbers from least to greatest, or
from greatest to least. Before you can order numbers, you have to compare the
digits in the two numbers that belong to the same place value.


A. Compare the numbers. Write or in the box.
1. 28 350 425 27 475 986
2. 275 875 126 275 758 326
3. 8 925 627 829 7 978 735 926
4. 75 826 425 139 75 826 425 142
5. 182 373 003 146 182 373 003 164

B. Arrange the numbers from least to greatest. Write 1 (least) to 3 (greatest) in the
blanks.

1. 8 328 427 826 8 328 438 725 8 328 426 125

2. 75 639 628 73 789 009 77 836 133

3. 427 828 536 132 427 938 125 002 427 828 635 143

C. Arrange the numbers from greatest to least. Write 1 (greatest) to 3 (least) in the
blanks.

1. 89 426 872 89 426 982 89 628 135

2. 5 825 326 143 5 825 145 009 5 825 427 182

3. 78 926 008 426 87 375 009 412 87 872 007 325

10 Math for Smart Kids 5

The search for the World’s Top Singing Idol was launched in the highest rated
television station. The top 5 singing idol hopefuls garnered the following text votes all
over the world.
Fill in the blanks with a digit that will make the ranking of the idols correct.
1. Singing Idol A 92 3 __ __ 405 302
2. Singing Idol D 92 3 __ __ 139 406
3. Singing Idol E 87 __ __ 3 439 127
4. Singing Idol C 87 __ __ 3 203 725
5. Singing Idol B 87 __ __ 8 302 905

A. Compare the numbers. Write or in the box.
1. 39 426 104 39 462 139 4. 32 926 428 926 32 926 428 629
2. 526 927 324 526 927 423 5. 625 829 327 412 625 828 327 415
3. 9 825 326 008 9 825 326 800

B. Arrange the numbers from least to greatest. Write 1 (least) to 3 (greatest) in the
blanks.

1. 8 723 142 7 829 602 7 826 342

2. 104 602 008 104 602 080 104 602 800

3. 92 829 523 92 826 439 92 838 426

4. 33 402 926 134 33 402 962 426 33 402 923 246

5. 825 302 411 321 832 412 142 008 894 142 602 000

Whole Numbers, Number Theory, and Fractions 11

Lesson 3 Rounding off Numbers

Look at the table below.
Top 10 Most Populated Countries in the World

Rank Country Population
1 China 1 330 044 605
2 India 1 147 995 898
3 United States 303 824 646
4 Indonesia 237 512 355
5 Brazil 191 908 598
6 Pakistan 167 762 040
7 Bangladesh 153 546 901
8 Russia 140 702 094
9 Nigeria 138 283 240
10 Japan 127 288 419
Source: Internet World Stats. http://www.internetworldstats.com/stats8.htm
(accessed 20 May 2009)

What is China’s population rounded off to the nearest millions?
To answer this question, you need to round off the population of China. Study the
figure below.
Round up if the digit to the right of the place
value you are rounding off to is 5 or more.
1. Add 1 to the digit in the place value you are
rounding off to.
2. Replace the remaining digits to the right
with zeros.

0 1 2 3 4 5 6 7 8 9 10
Round down if the digit to the right of the
place value you are rounding off to is 4 or less.
1. Retain the digit in the place value you
are rounding off to.
2. Replace the remaining digits to the right
with zeros.


Round off 1 330 044 605 to the nearest
millions 1 330 044 605 1 330 000 000
ten millions 1 330 044 605 1 330 000 000
hundred millions 1 330 044 605 1 300 000 000
billions 1 330 044 605 1 000 000 000

Therefore, there are about 1 330 000 000 people living in China.

1. When do you round up? round down?
2. How do you round off numbers through millions?

In rounding off a number, look at the digit in the place value you are rounding
off and identify the digit to its right. If the digit to its right is 5 or more, round up. Add
1 to the digit in the place value you are rounding off, then replace the remaining
digits to the right with zeros. If the digit to the right is 4 or less, round down. When
you round down a digit, retain the digit in the place value you are rounding off.
Then, replace the remaining digits to the right with zeros.

Round off each number to the nearest place value indicated.
Number Millions Hundred Thousands Ten Thousands
1. 426 945 126
2. 824 136 621
3. 758 247 829
4. 566 729 124
5. 324 846 439


Number Millions Hundred Thousands Ten Thousands
6. 748 982 005
7. 834 193 448
8. 136 708 136
9. 493 625 278
10. 574 384 352

A house owner, who is leaving for abroad, plans to sell his house including all his
appliances. The selling price of the house is P 5 534 000. The appliances are marked as
follows:
refrigerator — P 16 700
television — P 24 300
aircon — P 11 500
Without using paper and pencil, or a calculator, how would you compute for the
total amount instantly? What is the approximate total amount? Is it a good estimate?

Round off each number to the indicated place value.
1. 9 805 thousands
2. 12 302 hundreds
3. 156 235 ten thousands
4. 3 845 364 millions
5. 36 081 426 ten millions
6. 785 320 437 ten millions
7. 8 829 009 621 billions
8. 15 826 435 085 ten billions
9. 159 927 390 713 hundred billions
10. 304 038 592 004 billions


Lesson 4 Adding Whole Numbers

Properties of Addition
Look at each set of equations below. Write down what you observe.

Set A Set B Set C

125 + 206 = 206 + 125 (583 + 926) + 428 = 583 + (926 + 428) 5 496 + 0 = 5 496
331 = 331 1 509 + 428 = 583 + 1 354 5 496 = 5 496
1 937 = 1 937

Look at set A. Focus on the order of the addends. What happened to the order of
the addends? Did the sum change? Set A illustrates the commutative property of
addition. This property states that changing the order of the addends does not change
the sum.

Look at set B. How are the addends grouped? Did the sum change? Another property
of addition is illustrated here—the associative property of addition. The associative
property of addition states that changing the grouping of the addends does not affect
the sum.

Look at set C. What happens when zero is added to a number? Zero is known as the
identity element for addition. When you add zero to a number, the identity property
of addition is observed. This property states that the sum of a number and zero is the
number itself.

Adding Numbers through Millions
In a combined effort of two organizations, a fund-raising project was launched to
raise funds for typhoon victims. Organization A was able to raise P 4 362 890, while
Organization B was able to raise P 5 471 008. How much money was raised by the two
organizations?


To find the answer, follow these steps in adding numbers.

Step 1. Align the digits of the addends Step 3. Add the digits from right to
according to their place values. left in the thousands period.
Regroup if needed.
1
4 362 890 4 362 890
+ 5 471 008 + 5 471 008
833 898

Step 2. Add the digits in the units period Step 4. Add the digits from right to
starting from the rightmost place left in the millions period.
value (ones). Write the sum Regroup if needed.
below the digits being added.
Regroup if needed.
1
4 362 890 4 362 890
+ 5 471 008 + 5 471 008
898 9 833 898

1. From what place value did you start adding?
2. What is the period of the digits that you added first?
3. If the sum of the digits is more than 9, what do you do?

The properties of addition are the commutative property, the associative property,
and the identify property. Zero is known as the identity element of addition.
In adding numbers, align the digits of the addends according to their place
values. Add the digits in the units period starting from the ones digit. Then, add the
digits, from right to left, in the thousands and millions period. If the sum of the digits
is more than 9, regroup to the next place value.


A. Identify the property of addition shown in each equation.
1. 3 654 + 0 = 3 654
2. 4 351 + 3 610 = 3 610 + 4 351
3. (3 629 + 4 317) + 4 216 = 3 629 + (4 317 + 4 216)
4. 6 211 + (4 652 + 9 275) = (6 211 + 4 652) + 9 275
5. 0 + 1 295 602 408 = 1 295 602 408

B. Fill in the box with the correct number to show a property of addition. Then,
identify the property of addition illustrated in each.
1. 486 + 3 749 = + 486
2. 9 357 + = 9 357
3. + 5 982 = 5 982 + 9 285
4. 4 398 = 4 398 +
5. 6 743 + = 1 925 + 6 743
6. 982 + (697 + 528) = (982 + 697) +
7. 629 + (3 456 + 928) = (629 + ) + 928
8. ( + 8 299) + 4 673 = 4 664 + (8 299 + 4 673)
9. (9 465 + ) + 5 374 = 9 465 + (9 825 + 5 374)
10. 4 679 + (8 263 + 9 489) = (8 263 + 9 489) +

C. Find the sum.
1. 38 726 630 3. 12 743 981 5. 900 170 156
+ 90 721 819 + 57 914 162 + 25 107 253

2. 210 725 351 4. 94 026 357
+ 324 186 415 + 15 008 438


Solve the following problems.
1. In a nearby province, a landslide damaged a school building, a bridge, and
a barangay center. The province needs P 1 512 650 to build another school
building, P 3 421 540 to repair the bridge, and P 1 320 000 to rebuild the
barangay center. How much money is needed to rebuild the structures that the
landslide damaged?

2. Two teams of players are going to play volleyball. For the first game, team A
has 10 players while team B has 11. For the second game, team A has 11 players
while team B has 10. Considering both games, is the number of players fair to
the two teams? Why?

A. Find the sum.
1. 194 281 3. 927 014 5. 57 263 455
+ 625 132 + 356 820 + 18 900 516

2. 96 502 291 4. 658 166 800
+ 12 816 589 + 132 823 918

B. Complete the addition equations below and identify the property shown.
1. 325 + 391 = + 325
2. 4 635 + 0 =
3. (928 + 425) + 391 = 982 + ( + 391)
4. + (7 928 + 9 284) = (8 276 + 7 928) + 9 284
5. 17 829 + 0 =


Lesson 5 Subtracting Whole Numbers

A country has 75 826 427 registered
voters. Of this number, 64 936 316
exercised their right to vote during the
presidential election. How many failed
to vote?

In the problem above, you need to subtract to find the answer. Study the following
steps to perform subtraction.

Step 1. Write the subtrahend below Step 3. Subtract the digits from right
the minuend. Align the digits to left in the thousands period.
with the same place value. Regroup if needed.
4 17 12
75 826 427 75 826 427
– 64 936 316 – 64 936 316
890 111

Step 2. Subtract the digits in the Step 4. Subtract the digits from right
units period starting from the to left in the millions period.
rightmost place value (ones). Regroup if needed.
Write the difference under 4 17 12
the digits being subtracted. 75 826 427
Regroup if needed. – 64 936 316
10 890 111
75 826 427
– 64 936 316
111

Therefore, 10 890 111 failed to vote.


Here is another example.
Step 1: Step 2: Step 3: Step 4:
1 18 3 11 18 3 1118
9 342 827 9 342 827 9 342 827 9 342 827
– 7 213 925 – 7 213 925 – 7 213 925 – 7 213 925
902 128 902 2 128 902

Hence, the difference is 2 128 902.

1. What is the period of the digits that you subtracted first?
2. If the subtrahend is less than the minuend, what should you do?

In subtracting numbers, write the subtrahend below the minuend and align the
digits with the same place value. Start subtracting from the ones digits in the units
period. Then, continue subtracting the digits from right to left. If the subtrahend
is greater than the minuend, rename the digit in the next higher place value and
regroup.

A. Subtract.
1. 82 782 837 2. 738 000 136 3. 394 826 927
– 51 653 747 – 259 345 024 – 135 908 918

4. 92 343 983 5. 523 925 321
– 51 814 862 – 121 686 410


B. Find the missing numbers.
1. 6 0 0 0 3 4.
– – 8 2 9 6 2 5 1 3 5
5 6 3 5 2 2 1 6 8 9 9 8 6 3

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

3. 3 2 4 8 9 6
– 3 2 5 6 7

You are asked to prepare an inventory of unsold items in a boutique. How would
you do this without counting the unsold items? Here are the data.

Items Total Number of Pieces Number of Pieces Sold
Dresses 375 892 286 391
Pants 469 527 358 792
Blouses 569 375 387 453

Find the difference.
1. 398 295 2. 87 926 425 3. 895 629 821
– 326 132 – 23 489 514 – 326 783 950

4. 826 713 5. 926 532 491
– 314 824 – 714 856 590


Lesson 6 Multiplying Whole Numbers

Properties of Multiplication
Look at the sets of multiplication equations below. Note what you observe.

Set A Set B Set C
392 × 685 = 685 ×392 (683 × 421) × 5 = 683 × (421 × 5) 8 620 × 0 = 0
268 520 = 268 520 287 543 × 5 = 683 × 2 105 0 × 8 620 = 0
1 437 715 = 1 437 715

Set D Set E
942 × (8 +2) = (942 × 8) + (942 × 2) 90 420 ×1 = 90 420
942 × 10 = (942 × 8) + (942 × 2)
= 7 536 + 1 884
9 420 = 9 420

Look at set A. How are the factors arranged on both sides of the equation? What
happened to the product? In set A, you can observe one property of multiplication called
the commutative property of multiplication. This property states that changing the
order of the factors does not change the product.
Look at set B. How are the factors grouped on each side of the equation? What
happened to the product? In set B, you see that changing the grouping of the factors does
not change the product. This shows the associative property of multiplication.
Look at set C. What happens when a factor is multiplied by zero? This set shows
the zero property of multiplication. This property states that when 0 is a factor, the
product is 0.
How many operations are involved in set D? What are these operations? In set D,
where the distributive property combines multiplication and addition, you can observe
the distributive property of multiplication over addition. This property states that
multiplying a sum by a number gives the same result as multiplying each addend by
the number and then adding their products.
In set E, when 1 is multiplied by a number, the product is that number. This shows
the identity property of multiplication. One (1) is known as the identity element for
multiplication.

Math for Smart Kids Gr.5

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