2. Lesson 1: Visualizing Numbers up to 10 000 000 with emphasis on
numbers 100 001 – 10 000 000
A group of doctors donated a total of 234 534 kilograms of rice to the
earthquake victims. Can you imagine how big the number 234 534 is? One way
you can imagine is to think of discs to represent the number shown below.
Two 100 000 Three 10 000 Four 1 000 Five 100 Three 10 Four 1
200 000 30 000 4 000 500 30 4
234 534
What number is represented by these number discs? Write your answer in your
notebook.
100 000
100 000
10 000
10 000
10 000 1 000
1 000
1 000
1 000
100
100
100
100
100
10
10
10
1
1
1
1
100 000
100 000
10 000
10 000
100 000
1 000
1 000
1 000
1 000
100
100
100
1 000
10 000
10
10
100
1
1
10
10
Get Moving!
Explore and Discover!

3. Use number discs to show the following numbers.
1. 10 345 789
2. 389 456
3. 4 234 789
4. 123 098
5. 4 456 678
Read the following items. Then, write your answer to each item in your notebook.
1. There were 345 895 children and adults who watched the football game. Draw number
discs to show the given number.
2. How will you show the number 2 345 789 using number discs?
3. There were 1 234 897 tree seedlings distributed to the barangays by the Department of
Environment and Natural Resources. Draw number discs to show the given number.
4. Mrs. Angeles bought some appliances worth Php 156 907. Represent this amount using
number disc.
5. Which number is 100 000 smaller than 234 456?
a. 134 456
b. 334 456
c. 34 456
d. 434 456
Keep Moving!
Apply Your Skills!

4. Lesson 2 : Reading and Writing Numbers up to 10 000 000 in Symbols
and in Words
EXPLORE AND DISCOVER!
The Department of Agriculture distributed 1 456 897 eggplant seedlings in some farmers
in Region IVA.
How do you read and write the number 1 456 897 in words and in symbol?
The number 1 456 897 is read as “ one million four hundred fifty-six thousand eight
hundred ninety-seven”. In symbol it is 1 456 897.
GET MOVING!
A. Write the following numbers in words.
1. 10 345 897
2. 7 456 902
3. 234 713
4. 678 345
5. 1 097 234
B. Rewrite the numbers correctly by putting a space in the correct places in column A. In
column B, write the numbers in words.
Number Column A Column B
1234678
6578234
10234123
123987
23908

5. KEEP MOVING!
Write the numbers in symbols.
1. Three million seven hundred eighty-six thousand four hundred 10
2. Five hundred ten thousand twenty- six
3. Ten million two hundred seven thousand one hundred seven
4. Six million eighty-three
5. Four million seven hundred twenty thousand three hundred eight
APPLY YOUR SKILLS!
A. 1. What is largest 7- digit number having different digits?
Write it in symbols and in words ___________________________________
2. What is the number next to 234 456?
Write it in symbol and in words ___________________________________
3. What is the smallest 6-digit number having different digits?
Write it in symbol and in words ___________________________________
4. What is the number before 1 567 678?
Write it in symbol and in words ___________________________________
5.What is the number between 890 789 and 890 791?
Write it in symbol and in words ____________________________________

6. 1
Lesson 3 : Rounding Numbers to the Nearest Hundred Thousands and Millions
EXPLORE AND DISCOVER!
The circumference of the earth is 40 053 840 meters.
About how many million meters is the circumference of the earth?
You can find the answer by rounding 40 053 840 to the nearest millions.
Study these examples.
Number
Rounded to
Millions Hundred thousands
3 456 789 3 000 000 3 500 000
14 578 907 15 000 000 14 600 000
2 389 897 2 000 000 2 400 000
1 345 890 1 000 000 1 300 000
7 567 079 8 000 000 7 600 000
GET MOVING!
A. Round each number to the place value of the underlined digit.
1. 12 234 556 ______________________
2. 3 456 871 ______________________
3. 5 098 678 ______________________
4. 9 789 123 ______________________
5. 6 234 189 ______________________
B. Complete the table.
Number
Round to the nearest
Millions Hundred thousands
4 496 709
13 508 807
5 889 897
6 045 890
7 527 099

7. 2
KEEP MOVING!
A. Encircle the numbers that can be rounded to the given number.
1. 8 900 000
8 923 567 8 984 456 8 908 671 8 938 123 8 971 109
2. 7 000 000
6 821 507 7 284 471 7 578 653 6 968 123 7 771 178
3. 4 500 000
4 493 517 4 381 426 4 508 671 4 561 126 4 478 120
4. 3 200 000
3 194 100 3 283 401 3 221 611 3 138 170 3 201 156
5. 6 000 000
5 622 160 6 904 431 5 998 601 6 868 173 6 001 101
B. Round to the highest place value.
1. 345 789 ______________________________
2. 1 563 890 ______________________________
3. 3 213 456 ______________________________
4. 9 912 123 ______________________________
5. 312 465 ______________________________

8. 3
APPLY YOUR SKILLS!
A. Complete the table.
Greatest Number Rounded Number Least Number
3 000 000
2 400 000
10 200 000
6 300 000
1 000 000
B. Read the problem and complete the table below.
Mrs. Angeles bought the following cars for her 5 children.
Honda Civic – Php 1 289 980
Limousine - Php 5 561 003
Ferrari - Php 8 123 315
Jaguar - Php 4 123 980
BMW - Php 3 901 312
Copy the price of each item. Then, round each to the nearest hundred thousand and
millions.
Item Price Hundred Thousands Millions
Honda Civic Php 1 289 980
Limousine Php 5 561 003
Ferrari Php 8 123 315
Jaguar Php 4 123 980
BMW Php 3 901 312

9. Lesson 4: Using Divisibility rules for 2, 5 and 10 to find common
factors
Explore and Discover!
Jaira has 70 collected stamps. Can she shared that equally to 2 friends? 5 Friends? 10
friends?
To solve the problem, you need to know if 2 5 and 10 are factors of 70 or if 70 is divisible
by 2 5 and 10.
To see if the number is divisible by 2, 5 or 10 ,test by checking the ones digit
DIVISIBILITY TEST 70
By 2: Is the ones digit 0,2,4,6 or 8? YES
By 5: Is the ones digit 0 or 5? YES
By 10: Is the ones digit 0? YES
Get Moving!
A. Put a check under each column to identify whether each number is divisible by 3, 6 or 9.
2 5 10
54180
912
2700
5605
568
3765
233
80

10. B. Determine whether the first number listed is divisible by the second number. Write Yes
or No on the blank.
Yes or No
1. 45 ; 2 ____________
2. 5080 ; 5 ____________
3. 90 ; 5 ____________
4. 1180 ; 10 ____________
5. 6998 2 ____________
Keep Moving!
Underline the answer that makes each sentence correct.
1. Twenty is (divisible/not divisible) by ten.
2. Three hundred is (divisible/not divisible) by five.
3. Nine hundred ninety is (divisible/not divisible) by two.
4. One hundred six is (divisible/not divisible) by five.
5. Ten thousand four hundred two is (divisible/not divisible) by two.
Apply Your Skills!
Use divisibility rules to help you solve the following problems.
1. Frances has a collection of 672 stamps. She wants to place the stamps in 2 envelopes
.Can she place the same number of stamps in each envelope?
2. The number of books in Karla’s collection is divisible by 2,5 and 10. She has more than
11 books and fewer than 25 books. How many books does Karla have?

11. Lesson 5: Using Divisibility rules for 3, 6 or 9 to find common factors
Explore and Discover!
Nena’s garden has 414 bougainvillea plants. She wants to arrange them in either rows
of 3, 6 or 9. Which are the possible arrangements of the plants?
To solve the problem, you need to know if 3 6 or 9 are factors of 414 or if 414 is divisible
by 3, 6 or 9.
Recall the rules:
 Divisible by 3: sum of digits of the number is divisible by 3
414= 4+1+4=9, 9 is divisible by 3,therefore 414 is divisible by 3
 Divisible by 6: number is divisible by both 2 and 3
414 is divisible by 2 and 3, therefore 414 is divisible by 6
 Divisible by 9: sum of the digits of the number is divisible by 9
414= 4+1+4=9, 9 is divisible by 9, therefore 414 is divisible by 9
Get Moving!
A. Put a check under each column to identify whether each number is divisible by 3, 6 or 9.
3 6 9
528
1242
3456
624
852
2547
324
120
B. Write the letter of the correct answer on your notebook.
1. Which of the following is divisible by 3?
a.11 b.36 c.23
2. Which of the following is divisible by 9?
a. 108 b.100 c.124
3. Which of the following is divisible by 6?
a. 71 b.134 c.234
4. 3 is a factor of
a.272 b.153 c.92
5. 6 is a factor of
a.84 b.75 c.53

12. Keep Moving!
A . Tell whether each number is divisible by 6 or not.
1.906
2.1322
3.4714
4.5166
5.84510
B. Tell whether each number is divisible by 9 or not.
1.89019
2.4617
3.48753
4.1404
5.75834
Apply Your Skills!
Answer the following questions and write the answers on your notebook.
1. Are all numbers divisible by 9 also divisible by 3?
2. How many numbers between 50 to 100 are divisible by 3?
3. Marjorie wants to arrange her 186 books in three rows. Would it be possible?

13. Lesson 5: Using Divisibility rules for 4, 8, 12 and 11 to find common factors
Explore and Discover!
The school auditorium has 372 chairs. Mrs.Cruz, the principal wants to align them in
either rows of 4,8,12 or 11. Which are the possible alignments of the chairs?
To solve the problem, you need to know if 4, 8,12 or 11 are factors of 372 or if 372 is
divisible by 4, 8, 12 or 11.
Recall the rules:
 Divisible by 4: if the last two digits form a number that is divisible by 4.Also, numbers
ending in two zeros are divisible by 4
372→ 72 ÷ 4 = 18, therefore 372 is divisible by 4,chairs can be aligned by 4
 Divisible by 8: if the number formed by the last 3 digits is divisible by 8.Also, a number
ending in three zeros are also divisible by 8
372 ÷4 = 93,therefore 372 is divisible by 8,chairs can be aligned by 8
 Divisible by 12: if the sum of the digits of the number is divisible by 2 and 3
372= 3+7+2=12, 12 is divisible by 2 and 3, therefore 372 is divisible by 12
 Divisible by 11:if the sum of the digits in the odd places and the sum of the digits in the
even places are equal or their difference is a multiple of 11.
372→(3+2)-7= -2,therefore 372 is not divisible by 11,chairs CANNOT be aligned by 11
Get Moving!
A. Encircle 4,8,11 and 12 if these are factors by these numbers.
1. 1572 - 4 8 11 12
2. 88 - 4 8 11 12
3. 160 - 4 8 11 12
4. 642 - 4 8 11 12
5. 2400 - 4 8 11 12
B. Write on the blank before each number whether it is divisible by 4,8,11 or 12.
_____1. 500
_____2. 3000

14. _____3. 121
_____4. 492
_____5. 648
Keep Moving!
Supply the missing number to make the number divisible by the number opposite it.
1.273_ - 4
2.216_ - 8
3.91_ - 12
4.26_ - 11
5.38_ - 12
Apply Your Skills!
Answer the following questions and write the answers on your notebook.
1. Can 88 stamps be shared equally by 4 friends? 8 friends? 11 friends? 12 friends?
2. Annie wants to distribute his 276 marbles to 12 children. Will each child receive the
same number of marbles?

15. Lesson 7: Solving Routine and Non – Routine Problems Involving Factors, Multiples and
i
Divisibility Rules for 2,3,4,5,6,7,8,9,10,11 and 12
Explore and Discover!
Read the problems below.
Can you solve the problem?
Here are the steps in analyzing and solving word problems.
Study the solution below.
Problem 1:
Michael has to split 60 students in
his class into different groups with
equal number of students each .Not
all students can be in one group and
each group has to have more than
one student. In how many ways can
he form these groups?
Mila baked cookies for her 3 sons
and 2 daughters. If she baked 45
cookies only. How many cookies did
her 3 sons have and 2 daughters
have?
 Understand
Know what is asked: In how many ways can he form these groups?
Know the given facts: 60 students, different groups with equal number of students
each
 Plan
Determine the way/s to be used: factoring, finding the multiples and divisibility rules
 Solve
Show your solution:
A. Find the factors of 60.
60= 4X15= 2X2X3X5
B. Get the numbers(once) on the given factors. Then find the multiples of each
number till you reach 60.
 Multiples of 2 = 2, 4, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40,
42, 44, 46, 48, 50, 52, 54, 56, 58, 60
 Multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60
 Multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60
Members 2 3 4 5 6 10 12 15 20 30
Groups 30 20 15 12 10 6 5 4 3 2
Ways 10 ways
C. Identify the number that can divide 24 equally.
Answer: There are 10 ways to form a group.

16. Problem 2:
Which of the problems do you think is easier to solve? Why?
Can you try solving this problem?
Do this with your partner.
1. A farmer in Calauan gathered 96 pineapples. If he would want to distribute those 96
pineapples in baskets, how many ways can he distribute when each basket has the
same number of pineapples?
Get Moving!
Solve the following problems:
1. Josephine planted 600 onions equally in 20 rows. How many onions were planted in
each row?
Understand:____________________________________________________________
Plan:__________________________________________________________________
Solve:_________________________________________________________________
Check and Look back:____________________________________________________
2. The product of numbers is 138.If one factor is 2, hat is the other factor?
Understand:____________________________________________________________
Plan:__________________________________________________________________
Solve:_________________________________________________________________
Check and Look back:____________________________________________________
 Understand
Know what is asked: How many cookies did her 2 sons have? How many cookies
did they have individually?
Know the given facts: 3 sons and 12 daughters ,30 cookies
 Plan
Determine the way/s to be used: Finding Multiples
 Solve
Find the multiples of 2 and 1 till you get the sum of 30.
3 sons 3 6 9 12 15 18
2 daughters 2 4 6 8 10 12
sum 5 10 15 20 25 30
Answer: There are 18 cookies for her 3 sons and 12 cookies for her 3 daughters.
Show your solution:
 Check and Look back
Review and recheck your answer:

17. 3. How many 5,00 are there in 50,000?
Understand:____________________________________________________________
Plan:__________________________________________________________________
Solve:_________________________________________________________________
Check and Look back:____________________________________________________
Keep Moving!
Solve the following word problems
1. Joel and Harry love playing marbles. Joel has 60 marbles while Harry has 80 marbles .
They plan to keep their marbles in a clay jar. How many clays are there?
2. What is the smallest number of avocados that can be placed in baskets with 50 and 75
pieces ?
3. Joseph has some chocolates. If he shares them equally among 4 friends or 5 friends,
there are always 2 exrta chocolates left. What is the possible number of chocolates
Joseph Could have?
Apply Your Skills!
Try solving more problems.
Read and analyze more problems. Then write your solutions and answers in your
notebook.
1. Lorna earns Php 10,000 a month. How much does she earn in 2 years?
2. A farmer planted 180 monggo seeds equally in 3 big bowls. How many monggo were
planted each row?

19. Lesson 8: Creates Problem (with Reasonable Answers) Factors, Multiples and Divisibility
Rules
Explore and Discover!
Arrange the sentences to form a word problem
Get Moving!
Use the data inside the box to complete the problems below. Solve the problem in your
notebook.
1. The average of five consecutive odd numbers is 155.
What are the five odd numbers?
2. I am thinking of a number . Twice the number plus . What is my number?
3. How many are there on a n 8-by-8 checkerboard?
4. The average of consecutive numbers is 112. What is the largest number?
and the total payment of T-shirts
Ronald paid P155.00 each T-shirts for his indigent students for the to use as a
uniform and the total payment of T-shirts he bought is P1,240.00. How many T-
shirts did he buy?
How many T-shirts did he buy?
P
he bought is P1,240.00.
Ronald paid P155.00 each T-shirts
Answer:
for his indigent students for the to use as a uniform
115 equals 52 squares 5 even

20. Keep Moving!
Study the story problems given below. Complete each problem by creating a question for what
is asked. Then, solve the problem.
1. Karen collected 28 recipes of salad. She devided them equally into different categories.
Fruit salad is her favourite category.
Question:_____________________________________________________________?
Solution and Answer:
2. Mike 42 seashells. He places an equal number of seashells in different boxes.
Question:_____________________________________________________________?
Solution and Answer:
3. Sarah wants to invite her cousins to see a volleyball game with her. She has
Php500.Each ticket costs Php85
Question:_____________________________________________________________?
Solution and Answer:
Apply Your Skills!
Create a problem using the given data. Then solve the problem.
1. Given : 31 sachet of shampoos / month
1 year
Asked: total number of sachet of shampoos in 1year
Problem:_____________________________________________________
Solution and answer:
2. Given : 12 apples / day
3 months
Asked: total number of apples in 3 months
Problem:_____________________________________________________
Solution and answer:
3. Given : 36 red mangoes for Anny
40 green mangoes for Sonia
4 mangoes a day
Asked: total number of mangoes mangoes last individually.
Problem:_____________________________________________________
Solution and answer:

21. Lesson 9: Stating, Explaining, and Interpreting Parenthesis, Multiplication, Division,
Addition, Subtraction (PMDAS) or Grouping, Multiplication, Division, Addition,
Subtraction (GMDAS) rule.
Explore and Discover!
Study the rules in the order of operations.
Examples:
1. 12 ÷ 6 × 2 + 4 – 5
2. 6 ÷ 3 × 2 + 5 – 1
3. 3 ×[4 - 2 × (10 -8) + 12 ÷ 6 × 1]
How will you solve the order of operations?
To solve the order of operations, follow the rules listed above.
Solutions:
1. 12 ÷ 6 × 2 + 4 – 5
2 × 2 + 4 – 5
1. Perform the operations within each pair of grouping symbols
(parenthesis, brackets, and braces) beginning with the innermost pair.
2. Simplify the expression with exponents.
3. Perform multiplication and division as they occur from left to right.
4. Perform addition and subtraction as they occur from left to right.
4 + 4 -5
8 - 5
3
Rule 3
Rule 3
Rule 4

22. 2. 6 ÷ 3 × 2 + 5 – 1
3. 3 ×[4 - 2 × (10 -8) + 12 ÷ 6 × 1]
2× 2 + 5 – 1
4+ 5 – 1
9– 1
8
Rule 3
Rule 3
Rule 4
3 × 2
3 × [4 - 2 × 2 + 12 ÷ 6 × 1]
3 × [4 – 4 + 12 ÷ 6 × 1]
3 × [4 – 4 + 2 × 1]
3 × [4 – 4 + 2]
3 × [o + 2]
3 × 2
6
Rule 1
Rule 3
Rule 3
Rule 4
Rule 1
Rule 3

23. Get Moving
Solve the expression below by following the rules.
1. 32 ÷ 2 × 2
2. 6 ÷ 2 +1 × 4
3. (15 – 6) + (4 – 1) × 23
4. 3 × [3 + 2 × (10 -3)]
5. 12 + 3 × 3 {3 × [4 +(9 – 8) – 2] – 3}
Keep Moving
Perform the indicated operations.
1. (12 + 3) – 7 = N
2. 4 (6 + 8) = N
3. 25 ÷ 5 + 9 = N
4. (18 – 4) + (5 + 3) = N
5. (6 ÷ 3) + (10 × 3) = N
Apply Your Skills!
A. Place parenthesis in the equation so that each equation will be a true statement.
1. 16 – 7 + 8 = 1
2. 3 × 5 – 4 = 3
3. 18 ÷ 6 × 3 = 1
4. 16 – 7 + 8 = 17
5. 12 ÷ 2 + 4 = 2
B. Use the numbers 3, 4, 6, and 8 once in each exercise to make a true statement.
( ______ × _______ ) ÷ ( _____ + ______ ) = 2
______ × _______ ÷ ( ______ × _______ ) = 1
______ × ( ______ × ______ ) ÷ _______ = 18
______ ÷ ( _______ - _______ ) + _______ = 14

24. Lesson 10 : Simplifies a series of operations on whole numbers
involving more than two operations using the PMDAS or GMDAS rule.
Explore and Discover!
A lot of problems in mathematics involve more than one operation. Some may contain a
series of operations and different grouping symbols such as parenthesis ( ),
brackets[ ], and braces { }
 Simplify 4 + 32
.
You need to simplify the term with the exponent before trying to add in the 4:
4 + 32
= 4 + 9 = 13
 Simplify 4 + (2 + 1)2
.
You have to simplify inside the parentheses before I can take the exponent through.
Only then can I do the addition of the 4.
4 + (2 + 1)2
= 4 + (3)2
= 4 + 9 = 13
 Simplify 4 + [–1(–2 – 1)]2
.
You shouldn't try to do these nested parentheses from left to right; that method is simply
too error-prone. Instead, you will try to work from the inside out. First you will simplify
inside the curvy parentheses, then simplify inside the square brackets, and only then
take care of the squaring. After that is done, then you can finally add in the 4:
4 + [–1(–2 – 1)]2
= 4 + [–1(–3)]2
= 4 + [3]2
= 4 + 9
= 13

25. Get Moving!
A. Simplify the following expressions in your notebook following the PEMDAS or
GEMDAS rule.
1. 3 + 52
– 2
2. (2 – 5 ) + 42
– 10 ÷ 5
3. ( 4 – 1 )2
+ 15 ÷ 3 – 1
4. 10 – ( 3 + 4 ÷ 2 )2
+ 15
5. (3 + 2 ) + 8 ÷ 2 x 4
B. Complete the table. Write the order of operations to simplify the given
expression.
Expression Operation first to perform Answer
6) 6 + 5 x 3 – 7
7 ) 6 + 15 - 7
8) 14 – 7 + 18 ÷ 3
To simplify series of operations on whole numbers involving more than two
operations using PEMDAS or GEMDAs rule.
 First, perform operations inside the innermost grouping symbols,
if any.
 Next, evaluate powers.
 Then, perform multiplication/division from left to right.
 Finally, perform addition/subtraction from left to right.

26. 9) 7 + (15 - 6 x 2)
10) 2 ( 2 + 5 ) 2
Keep Moving!
A. Simplify the following expressions in your notebook following the PEMDAS or
GEMDAS rule.
1. (25 + 5) x 3 – 13
2. 19 + 12 x 3 – 5
3. 54 + 23 ÷ 2
4. 21 + 34 x 3 ÷ 2
5. (5 + 3) x (3 + 8)
6. 2 + 16 x 7 – 3
7. 11 + 5 x 5 – 10
8. 31 – 24 ÷ 1 -7
9. 21 + 16 – 8 ÷ 9
10.(19 – 3 ) x 3
Apply Your Skills!
A. Write the expression and solve the given problems.
1. Peter has 450 pesos. He spends 210 pesos on food. Later he divides all the
money into four parts out of which three parts were distributed and one part he

27. keeps for himself. Then he found 50 pesos on the road. Write the final
expression and find the money he has left?
2. You pay 10.00 pesos to buy a package of paper napkins costing 640 pesos. How
much change will you get back? Give the expression also.
3. Liza has 1,000 pesos to be distributed among two groups equally. Later, the first
part is divided among five children and second part is divided among two
brothers. Give the expression that represents how the money distribution
between two groups was dispersed?
Challenge yourself with this problem!
Read the problem and then write the answers to the questions in your notebook.
1) Flora bought 3 notebooks for 10 pesos each, a box of pencils for 21.00
pesos, and a box of pens for 35.00 pesos.
2) Darwin had 35 500 pesos and withdrew 5200 pesos from his bank account.
He bought a pair of trousers for 340 pesos, 2 shirts for 360 pesos each, and 2
pairs of shoes for 540 each. Give the final expression, and determine how
much money Darwin had at the end of the shopping day.

28. 1
Lesson 11: Finding the common factors and the GCF of two - four
numbers using continuous division
You find the common factors and Greatest Common Factor or (GCF) of 24, 30 and 42.
Study the solution below.
Using Continuous Division
2 24 30 42
3 12 15 21
3 5 7
A. Find the common factors and GCF of the following numbers.
1) 24 32 4) 4 6 20
2) 12, 30 42 5) 8 56 84 112
3) 28 32 40
Mrs. Ragas bought 24 mangoes, 30 apples and 42
bananas. If she is going to group these equally, what is
the greatest number of mangoes, apples and bananas in
each group?
GCF: 2 x 3 = 6
The greatest number of mangoes, apples and
bananas is 6.
Explore and Discover!
Get Moving!

29. 2
B. Find the GCF of the following problems using continuous division.
1) Miss Dela Cueva has to prepare number of exercises for her lesson for the
day. She has three classes. One class has 48 pupils, another class has 50,
and another has 46. What must be the largest number of exercises she
should prepare so that each class will have the same number of pupils
working on different problems?
2) There are 10 green, 14 blue, 20 red and 24 yellow bulbs to be used for a
birthday party. They are to be placed in plastic bags so that each bag
contains the same number of green, blue, red and yellow bulbs. What is the
largest number of plastic bags that will be needed?
A. Determine the GCF.
1) 3 72 99 126 GCF = ________
3 24 33 42
8 11 14
2) 5 90 135 180 195 GCF = ________
8 17 36 39
3) 2 42 56 70 98 GCF = ________
7 21 28 35 49
3 4 5 7
A. Find the common factors and GCF the following word problems.
4) The mathematics teacher in a certain elementary school is planning to have
an educational tour for four grade levels with 800 pupils in Grade II, 560
Keep Moving!

30. 3
pupils in Grade III, 480 pupils in Grade IV and 400 pupils in Grade V. What is
the largest number of pupils in a group in each grade level so that each group
has the same number of pupils?
5) A group of 45 dancers will march behind a group of 30 clowns in a parade.
You want to arrange the two groups in rows with the same number of people
in each row, but without mixing the group. What is the greatest number of
people you can have in each row?
6) A group of 45 dancers will march behind a group of 30 clowns in a parade.
You want to arrange the two groups in rows with the same number of people
in each row, but without mixing the group. What is the greatest number of
people you can have in each row?
Read and solve the common factors and GCF the following word problems.
1) The mathematics teacher in a certain elementary school is planning to have
an educational tour for four grade levels with 800 pupils in Grade II, 560
pupils in Grade III, 480 pupils in Grade IV and 400 pupils in Grade V. What is
the largest number of pupils in a group in each grade level so that each group
has the same number of pupils?
2) The parents are making sandwiches for the class picnic. They have 72 ham
slices, 48 cheese slices, and 96 tomato slices. What is the greatest number of
sandwiches they can make if each sandwich has the same filling?
Apply Your Skills!

31. 4
Challenge yourself with this problem!
Read the problem and then write the answers to the questions in your notebook.
1) Gina has two pieces of cloth. One piece is 72 inches wide and the other piece
is 90 inches wide. She wants to cut both pieces into strips of equal width that
are as wide as possible. How wide should she cut the strips?
2) The GCF of 40 and a number 8. What is the number if it lies between 70 and
80?
3) Two numbers have a GCF of 6. If their difference is also equal to their GCF,
what are the numbers?
4) Find three consecutive even numbers such that their GCF is the lowest of the
three even numbers.

32. Lesson 12: Finds the common multiples and LCM of two - four
numbers using continuous division
Explore and Discover!
To solve the problem, you need to find the least common multiple or LCM of 6
and 9 using continuous division.
Here’s how you do it.
3 6 9
2 3
LCM = 2 x 3 x 3 = 18
 Notice that 18 is the least common multiple or LCM of 6 and 9. Therefore, the
smallest number of roses and daisies that she will need for her bouquets is 18.
 You do not include when dealing with common multiples.
Get Moving!
A. Write the letter of the correct answer in your notebook.
1) The common multiples of 6 and 4 are
a. 2, 3, 4 b. 4, 6, 8 c. 8, 12, 16 d. 12, 24, 36
2) The common multiples of 4, 5 and 8 are
a. 8, 10, 16 b. 16, 20, 24 c. 40, 80, 120 d. 50, 90, 100
Lovelyn and Zerma are going to
prepare bouquets with 6 roses to a bouquets
and with 8 daisies to a bouquet. What will be
the smallest number of roses and daisies that
she will need for their bouquets?

33. 3) The LCM of 15 and 9 is
a. 3 b. 15 c. 45 d. 135
4) The LCM of 2, 3, 4 and 5 is
a. 20 b. 30 c. 50 d. 60
5) A common multiple of 3, 5, 9 and 10 is
a. 30 b. 50 c. 90 d. 100
B. Determine the LCM.
1) 3 9 12 LCM = ________
3 4
2) 5 5 10 30 45 LCM = ________
1 2 6 9
3) 2 12 16 20 28 LCM = ________
2 6 8 10 14
3 4 5 7
Keep Moving!
A. Find the Least Common Multiple (LCM) of the given sets of numbers use
continuous division.
1) 3 15 21 LCM = ________
2) 4, 8, 16, 20 LCM = ________
3) 5, 10, 25, 30 LCM = ________
4) 6, 12, 15, 60 LCM = ________
5) 2, 6, 10, 14, LCM = ________

34. B. Find the Least Common Multiple (LCM) the following word problems using
continuous division.
1. What is the least number of candies that can be divided equally among 8, 9,
and 12 children?
2. You bring the drinks for your basketball team every sixth game. Every third
game is a home game. When will you first bring the drinks to a home game?
If there are 20 games in an annual sportsfest, how many times will you bring
the drinks to a home game?
Apply Your Skills!
Read and solve the common factors and GCF the following word problems.
1. Chill water her petchay every 2 days, and her cabbage every 3 days. Not
counting the first day, when is the first time both plants are waters on the same
day? When is the next time?
2. Efren goes home every other day. His wife Fely goes home every 4 days. His
daughter Edlin goes home every 6 days. If they see each other today, when will
they see each other again at home?
Challenge yourself with this problem!
Read the problem and then write the answers to the questions in your notebook.
A. Mr. Placido, a security guard, has 3 successive night duties a week? His wife
who is a nurse has 2 successive night duties. When will they see again if they
are together now?

35. B. Two films were played at the same time. But the length of time of each film is
different from each other. Film A took 120 minutes while Film B took minutes.
After how many minutes will the two films be played at the same time again?
C. Find the LCM of each pair of number.
a. 33
and 23
b. 23
and 42
c. 8, 16, 24 and 48
d. 12, 30, 42, and 66

36. Lesson 13: Solves real life problems involving GCF and LCM of 2-3 given numbers.
Explore and Discover!
Elmer wants to cut as many pieces of wood of equal lengths from three pieces with
lengths 35dm, 49dm, and 56 dm. What is the longest that he could cut each piece? How will you
solve for the answer to the problem?
You can use the 4 - step plan in solving for the answer.
Understand:
What does the problem? The longest that he could cut each piece.
What facts are given? 35 dm, 49 dm, 56 dm of wood.
Plan
How will you solve the problem? By finding the Greatest Common Factor (GCF) and Least
Common Multiple (LCM)
Solve:
How is the solution done? By listing the factors
35: 1, 5, 7, 35
49: 1, 7, 49
56: 1, 2, 4, 7, 8, 14, 28, 56
GCF: 7
By Prime Factorization
35: 5 x 7
49: 7 x 7
56: 7 x 2 x 2 x 2
GCF: 7
Check and Look Back:
What is the answer to the
problem?
7 dm is the longest cut can be done in the wood.
Can you think of other ways to solve the problem?
Get Moving!
Read each problem and answer the questions that follow. Write your answers in your notebook.
1. Mr Ramos has to prepare a number of exercises for his lesson for the day. He has three
classes. One class has 48 students, another class has 50, and another has 46. What
must be the largest number of exercises he should prepare so that each class will have
the same number of students working on different problems?
a. What is asked in the problem? __________________________________________
b. What facts are given? _________________________________________________
c. How will you solve the problem? _________________________________________
d. What is the answer to the problem? ______________________________________

37. 2. Yesterday Tony bought 4 “Monay Bae” for Php. 5.00. He sold 3 “Monay Bae” for Php.
5.00. How many “Monay Bae” did he have to sell in order to make a profit of Php. 5.00.?
a. How will you solve the problem? ______________________________________
b. What is the answer to the problem? ___________________________________
Keep Moving!
Read and solve each problem. Write the solution in your notebook.
1. There are 14 blue and 20 red bulbs to be used for a birthday party. They are to be
placed in the plastic bags so that each bag contains the same number of blue and red
bulbs. What is the largest number of plastic bags that will be needed?
2. Liza has bought eight hair clips for Php. 10.00 and has sold them at 6 clips for
Php.10.00. How many hair clips have to be sold to make a profit of Php. 10.00?
Apply Your Skills!
Challenge yourself by solving these problems. Write your answers in your notebook.
1. Serena wants to create snack bags for a trip she is going on. She has 6 granola bars
and 10 pieces of dried fruit. If the snack bags should be identical without any food left
over, what is the greatest number of snack bags Serena can make?
2. Evelyn is packing equal numbers of apple slices and grapes for snacks. Evelyn bags the
apple slices in groups of 18 and the grapes in groups of 9. What is the smallest number
of grapes that she can pack?

38. Lesson 14: Create problems (with reasonable answers) involving GCF and LCM of 2-3
given numbers.
Explore and Discover!
How will you create a problem involving Greatest Common Factor (GCF) or Least Common
Factor (LCM) given the following information?
You can create a problem by following this guide:
 Familiarize yourself with the concepts of GCF and LCM and their application to real life
situations.
 Think of the type of problem you want to create.
 Read some problems and study their solutions.
Problem 1:
Arielle is making flower arrangements. She has 7 roses and 14 daisies. She wants to
make all the arrangements identical and have no flowers left over.
Problem 2:
Tayli wishes to advertise her business, so she gives packs of 13 red flyers to
each restaurant owner and sets of 20 blue flyers to each clothing store owner. Tayli
realizes that she gave out the same number of red and blue flyers.
Study the following problems as examples for the above information.
Problem 1 involves finding the Greatest Common Factor (GCF)
Arielle is making flower arrangements. She has 7 roses and 14 daisies. She
wants to make all the arrangements identical and have no flowers left over.
What is the greatest possible number of flower arrangements she can make?
Problem 2 involves finding the Least Common Multiple (LCM)
Tayli wishes to advertise her business, so she gives packs of 13 red flyers to each
restaurant owner and sets of 20 blue flyers to each clothing store owner. Tayli realizes that she
gave out the same number of red and blue flyers.
What is the minimum number of flyers of each color she distributed?
Can you make another problem similar to these examples?

39. Get Moving!
A. Write a question to complete each item. Then, solve each problem.
1. Nathan is stocking bathrooms at the hotel where he works. He has 18 rolls of toilet paper and
9 bars of soap. If he wants all bathrooms to be stocked identically, with the same combination of
supplies in each one and nothing left over, what is the greatest combination of bathrooms
Nathan can stock?
2. Sarah’s Shipping and Ryan’s Mail Services both ship packages. Sarah’s trucks will only carry
loads of 18 packages. In contrast, Ryan’s trucks will only carry loads of 11 packages. If both
businesses ended up shipping the same number of packages this morning, what is the
minimum number of packages each must have shipped?
B. Create some problems involving GCF and LCM.
Keep Moving!
Write a problem for the numbers and phrases in the box.
1.
2.
3.
4.
Apply Your Skills!
Create problems involving GCF and LCM based on the following situations:
1. Alaiza arranged the fruits in a box.
2. Luisa shared her toys with her playmates.
3. Anthony is selling newspaper every morning.
4. Volunteers gives clothes in the orphanage.
5. Joey is preparing cake for her friends.
30 and 48 cookies Biggest numberPut inside the
identical container
32, 24, 16 balloons Arranged in a table Greatest number
5, 10 and 15
pictures
Smallest numberCollected
photographs
5, 6, 30 books Least numberNumber of books in
a shelve

40. Lesson 15: Adds fraction and mixed fraction without and with regrouping
Explore and Discover!
A. Musician practiced very well for the concert. He practiced hours yesterday and hours
today. How many hours did he practice in two days?
The mathematical sentence is: + = n
Step 1 Step 2 Step 3
Find the LCD of the add the fractional Add the whole numbers
Given fractions and rename parts reduce in lowest term
These to similar fraction if needed
= = = 5 + 5 +
= = = = =
______ ______
= 5
Will become
While
Will become the answer will be

41. And will become in lowest term.
Get Moving!
Find the sum. Express your answer in lowest terms if possible.
1. 2. 18 3. 4. 5.
+ + + + +
_________ _________ _________ _________ _________
Keep Moving!
Add. Rename the sum in lowest terms if possible.
1. 2. 3. 4. 5.
+ 216 + + + +
_________ _________ _________ _________ _________
Apply Your Skills!
Read each Problem then, answer the question that follow.
1. Joshua picked 2½ buckets of strawberries. Joe picked 3½ buckets of strawberries. How
many buckets of strawberries did the two boys pick?
a. What is asked?
b. What are given?
c. What is the operation to be used?
d. What is the number sentence?
e. How is the solution done? Show your solution?
f. What is the complete answer?

42. 2. Mrs. Gonzales used 2¼ cups of flour to make a plain cake, 3 ½ cups to make brownies and
2 ¾ cups to make doughnuts. How many cups of flour did she use?
a. What is asked?
b. What are given?
c. What is the operation to be used?
d. What is the number sentence?
e. How is the solution done? Show your solution?
f. What is the complete answer?
3. Three hogs weigh respectively kilograms, kilograms and kilograms. What
is the total weight?
a. What is asked?
b. What are given?
c. What is the operation to be used?
d. What is the number sentence?
e. How is the solution done? Show your solution?
f. What is the complete answer?

43. Lesson 16: Subtracting Fractions and Mixed fractions without and with Regrouping
Explore and Discover!
Karen bought kilogram of lanzones and kilograms of banana.
How many more kilograms of lanzones than banana did she buy?
How will you answer the question in the problem?
To answer the question, subtract .
The number sentence is
= Change and to similar fractions by first finding
= the LCD OF 2 and 4.
Subtract the numerators.
Write the sum over the least common denominator
So, Karen bought kilogram more lanzones than banana.
Get Moving
Subtract the following fractions
1.
2.
3.
4.
5.
6.
7.
8.
LCD
LCD

44. Keep Moving
Subtract. Reduce the difference to lowest terms whenever possible.
1. 2. 3. 4. 5.
Apply Your Skills!
A. Read and solve each problem.
1. Jerrie can repair her car in hours. A mechanic can do the same job in 8/12 hour.
How much longer does it take Jerrie to do the job?
2. Pia spent hours in her grandparents’ house. This was of an hour more than
the time she spent at the mall. How much time did she spend at the mall?
3. Ana bought kg of grapes for her younger sister. They ate kg of it. How many
kilograms of grapes were left?
B. Read and solve each problem
1. Lailani has meters of yarn on a ball. After meters were unwound, how many
meters of yarn remained on the ball?
2. There are 5 pitchers of fruit juiced arranged in a row. The first pitcher contains
cups of juice. If each pitcher has cups less juice than the one before it, find the
amount of juice in each of the other pitchers.
3. Alvin weeded the garden in hours and watered the plants in hours. How
much longer did he spend weeding the garden than watering the plants?

45. First Quarter- Week 6
Lesson 17:Solving Routine and non routine problems involving addition and/ or subtraction of
fraction using appropriate problem solving strategies and tools.
EXPLORE AND DISCOVER
Darwin painted his room using 5/6 liter of blue paint and 2/3 liter of white paint.
What color of paint was used more than the other? How much more of it was used than
the other?
You can solve the problem using the following steps.
Understand:
 Know what is asked : The paint color that was used more and by
by how much more
 Know the given facts: 4/6 liter of blue paint; 4/5 liter of white paint
Plan:
 Draw a picture :
5/6
2/3 = 4/6 1/6
 Identify the operation to be use: Subtraction
 Write the number sentence : 5/6 – 2/3 = n
5/6 – 2/3= 5/6 – 4/6 = 1/6
Solve:
 Solution:
Check and Look back:
 Answer: a. more blue paint was used
b. by 1/6 liter
Get Moving

46. Read each problem carefully and then solve.
1. Mark wash his car in 4/5 of an hour, cleaned the garage in 2/6 of an hour, and painted
the garden fence in 2 hours. How long did it take him to do all the tasks?
2. Anthony walked ¼ of a kilometer to Jane’s house and 7/8 of a km to the park. How
far did he walk?
3. Jenny spends ¼ of her daily allowance for snacks, ½ for lunch, 1/8 for transportation,
and saves the rest. What part of her daily allowance does she save?
Keep Moving
Read and solve each problem.
1. Liza spent ¾ hour preparing the soil and 2/3 hour planting. How much time did she
spend in the garden?
2. In a fruit basket, 4/5 of the fruits are bananas and 3/8 are mangoes, which are more-
the bananas or the mangoes?
3. Ronnie had 7/8 gallon of paint. He used 4/5 of it. What fraction of a gallon of paint was
used? How much paint remained?
Apply Your Skills
1. Four fifths of a group of rallyists were students, If 3/8 of the students were female,
what part of the rallyists were male students?
2. Angie covered ¾ of a bulletin board with white paper. Then, she made a Math poster
on 2/3 of the white paper. What part of the board was covered by the poster?
3. A water tank was 20/21 full. If 3/5 of it was used to water the plants, what fraction of
the tank was used in watering the plants?

47. Lesson 18: Creating problems (with reasonable answers) involving addition and/or subtraction
Fraction using appropriate problem strategies
Explore and Discover
How do you create a word problem involving addition, subtraction, or addition and
subtraction of fractions?
You can create a word problem by observing the following guide:
 Familiarize yourselves with the concept of addition and subtraction of fractions
and their application to real- life situations
 Think of the problem you want to write.
 Read some problems and study their solutions.
You also consider the following when creating a problem:
a. Characters
b. Situation/ setting
c. Data
d. Key question
Study the table below:
Name Ribbons used/ left Quantity Unit
Liza White ribbon 2/8 meter
yellow 2/4 meter
Ribbons left 3/6 meter
Study the problem as an example for the given data.
Liza bought 2/8 m of white ribbon and 2/4 m of yellow ribbon to make flowers.
After making 5 flowers, she found out she had 3/6 m of ribbons left. How many
meters of ribbon did she use for the flowers?
Get Moving!
Use the data below to create a one-step word problem involving subtraction of
fractions.
Name Color of Paint used in the
classroom
Quantity Unit
James Blue 4/5 liter
Harold Green 3/6 liter

48. Keep Moving!
Use the data below to create a two- step word problem involving addition and
subtraction of fractions.
Name Quantity Unit Color of cloth
needed in science
project
Carmen 3/5 Meter White
Rowena 2/8 Meter Red
Aileen 5/6 Meter yellow
Apply Your Skills!
Using the table below, create a problem for each of the following.
1. One- step word problem involving addition of fractions (Group 1)
2. One-step word problem involving subtraction of fractions (Group 2)
3. Two-step word problem involving addition and subtraction of fractions (Group 3)
Name Quantity Quantity ( in kilograms)
Sharon Grapes 1/2
Vilma Papaya 3/4
Nora Lansones 4/8

49. 1 x 1 = 1
5 2 10
LESSON 19 Visualizes Multiplication Of Fractions Using Models
Explore and Discover!
Darwin had a piece of plastic cover ½ meter long. He used 1/5 of it to cover his book. What fractional
part of the plastic cover did he use?
To find the answer, get 1of 1.
5 2
The diagram below will help you find the answer.
If the rectangle represents 1 metre, the shaded part represents the plastic cover of Darwin.
To find 1 of 1 , we have to divide the shaded
parts into 5 equal parts and take 1 of the equal
parts 5 2
The part of the whole representing the product of 1/5 x 1/2 is the region that has been shaded twice.
Notice that the rectangle has been divided into 10 equal parts and 1 part is shaded twice so 1/5 x 1/2
must be 1/10, Darwin used 1/10 metre of plastic cover.
STUDY THESE EXAMPLE
Let us use a region to find 1/3 of 2/5 or 1/3 x 2/5.
The region at the left has been divided vertically into 5 parts of
the same size. Therefore each parts is 1/5 of the whole region.
Ask. How many fifths are shaded? What part of the region is
shaded?

50. Thus: 1 2122
3 of 5 equals 3 x 5 = 15
Get Moving
1. Write a multiplication equation for each visualization and find the answer
a
X = =
_________
b.
X =
= _________
c.
x = =
____
Next picture 2 horizontal lines have been drawn to divide
each 1/5 into 3 equal parts.
Into how many equal parts is the region now divided?
How many small regions are shaded?
Next , 1/3 of the shaded part has been shaded in
another direction. How many of the small regions are
now shaded twice ? What part of the whole region is
shaded twice?

51. Keep Moving !
Match the picture in Column A with the multiplication sentence in Column B. Write your answer
in your notebook.
A B
Apply Your Skills:
Write the multiplication equation for each and find the answer.
a. 11
2 of 4
b. 13
3 of 4
c. 11
5 of 2

52. x =
=
A. Illustrate and Find the Product
1 323 5. 31
3 45 4 5 4
x =
x
=
x
= x =
=
3.
4.
1.
2.

53. LESSON 20: MULTIPLIES A FRACTION AND A WHOLE NUMBER AND
ANOTHER FRACTION
First Quarter : Week 7
EXPLORE AND DISCOVER!
A. Michelle bought kg of carrots for her pet rabbit. Her
pet ate of this how many carrots did her pet
eat?
To know the amount of carrots the rabbit ate, let us compute of . How do we right of
in a number sentence ? x = N
Let’s solve this in two ways
First let us use an illustration to get the product.
3
4
133 1
3 4 12 4
The double shaded parts show the parts of the carrot eaten by the rabbit. .When reduced to
lowest terms is
So, Michelle’s pet ate ¼ of the carrots.
1
3
1
3
1
3
3
4
3
4
x = =
3
12
1
4
1
3
3
4

54. 3
12
Let’s solve x again, this time through computations.
Multiply the numerators x =
Multiply the denominators x =
Reduce the product to lowest terms ÷ =
Study these other examples:
of == or x 8
If we have a whole number like 8 we can also write it as a fraction with a denominator of 1.
1
4 because 0r 2
GET MOVING
Direction : Cut each strip. You may paste them in a chipboard or cartolina to make them harder.
STEP 1
1
3 3
4
3
STEP 2
1
3
3
4
3
12
STEP 3
3
3
1
4
1
4 8
1
4
1
4
1 8
so4 x 8 = 4 = 2
X 8 = 2,
1
4
x 8
1
= 8
4

55. KEEP MOVING !
Read the problem:
A. Ramona has yard of a beautiful lace. She gave of it to her friend for their project.
If 2/3 of the apples above are green, then how many are green apples ? Color them to
show your answer ?
If 6/7 of the socks are black,
then how many pieces are
not black ? Color them to
show
ofof the pizza shown will be
given to my classmate. Draw line/s on it to
show the part to be given.
1
2
3
4
4
6
2
3

56. What part of the lace did Ramona give to her friend ?
B. Bon bought 8 kilos of mango in the market. He shared of it with his relatives for their
outing. How many kilos of mango did he share?
C. One fifth of the 40 pupils of Miss Ramos are Math Club members. How many pupils are
Math Club members ?
APPLY YOUR SKILLS !
Find the product. Always express answers in lowest terms.
1. What is 2,andof ? __________________
2. What is the product of 3/4 ,4 and ½ ? __________
3. What is the value of N ? 5 x ¾ x = N
4. Multiply, and 4.The answer is ________.
5. What is the product of 5 , ½ and? _________
2
3
5
6
3
4
2
4
3
4
2
6
3
6

57. Lesson 21: Multiplying Mentally Proper Fractions with the Denominators Up to 10.
Explore and Discover!
Nelia has ½ piece of a cartolina. She shares 1/3 of it to Joe who needs it very badly for his
Science project. What part of the cartolina did Nelia Share?
How will you solve for the product of the fractions mentally?
Here’s how to do it.
Do these steps in your mind.
11 1x1 =1
2 x 3 = 2x3 =6 Step 1: Multiply the numerators.
1x1 = 1
Step 2: Multiply the denominators.
2x3 = 6
Step 3: Express the answer in lowest terms if needed.
Get Moving!
Solve the following mentally.
13318541 2 1
5 × 4 = 7 × 3 = 10 × 6= 9× 2 = 8×3 =
A. Find the product mentally.
7461256379
9 x 5 = 10x6 = 3x9 = 8x 5 = 8x 10 =
B. Find the value of N.
1237548134
2 x 3 =N 5 x 8 =N 6 x 9 =N 9 x 3 =N 7 x 5 =N
Apply Your Skills!
A. Read the problems, then solve them mentally.
1. Roy harvested 5/6 crates of mangoes. He sold 4/5 of them. What part of the crate of
mangoes was sold?
2. Dino bought ½ L of white paint. He used ½ of this to paint the doghouse that he
made. How many litres of paint did he use?
B. Understand the equations carefully, then answer it.

58. 1. In the equation 2/3 x ½ =N, what is the value of N?
2. If you multiply ¼ and 2/3, what will be the product?
3. Multiply 2/5, 3/4 and 4/5. It will give a product of_____.
4. What is the product of 2/7,3/8, and ½ ? _______
5. Multiply 2/3, 5/6 and ¾. The answer is _______

59. 1kilo
o
1kilo
o
1kilo
o
1kilo
o
1kilo
o
1kilo
o
1kilo
o
Lesson 22: Solving Routine or Non-routine Problems Involving
Multiplication Without or With
 Addition or Subtraction of Fractions and Whole Numbers
Using Appropriate Problem Solving Strategies or Tools.
Explore and Discover!
Problem 1: Lucy’s mother worked in her boutique for 2 ½ hours each day for 2 weeks.
How many hours did she work in all?
You can solve the problem using the steps below:
 Understand
Know what is asked: No. of hours she worked in all.
Know the given facts: 2 ½ hours and 14 days = 2 weeks
 Plan
Determine the operation to be used: Multiplication
Write the number sentence: 2 ½ × 14 = N
 Solve: Show your solution:
2 ½ × 14 = N
2 ½ × 14/1 = N
5/2 × 14/1 = N
1 7
5/1 × 7/1 = 35/1 or 35 hours
Lucy’s mother worked 35 hours in 2 weeks or 14 days.
 Check and look back: Did I do the operation correctly?
Is my answer reasonable? Did I write my answer in a complete sentence?
Problem 2: Jose harvested 15 kilograms of guavas from the orchard. He gave 2/5 of them to
his neighbors. How many kilograms of guavas did he share to his neighbors?
 Understand
Know what is asked: No. of kilograms he shared to his neighbors.
Know the given facts: 15 kilograms of guava and 2/5 part shared to his
neighbors
 Plan: Make a diagram or drawing.
 Draw 15 kilograms of guavas

60. 1kilo
o
1kilo 1kilo
1kilo
o
1kilo
o
1kilo
o
1kilo
o
1kilo
o
1kilo
o
1kilo
o
1kilo
o
Divide 15 kilograms of guavas into group of 5.
There will be 3 kilograms per group.
Circle 2 parts of it to show 2/5.
Each group shows 1/5 or 3 kilograms of guavas. There are 2 groups of 1/5 and
this shows 2/5.
So, 2 × 3 = 6 kilograms of guavas he shared to his neighbors.
 Look back: To check 15 × 2/5 = 6 kilograms.
15 × 2/5 = 3 15 × 2 =6 or 6 kilograms of guavas
1 5 1 1
Answer: He shared 6 kilograms of guavas to his neighbors.
Can you try solving the following problems?
Do this with your partner.
1. One – fourth meter of the cloth was left from Evelyn’s uniform. Her friend asked
3/5 of it to her Science project. What part of the cloth did her friend get?
2. Mrs. Albano gathered 50 eggs from her poultry. She gave 4/8 of these to her co-
teachers. How many eggs were given to her co-teachers?
Get Moving!
Solve the following problems.
1. A recipe for doughnuts needs 1 ¼ cups of flour. Mother will prepare 1 1/3 times the
recipe. How much flour will she need?
2. On Jenny’s birthday, her mother prepared 48 cupcakes. If ¾ of the cupcakes were
served, how many cupcakes were served?
1kilo
3 kilo
1kilo1kilo1kilo
o
1kilo 1kilo
3 kilo 3 kilo
3 kilo 3 kilo
3 kilo
1kilo 1kilo 1kilo 1kilo 1kilo 1kilo 1kilo 1kilo1kilo

61. 3. Nelson had 3 ½ liters of paint. He used 2/3 of it to paint their fence. What part of the
paint did he used?
4. Remy had ¾ meter long of lace. She gave 1/3 of it to her classmate to decorate her
Science project. What part of the lace was given to her classmate?
5. Everyday Alvin spends 3 ½ hours reading books. How many hours does he spend in a
week reading books?
Keep Moving!
Solve the following problems. Write your solutions and answers in your notebook.
1. Anselmo spent 6/8 of his time in morning studying Math and Science. He spent ¼ of this
time studying Science. What fraction of the total time did he spend studying Science?
2. What is the area of a rectangle whose length is 8/10 m and width is 2/3 m?
3. Aling Aning planted vegetables on 4/7 of her vacant lot. Two thirds of it was planted with
pechay. What fraction of the vacant lot had pechay?
4. Lorna had 2 ½ liters of beef broth. She used 3/5 of it to make soup. How much beef
broth did she use to make soup?
5. A recipe calls for 1 1/3 litres of milk. How many litres of milk do you need to make 2
recipes?
Apply Your Skills!
Try solving more problems!
1. Joanne signed up for 24 dancing lessons. She took ¾ of them by April. How many
dancing lessons did she take by April?
2. Father’s monthly salary amounts to ₱ 20,500. Every time he receives his salary, he
deposits 1/5 of it. How much is father’s monthly savings? How much is his annual
savings?
3. How far can father go in 8 ½ hours if he travels at an average speed of 15 kilometers an
hour?
4. Six pitchers each filled with ¾ liter of juice were served to Luisa’s visitors. Find the total
amount of juice served.
5. The school Home Economics Club had a buko pie sale. The member sold 2/3 of their
pies in the morning and 1/6 of their pies in the afternoon. If 150 pies were left, how many
pies had been sold?

62. Alice used ¼ teaspoon of salt on her cake recipe. How much salt is needed if
she will make 8 recipes of it?
Lesson 23: Creating Problems (with reasonable answer) Involving
Multiplication of Fractions
Explore and Discover!
How do you create a word problem involving multiplication of fractions?
Observe the guide below in creating a word problem:
 Familiarize yourselves with the different Mathematical concepts
especially multiplication of fractions.
 Analyse the data and think of the type of problems you
want to create.
 Read and study some sample problems and be familiar with
the organization of data on the problem.
 Learn about the basic terms or word clues often used in mathematical problems.
The following data is important to be considered in creating a problem:
 Name/Character
 Situation/Setting
 Data
 Unit
 Key question
Study the given data below:
teaspoon of salt for cake recipe
8 recipes to make
Study the problem as an example for the data given.

63. Get Moving!
Create a one-step word problem involving multiplication of fractions using the
data given.
1. – part of Mrs. Marco’s class who joined the field trip
30 - total number of pupils
2. kg- mixed nuts bought by Ana
of it given to Alma
3. 12 km- distance of Elmo’s house to school
of total distance is being travelled by jeepney
4. Php 24,000- amount raised by PTA of Del Mundo Elementary School
spend for completing the school’s fence
5. 5 kg of rice bought by mother
cooked for dinner
Keep Moving
Create a one-step word problem involving multiplication of fractions using the data given.
1. kg meat bought by Mr. Guanson
kg cooked
2. Php 420 earnings of Gab every day in working at a printing house
for transportation allowance
3. 5 m ribbon bought by Kim for gift wrapping
used in wrapping square boxes
used in wrapping rectangular boxes

64. Apply Your Skills!
Create a word problem involving multiplication of fractions for each of the following. Use the
data below.
1. One-step word problem involving multiplication of fractions.
2. Two or more step word problem involving multiplication of fractions.
Name Monthly Salary Monthly Savings
Randy Php 12,000
Ryan Php 15, 000

65. 4 x
𝟏
𝟒
=
𝟒
𝟏
x
𝟏
𝟒
=
𝟒
𝟒
= 1
𝟏
𝟐
𝟑
x
𝟑
𝟓
=
𝟓
𝟑
x
𝟑
𝟓
=
𝟏𝟓
𝟏𝟓
= 1
𝟑
𝟕
x 𝟐
𝟏
𝟑
=
𝟑
𝟕
x
𝟕
𝟑
=
𝟐𝟏
𝟐𝟏
= 1
Lesson 24: Show that multiplying a fraction by its
reciprocal is equal to 1
Explore and Discover!
Janice shared the pizza she made among her 5 friends.
Each one received of the pizza. Nothing was left for her.
How much pizza did she make?
Find: 5 x
Solution:
5 x = x = = 1
Answer: Janice made 1 whole pizza divided among her five friends.
Study some more examples:
 Two numbers whose product is 1 are reciprocals of each other.
 To find the reciprocal of a fraction make the numerator of the
fraction the denominator of the reciprocal and the denominator of
the fraction the numerator of the reciprocal.
 Mixed fractions must first be converted to improper fractions
before the method can be applied.
 Write the reciprocal of a whole number as fraction.

66. Get Moving!
Give the reciprocal of each number.
1. 51 6. 11. 2
2. 3 7. 12. 35
3. 8. 13. 5
4. 9. 14. 12
5. 24 10.5 15.
Keep Moving!
Write the missing factor.
1. x = 1 6. x = 1
2. x = 1 7. x = 1
3. x = 1 8. 5 x = 1
4. 15 x = 1 9. x = 1
5. x = 1 10. x = 1
Apply Your Skills!
1. The reciprocal of a number is , and their product is 1. What is the number?
2. Two numbers are reciprocals of each other. One number is 36 times as large as the other. What
are the numbers?
X = 1
3. What is the reciprocal of a number whose numerator is 8 times as great as 3 and the
denominator is half the numerator?

67. Lesson 25: Visualizing Division of fraction
Explore and Discover!
Janella helps her mother cut meter long ribbon from meter ribbon. How many strips can she
cut? Study the illustration and solution below.
Number sentence :
÷ = __________
From the figure we can see that there are twenty-eight in . Therefore, the answer is 28.
Get Moving!
Use illustrations to answer each question.
1. How many are there in 9?
2. How many s are there in ?
3. How many are there in 24?
4. How many are there in ?
5.
6. How many are there in ?
Keep Moving!
Illustrate to find the quotient.
1. ÷ = 2. ÷ = 3. ÷ = 4. ÷ = 5. ÷ =

68. Apply your Skills!
Use a model to solve the problem.
1. Camille needs m of cloth for a banner. If she has m of cloth, how many banner can
she make?
2. Mrs. Dator needs4 liters of fresh milk. The store has liter packs of fresh milk. How
many packs does she need to buy?
3. Mario has 8 logs to be cut in fourths to make fence posts. How many fence post can he
make?
4. Mother bought 5 apples. She divided them into halves. How many pieces of apple
were there?

69. Lesson 26: Dividing Simple Fractions and Whole Number and a Fraction
Explore and Discover!
Nathan wants to share his buko pie with his friends. He has of the buko pie, and he wants to
give each friend of the buko pie. How many friends can Nathan feed?
Solution:
Without any illustration we can solve the problem following the steps in dividing simple fraction.
Solution:
÷ = n
÷ = n Write the reciprocal of the divisor
x = n Change the division sign into multiplication sign
x = Multiply the numerators then the denominators
÷ = 4 Express in lowest term if necessary
Therefore ÷ = 4
Nathan can feed 4 of his friends.
Problem 2.
Jane received 3 guavas from her friend. She cut it into pieces. How many halves did she
have?
Use real guava to solve the problem
Problem 3
Lina has of a chocolate bar. It will be divided equally among 4 persons, what part of the
chocolate bar will each one get?
We can solve the problem following these steps:
Step 1. Write thee number sentence. ÷ 4 = ___
Step 2. Rename the whole number in fraction form ÷ = ___
Step 3. Get the reciprocal of the divisor then proceed to
Multiplication of fractions. ÷ =

70. Step 4. Write the product of the numerators over the product
of the denominators; and reduce the fractions if needed
Get Moving!
Read and analyze each question then solve.
1. What is the quotient of and ?
2. If you divide by times, what is the answer?
3. What is the quotient of divided by ?
4. ÷ =
5. ÷ = n
Find the quotient. Show your solution.
1) 6  = n
2) 16  = n
3) 14  = n
4) 8 = n
5) 30 = n
Keep Moving!
Find each quotient.
1. ÷ = n 2. ÷ = n 3. ÷ = n 4. ÷ = n 5. ÷ = n
Find the quotient.
1. 10 = n 2. 15 = n 3. 45 = n 4. 36 = n 5. 28 =
Apply your Skills!
1. Find the number of eights ( ) in
2. AlingNarda repacked kg of pepper into kg bags . How many bags of pepper can she
make?
6
5
4
3
7
2
8
1
4
3

71. 3. There are 4 kilograms of rice. Each girl scout can consume kg of rice per meal. For
how many girl scouts is the rice enough for a meal?
5
1

72. Lesson 27 : Solving Routine or Non-Routine Problems Involving Division Without or With
Any of the Other Operations of Fractions and Whole Numbers Using
Appropriate Problem Solving Strategies and Tools.
Explore and Discover!
Read the problems below.
Can you solve the problem?
Here are the steps in analyzing and solving the problems.
Study the solution below.
Problem 1
A 4-meter piece of wood is to be
divided into pieces, each 2/3 m long.
How many pieces can be cut from it?
Nicole has 42 meters of ribbon. She
uses meters for every box she
makes. How many boxes can she
make from the ribbon?
 Understand
Know what is asked: Number of pieces that can be cut
Know the given facts: 4 meter piece of wood
m each long
 Plan
Determine the operation to be used: Division
Write the number sentence: 4 = n
 Solve
Show your solution:
4 = n x = or 6
 Check and Look back
Review and check your answer
Answer: There were 6 pieces of wood
 Check and look back
Did I do the operations correctly?
Is my answer reasonable?

73. Problem 2
Can you try solving the following problems?
Do this with your partner.
1. If a skirt requires 1 ¼ meters of cloth, how many skirts can be made from 21
meters of cloth?
2. Linda made a trip of 112 kilometers in 2 ½ hours. What was her average
speed?
Get Moving!
Solve the following problems.
1. A log 5 ¾ meters long will be cut into 6 equal pieces. How long will each piece
be?
 Understand: _____________________________________________
 Plan:___________________________________________________
 Solve: __________________________________________________
 Look back: ______________________________________________
2. Andrea has 35 meters of cloth. How many aprons can she make if each
apron requires meters?
 Understand: _____________________________________________
 Plan:___________________________________________________
 Solve: __________________________________________________
 Look back: ______________________________________________
 Understand
Know what is asked: The number of boxes that can be make from the ribbon
Know the given facts: 42 meters of ribbon
meters for each box
 Plan
Determine the operation to be used: Division
Write the number sentence: 42 = n
Solve
Show your solution:
42 = n = = = 18
Using cancellation
= x = or 18
 Check and Look back
Review and recheck your answer:
Answer: Nicole can make 18 boxes from the ribbon.
 Check and Look back
Review and check your answer
Answer: There were 6 pieces of wood

74. 3. Calix needs to divide cups of flour among 3 recipes. How many cups
of flour does each recipe need?
 Understand: _____________________________________________
 Plan:___________________________________________________
 Solve: __________________________________________________
 Look back: ______________________________________________
4. The class of Lora is repacking goods for the outreach. How many kg
packs of sugar can they make out of a bag that contains 15 kg of sugar?
 Understand: _____________________________________________
 Plan:___________________________________________________
 Solve: __________________________________________________
 Look back: ______________________________________________
5. Nikki has 8 meters of fabric to make shirts. If each shirt requires m of
fabric, how many shorts can she make?
 Understand: _____________________________________________
 Plan: ___________________________________________________
 Solve: __________________________________________________
 Look back: ______________________________________________
Keep Moving!
Solve the following problems. Write your solutions and answers in your notebook.
1. Jeff has meters of rope. He wants to make 3 pieces of clothes hanger
out of it. How long will each clothes hanger be?
2. A farmer bought kilograms of fertilizer for his rice, camote, and potato
crops. If the fertilizer will be used equally on the three crops, how much
will be used for each crop?
3. Mr. Reyes has a coconut plantation that measures hectares. If the
entire plantation is to be subdivided among 36 tenants, how much would
be each tenant’s share ?
4. Donald was able to harvest 2 ¼ kg of tomatoes from each of 4 plots.
Then he divided them equally into 6 piles. How many kilograms of
tomatoes did each pile have?
5. Lucy equally poured 4/5 liter of lemonade into 6 cups. How much
lemonade did each cup have?
Apply Your Skills!
Try solving more problems!
Read and analyze the following problems. Solve them in any method you like
1. Mrs. Gibe has to pack 50 kg of rice. How many plastic bags are needed if
each bag can contain kg of rice?
2. Merllie makes hand towels for sale. How many hand towels can she make
from 6 meters of cloth if meter is used for 1 hand towel?
3. How many meter of cloth can be cut from 40 meters of ribbon?

75. 4. On their trip to Laguna, Mr. Santos’ family bought 4 baskets of lanzones.
Each basket contained kg of lanzones. The family shared the 4 baskets of
lanzones among 8 persons. How many kilograms of lanzones did each one
receive?
5. How many m pieces of ribbon can be cut from a 50 m of ribbon?

76. Lesson 28 : Creating Problems (with reasonable answers) Involving Division or With Any
of Other Operations of Fractions and Whole Numbers
Explore and Discover!
How do you create a word problem involving division or with any of other operations of
fractions and whole number?
You can create a word problem by observing the following guide:
 Familiarized yourselves with the concepts division with other operation of
fractions and whole number and their application to real-life situations.
 Think of the problem you want to write.
 Read some problems and study their solutions.
You also consider the following when creating a problem:
a. Characters
b. Situation/Setting
c. Data
d. Key Question
Look at the given data below.
 60 kilograms of rice
 kg of rice in each plastic bag
 Number of plastic bags needed for repacking
Can you now complete the word problem below and solve for the correct answer, too?
Do this in your notebook.
Mrs. Ana has to pack _____________. How many plastic bags are needed if each bag
can contain______________?
Get Moving!
Create a word problem from the given data below.
Solve the problem in your notebook.
1. 12 kilograms of flour
¾ kilograms in each plastic bag
Number of plastic bags used
Problem:______________________________________________________________
Solution and Answer:____________________________________________________
2. 30 meters long of wood
1 ½ meters long for each piece
The pieces of woods that can be cut
Problem:______________________________________________________________
Solution and Answer:____________________________________________________

77. 3. meters long of electric wire
6 equal pieces
The length of each wire
Problem:______________________________________________________________
Solution and Answer:____________________________________________________
4. of a pie
4 persons
The part of the pie each one gets
Problem:______________________________________________________________
Solution and Answer:____________________________________________________
5. 5 meters of cloth
2/3 meters for each scarf
The number of scarves that can be make
Problem:______________________________________________________________
Solution and Answer:____________________________________________________
Keep Moving!
Complete each problem by creating a question for what is asked. Then, solve the problem.
1. Ella has 15 kilograms of rice for sale. She placed these in plastic bags. Each bag
contains kilograms.
Question:________________________________________________________
Solution and Answer: ______________________________________________
2. A log meters long will be cut into 7 equal pieces.
Question:________________________________________________________
Solution and Answer: ______________________________________________
3. Nene has cake. She divided it among her 6 friends.
Question:________________________________________________________
Solution and Answer: ______________________________________________
4. Michelle needs to divide cups of flour among 3 recipes.
Question:________________________________________________________
Solution and Answer: ______________________________________________
5. A tailor has a bolt of cloth 20 meters long. Each uniform needs meters of cloth.
Question:________________________________________________________
Solution and Answer: ______________________________________________

78. Apply You Skills!
Create a problem using the given data. Then, solve the problem.
1. Given: 50 kilograms of cholcolate
kilograms packed in each box
Asked: Number of boxes used
Problem: _____________________________________________________
Solution and Answer: ___________________________________________
2. Given: 12 meters of fabric
meters for each shirt
Asked: Number of shirts that can make
Problem: _____________________________________________________
Solution and Answer: ___________________________________________
3. Given: 15 meters of cloth
meters for each cloth
Asked: Numbe