Download Rule 1. - Thutong

Common fractions.
§1. Factors and multiples.
4 × 3
= 12
12 ÷ 3
factor × factor = product
Rule 1.
= 4
dividend ÷ divisor = quotient
A factor is a number that divides into another number without a remainder.
Example. Factor of 12 is 1, 2, 3, 4, 6, 12 because
12 ÷ 1 = 12
12 ÷ 2 = 6
12 ÷ 3 = 4
12 ÷ 4 = 3
12 ÷ 6 = 2
12 ÷ 12 = 1
A factor pair is two numbers that have the product of a chosen number.
Example. Factors pairs of 12: 1 × 12, 2 × 6, 3 × 4 ( 1, 2, 3, 4, 6, 12 )
Rule 2.
The multiples of a number are all the numbers that are products of a given number
and any other numbers.
Example. The multiples of 3 are 3, 6, 9, 12, 15, 18, . . . , because
3×1=3
3×2=6
3×3=9
3 × 4 = 12
3 × 5 = 15
3 × 6 = 18
Exercise 1. Is it correct? Explain.
Example.
a) 6 is a factor of 24
b) 6 is a factor of 26
c) 27 is a multiple of 3
d) 28 is a multiple of 3
Answer.
a) 6 is a factor of 24. It is correct because 24 ÷ 6 = 4 (no reminder)
b) 6 is a factor of 26. It is not correct because 26 ÷ 6 = 4 (reminder 2)
c) 27 is a multiple of 3. It is correct because 3 × 9 = 27
d) 28 is a multiple of 3. It is not correct because 3 × 9 = 27 and 3 × 10 = 30
1)
4 is a factor of 44
8)
3 is a factor of 69
15)
23 is a multiple of 2
2)
17 is a multiple of 7
9)
11 is a factor of 47
16)
150 is a multiple of 3
3)
28 is a multiple of 2
10)
49 is a multiple of 5
17)
2 is a factor of 748
4)
9 is a factor of 72
11)
56 is a multiple of 7
18)
34 is a multiple of 17
5)
100 is a multiple of 10
12)
5 is a factor of 625
19)
67 is a multiple of 5
6)
12 is a factor of 49
13)
20 is a multiple of 4
20)
10 is a factor of 1 000
7)
12 is a multiple of 4
14)
3 is a factor of 702
21)
125 is a multiple of 5
Exercise 2. We have several boxes with 6 spoons in a box. Can you get 42 spoons without
opening any of the boxes?
Exercise 3. We have several boxes with 8 spoons in a box. Can you get 66 spoons without
opening any of the boxes?
-3-
Exercise 4. The area of a rectangular garden is 24 square metres.
List all possible whole number dimensions the garden can have.
Exercise 5. Write down all the factors of each number.
1)
12
6)
26
11)
15
16)
32
21) 46
2)
25
7)
20
12)
44
17)
22
22) 80
3)
8
8)
100
13)
16
18)
14
23) 60
4)
36
9)
40
14)
50
19)
55
24) 150
5)
17
10)
30
15)
48
20)
10
25) 200
Exercise 6. Find the multiples for each number as indicated.
1)
first six multiples of 4
7)
first six multiples of 20
2)
first five multiples of 6
8)
first nine multiples of 30
3)
first ten multiples of 3
9)
first six multiples of 8
4)
first four multiples of 12
10)
first five multiples of 10
5)
first six multiples of 5
11)
first four multiples of 23
6)
first five multiples of 13
12)
first three multiples of 36
Exercise 7. Solve these problems.
1)
Write down 2 factors of 18.
2)
Write down all the factor pairs of 18.
3)
Which number is a factor of any number?
4)
Copy and circle the multiples of 4:
5)
Write down the first eight multiples of 11.
6)
Find all the factors of number 24.
7)
What is the second multiple of 7?
8)
What is the ninth multiple of 12?
9)
Copy and circle the factors of the first number.
4, 8, 48, 0, 11, 9, 18, 26, 32.
45: 1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 45.
10) Write down the first six multiples of 20 in ascending order.
11) Write down first four multiples of 30 in descending order.
12) Write down all multiples of 10 greater than 100 and less than 220.
13) List all factors of 60.
14) Find all factors of 36 which are more than 7 but less than 12.
15) I am a multiple of 4. I am divisible by 3 and also by 6. What is the smallest number
I could be?
16) Find the sum of the second and fourth multiples of 6.
17) List all the multiples of 3 less than 29.
18) Write 100, 64, 25 and 36 as a product of two equal factors.
19) Find the numbers less than 31 that have 3 factors.
20) Choose from numbers 15, 4, 6, 8, 10, 12, 14, 16 which numbers are:
a) factors of 20
b) multiples of 4
c) factors of 16 and multiples of 4
d) multiples of 3 and factors of 18.
-4-
§2. Highest common factor (HCF). Least common multiple (LCM).
Rule 1.
The highest common factor (HCF) of two or more numbers is the largest number
that is a factor of all of the given numbers.
In mathematics, the highest common factor (HCF) also known as the greatest common factor (GCF).
Example. Find the HCF of 15, 30 and 105.
Questions.
Solution.
1) Find all factors of the given numbers.
1) Factors of 15: 1, 3, 5, 15.
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30.
Factors of 105: 1, 3, 5, 7, 15, 21, 35, 105.
2) Select the ones that are common to
the given numbers.
2) Common factors of 15, 30, 105: 1, 3, 5, 15.
3) Choose the largest number from
common factors.
3) Answer.
Rule 2.
HCF is 15.
The least common multiple (LCM) is simply the smallest of the common multiples.
In mathematics, the least common multiple (LCM) also known as the lowest common multiple.
Example. Find the LCM of 3 and 5.
Method 1.
Questions.
1) Find several multiples
of the given numbers.
Solution.
1) Multiples of 3: 3, 6, 9, 10, 12, 15, 18, 21, 24,
27, 30, 33, . . .
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, . . .
Method 2.
(simple
and
short
method)
2) Find common
multiples.
2) Common multiples of 3 and 5: 15, 30, . . .
3) Choose the smallest
number from common
multiples.
3) LCM is 15.
Find the multiples of the 5 > 3
larger number until one Multiples of 5: 5, 10, 15, . . .
of them is also a multiple
15 is also a multiple of 3, so the L.C.M of 3 and 5 is 15.
of the smaller number.
Exercise 8. Solve these problems.
1)
Find the Least Common Multiple of 8 and 10.
2)
Find the Lowest Common Multiple of 8 and 12.
3)
Find the Highest Common Factor of 8 and 12.
4)
Find the GCF of 10 and 15.
5)
Find LCM of 6 and 8.
6)
Find four common factors of 12 and 30.
7)
Find LCM of 12 and 15.
8)
Find greatest common factor of 9 and 12.
9)
Find HCF of 24 and 36.
10) Find LCM of 24 and 36.
-5-
11) Find LCM of 5 and 8.
12) Find the least common multiple for 4, 6, and 8
13) Find four common multiples of 4 and 8.
14) Find the Highest Common Factor of 24 and 48.
15) Find LCM of 2, 5 and 8.
16) Find the Highest Common Factor of 24 and 60.
17) Find the GCF of 56 and 63.
18) Find LCM of 20 and 36.
19) Find the GCF of 63 and 35.
20) Find the Highest Common Factor of 45 and 100.
Exercise 9. During the summer months, one ice cream truck visits Jeannette's neighbourhood
every 4 days and another ice cream truck visits her neighbourhood every 5 days. If both trucks
visited today, when is the next time both trucks will visit on the same day?
Exercise 10. Samantha has two pieces of cloth. One piece is 72 cm wide and the other piece is
90 cm 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?
Exercise 11.
Ben exercises every 12 days and Isabel every 8 days. Ben and Isabel both
exercised today. How many days will it be until they exercise together again?
Exercise 12. Rosa is making a game board that is 16 cm by 24 cm. She wants to use square
tiles. What is the largest tile she can use?
Exercise 13. Sean has 8 cm pieces of a toy train track and Ruth has 18 cm pieces of a train
track. How many of each piece would each child need to build tracks that are equal in length?
Exercise 14. I am planting 50 apple trees and 30 peach trees. I want the same number and type
of trees per row. What is the maximum number of trees I can plant per row?
§ 3. Divisibility criteria.
1) by 2.
A number is divisible by 2, if the last digit is 0, 2, 4, 6, or 8.
Numbers, which are divisible by 2 are called even numbers.
Otherwise, numbers are called odd numbers.
2) by 3. A number is divisible by 3, if the SUM of its digits is
divisible by 3.
3) by 5. A number is divisible by 5, if its last digit is 0 or 5.
Exercise 15. Write down three numbers which are:
1)
divisible by 2
4)
divisible by 2 and by 5
2)
divisible by 5
5)
even and divisible by 5
3)
not divisible by 2 and by 5
6)
odd and divisible by 5
-6-
Exercise 16. Determine whether 7168 is divisible by 2, 3, 5, 6.
Example.
Determine whether 225 is divisible by 2, 3, 5, 6.
Solution.
225 is not divisible by 2 since the last digit is not 0, 2, 4, 6 or 8.
225 is divisible by 3 since the sum of the digits is 9 ( 2 + 2 + 5 = 9 ),
and 9 is divisible by 3.
225 is divisible by 5 since the last digit is 5.
225 is not divisible by 6 since it is not divisible by both 2 and 3.
Answer.
225 is divisible by 3 and 5.
Exercise 17. Determine whether 9 042 is divisible by 2, 3, 5, 6.
Exercise 18. Determine whether 35 120 is divisible by 2, 3, 5, 6.
Exercise 19. Solve these problems.
1)
Arrange these digits to make four different numbers that are all divisible by 5: 8, 0, 5, 4, 1.
2)
Write the smallest and the largest numbers divisible by both 3 and 5. All answers must be
between 100 and 1 000, using only the digits 2, 5 and 8.
3)
What is the smallest number that must be added to 78 223 to make it divisible by 3?
4)
Name the digit which must replace a, so that the following number will be divisible by 3:
7 a 34.
5)
Name the digits which must replace a and m , so that the following numbers will be divisible
by 3. (Find two values each for a and m).
a) 4a12
b) m415
6)
Arrange these digits to make four different numbers that are all divisible by 5: 7, 0, 5, 3.
7)
Write the smallest and the largest numbers divisible by both 2 and 3. All answers must be
between 100 and 1 000, using only the digits 2, 5 and 8.
8)
What is the smallest number that must be added to 61 010 to make it divisible by 3?
9)
Name the digit which must replace a, so that the following number will be divisible by 3:
3 12a.
10) Name the digits which must replace a so that the following number will be divisible by 3.
(Find two values for a): 2 61a.
11) Write the smallest 3-digit number which divisible by 5.
12) Write the smallest even 4-digit number.
13) Write the largest odd 4-digit number.
14) Write the largest 5-digit number which divisible by 5.
15) Write the smallest even 6-digit number.
16) Write the largest even 6-digit number.
17) Write the smallest even 6-digit number which divisible by 3.
18) Write the largest odd 5-digit number which divisible by 3.
-7-
§ 4. Understanding and naming fractions.
Problem 1.
We want to share 1 pizza between 4 people. How can we
do it?
We have to cut (divide) the pizza into 4 equal pieces so that
everyone can get one – fourth of the pizza.
Problem 2.
If we want to share a pizza between 7 people we have to
cut (divide) the pizza into 7 equal pieces so that everyone
will get one-seventh of the pizza.
Problem 3.
We cut a pizza into 5 equal pieces. If during dinner we eat
3 pieces, which part of the pizza did we eat?
We eat three-fifths of the pizza.
A number that can represent part of a whole unit we call it a common fraction ( or we can
just say – fraction).
We write the fraction one-fourth as
3
1
4
, the fraction one-seventh as
1
7
and fraction three-fifth
as .
5
In the fraction
3
5
the top number 3 we call the numerator and the bottom number 5 we call it
the denominator.
Rule 1.
The denominator (the bottom number) of a fraction tells how many equal pieces
an object was divided into.
Rule 2.
The numerator (the top number) of a fraction tells us how many of the parts in the
unit (object, whole) are to be taken.
The line between the numerator and denominator is known as the fraction bar.
It is also called the division bar.
𝟑
𝟒
The fraction
(three-fourths) tells us that the whole unit is divided
into 4 equal parts and that we have taken 3 of those parts.
Representing, writing and naming the fraction.
Exercise 20. Read the following common fractions. Name the numerator and denominator.
1)
2)
3)
2
7
3
9
1
10
4)
5)
6)
7
10
4
11
7
12
7)
8)
9)
1
18
1
4
5
5
10)
11)
12)
-8-
10
9
4
15
2
5
13)
14)
15)
3
6
7
21
3
2
16)
17)
18)
3
8
4
7
2
2