1. Index Divisibility Fractions Simplest form Operations Exercises
Factors and Fractions
Matem´ticas 2o E.S.O.
a
Alberto Pardo Milan´s
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2. Index Divisibility Fractions Simplest form Operations Exercises
1 Divisibility
2 Fractions
3 Simplest form of a fraction
4 Operations
5 Exercises
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3. Index Divisibility Fractions Simplest form Operations Exercises
Divisibility
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4. Index Divisibility Fractions Simplest form Operations Exercises
Divisibility
Factors and multiples
A factor of a number n, is a number d which divides n.
Read ⇐⇒ if and only if.
d is a factor of n ⇐⇒
d is a divisor of n ⇐⇒
d divides n ⇐⇒
n is divisible by d ⇐⇒
n is a multiple of d.
Examples:
−7 divides 14 ⇐⇒
−7 is a factor of 14 ⇐⇒
14 is divisible by −7 ⇐⇒
14 is a multiple of −7.
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5. Index Divisibility Fractions Simplest form Operations Exercises
Divisibility
Primes
A prime number is a positive number that has only two positive
factors 1 and the number itself (1 is not considered a prime
number as it only has one positive factor). A number with more
than two positive factors it is called composite number.
Examples: 3 is a prime number because has only two positive fac-
tors (1 and 3). 6 is a composite number because has four positive
factors (1, 2, 3 and 6).
Two numbers are relatively prime if they have no common positive
divisors except 1.
Example: 6 and 25 are relatively prime because positive factors
of 6 are 1, 2, 3, 6 and positive factors of 25 are 1, 5, 25.
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6. Index Divisibility Fractions Simplest form Operations Exercises
Divisibility
Prime decomposition
Prime decomposition is to find the set of prime factors of an
integer: To factorize a number you have to express the number as
a product of its prime factors.
To factorize negative numbers use also −1.
Examples:
• 90 = 2 · 45 = 2 · 3 · 15 = 2 · 3 · 3 · 5 = 2 · 32 · 5.
• −25 = −1 · 25 = −1 · 5 · 5 = −1 · 52 .
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7. Index Divisibility Fractions Simplest form Operations Exercises
Divisibility
GCD and LCM
The Greatest Common Divisor (GCD) is the highest number that
is a common factor of two or more numbers. It is clear that if
GCD(a, b) = 1, a and b are relatively prime.
Examples: GCD(42, 110) = 2, because positive factors of 42
are 1, 2, 3, 6, 7, 14, 21, 42, and positive factors of 110 are
1, 2, 5, 10, 11, 22, 55, 110.
12 and 35 are relatively prime, because GCD(12, 35) = 1.
The Least Common Multiple (LCM) is the lowest positive number
that is a common multiple of two or more numbers.
Example: LCM(6, 9) = 18, because positive multiples of 6 are
6, 12, 18, 24, . . . and positive multiples of 9 are 9, 18, 27, . . .
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8. Index Divisibility Fractions Simplest form Operations Exercises
Fractions
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9. Index Divisibility Fractions Simplest form Operations Exercises
Fractions
What’s a fraction?
A fraction is a number that represents a part of something.
a
Fractions are written in the form , where a and b are integers,
b
a
and the number b is not zero. Read a over b.
b
The number a is called numerator, and the number b is called
denominator.
13
Example: , 13 is the numerator and 25 is the denominator.
25
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10. Index Divisibility Fractions Simplest form Operations Exercises
Fractions
Proper fractions and improper fractions
A proper fraction is a fraction that is less than one. A fraction
greater than one is called an improper fraction.
12 23
Examples: is a proper fraction. is an improper fraction.
17 15
Equivalent fractions
Equivalent fractions are different fractions that name the same
amount.
3 6
Example: =
7 14
because they represent the same amount.
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11. Index Divisibility Fractions Simplest form Operations Exercises
Fractions
Amplify and reduce fractions
To find equivalent fractions, multiply or divide the numerator and
the denominator by the same number. If you multiply by the same
number, you amplify the fraction. If you divide by the same factor,
you reduce the fraction.
Examples:
6
• To amplify we
9
6 60
multiply the numerator and the denominator by 10: = .
9 90
6
• To reduce we
9
6 2
divide the numerator and the denominator by 3: = .
9 3
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12. Index Divisibility Fractions Simplest form Operations Exercises
Simplest form of a fraction
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13. Index Divisibility Fractions Simplest form Operations Exercises
Simplest form of a fraction
Lowest Terms Fraction
A lowest terms fraction is a fraction that can not be reduced
anymore. To write a fraction in the simplest form find the lowest
terms fraction. To reduce a fraction to the lowest terms fraction,
we can use two methods:
• Divide the numerator and the denominator by the Greatest
Common Factor.
• Divide the numerator and the denominator by any common
factor and keep dividing until they are relatively prime.
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14. Index Divisibility Fractions Simplest form Operations Exercises
Simplest form of a fraction
Examples:
70
• To obtain the lowest terms fraction of we can divide
105
70 2
70 and 105 by GCD(70, 105) = 35, so = .
105 3
70
• To reduce to the lowest terms fraction we can di-
105
vide by 5 and keep dividing until there are no common factors
70 14 2
but one = = , (note 2 and 3 are relatively prime ).
105 21 3
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15. Index Divisibility Fractions Simplest form Operations Exercises
Operations
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16. Index Divisibility Fractions Simplest form Operations Exercises
Operations
Add and subtract
To add or subtract fractions with like denominators, add the
numerators and keep the same denominator. To add or subtract
fractions with unlike denominators, first amplify these fractions to
obtain like denominators (using the LCM or any multiple).
Simplify, if possible.
Examples:
4 8 12 4
• + = = .
15 15 15 5
3 1 3 1 6 5 11
• To add + using the LCM: + = + = .
15 6 15 6 30 30 30
Because LCM(15, 6) = 30.
• Using another common multiple of 15 and 6, for example 60:
3 1 12 10 22 11
+ = + = = .
15 6 60 60 60 30
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17. Index Divisibility Fractions Simplest form Operations Exercises
Operations
Multiply and divide
To multiply fractions, multiply the numerators and multiply the
denominators then simplify.
12 3 12 · 3 36 18
Example: · = = = .
5 14 5 · 14 70 35
You can get the reciprocal of a fraction by switching its numerator
and denominator.
14 21
Example: The reciprocal of is .
21 14
To divide by a fraction,multiply by its reciprocal and simplify,if
possible.
2 4 2 17 2 · 17 34 17
Example: : = · = = = .
8 17 8 4 8·4 32 16
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18. Index Divisibility Fractions Simplest form Operations Exercises
Exercises
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19. Index Divisibility Fractions Simplest form Operations Exercises
Exercises
Exercise 1
There are 28 students in our class and we want to divide it into
groups with equal number of students. How many ways can the
class be divided into groups? What are they?
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20. Index Divisibility Fractions Simplest form Operations Exercises
Exercises
Exercise 2
Mary wants to serve hot dogs for 48 people. Sausages come in
packages of 8 and hot dog buns come in packages of 12. She wants
to have enough to serve everyone and have none left over, how
many packages of sausages and hot dog buns should she purchase?
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21. Index Divisibility Fractions Simplest form Operations Exercises
Exercises
Exercise 3
Peter works in a florist. Today He is making identical floral
arrangements for a bridal party. He has 84 daisies, 66 lilies, and 30
orchids. He wants each arrangement to have the same number of
each flower. What is the greatest number of arrangements that he
can make if every flower is used?
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22. Index Divisibility Fractions Simplest form Operations Exercises
Exercises
Exercise 4
Samantha loves the sea. She has kayaking lessons every fifth day
and diving lessons every seventh day. If she had a kayaking lesson
and a diving lesson on June the sixth, when will be the next date
on which she has both kayaking and diving lessons?
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23. Index Divisibility Fractions Simplest form Operations Exercises
Exercises
Exercise 5
There are two flashing neon lights. One blinks every 4 seconds and
the other blinks every 6 seconds. If they are turned on exactly at
the same time, how many times will they blink at the same time in
a minute?
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24. Index Divisibility Fractions Simplest form Operations Exercises
Exercises
Exercise 6
Peter sells books. He made 240e selling children’s books, 140e
from cookbooks, and 280e from paperback books. He gets exactly
the same benefit from each book. What is the most that Peter
could get for each book? How many books could Peter have sold
then?
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25. Index Divisibility Fractions Simplest form Operations Exercises
Exercises
Exercise 7
Complete:
3
=
4
A half= One and a half=
Three fifths=
5 1
= 1 4 =
2 = 3
5
Two thirds= Two and a quarter=
Two quarters=
1 2
= 1 4 =
3 = 5
7
A quarter= Two and four tenths=
An/one eighth=
1
3 1 =
= 12
6
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26. Index Divisibility Fractions Simplest form Operations Exercises
Exercises
Exercise 8
Write a fraction in simplest form beside each sentence:
There are forty-two girls out of seventy-two people:
The baby snake was just four and three quarters inches long:
Fifteen out of the twenty students has dogs as pets:
I walk five and a quarter miles a day:
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27. Index Divisibility Fractions Simplest form Operations Exercises
Exercises
Exercise 9
An orchard has 72 peach trees 15 apple trees, and 23 lemon trees.
In simplest form, what fraction of the trees are peach trees?
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28. Index Divisibility Fractions Simplest form Operations Exercises
Exercises
Exercise 10
George swims seven eighths miles and Mat swims ten twelfths
miles. Who swims farther?
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29. Index Divisibility Fractions Simplest form Operations Exercises
Exercises
Exercise 11
Compare these fractions using the least common denominator:
2 3 9 7 −1 −2 13
, , , , , , .
−6 4 15 5 −3 10 12
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30. Index Divisibility Fractions Simplest form Operations Exercises
Exercises
Exercise 12
Lance and Frank ate seven twelfths of a pizza. If Frank ate a third
of the pizza, how much of the pizza did Lance eat?
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31. Index Divisibility Fractions Simplest form Operations Exercises
Exercises
Exercise 13
Tangram means seven boards of skill.
The Tangram is a Chinese
puzzle consisting of seven flat
shapes (called tans) which are
put together to form different
shapes. Study the tans at the
right forming a square. The
side of the square is 1 cm, so
the area is 1 cm2 . Find the
fractional value of each piece.
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32. Index Divisibility Fractions Simplest form Operations Exercises
Exercises
Exercise 14
Of the 20 students, a quarter of the students wear jeans, a fith
wear shorts and a tenth wear a skirt. How many students wear
something else?
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33. Index Divisibility Fractions Simplest form Operations Exercises
Exercises
Exercise 15
My car´s gas tank was empty and now is full but I paid forty euros
for it. To go to Marbella I need two fifths of the tank, how much
money is this fraction of the tank?
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34. Index Divisibility Fractions Simplest form Operations Exercises
Exercises
Exercise 16
Peter cut six apples into quarters, how many pieces of apple did he
have?
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35. Index Divisibility Fractions Simplest form Operations Exercises
Exercises
Exercise 17
1
Mike cleaned of the 24 m2 yard . How many are left?
4
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36. Index Divisibility Fractions Simplest form Operations Exercises
Exercises
Exercise 18
3
Thomas drunk cup of milk, and now he has 24 cl left. How
5
many ml did the cup have originally?
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37. Index Divisibility Fractions Simplest form Operations Exercises
Exercises
Exercise 19
2
Eric is reading a book with 195 pages. He read of the pages
5
today. How many are left?
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38. Index Divisibility Fractions Simplest form Operations Exercises
Exercises
Exercise 20
1
Of Paul’s stone collection are white stones. He has total of 75
5
stones. How many of them are not white?
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39. Index Divisibility Fractions Simplest form Operations Exercises
Exercises
Exercise 21
1
Mom baked cookies, and gave of them to Beth, and of the
2
2
remaining ones, she gave to Seth. The rest, which was 4
3
cookies, she ate herself. How many did she bake originally?
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40. Index Divisibility Fractions Simplest form Operations Exercises
Exercises
Exercise 22
1
This time Mom baked again cookies, and gave of them to Beth,
2
1
and of the remaining ones she gave to Seth. There were 2
3
cookies left. How many did she bake originally?
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