The document is about fractions and contains sections on definition, proper and improper fractions, equivalent fractions, and operations. It defines what a fraction is, discusses proper and improper fractions, equivalent fractions through amplification and reduction, and how to add, subtract, multiply and divide fractions.

1. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Fractions
Matem´ticas 1o E.S.O.
a
Alberto Pardo Milan´s
e
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2. Indice Definition Proper fractions Equivalent fractions Operations Exercises
1 Definition
2 Proper fractions and improper fractions
3 Equivalent fractions
4 Operations
5 Exercises
Alberto Pardo Milan´s
e Fractions

3. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Definition
Alberto Pardo Milan´s
e Fractions

4. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Definition
What is 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 naturals,
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.
Example:
3
9
3 is the numerator,
9 is the denominator.
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e Fractions

5. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Proper fractions and improper
fractions
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e Fractions

6. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Proper fractions and improper fractions
Proper fraction
a
A proper fraction is a fraction such that< 1.
b
(The numerator is less than the denominator)
Improper fraction
a
A fraction > 1 is called an improper fraction.
b
(The numerator is greater than the denominator)
Examples:
2 9
is a proper fraction. is an improper fraction.
7 7
2 is less than 7 9 is greater than 7
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e Fractions

7. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Equivalent fractions
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e Fractions

8. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Equivalent fractions
Definition
Equivalent fractions are different fractions which represent the
same amount.
Example:
2 4
= because they represent the same amount.
7 14
2 4
7 14
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e Fractions

9. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Equivalent fractions
Amplify and reduce fractions
To amplify a fraction we must multiply the numerator and the
denominator by any number.
To reduce a fraction we must divide the numerator and the
denominator by any common factor.
Examples:
2
To amplify we can multiply the numerator and the denominator
7
6
by 3 =⇒ .
21
15
To reduce we can divide the numerator and the denominator by
45
3
5 =⇒ .
9
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10. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Equivalent fractions
Simplest form of a fraction
A lowest terms fraction is a fraction that can not be reduced
anymore. If you reduce a fraction to the lowest terms fraction you
find the simplest form of the fraction.
To reduce a fraction to the simplest form, we can use two
methods:
• Divide the numerator and the denominator by any common
factor and keep dividing until there are no common factors
(only 1).
• Divide the numerator and the denominator by their Greatest
Common Factor.
Alberto Pardo Milan´s
e Fractions

11. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Equivalent fractions
How to obtain the simplest form of a fraction
Method 1
• Divide the numerator and the denominator by any common
factor and keep dividing until there are no common factors
(only 1).
Example:
18
To reduce to the lowest terms fractions we can divide by 2 and
60
keep dividing until there are no common factors (only 1)
18 9 3
= = , and GCF (3, 10) = 1.
60 30 10
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12. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Equivalent fractions
How to obtain the simplest form of a fraction
Method 2
• Divide the numerator and the denominator by their Greatest
Common Factor.
Example:
18
To reduce to the lowest terms fractions we can divide 18 and
60
18 3
60 by GCF (18, 60) = 6, so = .
60 10
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e Fractions

13. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Operations
Alberto Pardo Milan´s
e Fractions

14. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Operations
Add and subtract
To add or subtract fractions with the same denominator, we have
to add the numerators and keep the same denominator.
2 1 3 4 2 2
Examples: + = . − = .
5 5 5 7 7 7
To add or subtract fractions with different denominators, first we
have to amplify these fractions changing the denominators to the
same number (using the LCM or any multiple of the
denominators).
3 1 11
Examples: + = , because
15 6 30
3 6 1 5
LCM (15, 6) = 30 = =
15 30 6 30
3 1 6 5 11
+ = + = .
15 6 30 30 30
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15. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Operations
Multiply and divide
To multiply fractions, multiply the numerators and multiply the
denominators.
2 3 2·3 6 3
Example: · = = = .
5 4 5·4 20 10
You can get the reciprocal of a fraction by switching its numerator
and denominator.
4 7
Example: The reciprocal of is .
7 4
To divide by a fraction, multiply by its reciprocal.
1 4 1 7 1·7 7
Example: : = · = = .
8 7 8 4 8·4 32
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16. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Alberto Pardo Milan´s
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17. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 1
Complete:
4
A half= =
5
2 An/one eighth=
=
2
1
A/one third= =
6
2 Three quarters=
=
3
3
A quarter= =
7
2 Forty over twenty-three=
=
4
35
Two fifths= =
75
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18. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 1
Complete:
1
A half=
2
2
=Two halves/ Two over two
2
1
A/one third=
3
2
=Two thirds / Two over three
3
1
A quarter=
4
2
=Two fourths / Two over four
4
2
Two fifths=
5
Alberto Pardo Milan´s
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19. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 1
Complete:
1
A half=
2
2
=Two halves/ Two over two
2
1
A/one third=
3
2
=Two thirds / Two over three
3
1
A quarter=
4
2
=Two fourths / Two over four
4
2
Two fifths=
5
Alberto Pardo Milan´s
e Fractions

20. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 1
Complete:
1
A half=
2
2
=Two halves/ Two over two
2
1
A/one third=
3
2
=Two thirds / Two over three
3
1
A quarter=
4
2
=Two fourths / Two over four
4
2
Two fifths=
5
Alberto Pardo Milan´s
e Fractions

21. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 1
Complete:
1
A half=
2
2
=Two halves/ Two over two
2
1
A/one third=
3
2
=Two thirds / Two over three
3
1
A quarter=
4
2
=Two fourths / Two over four
4
2
Two fifths=
5
Alberto Pardo Milan´s
e Fractions

22. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 1
Complete:
1
A half=
2
2
=Two halves/ Two over two
2
1
A/one third=
3
2
=Two thirds / Two over three
3
1
A quarter=
4
2
=Two fourths / Two over four
4
2
Two fifths=
5
Alberto Pardo Milan´s
e Fractions

23. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 1
Complete:
1
A half=
2
2
=Two halves/ Two over two
2
1
A/one third=
3
2
=Two thirds / Two over three
3
1
A quarter=
4
2
=Two fourths / Two over four
4
2
Two fifths=
5
Alberto Pardo Milan´s
e Fractions

24. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 1
Complete:
1
A half=
2
2
=Two halves/ Two over two
2
1
A/one third=
3
2
=Two thirds / Two over three
3
1
A quarter=
4
2
=Two fourths / Two over four
4
2
Two fifths=
5
Alberto Pardo Milan´s
e Fractions

25. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 1
Complete:
1
A half=
2
2
=Two halves/ Two over two
2
1
A/one third=
3
2
=Two thirds / Two over three
3
1
A quarter=
4
2
=Two fourths / Two over four
4
2
Two fifths=
5
Alberto Pardo Milan´s
e Fractions

26. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 1
Complete:
4
=Four fifths / Four over five
5
1
An/one eighth=
8
1
=A sixth / One over six
6
3
Three quarters=
4
3
=Three sevenths / Three over seven
7
4
Forty over twenty-three=
23
35
=Thirty five over seventy-five
75
Alberto Pardo Milan´s
e Fractions

27. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 1
Complete:
4
=Four fifths / Four over five
5
1
An/one eighth=
8
1
=A sixth / One over six
6
3
Three quarters=
4
3
=Three sevenths / Three over seven
7
4
Forty over twenty-three=
23
35
=Thirty five over seventy-five
75

28. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 1
Complete:
4
=Four fifths / Four over five
5
1
An/one eighth=
8
1
=A sixth / One over six
6
3
Three quarters=
4
3
=Three sevenths / Three over seven
7
4
Forty over twenty-three=
23
35
=Thirty five over seventy-five
75

29. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 1
Complete:
4
=Four fifths / Four over five
5
1
An/one eighth=
8
1
=A sixth / One over six
6
3
Three quarters=
4
3
=Three sevenths / Three over seven
7
4
Forty over twenty-three=
23
35
=Thirty five over seventy-five
75
Alberto Pardo Milan´s
e Fractions

30. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 1
Complete:
4
=Four fifths / Four over five
5
1
An/one eighth=
8
1
=A sixth / One over six
6
3
Three quarters=
4
3
=Three sevenths / Three over seven
7
4
Forty over twenty-three=
23
35
=Thirty five over seventy-five
75
Alberto Pardo Milan´s
e Fractions

31. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 1
Complete:
4
=Four fifths / Four over five
5
1
An/one eighth=
8
1
=A sixth / One over six
6
3
Three quarters=
4
3
=Three sevenths / Three over seven
7
4
Forty over twenty-three=
23
35
=Thirty five over seventy-five
75

32. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 1
Complete:
4
=Four fifths / Four over five
5
1
An/one eighth=
8
1
=A sixth / One over six
6
3
Three quarters=
4
3
=Three sevenths / Three over seven
7
4
Forty over twenty-three=
23
35
=Thirty five over seventy-five
75

33. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 1
Complete:
4
=Four fifths / Four over five
5
1
An/one eighth=
8
1
=A sixth / One over six
6
3
Three quarters=
4
3
=Three sevenths / Three over seven
7
4
Forty over twenty-three=
23
35
=Thirty five over seventy-five
75

34. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 2
Calculate the following and write down how we read the answer:
3 1 3 2 1
+ = − · =
5 5 20 4 5
2 5 1 3
+ = 2− : =
4 6 5 10
1 1 4 1 1 3
· + = + · =
3 5 15 2 3 5
Alberto Pardo Milan´s
e Fractions

35. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 2
Calculate the following and write down how we read the answer:
3 1 4
+ = four over five
5 5 5
2 5 6 10 16 4
+ = + = = four over three
4 6 12 12 12 3
1 1 4 1 3 4 1 7 7
· + = · + = · =
3 5 15 3 15 15 3 15 45
seven over forty-five
3 2 1 3 2 1
− · = − = one over twenty
20 4 5 20 20 20
1 3 10 1 3 9 3 90
2− : = − : = : = =6 six
5 10 5 5 10 5 10 15
1 1 3 3 2 3 5 3 15 1
+ · = + · = · = = A half
2 3 5 6 6 5 6 5 30 2
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e Fractions

36. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 2
Calculate the following and write down how we read the answer:
3 1 4
+ = four over five
5 5 5
2 5 6 10 16 4
+ = + = = four over three
4 6 12 12 12 3
1 1 4 1 3 4 1 7 7
· + = · + = · =
3 5 15 3 15 15 3 15 45
seven over forty-five
3 2 1 3 2 1
− · = − = one over twenty
20 4 5 20 20 20
1 3 10 1 3 9 3 90
2− : = − : = : = =6 six
5 10 5 5 10 5 10 15
1 1 3 3 2 3 5 3 15 1
+ · = + · = · = = A half
2 3 5 6 6 5 6 5 30 2
Alberto Pardo Milan´s
e Fractions

37. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 2
Calculate the following and write down how we read the answer:
3 1 4
+ = four over five
5 5 5
2 5 6 10 16 4
+ = + = = four over three
4 6 12 12 12 3
1 1 4 1 3 4 1 7 7
· + = · + = · =
3 5 15 3 15 15 3 15 45
seven over forty-five
3 2 1 3 2 1
− · = − = one over twenty
20 4 5 20 20 20
1 3 10 1 3 9 3 90
2− : = − : = : = =6 six
5 10 5 5 10 5 10 15
1 1 3 3 2 3 5 3 15 1
+ · = + · = · = = A half
2 3 5 6 6 5 6 5 30 2
Alberto Pardo Milan´s
e Fractions

38. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 2
Calculate the following and write down how we read the answer:
3 1 4
+ = four over five
5 5 5
2 5 6 10 16 4
+ = + = = four over three
4 6 12 12 12 3
1 1 4 1 3 4 1 7 7
· + = · + = · =
3 5 15 3 15 15 3 15 45
seven over forty-five
3 2 1 3 2 1
− · = − = one over twenty
20 4 5 20 20 20
1 3 10 1 3 9 3 90
2− : = − : = : = =6 six
5 10 5 5 10 5 10 15
1 1 3 3 2 3 5 3 15 1
+ · = + · = · = = A half
2 3 5 6 6 5 6 5 30 2
Alberto Pardo Milan´s
e Fractions

39. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 2
Calculate the following and write down how we read the answer:
3 1 4
+ = four over five
5 5 5
2 5 6 10 16 4
+ = + = = four over three
4 6 12 12 12 3
1 1 4 1 3 4 1 7 7
· + = · + = · =
3 5 15 3 15 15 3 15 45
seven over forty-five
3 2 1 3 2 1
− · = − = one over twenty
20 4 5 20 20 20
1 3 10 1 3 9 3 90
2− : = − : = : = =6 six
5 10 5 5 10 5 10 15
1 1 3 3 2 3 5 3 15 1
+ · = + · = · = = A half
2 3 5 6 6 5 6 5 30 2

40. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 2
Calculate the following and write down how we read the answer:
3 1 4
+ = four over five
5 5 5
2 5 6 10 16 4
+ = + = = four over three
4 6 12 12 12 3
1 1 4 1 3 4 1 7 7
· + = · + = · =
3 5 15 3 15 15 3 15 45
seven over forty-five
3 2 1 3 2 1
− · = − = one over twenty
20 4 5 20 20 20
1 3 10 1 3 9 3 90
2− : = − : = : = =6 six
5 10 5 5 10 5 10 15
1 1 3 3 2 3 5 3 15 1
+ · = + · = · = = A half
2 3 5 6 6 5 6 5 30 2

41. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 2
Calculate the following and write down how we read the answer:
3 1 4
+ = four over five
5 5 5
2 5 6 10 16 4
+ = + = = four over three
4 6 12 12 12 3
1 1 4 1 3 4 1 7 7
· + = · + = · =
3 5 15 3 15 15 3 15 45
seven over forty-five
3 2 1 3 2 1
− · = − = one over twenty
20 4 5 20 20 20
1 3 10 1 3 9 3 90
2− : = − : = : = =6 six
5 10 5 5 10 5 10 15
1 1 3 3 2 3 5 3 15 1
+ · = + · = · = = A half
2 3 5 6 6 5 6 5 30 2

42. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 3
Fill in the blank with a number so the fractions are equivalent:
16
=
7 14
11
=
27 9
1
=
5 25
11
=
22 6
144 12
=
60
2 22
=
5
Alberto Pardo Milan´s
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43. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 3
Fill in the blank with a number so the fractions are equivalent:
8 16
=
7 14
33 11
=
27 9
1 5
=
5 25
11 3
=
22 6
144 12
=
60 5
2 22
=
5 55
Alberto Pardo Milan´s
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44. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 4
Order these fractions from least to greatest:
4 3 9 7 1 13
, , , , , .
6 2 15 5 3 6
Alberto Pardo Milan´s
e Fractions

45. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 4
Order these fractions from least to greatest:
4 3 9 7 1 13
, , , , , .
6 2 15 5 3 6
LCM (2, 3, 5, 6, 15) = 2 · 3 · 5 = 60
4 20 3 45 9 18 7 42 1 10 13 65
= , = , = , = , = , =
6 30 2 30 15 30 5 30 3 30 6 30
1 9 4 7 3 13
< < < < < .
3 15 6 5 2 6
Alberto Pardo Milan´s
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46. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 4
Order these fractions from least to greatest:
4 3 9 7 1 13
, , , , , .
6 2 15 5 3 6
LCM (2, 3, 5, 6, 15) = 2 · 3 · 5 = 60
4 20 3 45 9 18 7 42 1 10 13 65
= , = , = , = , = , =
6 30 2 30 15 30 5 30 3 30 6 30
1 9 4 7 3 13
< < < < < .
3 15 6 5 2 6
Alberto Pardo Milan´s
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47. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 4
Order these fractions from least to greatest:
4 3 9 7 1 13
, , , , , .
6 2 15 5 3 6
LCM (2, 3, 5, 6, 15) = 2 · 3 · 5 = 60
4 20 3 45 9 18 7 42 1 10 13 65
= , = , = , = , = , =
6 30 2 30 15 30 5 30 3 30 6 30
1 9 4 7 3 13
< < < < < .
3 15 6 5 2 6
Alberto Pardo Milan´s
e Fractions

48. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 5
Find the lowest terms fraction dividing the numerator and
denominator by common factors until the only common factor is 1:
14 60
= =
6 80
18 22
= =
45 44
Alberto Pardo Milan´s
e Fractions

49. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 5
Find the lowest terms fraction dividing the numerator and
denominator by common factors until the only common factor is 1:
14 7
= 60 30 3
6 6 = =
80 40 4
dividing by
dividing by
2
2 and 10
18 6 2
= = 22 11 1
45 15 5 = =
44 22 2
dividing by
dividing by
2 and 3
2 and 11
Alberto Pardo Milan´s
e Fractions

50. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 5
Find the lowest terms fraction dividing the numerator and
denominator by common factors until the only common factor is 1:
14 7
= 60 30 3
6 6 = =
80 40 4
dividing by
dividing by
2
2 and 10
18 6 2
= = 22 11 1
45 15 5 = =
44 22 2
dividing by
dividing by
2 and 3
2 and 11
Alberto Pardo Milan´s
e Fractions

51. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 5
Find the lowest terms fraction dividing the numerator and
denominator by common factors until the only common factor is 1:
14 7
= 60 30 3
6 6 = =
80 40 4
dividing by
dividing by
2
2 and 10
18 6 2
= = 22 11 1
45 15 5 = =
44 22 2
dividing by
dividing by
2 and 3
2 and 11
Alberto Pardo Milan´s
e Fractions

52. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 5
Find the lowest terms fraction dividing the numerator and
denominator by common factors until the only common factor is 1:
14 7
= 60 30 3
6 6 = =
80 40 4
dividing by
dividing by
2
2 and 10
18 6 2
= = 22 11 1
45 15 5 = =
44 22 2
dividing by
dividing by
2 and 3
2 and 11
Alberto Pardo Milan´s
e Fractions

53. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 5
Find the lowest terms fraction dividing the numerator and
denominator by common factors until the only common factor is 1:
14 7
= 60 30 3
6 6 = =
80 40 4
dividing by
dividing by
2
2 and 10
18 6 2
= = 22 11 1
45 15 5 = =
44 22 2
dividing by
dividing by
2 and 3
2 and 11
Alberto Pardo Milan´s
e Fractions

54. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 5
Find the lowest terms fraction dividing the numerator and
denominator by common factors until the only common factor is 1:
14 7
= 60 30 3
6 6 = =
80 40 4
dividing by
dividing by
2
2 and 10
18 6 2
= = 22 11 1
45 15 5 = =
44 22 2
dividing by
dividing by
2 and 3
2 and 11
Alberto Pardo Milan´s
e Fractions

55. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 5
Find the lowest terms fraction dividing the numerator and
denominator by common factors until the only common factor is 1:
14 7
= 60 30 3
6 6 = =
80 40 4
dividing by
dividing by
2
2 and 10
18 6 2
= = 22 11 1
45 15 5 = =
44 22 2
dividing by
dividing by
2 and 3
2 and 11
Alberto Pardo Milan´s
e Fractions

56. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 5
Find the lowest terms fraction dividing the numerator and
denominator by common factors until the only common factor is 1:
14 7
= 60 30 3
6 6 = =
80 40 4
dividing by
dividing by
2
2 and 10
18 6 2
= = 22 11 1
45 15 5 = =
44 22 2
dividing by
dividing by
2 and 3
2 and 11
Alberto Pardo Milan´s
e Fractions

57. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 6
Find the lowest terms fraction using prime decomposition of the
numerator and denominator:
18
=
30
63
=
42
14
=
21
22
=
33
Alberto Pardo Milan´s
e Fractions

58. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 6
Find the lowest terms fraction using prime decomposition of the
numerator and denominator:
18 2 · 3 ·3 3
= =
30 2 · 3 ·5 5
63 3 ·3· 7 3
= =
42 2· 3 · 7 2
14 2· 7 2
= =
21 3· 7 3
22 2 · 11 2
= =
33 3 · 11 3

59. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 6
Find the lowest terms fraction using prime decomposition of the
numerator and denominator:
18 2 · 3 ·3 3
= =
30 2 · 3 ·5 5
63 3 ·3· 7 3
= =
42 2· 3 · 7 2
14 2· 7 2
= =
21 3· 7 3
22 2 · 11 2
= =
33 3 · 11 3

60. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 6
Find the lowest terms fraction using prime decomposition of the
numerator and denominator:
18 2 · 3 ·3 3
= =
30 2 · 3 ·5 5
63 3 ·3· 7 3
= =
42 2· 3 · 7 2
14 2· 7 2
= =
21 3· 7 3
22 2 · 11 2
= =
33 3 · 11 3

61. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 6
Find the lowest terms fraction using prime decomposition of the
numerator and denominator:
18 2 · 3 ·3 3
= =
30 2 · 3 ·5 5
63 3 ·3· 7 3
= =
42 2· 3 · 7 2
14 2· 7 2
= =
21 3· 7 3
22 2 · 11 2
= =
33 3 · 11 3

62. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 6
Find the lowest terms fraction using prime decomposition of the
numerator and denominator:
18 2 · 3 ·3 3
= =
30 2 · 3 ·5 5
63 3 ·3· 7 3
= =
42 2· 3 · 7 2
14 2· 7 2
= =
21 3· 7 3
22 2 · 11 2
= =
33 3 · 11 3

63. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 7
Find the lowest terms fraction dividing the numerator and
denominator by the GCF:
10
=
6
30
=
60
18
=
24
23
=
46
Alberto Pardo Milan´s
e Fractions

64. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 7
Find the lowest terms fraction dividing the numerator and
denominator by the GCF:
10 5
= dividing by GCF(10,6)=2
6 3
30 1
= dividing by GCF(30,60)=30
60 2
18 3
= dividing by GCF(18,24)=6
24 4
23 1
= dividing by GCF(23,46)=23
46 2

65. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 7
Find the lowest terms fraction dividing the numerator and
denominator by the GCF:
10 5
= dividing by GCF(10,6)=2
6 3
30 1
= dividing by GCF(30,60)=30
60 2
18 3
= dividing by GCF(18,24)=6
24 4
23 1
= dividing by GCF(23,46)=23
46 2

66. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 7
Find the lowest terms fraction dividing the numerator and
denominator by the GCF:
10 5
= dividing by GCF(10,6)=2
6 3
30 1
= dividing by GCF(30,60)=30
60 2
18 3
= dividing by GCF(18,24)=6
24 4
23 1
= dividing by GCF(23,46)=23
46 2
Alberto Pardo Milan´s
e Fractions

67. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 7
Find the lowest terms fraction dividing the numerator and
denominator by the GCF:
10 5
= dividing by GCF(10,6)=2
6 3
30 1
= dividing by GCF(30,60)=30
60 2
18 3
= dividing by GCF(18,24)=6
24 4
23 1
= dividing by GCF(23,46)=23
46 2
Alberto Pardo Milan´s
e Fractions

68. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 7
Find the lowest terms fraction dividing the numerator and
denominator by the GCF:
10 5
= dividing by GCF(10,6)=2
6 3
30 1
= dividing by GCF(30,60)=30
60 2
18 3
= dividing by GCF(18,24)=6
24 4
23 1
= dividing by GCF(23,46)=23
46 2
Alberto Pardo Milan´s
e Fractions

69. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 8
12 11
Alice runs miles on Monday. On Wednesday, she runs miles.
5 6
How many miles does Alice run on both days?
Alberto Pardo Milan´s
e Fractions

70. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 8
12 11
Alice runs miles on Monday. On Wednesday, she runs miles.
5 6
How many miles does Alice run on both days?
Data: She runs:
12 11
m on Monday. m on Wednesday.
5 6
12 11 72 55
+ = + =
5 6 30 30
122 61
= =
30 15
Using: GCF (5, 6) = 30 and
GCF (30, 122) = 2 Answer: Alice
61
runs miles.
15
Alberto Pardo Milan´s
e Fractions

71. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 8
12 11
Alice runs miles on Monday. On Wednesday, she runs miles.
5 6
How many miles does Alice run on both days?
Data: She runs:
12 11
m on Monday. m on Wednesday.
5 6
12 11 72 55
+ = + =
5 6 30 30
122 61
= =
30 15
Using: GCF (5, 6) = 30 and
GCF (30, 122) = 2 Answer: Alice
61
runs miles.
15
Alberto Pardo Milan´s
e Fractions

72. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 8
12 11
Alice runs miles on Monday. On Wednesday, she runs miles.
5 6
How many miles does Alice run on both days?
Data: She runs:
12 11
m on Monday. m on Wednesday.
5 6
12 11 72 55
+ = + =
5 6 30 30
122 61
= =
30 15
Using: GCF (5, 6) = 30 and
GCF (30, 122) = 2 Answer: Alice
61
runs miles.
15
Alberto Pardo Milan´s
e Fractions

73. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 9
Find:
1 My soccer team wins 3 games and lost 5.
Write down a fraction of the games they win:
3
2 Tim sells of his cookies, and now he has 14 left.
5
How many did he have originally?
1
3 Of Tim’s stone collection are white stones. He has total of
5
75 stones.
How many of them are not white?
Alberto Pardo Milan´s
e Fractions

74. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 9
Find:
1 My soccer team wins 3 games and lost 5.
3
Write down a fraction of the games they win:
8
3
2 Tim sells of his cookies, and now he has 14 left.
5
How many did he have originally?
2 1
of them all is 14 , so of them all is 7, so He had
5 5
originally 35.
1
3 Of Tim’s stone collection are white stones. He has total of
5
75 stones. How many of them are not white?
1 1 75
of 75 = · 75 = = 15.
5 5 5
75 − 15 = 60 60 of the stones are not white.

75. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 9
Find:
1 My soccer team wins 3 games and lost 5.
3
Write down a fraction of the games they win:
8
3
2 Tim sells of his cookies, and now he has 14 left.
5
How many did he have originally?
2 1
of them all is 14 , so of them all is 7, so He had
5 5
originally 35.
1
3 Of Tim’s stone collection are white stones. He has total of
5
75 stones. How many of them are not white?
1 1 75
of 75 = · 75 = = 15.
5 5 5
75 − 15 = 60 60 of the stones are not white.

76. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 9
Find:
1 My soccer team wins 3 games and lost 5.
3
Write down a fraction of the games they win:
8
3
2 Tim sells of his cookies, and now he has 14 left.
5
How many did he have originally?
2 1
of them all is 14 , so of them all is 7, so He had
5 5
originally 35.
1
3 Of Tim’s stone collection are white stones. He has total of
5
75 stones. How many of them are not white?
1 1 75
of 75 = · 75 = = 15.
5 5 5
75 − 15 = 60 60 of the stones are not white.

77. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 9
Find:
1 My soccer team wins 3 games and lost 5.
3
Write down a fraction of the games they win:
8
3
2 Tim sells of his cookies, and now he has 14 left.
5
How many did he have originally?
2 1
of them all is 14 , so of them all is 7, so He had
5 5
originally 35.
1
3 Of Tim’s stone collection are white stones. He has total of
5
75 stones. How many of them are not white?
1 1 75
of 75 = · 75 = = 15.
5 5 5
75 − 15 = 60 60 of the stones are not white.

78. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 10
When you write music in the stave you must write signs called
notes, and you need to say how long a note is. The note value is a
code which determines the note‘s duration. The length of a whole
note is equal to four beats (in 4/4 time), half notes are played
for one half the duration of the whole note, crotchet are played
for one quarter the duration of the whole note, quavers are each
played for one eighth the duration of the whole note, semiquaver
are each played for one sixteenth the duration of the whole
note,...
In this piece from Tchaikovsky‘s The Nutcracker draw a vertical
line every two beats.
Alberto Pardo Milan´s
e Fractions

79. Indice Definition Proper fractions Equivalent fractions Operations Exercises
Exercises
Exercise 10
When you write music in the stave you must write signs called
notes, and you need to say how long a note is. The note value is a
code which determines the note‘s duration. The length of a whole
note is equal to four beats (in 4/4 time), half notes are played
for one half the duration of the whole note, crotchet are played
for one quarter the duration of the whole note, quavers are each
played for one eighth the duration of the whole note, semiquaver
are each played for one sixteenth the duration of the whole
note,...
In this piece from Tchaikovsky‘s The Nutcracker draw a vertical
line every two beats.
Alberto Pardo Milan´s
e Fractions