Unit 2: Fractions. Mathematics 2nd E.S.O.

Teacher: Miguel Angel Hernández

UNIT 2: FRACTIONS.

Fractions:

A fraction is a number that expresses a part of a unit or a part of a quantity.

a
Fractions are written in the form where a and b are natural numbers, and b is not 0.
b
They can be written also in the form a/b.

The number a is called the numerator, it is always an integer, and the number b is called the
denominator, it can be any natural number except zero.

The denominator is the number, which indicates how many equal parts the unit is divided into.

The numerator of a fraction indicates how many equal parts of the unit are taken.

4
represents the shared portion of the rectangle
6

Reading fractions:

We use the cardinals to name the numerator and the ordinals for the denominator with two
exceptions, when the denominators are 2 or 4. For denominators larger than 10 we can say “over”
and do not use ordinal, so we read:

1 3 2
one half three halves two thirds
2 2 3

1 3 1
one quarter or one fourth three quarters or three fourths one fifth
4 4 5

2 3 5
two sixths three sevenths five ninths
6 7 9

6 12 17
six tenths twelve over fifteen or seventeen over
10 15 32
twelve fifteenths thirty-two

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Unit 2: Fractions. Mathematics 2nd E.S.O. Teacher: Miguel Angel Hernández

1. Write in words the following fractions:

5 1
five eighths one tenth
8 10
1 2
3 5
4 4
7 10
5 6
12 19
7 3
200 10
524 3456
1000 5461

2. Express in figures:

Six Sevenths Four Elevenths A half

Three quarters Two thirds Seven over twenty

Seventeen over three hundred and forty one Thirty-two over five hundred and twenty-two

Sixty-two over seventy-one

3. Draw using the geometric shapes, the portion that the fractions represent:

1 3 2
6 4 5

1 5 4
3 8 7

In this exercise, you can see the fractions as a part of the unit: the denominator tell us how many
equal parts we have in total, and the numerator, how many parts we take.

2
Unit 2: Fractions. Mathematics 2nd E.S.O. Teacher: Miguel Angel Hernández

4. Calculate the decimal number that each fraction represents:

1
=0,25 (1:4=0,25)
4

3 1 2
4 5 3

In this exercise, you can see the fractions as a quotient or division. Every fraction represents a
decimal number. We can calculate it dividing numerator by denominator.

5. Calculate:

3 5 3
of 28 of 30 of 150
4 6 10

In this exercise, you can see the fractions as an operator, the fraction is a part of a quantity.
Remember: the numerator multiplies this number, and the denominator divides it.

Equivalent Fractions:

Equivalent fractions are different fractions that name the same

amount 4
6
4 2
Example: and are equivalent as we can see in the
6 3
drawing on the right.

2
3

The rule is: The value of a fraction does not change multiplying or dividing its numerator and
denominator by the same number.

The process of dividing numerator and denominator by the same number is called reduction.

12 3
is equivalent to , because we have divided both the numerator and the denominator by
20 5
4.

1 2 3 100 512
The fractions , , , and are all equivalent fractions.
2 4 6 200 1024

We can test if two fractions are equivalent by cross-multiplying their numerators and denominators.
This is also called taking the cross-product.

3
Unit 2: Fractions. Mathematics 2nd E.S.O. Teacher: Miguel Angel Hernández

a c a c
Two fractions and are equivalent, and we write = , if a⋅d =b⋅c .
b d b d

12 24
So if we want to test if and are equivalent fractions:
20 40
The first cross-product of the first numerator and the second denominator: 12⋅40=480 .
The second cross-product is the product of the second numerator and the first denominator:
24⋅20=480 .
Since the cross-product are the same, the fractions are equivalent.

Simplest form:

When numerator and denominator have no common factors the fraction is in the simplest form or
in its lowest terms.

Example:

4 2 1
We know that = = .
12 6 3
4 1
4 and 12 have a common factor (4), so can be written as (Divide the top and the bottom
12 3
by 4).

2 1
2 and 6 have a common factor (2), so can be written as (Divide the top and bottom by 2).
6 3
1
However, 1 and 3 have no common factors, so is the simplest form of these fractions.
3

There are two methods of reducing a fraction to the lowest terms.

Method 1:
Divide the numerator and denominator by the HCF of both:

12 12 12 :6 2
Example: . The HCF of 12 and 30 is 6 so = = .
30 30 30 :6 5

Method 2:
Divide the numerator and denominator by any common factor. Keep on dividing until there are no
more common factors.

12 12 : 2 6 6:3 2
Example: = = = =
30 30: 2 15 15: 3 5

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Unit 2: Fractions. Mathematics 2nd E.S.O. Teacher: Miguel Angel Hernández

1. Write a sequence of equivalent fractions as in the example of the first line:

Equivalent Fractions
1 2 3 4 5 6 7
2 4 6 8 10 12 14
2
5
3
7
−1
3

2. Express these fractions in the simplest form:

22 26
40 52

29 70
58 119

84 42
119 238

20 56
48 84

22 34
66 68

52 54
117 117

39 20
234 160

72 324
272 720

720 284
284 720

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Unit 2: Fractions. Mathematics 2nd E.S.O. Teacher: Miguel Angel Hernández

3. Express as its simplest form a fraction that describes these situations:

I have 27 pens and 9 of them are black.

In our school 48 of the 84 teachers are women.

4. Count how many boys and girls are in our class of 2nd ESO and write down the fraction of
each compared with the total number of pupils.

Comparing and ordering fractions:

Fractions with the same denominator: When two fractions have the same denominator, the
greatest of them is the fraction with the greatest numerator.

5 2 2 5
Example: and ⇒ 
6 6 6 6

Fractions with the same numerator: When two fractions have the same numerator, the greatest of
them is the fraction with the smallest denominator.

2 2 2 2
Example: and ⇒ 
3 5 5 3

Fractions with different numerators and denominators:

If you want to order two fractions with different denominators, you have to reduce to lowest
common denominator:
1. Find the LCM of both denominators.
2. Rewrite the fractions as equivalent fractions with the LCM as the denominator.
3. Then, we will have two fractions with the same denominator, so we order the numerator.

3 4
Example: and
4 5

LCM (4,5)=20

3 15 4 16 3 4
= and = ⇒ 
4 20 5 20 4 5

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Unit 2: Fractions. Mathematics 2nd E.S.O. Teacher: Miguel Angel Hernández

Adding and subtracting fractions:

1. If the fractions have the same denominator:

The numerator of the sum is found by simply adding the numerators over the denominator.

Their difference is the difference of the numerators over the denominator.

We do not add or subtract the denominators! Reduce always when possible.

4 3 2 9 3 2 1 4 1
Examples:   =  − = =
5 5 5 5 8 8 8 8 2

2. If the fractions have different denominators:

First, reduce them to a common denominator.

Second, add the numerators and do not change the denominator.

Finally, reduce if possible.

3 1 3 18 4 9 11
Examples:  − =  − =
2 3 4 12 12 12 12

2 5 6 4 5 3 1
1− − = − − =− =−
3 6 6 6 6 6 2

Opposite fraction:

Two fractions are opposite if the sum of both fractions is 0.

a −a
The opposite fractions of is .
b b

1. Calculate:

1 7 5
a) − 
2 9 6

3 17
b) 5− 
4 10

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Unit 2: Fractions. Mathematics 2nd E.S.O. Teacher: Miguel Angel Hernández

2 5 8
c) − 
6 9 27

5 7 25
d) − 
26 26 39

1 1 4
e) 3− 1− −
3 5 15

2 3 13 5
f) − − 
5 10 35 14

2. Operate and reduce when necessary:

a) 1−  
7 4
− −
1
5 7 35

b)
5 1
 
3 1
 − − 3
3 2 2 4

c) 1−  
3 11

10 6

d) 11−  7 2
−
10 5

e) [ 
3
5
− 1−
1
3 ] [  ]
3 1
− 2− 
5 9

f) 1 [ 
3
2
 5−
1
4 ] [ 
3 1
− 3 
5 10
−
1
20 ]
g) − [ 
1
5
− 2−
1
6
 ] [ 
3
10
2 1
−  −
7
3 10 30 ]

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Unit 2: Fractions. Mathematics 2nd E.S.O. Teacher: Miguel Angel Hernández

Proper and Improper Fractions. Mixed Numbers:

1 3 2
Proper fractions have the numerator less than the denominator: , , ,…
2 4 5

15
Improper fractions have numerators that are larger than or equal to their denominators: ,
7
8 5
, ,…
3 4

Mixed numbers have a whole number part and a fraction part.

3 1
Example: 2 or 5 are mixed numbers meaning:
5 3

3 3 10 3 13 1 1 15 1 16
2 =2 =  = and 5 =5 =  = .
5 5 5 5 5 3 3 3 3 3

Converting improper fractions into mixed numbers:

To change an improper fraction into a mixed number, divide the numerator by the denominator. The
quotient is the whole part and the remainder is the numerator of the fractional part.

Example:
17
3
, 17 : 3⇒ {
quotient =5
remainder=2
, so
17
3
=5
2
3

Converting mixed numbers into improper fractions:

To change a mixed number into an improper fraction, multiply the whole number by the
denominator and add it to the numerator of the fractional part.

2 7· 32 23
Example: 7 = =
3 3 3

Note that converting mixed numbers into improper fractions is the same as adding natural numbers
and fractions.

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Unit 2: Fractions. Mathematics 2nd E.S.O. Teacher: Miguel Angel Hernández

1. Convert the mixed number to improper fractions, operate and convert the results to mixed
numbers.

1

2 4
a) 5 − 3  4
3 3 5
2
15 
1 13 1 1
b) 3  − 1 − 3
3 2 2 4 
5
12 
3 1 12
c) 7 2 −
9 3 27

Multiplying fractions:

a c a ·c
· =
b d b·d

When two fractions are multiplied, the result is a fraction with a numerator that is the product of the
fraction's numerators and a denominator that is the product of the fraction's denominators. Reduce
when possible.

7 2 14 7
Example: · = =
6 5 30 15

1. Operate and reduce when possible:

7 1 3 −5
a) · b) ·
3 14 5 13

c)
−5 −2
6
·  
15
d)
12
5
· 3·
5
27

10
Unit 2: Fractions. Mathematics 2nd E.S.O. Teacher: Miguel Angel Hernández

e)    
−26 −1 −9
3
·
2
·
39
f)   
1−
4 1 1
· 
7 3 2

g)  
2
7  5 25
−2 · 1− −
4 12

Reciprocal of a fraction:

a b
The reciprocal of a fraction is the fraction .
b a
The product of both fractions is 1.

Dividing fractions:

To divide fractions, multiply the first by the reciprocal of the second fraction.

We can also take the cross product.

a c a d a ·d a c a·d
: = · = or using the cross product: : =
b d b c b·c b d b·c

Example:

4 7 4 11 44 4 7 4 · 11 44
: = · = or simply taking the cross product : = =
5 11 5 7 35 5 11 5 ·7 35

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Unit 2: Fractions. Mathematics 2nd E.S.O. Teacher: Miguel Angel Hernández

Activities.

1. Calculate:

3 2 −1 6
a) : b) :
4 5 3 5

3 −1
5 3
c) d)
4 2
6 9

2. Calculate:

a)  
5 3 7
: 
4 2 4

b)   
2−
1 1 1
: 
12 3 2

c)  
2 1 11
− :
5 4 20

d)   
1 1 1 1
− : 
3 5 3 5

e)  
5 1 3 1
· − : :2
3 2 2 4

f) [  ] [  ]
2
5
: 1−
1
5
· 2− 2:
3
2

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Unit 2: Fractions. Mathematics 2nd E.S.O. Teacher: Miguel Angel Hernández

1
1
2
g)
3
2−
4

1 3
−
3 2 7
h) 
1 8
5
3

3
3. There are 300 passengers on a train. At a station, of the passengers get off. How many
5
people get off the train? How many people are left on the train?

2 1
4. Allan has 120 €. He decides to save of this and to spend on books. How much
5 6
does he save? How much does he spend on books? How much is left?

1
5. In a magazine there are three adverts on the same page. Advert 1 uses of the page,
4
1 1
advert 2 uses and advert 3 uses of the page. What fraction of the page do the
8 16
three adverts use?
3 1
An advert uses of the page, if the cost of an advert is 12 € for each of the page,
16 32
How much does it cost?

13
Unit 2: Fractions. Mathematics 2nd E.S.O. Teacher: Miguel Angel Hernández

3
6. A farmer owns 360 hectares of land. He plants potatoes on of his land and beans on
10
1
of the remainder. How many hectares are planted with potatoes? How many hectares
6
are planted with beans? How many hectares are left?

3 5
7. A journey is 120 miles. Richard has driven of this distance and in a second stage
5 6
of the rest. How much farther does he have to drive to complete the journey?

1 1
8. Sue bought a record with of the money and she spent to see a movie. Which part
4 8
of her money did she spend?

3
9. At a sale shirts are sold by of their original price and the sale price is 36 €. What was
5
the original price?

3 3
10. Joe spends of his salary on his own, gives of the remainder to his parents and
8 5
saves 450 €. What is his salary?

14
Unit 2: Fractions. Mathematics 2nd E.S.O. Teacher: Miguel Angel Hernández

Power of a fraction:

n
a an

a
b
a a a
= · · ·... · = n
b b b b b

The power of a fraction is the quotient between the power of the numerator and the power of
denominator.

Examples:

4 3
2 4 16 −13 −1
2
3
= 4=
3 81  
−1
5
= 3 =
5 125

Powers with negative exponent:

The powers of an integer with a negative exponent are equivalent to a fraction with:

Numerator = 1.

Denominator = the power with the positive exponent.

−n 1
Use the formula: a =
an

Examples:

−3 1 1 −4 1 1 −6 1 1
2 = = 3 = = 10 = =
23 8 3 4 81 10 1000 000
6

1 1 1 1 1
7−1= −5−2= = −5−3= =−
−5 25 125
2 3
7 −5

When the base of the power is a fraction and the exponent is negative, we use the formula:
−n n

 
a
b
=
b
a

You write the inverse or reciprocal fraction with the positive exponent.

Examples:

−2 2 −4 −3 3
4 2 16 −23
 
3
4
=
4
3
= 2=
3 9 
1
2
=24=16    
−5
2
=
−2
5
= 3 =−
5
8
125

15
Unit 2: Fractions. Mathematics 2nd E.S.O. Teacher: Miguel Angel Hernández

Square root of a fraction:

 a a
=
b b

The square root of a fraction is the quotient between the square root of the numerator and the square
root of the denominator.

Examples:

2 2

 1
=
 1 = 1 , because
25  25 5  1
5
12 1
= 2=
5 25
.
 9 9 3
= =
4 4 2
, because 
3
2
32 9
= 2=
2 4
.

2
Actually, you have to know that
2 2
 9
4
=±
 9 =± 3 because
4 2 
3
2
32 9
= 2=
2 4
and also

  −3
2
−3 9
= 2 = .
2 4

1. Calculate:
2 4 2 3

 4
5  
−1
3  
−3
5   −5
2

2. Calculate:

3−2 −5 −3 2−5 −2−4

−4 −1 −2 −4

 2
3  1
5  7
4   −10
3

3. Calculate:

 9
16  1
100  4
49  −
1
4

4. Calculate:
2 −2
a)   2 1

5 2
b)  
7
3
−1

3 −1
c)   5 3
−
4 2
d)  
2
7
9

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Unit 2: Fractions. Mathematics 2nd E.S.O. Teacher: Miguel Angel Hernández

Activities.

−1 3 −4 6 3
1. Order these fractions from highest to lowest: , , , , .
4 4 5 10 5

1 −5 −3 1 2 −4
2. Order these fractions from lowest to highest: , , , , , .
3 6 2 2 3 3

3. Calculate:

2 3
a) · −1
5 4

1 2
b) −2 :
4 3

c)   
1 1
− · 1
2 3
3
4

d)   
3−
1
2
: 2−
1
3

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Unit 2: Fractions. Mathematics 2nd E.S.O. Teacher: Miguel Angel Hernández

e)  
7 1 3
−  −2
2 5 2

f)
1 1 1
− · 1
4 4 2  
g)
2 2 7 3
− · −
3 3 4 2  

1 3
h) 3−3 · 1− 
5 10 
4. Calculate:

1
2
2
a)
1 5

3 2

2 1
1− 
3 4
b)
9 5 5
− −
2 6 3

2
5
c) 3−
1
2

3
1
1 4
d) 
2 2
2−
5

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Unit 2: Fractions. Mathematics 2nd E.S.O. Teacher: Miguel Angel Hernández

3 1
5. The Mathematics' book has of pages of theory, of pages of exercises and the
5 3
remainder is pages of graphs. What fraction of the total are pages of graphs?

2 1
6. A truck is carrying some boxes of fruit: are boxes of oranges, are boxes of apples,
5 3
and the remainder are boxes of bananas. What fraction of the total are boxes of bananas?

7. If the fruit of the track weighs 30 tons, how many tons of each kind of fruit is carrying?

2 3
8. In a school of the students are of 1st and 2nd of ESO, of the remainder are students
5 4
of 3rd and 4th ESO and the rest are students of Bachillerato. What fraction of the total the
students of Bachillerato represent?

9. If in this school, there are 1000 students, how many students of Bachillerato are there?

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Unit 2: Fractions. Mathematics 2nd E.S.O. Teacher: Miguel Angel Hernández

Keywords:

Fraction=Fracción
Numerator=Numerador
Denominator=Denominador
Equivalent fractions=Fracciones Equivalentes
reduction=reducción, simplificación
fraction in simplest form/in lowest terms= fracción reducida o simplificada
Opposite fraction=Fracción Opuesta
Reciprocal fraction=Fracción Inversa
Proper fraction=Fracción propia
Improper fraction=Fracción impropia
Mixed number=Número mixto
Cross-product=Producto en cruz
a<b a is less than b= a es menor que b
a>b a is greater than b= a es mayor que b
from highest to lowest(UK)/from greatest to least(USA)= de mayor a menor

20