Lesson 2: Fractions. First definitions (I)
October 16, 2010
1. Fractions
Definition: A fraction is an expression
a 
b 
where and are integers and can never be zero. We call numerator and denominator
NOTE: Fraction: (from the Latin fractus, broken)
Fractions have three different meanings.
1.1. Fraction as a part of a whole
The denominator represents the parts the whole is divided into and the numerator shows the number of parts are taken.
1.2. Fraction as a quotient
A fraction
a 
b 
expresses the quotient .
To work out the value of the fraction we divide numerator by denominator.
Example:
4 
4:8 = 0,5 
8 

1.3. Fraction as operator
A fraction can act as an operator over a number.
To work out the fraction of a number multiply the numerator by the number and divide by the denominator.
Example:
In a class ¾ of 24 pupils wear glasses:
¾ of 24 = (3 · 24) : 4 = 18 pupils wear glasses
1.4. Types of fractions
There are 4 types of fractions:
 Proper fractions: Numerator < denominator. Proper fractions nominator part smaller than the denominator part. They have values less than 1, for example , or
 Improper fractions: Numerator> denominator. Improper fractions have the nominator par grater than the denominator part. They have values greater than 1, for example .
 Unit fractions: Numerator = denominator. Unit fractions have value 1, for example
 Mixed fractions: Mixed fractions have a whole number plus a proper fraction 2
If you want to practise this concepts click on the picture
2. Equivalent fractions
Definition:
Two fractions
a 

b 
and
c 

d 
areequivalentand we write it if a · d = b · c and they are named crossing products (productos cruzados)
What equivalent means?
The wordEQUIVALENTmeans the same as EQUAL or, more precisely, of equal value.
We can observe the following facts:
1. Graphically, two equivalent fractions represent the same part of the whole.
For example, you can see that the colored part of each of the circles below is exactlythe same.
It must follow that = =
2. Two equivalent fractions have the same value
Since equivalent fractions represent the same part of the whole, therefore have the same value, the quotient is the same when we divide
3. If we multiply both numerator and denominator of a fraction by the same number (not zero) we obtain an equivalent fraction.
4. If we divide both numerator and denominator of a fraction by the same number (not zero) we obtain an equivalent fraction.
How can we obtain equivalent fractions?
There are to ways:
− Amplifying:Equivalent fractions are obtained by multiplying both the numerator and the denominator by the same number (not equal to zero)
− Simplifying:Equivalent fractions are obtained by dividing both the numerator and the denominator by the same number (not equal to zero)
Definition: We saya fraction is in the Simplest form or it is a reduced fractionif we can’t simplify it any more.
Example:
SOLUTION:
= =
Both sides are divided by 8, so the fraction is canceled by 8
But the result is not in the Simplest form. It still can be simplified:
= =
Both sides are divided by 2, so the fraction is reduced by 2
Now the result is in the Simplest Form.
You could get this result by reducing the fraction once only by 16. (The Highest Common Factor).
You can practise the concept of equivalent fractions and simplifying fractions through several exercises if you click on the picture.
Sometimes we have to find the missing number to have two equivalent fractions. We must apply the definition, the cross pruducts must be equal. Practise on this web:
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