3ºESO

1. DEPTO MATEMÁTICAS 3ºESO FRACCIÓN GENERATRIZ
FRACTIONS AND DECIMALS
All fractions can be written as decimals .Example:
If the denominator of a fraction only has the prime factors 2 or 5,
its decimal terminates ( stops after a certain number of digits).
1
 1 : 5  0.2
5
Other denominators produce recurring decimals
Recurring decimals contain a group of repeating digits that “go for ever”.
They are shown by:
 A single dot above a single recurring digit
 A dot above the first and last digit of a set of recurring digits.
For example: 1 
 0.3333333333..........  0. 3
3
123  
 0.123123123.....  0.1 2 3
999
Changing terminating decimal as a fraction:
You can write a terminating decimal as a fraction.
Remember: a terminating decimal ends after a definite number of digits
Divide the number without the decimal point by the power of 10 that corresponds to
the number of digits in the decimal part.
Examples:
342 171
3.42  
100 50
325 65 13
0.0325   
1000 200 40
Changing recurring decimals to fractions
Recurring decimals can be converted into fractions
Let x= the recurring decimal.
Multiply both sides by the power of 10 that corresponds to the number of digits in
the recurring pattern.
E.g. by 101 =10 if only 1 digit recurs,
by 102  100 if 2 digits recur,
by 103  1000 if 3 digits recur,
IES OSTIPPO SECCIÓN BILINGÜE 1

2. DEPTO MATEMÁTICAS 3ºESO FRACCIÓN GENERATRIZ
 Subtract the original equation from the new equation.
 Solve the resulting equation for x.
 Make sure that the answer is a fraction in its simplest form.
Example:
Write down the fraction, in its simplest form, which is equal to these recurring
decimals.

a) 0. 4
x  0.444444444.......
Only 1 digit recurs, so, multiply both sides by 10
10 x  4.44444444444.......
Subtract the original equation from the new equation.
10 x  4.44444444444.......
 x  0.444444444444...
9x  4
4
x
9
 4
0.4 
9
b) x  0.575757.......
2 digits recur, so, multiply both sides by 100
100 x  57.575757.......
Subtract the original equation from the new equation.
100 x  57.575757.......
x  0.575757.......
99 x  57
Divide both sides by 99.
57 19
x  (in its simplest form)
99 33
  19
0.57 
33
IES OSTIPPO SECCIÓN BILINGÜE 2

3. DEPTO MATEMÁTICAS 3ºESO FRACCIÓN GENERATRIZ
REAL NUMBERS
Real numbers are either rational or irrational.
All terminating and recurring decimals are rational numbers.
An irrational number cannot be written as a fraction,
Irrational numbers include:
 Square roots of non-square numbers
 Cube roots of non-cube numbers.
Examples of irrational numbers are:
3
2 7  13
Examples:

1.- Show that 0.425 and 0. 2 are rational numbers.
0.425 is a terminating decimal.
425 17
0.425  
1000 40

0. 2 is a recurring decimal. x  0.2222222.....
10 x  2.2222...
 x  0.22222.......
9x  2
2
x
9
a
All terminating and recurring decimals can be written in the form so
b
they are all rational numbers.
2.- State whether each of the following are rational or irrational numbers.
Where a number is rational write it as a fraction in its simplest form.
3
0.6  7 36
6 3
a) 0.6   ……………………..rational
10 5
a
A rational number expressed in the form is in its simplest form if a and b
b
have no common factor.
IES OSTIPPO SECCIÓN BILINGÜE 3

4. DEPTO MATEMÁTICAS 3ºESO FRACCIÓN GENERATRIZ
b)   3.141592654... irrational
 is a non-recurring decimal and has no exact value.
3
c) 7  1.91293118..... irrational
3
7 is a non-recurring decimal and has no exact value.
d) 36  6 rational
IES OSTIPPO SECCIÓN BILINGÜE 4