FRACTIONS AND DECIMALS

                                                                       

Fractions and decimals are just different ways of writing the same numbers

 

 

·        Every fraction can be written as a decimal.

·        Every decimal can be written as a fraction.

 

 

 

FRACTION

 

DECIMAL

 

0.5

0.3333 

0.25

0.2

0.1666

0.4285

0.125

0.111

0.1

Figure 1. The decimal and fraction form for some numbers.

For example =0.75 and  = 0.8571

 

(the string 285714 keeps repeating itself all the time).

 

 

Normally a fraction produces a recurring decimal

in which a string of numbers keeps repeating itself

and never ends. For example  = 0. 

 

(a bar on top of the digit indicates where the recurring

string starts and ends).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

·        If 2 and 5 are the only prime factors of the denominator è non-recurring decimal

·        If the denominator has prime factors other than 2 and 5 è recurring decimal

 

Example 1. Which of the fractions 4/15, 2/25, 8/35 will produce recurring decimals are which will not?

Answer. 3 is a prime factor of 15 è 4/15 recurring decimal. 

5 is the only prime factor of 25 è 2/25 ending decimal.

7 is a prime factor of 35 è 8/35 recurring decimal. §

 

Decimals from fractions

 

Fraction means dividing.

 

To find the decimal for any fraction, just divide

You need to get enough digits to see what the recurring string is going to be.

 

Fractions from decimals

 

Non-recurring decimals

 

To go from decimal to fraction if the decimal is non-recurring just multiply and divide by 10, 100, 1000… and simplify

 

 

Example 2. Find the fraction form of 2.75.

Answer. If we multiply by 100 we get 275, which is a counting number and good for the numerator. Therefore,

2.75 =  = §

 

Example 3. Find the fraction form of 67.987.

Answer. If we multiply by 1000 we get 67987, which we use for the numerator. Therefore,

67.987 = . This fraction cannot be simplified§

 

Pure recurring decimals

 

For pure recurring decimals, i.e. decimals numbers with no terminating part the rule is:

 

 

0. ecurrin

 

 

=

 

 

recurring

As many 9s as digits are there in ‘recurring’

 

 

 

 

Example 4.    Write 0. as a fraction.

 

Answer. 0.   =  (Check it!) §

 

Example 4.    Write 0.0 as a fraction.

 

Answer. 0.0   =  (Check it!) §

 

Example 5.    Write 0. as a fraction.

Answer. 0.   =  (Check it!) §

 

Example 6.    Write 0.0 as a fraction.

Answer. 0.0   =  (Check it!) §

 

Example 7.    Write 0. as a fraction.

 

Answer. 0. =  (Check it!) §

 

Example 8.    Write 0.6 as a fraction.

 

Answer. 0.6 =  (Check it!) §

 

Recurring decimals with a non-recurring part

 

If the decimal is recurring but it has a non-recurring part to find the fraction do first the recurring part,

then the non-recurring part and add them together..

 

Example 9. Find the fraction form of 1.5.

Answer:  1.5 = 1.5 + 0.0=  +  =  +  = . §

 

Example 10. Find the fraction form of 8.94.

Answer:  8.94 = 8.9 + 0.04 =  +  = . §

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