Fractions and Decimals

Fractions and Decimals

Class Notes

Repeating or Terminating?

 $\frac{1}{2}=\ \ \ \ \ \ \frac{1}{3}=\ \ \ \ \ \ \ \frac{1}{4}=\ \ \ \ \ \ \frac{1}{5}=\ \ \ \ \ \ \frac{1}{6}=$ 

Repeating or Terminating?

 $\frac{5}{9}=\ \ \ \ \ \frac{24}{99}=\ \ \ \ \ \frac{817}{999}=\ \ \ \ \ \frac{2}{999}=\ \ \ \ \ \frac{71}{999}=\ \ \ \ \$  

Repeating or Terminating?

 $\frac{2}{7}=\ \ \ \ \ \ \frac{3}{7}=\ \ \ \ \ \ \frac{4}{7}=\ \ \ \ \ \ \frac{5}{7}=\ \ \ \ \ \ \frac{6}{7}=\ \ \ \$  

Terminating/Repeating Decimals

The decimal expansion of a fraction in simplest form will terminate only if the denominator contains no prime factors other than 2 or 5.

Examples:

Figure out which of the following can be represented by terminating decimals before checking your answers with a calculator: 


 $\frac{1}{30}\ \ \ \ \ \ \frac{7}{8}\ \ \ \ \ \ \frac{3}{125}\ \ \ \ \ \ \frac{5}{24}\ \ \ \ \ \ \frac{7}{128}\ \ \ \ \ \ \frac{3}{48}\ \ \ \ \ \$ 

Ninths, Ninety-Ninths, etc.:

The repeating block of a fraction whose denominator is 9, 99, 999, etc. in the denominator will be the numerator. Use leading zeros where necessary.

 $\frac{5}{9}=0.\overline{5}\ \ \ \ \ \ \ \ \ \frac{24}{99}=0.\overline{24}\ \ \ \ \ \ \ \ \ \frac{817}{999}=0.\overline{817}\ \ \ \ \ \ \ \ \$ 

 $\frac{1}{999}=0.\overline{001}\ \ \ \ \ \ \ \ \ \ \frac{7}{11}=\frac{63}{99}=0.\overline{63}\ \ \ \ \ \ \ \ \ \ \frac{4}{33}=\frac{12}{99}=0.\overline{12}\ \ \ \ \ \ \ \ \ \$  

Practice: Express as a decimal (without usinjg a calculator):

 $\frac{41}{99}=\ \ \ \ \ \ \ \ \ \ \frac{2}{99}=\ \ \ \ \ \ \ \ \ \ \frac{71}{999}=\ \ \ \ \ \ \ \ \ \$  


 $\frac{2}{33}=\ \ \ \ \ \ \ \ \ \ \frac{5}{111}=\ \ \ \ \ \ \ \ \ \ \frac{2}{37}=\ \ \ \ \ \ \ \ \ \$  

Converting a repeating decimal into a fraction

Example: 

 $x=0.\overline{5}\$  

 $10x=5.\overline{5}$  
 $10x-x=5.\overline{5}-0.\overline{5}=5$ 
 $9x=5$  
 $x=\frac{5}{9}$   

Examples: Convert the following to common fractions:

 $x=0.\overline{24}$  
 $x=0.1\overline{25}$  
 $x=0.01\overline{3}$  

Sevenths

There is a nice pattern to the repeating block of the sevenths. 

 $\frac{1}{7}=0.\overline{142857}\ \ \ \ \ \ \ \ \ \frac{2}{7}=0.\overline{285714}\ \ \ \ \ \ \ \ \ \frac{3}{7}=0.\overline{428571}\ \ \ \ \ \ \ \ \$  

 $\frac{4}{7}=0.\overline{571428}\ \ \ \ \ \ \ \ \ \frac{5}{7}=0.\overline{714285}\ \ \ \ \ \ \ \ \ \frac{6}{7}=0.\overline{857142}\ \ \ \ \ \ \ \ \$  

There are web pages dedicated to 142,857, which is called a cyclic number. I encourage you to do some research on the properties of 142,857 and other cyclic numbers.