REDUCTION OF CIRCULATING DECIMALS. 9^ 



2. If there be integral figures in the circulate, as many 

 cyphers must be annexed to the numerator, as the highest 

 place of the repetend is distant from the decimal point. 



EXAMPLES. 



1. Required the least vulgar fractions equal to '6 and * 1 23. 



•6=|=i ; and •I23=-^|=^^3V ^ns. 



2. Reduce '3 to its equivalent vulgar fraction. Ans. j. 



3. Reduce 1*62 to its equivalent vulgar fraction. 



Alls. --5P5-. 



4. Required the least vulgar fraction equal to '769230. 



CASE 11. 

 21? reduce a mixed repetetid to its equivalent vulgar fraction. 



RULE.* 



I. To as many nines as there are figures in the repetend, 

 annex as many cyphers as there are finite places, for a de- 

 nominator. 



2. Multiply 



Again, -j'-y, or -^^ being reduced to decimals, makes '0101 or, 

 &c. or -001001, &c. ad infinitum =:-oi or '00 1 ; that is, -j^zs 

 •oi',and^-5-^-5^=:*ooi ; consequently -^^p^'oz, -5^V='03, &c. and 

 -5^-5=-oo2, ■5j^='oo3, &c. and the same will hold universally. 



* In like manner for a mixed circulate ; consider it as divisible 

 ioto its fipite and circulating parts, and the same principle will be 



seen to run through thcnj also : thus, the mixed circulate '16 is 



divisible into the finite decimal •!, and the repetend -06 ; but 



*i=To»and -06 would be=|, provided the circulation began imme- 

 diately after the place of units ; but as it begins after the place of 



