the sum of numerator and denomenater of a positive fraction is 16. if three is added to both the numerator anf denimeneter the fraction is increased by 1/18. find the fraction
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Answer:
Step-by-step explanation :
L e t space x space b e space t h e space n u m e r a t o r space a n d space y space b e space t h e space d e n o m i n a t o r space o f space t h e space f r a c t i o n. T h u s space t h e space o r i g i n a l space f r a c t i o n equals x over y F r o m space t h e space g i v e n space s t a t e m e n t comma space w e space h a v e comma x plus y equals 16 rightwards double arrow x equals 16 minus y... left parenthesis 1 right parenthesis G i v e n space t h a t space 3 space i s space a d d e d space t o space b o t h space t h e space n u m e r a t o r space a n d space t h e space d e n o m i n a t o r comma space rightwards double arrow fraction numerator x plus 3 over denominator y plus 3 end fraction G i v e n space t h a t space t h e space n e w space f r a c t i o n space i s space 1 over 18 space m o r e space t h a n space t h e space o r i g i n a l space f r a c t i o n. fraction numerator x plus 3 over denominator y plus 3 end fraction equals x over y plus 1 over 18 rightwards double arrow fraction numerator x plus 3 over denominator y plus 3 end fraction equals fraction numerator 18 x plus y over denominator 18 y end fraction rightwards double arrow 18 y open parentheses x plus 3 close parentheses equals open parentheses 18 x plus y close parentheses open parentheses y plus 3 close parentheses rightwards double arrow 18 x y plus 54 y equals 18 x y plus 54 x plus y squared plus 3 y rightwards double arrow 54 x plus y squared plus 3 y minus 54 y equals 0 rightwards double arrow 54 x plus y squared minus 51 y equals 0 rightwards double arrow 54 open parentheses 16 minus y close parentheses plus y squared minus 51 y equals 0 space space space space space space space open square brackets because f r o m space e q u a t i o n space left parenthesis 1 right parenthesis close square brackets rightwards double arrow y squared minus 51 y minus 54 y plus 864 equals 0 rightwards double arrow y squared minus 105 y plus 864 equals 0 rightwards double arrow y squared minus 9 y minus 96 y plus 864 equals 0 rightwards double arrow y open parentheses y minus 9 close parentheses minus 96 open parentheses y minus 9 close parentheses equals 0 rightwards double arrow open parentheses y minus 96 close parentheses open parentheses y minus 9 close parentheses equals 0 rightwards double arrow y minus 96 equals 0 space o r space y minus 9 equals 0 rightwards double arrow y equals 96 space o r space y equals 9 S i n c e space x space a n d space y space a r e space p o s i t i v e space a n d space x plus y equals 16 comma space y space c a n n o t space b e space 96 T h u s comma space y equals 9 F r o m space x equals 16 minus y comma space w e space h a v e comma x equals 16 minus 9 equals 7 T h e r e f o r e space t h e space o r i g i n a l space f r a c t i o n space i s space x over y equals 7 over 9