Question

the sum of the numerator and the denominator of a certain fraction is 8 . if 2 is added to the numerator and denominator,the value is increased by 4/35.findthe fraction.

Answer 1

let fraction = x/y

x + y = 16  ---->  x = 16 - y  ----  (eqn 1)

new fraction = (x + 3)/(y + 3)

According to question :--------  (x + 3)/(y + 3)  -  x/y  =  1/18 

-----> {y(x + 3) - x(y + 3)} / y(y + 3)  =  1/18

----->  (xy + 3y - xy - 3x) / (y2 + 3y) = 1/18

----->  18(3y - 3x) = y2 + 3y  ------->  54y - 54x = y2+ 3y 

----->  51y - y2  =  54x  ------>  x  =  (51y - y2)/54  -------(eqn 2)

As,  x = x

---->  (51y - y2) / 54  =  16 - y

---->  51y - y2  =  864 - 54y

---->  y2 - 105y + 864 = 0

---->  y2 - 9y - 96y + 864 = 0

----->  y(y - 9) - 96(y - 9) = 0

----->  y = 9 or 96

----->  y = 9  [As sum of numerator and denominator is 16 and the fraction is a positive fraction]

As,  x + y = 16  ---->  x = 16 - y  -----> x = 16 - 9  ----->  x = 7

So , Required fraction = x/y = 7/9 .  (ans) .