Question

In a certain positive fraction, the denominator is greater than the numerator by 3. If 1 is subtracted from both the numerator and denominator, the fraction is decreased by `(1)/(14)`. Find the fraction.

Answer 1

Let the numerator of a fraction = x
then denominator = x + 3
then fraction = `(1)/(14)`
Now according to the condition,

new fraction `(x - 1)/(x + 3  1) = (x)/(x + 3) - (1)/(14)`

⇒ `(x - 1)/(x + 2) = (14x - x - 3)/(14(x + 3)`

⇒ `(x - 1)/(x + 2) = (13x - 3)/(14x + 42)`

⇒ (x - 1)(14x + 42) = (13x - 3)(x + 2)
⇒ 14x² + 42x - 14x - 42 = 13x² + 26x - 3x  6
⇒ 14x² + 28x  42 - 13x² - 23x + 6 = 0
⇒ x² + 5x - 36 = 0
⇒ x² + 9x - 4x - 36 = 0
x(x + 9) -4(x + 9) = 0
⇒ (x + 9)(x - 4) = 0
Either x + 9 = 0, 
then x = -9,
but it is not possible as the fraction is positive.
or
x - 4 = 0,
then x = 4

∴ Fraction = `(x)/(x + 3) = (4)/(4 + 3) = (4)/(7)`.