Question

The numerator of a fraction is 3 less than its denominator. If 1 is added to the denominator, the fraction is decreased by $\frac{1}{15}$. Find the fraction.

Answer 1

SOLUTION :
Let the denominator of fraction be x
& numerator be x - 3
Fraction = numerator / denominator
Fraction = (x - 3) / x……….(1)
ATQ
(x - 3) /(x+1)= (x -3 / x ) - 1/15
(x - 3) /(x+1) = (15 ( x- 3 ) - 1 × x) / 15x
(x - 3) /(x +1) =(15x - 45 - x) /15x
15x (x-3)/(x+1) = (14x - 45)
15x² - 45x = (14x - 45) (x +1)
15x² - 45x = x(14x - 45) + 1 (14x - 45)
15x² - 45x = 14x² - 45x + 14x - 45
15x² - 14x² - 45x + 45x - 14x + 45
x² - 14 x + 45 = 0
x² - 9x - 5x + 45 = 0
[By Middle term splitting]
x(x- 9) - 5(x - 9) = 0
(x - 9) or (x-5)= 0
x= 9 or x = 5
If x = 9
Fraction = (x - 3) / x = (9-3)/9 = 6/9
If x= 5, then
Fraction = (x - 3) / x = (5-3)/5 = 2/5

Hence, the fraction is 6/9 or 2/5

Read