The numerator of a fraction is 3 less than its denominator. If 1 is added to the denominator, the fraction is decreased by 
. Find the fraction.
Let the denominator of required fraction be x
Numerator of required fraction be = x – 3
Original number = 
If 1 is added to the denominator, then the new fraction will become 
According to the given condition,
x2 + x = 15x – 45
x2 – 14x + 45 = 0
Using the splitting middle term – the middle term of the general equation 
is divided in two such values that:
Product = a.c
For the given equation a = 1 b = – 14 c = 45
= 1.45 = 45
And either of their sum or difference = b
= – 14
Thus the two terms are – 9 and – 5
Sum = – 9 – 5 = – 14
Product = – 9. – 5 = – 45
x2 – 14x + 45 = 0
x2 – 9x – 5x + 45 = 0
x(x – 9) – 5(x – 9) = 0
(x – 9)(x – 5) = 0
x = 9 or x = 5
Case I: x = 5
Case II: x = 9
(Rejected because this does not satisfy the condition given)
Hence the required fraction is 