The denominator of a fraction is 3 more than its numerator. The sum of the fraction and its reciprocal is . Find the fraction.
Let the numerator be x
Denominator = x + 3
Original number =
On taking the LCM
{ using (a + b)2 = a2 + 2ab + b2}
29x2 + 87x = 20x2 + 60x + 90
9x2 + 27x – 90 = 0
9(x2 + 3x – 10) = 0
x2 + 3x – 10 = 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 = 3 c = – 10
= 1. – 10 = – 10
And either of their sum or difference = b
= 3
Thus the two terms are 5 and – 2
Difference = 5 – 2 = 3
Product = 5. – 2 = – 10
x2 + 5x – 2x – 10 = 0
x(x + 5) – 2(x + 5) = 0
(x + 5)(x – 2) = 0
(x + 5) = 0 or (x – 2) = 0
x = 2 or x = – 5
x = 2 (rejecting the negative value)
So numerator is 2
Denominator = x + 3 = 2 + 3 = 5
So required fraction is 2/5