This ques has been taken from Book:- ML aggarwal, Avichal publication, class10th, quadratic equation in one variable, chapter 5, exercise 5.5
This is an important ques and asked in 2015 exam.
Sum of two natural numbers is 8 and the difference of their reciprocal is 2/15. Find the numbers.
Question no.5 , ML Aggarwal, chapter 5, exercise 5.5, quadratic equation in one variable, ICSE board,
Let us consider two numbers as ‘x’ and ‘y’
So according to the question,
1/x – 1/y = 2/15 ….. (i)
It is given that, x + y = 8
So, y = 8 – x … (ii)
Now, substitute the value of y in equation (i), we get
1/x – 1/(8 – x) 2/15
By taking LCM,
[8 – x – x] / x(8 – x) = 2/15
(8 – 2x) / x(8 – x) = 2/15
By cross multiplying,
15(8 – 2x) = 2x(8 – x)
120 – 30x = 16x – 2x2
120 – 30x – 16x + 2x2 = 0
2x2 – 46x + 120 = 0
Divide by 2, we get
x2 – 23x + 60 = 0
let us factorize,
x2 – 20x – 3x + 60 = 0
x(x – 20) – 3 (x – 20) = 0
(x – 20) (x – 3) = 0
So,
(x – 20) = 0 or (x – 3) = 0
x = 20 or x = 3
Now,
Sum of two natural numbers, y = 8 – x = 8 – 20 = -12, which is a negative value.
So value of x = 3, y = 8 – x = 8 – 3 = 5
∴ The value of x and y are 3 and 5.