The difference of two natural number is 5 and the difference of their reciprocal is 1/10

Question 5 Quadratic Equations in One Variable Exercise 5.5

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Answer:

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)

\begin{array}{l} 120-30 x=16 x-2 x^{2} \\ 120-30 x-16 x+2 x^{2}=0 \\ 2 x^{2}-46 x+120=0 \end{array}

Divide by 2, we get

x^{2}-23 x+60=0

let us factorize,

x^{2}-20 x-3 x+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.

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