The difference of two numbers is 5 and the difference of their reciprocals is110 .find the numbers

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The difference of two natural number is 5 and the difference of their reciprocals is 1 / 10. Find the numbers.

Solution

Let the two natural numbers be x and y such that x > y

Given:

Difference between the natural numbers = 5 ⇒ x - y = 5 ------ (i)

Difference of their reciprocals = $\frac{1}{10}$

⇒ $\frac{1}{y}-\frac{1}{x}=\frac{1}{10}$

⇒$\frac{x-y}{xy}=\frac{1}{10}$

⇒$\frac{5}{xy}=\frac{1}{10}$

⇒ xy = 50 ------ (ii)

Putting the value of x from equation (i) in equation (ii), we get

$(y+5)y=50$

⇒ $y^2+5y-50=0$

⇒ $y^2+10y-5y-50=0$

⇒ $y(y+10)–5(y+10)=0$

⇒ $(y-5)(y+10)=0$

⇒ y = 5 or -10

As y is a natural number, therefore y = 5

Other natural number = y + 5 = 5 + 5 = 10

Thus, the two natural numbers are 5 and 10.

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