Q21 of 47 Page 1

The difference of two natural numbers is 5 and the difference of their reciprocals is . Find the numbers.

Let the larger natural number be x and the smaller natural number be y.

Given, x – y = 5 … (1)


And … [∵ If x > y, then 1/x < 1/y]


⇒


From (1),


⇒


Cross multiplying, we get


⇒ 50 = xy


⇒ y = 50/x … (2)


Substituting (2) in (1),


⇒ x – 50/x = 5


⇒ (x2 – 50)/x = 5


⇒ x2 – 50 = 5x


⇒ x2 – 5x – 50 = 0


By factorization method,


⇒ x2 – 10x + 5x – 50 = 0


⇒ x (x – 10) + 5 (x – 10) = 0


⇒ (x – 10) (x + 5) = 0


⇒ x – 10 = 0 (or) x + 5 = 0


⇒ x = 10 (or) x = -5


∵ x is a natural number, x = 10


Putting the above x value in (2), we get


⇒ y = 50/10


⇒ y = 5


i.e. x, y = 10, 5


Ans. The numbers are 10, 5.


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