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The difference between two natural numbers is 5 and the difference of their reciprocals is $514$ . Find the numbers.

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-2, 7

2, -7

2, 7

-2, -7

answer is A.

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Detailed Solution

Given that the difference between two numbers is

5

and the difference between their reciprocals
is

514

.
Let one of the natural numbers be

x

and the other be

y

and assume that

y > x

.
It is given that the difference between the numbers is

5

.
So,

y − x = 5

y = x + 5

And the difference between the reciprocals is

514

. So,

1 x − 1 x + 5 = 514

By solving the above equation by take LCM.

x + 5 − x x (x + 5) = 514

x 2 + 5 x = 14

x 2 + 5 x − 14 = 0

This is the standard form of the quadratic equation.
Factorize the equation,

x 2 + 7 x − 2 x − 14 = 0

x (x + 7) − 2 (x + 7) = 0

(x + 7) (x − 2) = 0

So, when

x + 7 = 0

x = − 7

But -7 is not a natural number so this is not possible.
When

x − 2 = 0 x = 2

which is a natural number.
Now find another natural number by substituting the value of

x = 2

in any one of the equations,

y = x + 5

= 2 + 5 = 7

Hence, the required natural numbers are

2, 7

.
Hence, option 1) is correct.

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