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The difference of two natural numbers is 5 and the difference of their reciprocals is $\frac{5}{14}$ . Find the numbers.

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2 and 7

7 and 7

2 and 6

5 and 7

answer is D.

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Detailed Solution

Given, the difference of two natural numbers is 5 and the difference of their reciprocals is

\frac{5}{14}

.
Let’s assume one natural number is x.
Other natural number is 5+x.
According to question,

\frac{1}{x} - \frac{1}{x + 5} = \frac{5}{14}

\Rightarrow \frac{x + 5 - x}{x (x + 5)} = \frac{5}{14}

\Rightarrow \frac{5}{x^{2} + 5 x} = \frac{5}{14}

\Rightarrow x^{2} + 5 x = 14

\Rightarrow x^{2} + 5 x - 14 = 0

\Rightarrow x^{2}

x^{2}

-2x-14 = 0

\Rightarrow

x(x+7) – 2(x+7) = 0

\Rightarrow

(x+7) (x-2) = 0

\Rightarrow

x+7 = 0 or x-2 = 0

\Rightarrow

x = 2
Now,

x ≠ − 7

(Because -7 is not a natural number)
When x = 2, x+5= 2+5=7
Thus, the two numbers are 5 and 7.
Hence, option 4 is correct.

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