The difference of two natural numbers is 5 and the difference of their reciprocals is 5/14. Find the numbers.

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#### Solution

Let* a *and* b* be two natural number such that, a>b

Given,

`a−b=5 ...(1)`

`and 1/b−1/a=5/14`

`⇒(a−b)/(ab)=5/14`

`⇒5/(ab)=5/14`

`⇒ab=14 ...(2)`

Putting the value of a from equation (1) in equation (2), we get

(5+b)b=14

⇒5b+b^{2}=14

⇒b^{2}+5b−14=0

⇒b^{2}+7b−2b−14=0

⇒b(b+7)−2(b+7)=0

⇒(b−2)(b+7)=0

⇒b=2 or b=−7

Since, b is natural number, neglecting the value, b = −7,we get

b=2⇒a=b+5=2+5=7

So, the required natural numbers are 7 and 2.

Concept: Algebraic Methods of Solving a Pair of Linear Equations - Substitution Method

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