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The Difference of Two Natural Numbers is 3 and the Difference of Their Reciprocals Is 3/28 . Find the Numbers - Mathematics

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The difference of two natural numbers is 3 and the difference of their reciprocals is 3/28 . Find the numbers

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Solution

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

Given:
Difference of natural numbers = 3

∴x−y=3 ...(i)

Given:
Difference of their reciprocals = 3/28

`∴1/y−1/x=3/28`

`⇒(x−y)/(xy)=3/28`

`⇒3/(xy)=3/28`

`⇒xy=28 ...(ii)`

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

(3+y)y=28

⇒3y+y2−28=0

⇒y2+7y−4y−28=0

⇒y(y+7)−4(y+7)=0

⇒(y+7)(y−4)=0

⇒y=−7 or 4

As y is a natural number, therefore

∴ y = 4 (neglecting y = −7)

∴ Other natural number = y + 3 = 4 + 3 = 7

Thus, the two natural numbers are 4 and 7.

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