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