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Question
Two natural numbers differ by 3. Find the numbers, if the sum of their reciprocals is `7/10`.
Solution
Let the numbers be x and x + 3.
From the given information,
`1/x + 1/(x + 3) = 7/10`
`(x + 3 + x)/(x(x + 3)) = 7/10`
`(2x + 3)/(x^2 + 3x) = 7/10`
20x + 30 = 7x2 + 21x
7x2 + x – 30 = 0
7x2 – 14x + 15x – 30 = 0
7x(x – 2) + 15(x – 2) = 0
(x – 2)(7x + 15) = 0
`x = 2, (-15)/7`
Since, x is a natural number, so x = 2.
Thus, the numbers are 2 and 5.
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