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# Sum of two natural numbers is 8 and the difference of their reciprocal is 2/15. Find the numbers.

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This ques has been taken from Book:- ML aggarwal, Avichal publication, class10th, quadratic equation in one variable, chapter 5, exercise 5.5
This is an important ques and asked in 2015 exam.

Sum of two natural numbers is 8 and the difference of their reciprocal is 2/15. Find the numbers.
Question no.5 , ML Aggarwal, chapter 5, exercise 5.5, quadratic equation in one variable, ICSE board,

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1. Let us consider two numbers as â€˜xâ€™ and â€˜yâ€™

So according to the question,

1/x â€“ 1/y = 2/15 â€¦.. (i)

It is given that, x + y = 8

So, y = 8 â€“ x â€¦ (ii)

Now, substitute the value of y in equation (i), we get

1/x â€“ 1/(8 â€“ x) 2/15

By taking LCM,

[8 â€“ x â€“ x] / x(8 â€“ x) = 2/15

(8 â€“ 2x) / x(8 â€“ x) = 2/15

By cross multiplying,

15(8 â€“ 2x) = 2x(8 â€“ x)

120 â€“ 30x = 16x â€“ 2x2

120 â€“ 30x â€“ 16x + 2x2Â = 0

2x2Â â€“ 46x + 120 = 0

Divide by 2, we get

x2Â â€“ 23x + 60 = 0

let us factorize,

x2Â â€“ 20x â€“ 3x + 60 = 0

x(x â€“ 20) â€“ 3 (x â€“ 20) = 0

(x â€“ 20) (x â€“ 3) = 0

So,

(x â€“ 20) = 0 or (x â€“ 3) = 0

x = 20 or x = 3

Now,

Sum of two natural numbers, y = 8 â€“ x = 8 â€“ 20 = -12, which is a negative value.

So value of x = 3, y = 8 â€“ x = 8 â€“ 3 = 5

âˆ´ The value of x and y are 3 and 5.

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