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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 ‘x’ and ‘y’

So according to the question,

1/x – 1/y = 2/15 ….. (i)

It is given that, x + y = 8

So, y = 8 – x … (ii)

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

1/x – 1/(8 – x) 2/15

By taking LCM,

[8 – x – x] / x(8 – x) = 2/15

(8 – 2x) / x(8 – x) = 2/15

By cross multiplying,

15(8 – 2x) = 2x(8 – x)

120 – 30x = 16x – 2x2

120 – 30x – 16x + 2x2 = 0

2x2 – 46x + 120 = 0

Divide by 2, we get

x2 – 23x + 60 = 0

let us factorize,

x2 – 20x – 3x + 60 = 0

x(x – 20) – 3 (x – 20) = 0

(x – 20) (x – 3) = 0

So,

(x – 20) = 0 or (x – 3) = 0

x = 20 or x = 3

Now,

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

So value of x = 3, y = 8 – x = 8 – 3 = 5

∴ The value of x and y are 3 and 5.

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