The sum of two natural numbers is 9 and the sum of their reciprocals is 1/2. Find the numbers.

Let the required natural numbers x and (9 – x)

According to given condition,

taking LCM

cross multiplying

Using the splitting middle term – the middle term of the general equation is divided in two such values that:

Product = a.c

For the given equation a = 1 b = – 9 c = 18

= 1.18 = 18

And either of their sum or difference = b

= – 9

Thus the two terms are – 6 and – 3

Sum = – 6 – 3 = – 9

Product = – 6. – 3 = 18

x(x – 6) – 3(x – 6) = 0

(x – 6) (x – 3) = 0

(x – 6) = 0 or (x – 3) = 0

x = 6 or x = 3

Case I: when x = 6

9 – x = 9 – 6 = 3

Case II: when x = 3

9 – x = 9 – 3 = 6

Hence required numbers are 3 and 6.

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