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.