The sum of two numbers is 35. Their product is 150. Find the HCF of the sum of their reciprocals and the difference of the reciprocals.

Let first number =x

Let second number =y

x+y=35 

​x×y=150

The sum of the reciprocals of $x$ and $y$ is

​$$\frac{x + y}{x \times y} = \frac{35}{150} = \frac{7}{30}$$  

difference between number x-y =d

​$$(x + y)^2 - (x - y)^2 = 4xy$$​​

​$$35^2 - d^2 = 4 \times 150$$ 

$$d = 25$$   

The difference  of the reciprocals of$x$and$y$is​

​$$\frac{1}{x} - \frac{1}{y} = \frac{x - y}{x \times y}$$ 

=25/150

=1/6 

So hcf of  7/30 and 1/6 is 1/30
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