The sum of the reciprocals of the ages of two colleagues is nine times the difference of the reciprocals of their ages. If the ratio of the products of their ages to the sum of their ages is 100 : 9, Find their ages.

Correct option is A

Given: ​

Sum of the reciprocals of their ages =  nine times the difference of the reciprocals

The ratio of the product of their ages to the sum of their ages = 100 : 9 

Solution: ​

Let the age of two persons be x and y 

Now, from first condition 

​$$\frac1x+\frac1y = 9\left(\frac1x-\frac1y\right) \\ \ \\ \frac{x+y}{\cancel{xy}} = 9\left(\frac{y-x}{\cancel{xy}}\right) \\ \ \\ \ \implies \frac{ x}{y} = \frac{4}{5}$$ 

Let x = 4k, y = 5k 

From condition second;

​$$\frac{xy}{x+y} = \frac{100}{9}$$ 

Putting value of x and y;

​$$\frac{4k \times 5k}{4k+5k} = \frac{100}{9} \\ \ \\ \frac{20k^2}{9k} = \frac{100}{9} \\ \ \\ \implies 20k = 100 \\ \ \\ k = 5$$ 

Now, 

​$$x = 4\times 5 = 20 \ year \\ \ \\ y = 5 \times 5 = 25 \ year$$​​