Question

Divide 40 into two parts such that the sum of their reciprocals is 8/75

Answer 1

Let, the two parts be x and y

According to the 1st case :

$x + y = 40$

Again, according to the 2nd case :

$\frac{1}{x} + \frac{1}{y} = \frac{8}{75} \\ \\ = > \frac{y + x}{xy} = \frac{8}{75} \\ \\ = > \frac{40}{xy} = \frac{8}{75} \\ \\ = > xy = 375 \\ \\ = > x = \frac{375}{y}$

On putting the value of y in equation (1) :

$\frac{375}{y} + y = 40 \\ \\ = > {y}^{2} + 375 = 40y \\ \\ = > {y}^{2} - 40y + 375 = 0 \\ \\ = > {y}^{2} - 25y - 15y + 375 = 0 \\ \\ = > y(y - 25) - 15(y - 25) = 0 \\ \\ = > (y - 25)(y - 15) = 0$

So, the value of y = 15 and 25

When, the value of y = 15 then the value of x :

45 - 15 = 25

When, the value of y is 25 then the value of x :

45 - 25 = 15

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