Find two numbers whose sum is 25 and the sum of whose reciprocals is 1/6

10 and 15

Step-by-step explanation:

Let 'x' and 'y' are the numbers we need to find.

x + y = 25 (two numbers whose sum is 25)

(1/x) + (1/y) = 1/6 (the sum of whose reciprocals is 1/6)

The solutions of the this system of equations are the numbers we need to find.

x = 25 - y

1 / (25 - y) + 1/y = 1/6 multiply both sides by 6 (25-y) y

6y + 6 (25-y) = (25-y) y

6y + 150 - 6y = 25y - (y^2)

y^2 - 25y + 150 = 0 quadratic equation has 2 solutions

y1 = 15

y2 = 10

Thus we have:

First solution: for y = 15, x = 25 - 15 = 10

Second solution: for y = 10, x = 25 - 10 = 15

The first and the second solution are in fact the same one solution we are looking for: the two numbers are 10 and 15 (since the combination 10 and 15 is the same as 15 and 10).