Question

The sum of two numbers is 8 and sum of their reciprocal is 8/15. Find the numbers.​

Answer 1

The 2 numbers are 3 and 5.

Explanation:

x + y = 8

1/x + 1/y = 8/15

From first equation y = 8 - x so

1/x + 1 / (8-x) = 8/15

Multiply through by the LCM 15x(8-x):-

15(8 - x) + 15x = 8x(8-x)

120 - 15x + 15x = 64x - 8x^2

8x^2 - 64x + 120 = 0

x^2 - 8x + 15 = 0

(x - 3)(x - 5) = 0

x = 3 or 5

so y = 8-3 = 5 or 3.

Answer 2

x + y = 8
x = 8 - y
And,
1/x + 1/y = 8/15
y + x/xy = 8/15
8/xy = 8/15
xy = 15
( 8 - y)y = 15
8y - y^2 = 15
y^2 - 8y + 15 = 0
y^2 - 3y - 5y + 15 = 0
y(y - 3) - 5( y - 3) = 0
(y - 3) (y - 5) = 0
y = 3 & x = 5
Thus, the two numbers are 3 and 5.