The sum of two numbers is 9 and the sum of their reciprocals is 1/2. Find the numbers.

Let the first number be x. Then the second number = 9 - x. Their reciprocals = 1/x, 1/(9 - x)
Then according to the question
1/x + 1/(9 - x) = 1/2
=> (9 - x + x)/x(9 - x) = 1/2
=> 9/x(9 - x) = 1/2
=> 18 = x(9 - x)
=> x² - 9x + 18 = 0
=> x² - 6x - 3x + 18 = 0
=> x(x - 6) - 3(x - 6) = 0
=> (x - 3)(x - 6) = 0
=> x - 3 = 0, x - 6 = 0
=> x = 3, 6
First number = 3 => Second number = 9 - 3 = 6.