Divide 50 into two parts so that the sum of their reciprocals is 1/2.

Let us assume the first part is A then second part will be 50 - A.
According to question,
Sum of their reciprocals is 1/2.
1/A + 1/(50 - A) = 1/12
⇒ (50 - A + A)/(50 - A)A = 1/12
⇒ (50)/(50 - A)A = 1/12
⇒ (50) x 12 = 1 x (50 - A)A
⇒ 600 = (50 - A)A
⇒ 600 = 50A - A²
⇒ A² - 50A + 600 = 0
⇒ A² - 30A - 20A + 600 = 0
⇒ A(A - 30) - 20(A - 30) = 0
⇒ (A - 30)(A - 20) = 0
⇒ (A - 30) = 0 or (A - 20) = 0
⇒ A = 20 , 30