## 11. Typists

Two typists need to retype a certain text. First, one retyped half of it, and then the second retyped the second half; in total, they completed the work in 25 hours. How many hours would each typist need to complete the entire work on their own, if it is known that, working simultaneously, they would type the entire text in 12 hours?
11. Let $x$ be the number of hours required by the first typist, and $y$ by the second typist, to complete the entire work independently. To complete half of this work, the first typist would need $x / 2$ hours, and the second typist $y / 2$ hours; therefore,

$$
x / 2 + y / 2 = 25
$$

On the other hand, working together, they complete a portion of the entire work in one hour, which is

$$
1 / x + 1 / y = (x + y) / x y
$$

Therefore, working together, they will complete the entire work in

$$
x y / (x + y) = 12 \text{ (hours)}
$$

Thus,

$$
x + y = 50, \quad x y = 600
$$

The two numbers $x$ and $y$, whose sum is 50 and product is 600, are the roots of the quadratic equation

$$
z^{2} - 50 z + 600 = 0
$$

from which we obtain

$$
x = 20, \quad y = 30
$$

- for the first typist to complete the entire work independently, it will take 20 hours, and for the second typist, 30 hours.