## Task B-1.4.

Fran decided to paint the fence with the help of his friends Tin and Luke. They estimated that it would take Tin 3 hours more than Fran to paint the fence alone, and Luke 2 hours less than Fran to paint the fence alone. Working together, each at their own pace, they would paint the fence in 4 hours. How many hours would it take each of them to paint the fence alone?
## First Solution.

Let's denote the number of hours needed for each of the three friends to paint the fence alone.

Fran: $x$ hours, Tin: $x+3$ hours, Luka: $x-2$ hours. Note that $x$ must be greater than 2.

Considering how much each of them paints in one hour, we get the equation

$$
\frac{1}{x}+\frac{1}{x+3}+\frac{1}{x-2}=\frac{1}{4}
$$

We can rewrite the equation as

$$
\frac{1}{x}+\frac{1}{x-2}=\frac{1}{4}-\frac{1}{x+3}
$$

Bringing both sides of the equation to a common denominator, we get

$$
\frac{2 x-2}{x(x-2)}=\frac{x-1}{4(x+3)}
$$

or

$$
\frac{2(x-1)}{x(x-2)}=\frac{x-1}{4(x+3)}
$$

Since $x>2$, after dividing by $(x-1)$, we get

$$
\frac{2}{x(x-2)}=\frac{1}{4(x+3)}
$$

Then the following equalities follow

$$
\begin{gathered}
8(x+3)=x(x-2) \\
8 x+24=x^{2}-2 x \\
x^{2}-10 x-24=0
\end{gathered}
$$

By writing the middle term as $-12 x+2 x$ and appropriate grouping, the last expression can be written in the form of a product

$$
(x-12)(x+2)=0
$$

Only the factor $x-12$ can be equal to zero, so the solution to the initial equation is $x=12$. Then Fran needs 12, Tin needs 15, and Luka needs 10 hours to paint the fence alone.