## Task A-1.1.

Three tractors plow a field. If the first two tractors work together, it takes them 15 days to plow the entire field. The first and third tractors plow the field together in 8 days, and all three tractors together plow the field in 6 days. How many days does each tractor need to plow the entire field on its own?
## Solution.

Let $a, b$, and $c$ be the number of days each tractor needs to plow the entire field alone. In one day, they plow the $\frac{1}{a}$ part of the field, the $\frac{1}{b}$ part of the field, and the $\frac{1}{c}$ part of the field, respectively.

Therefore, we have the system:

$$
\begin{aligned}
\frac{1}{a}+\frac{1}{b} & =\frac{1}{15} \\
\frac{1}{a}+\frac{1}{c} & =\frac{1}{8} \\
\frac{1}{a}+\frac{1}{b}+\frac{1}{c} & =\frac{1}{6}
\end{aligned}
$$

Subtracting the first equation from the last, we get

$$
\frac{1}{c}=\frac{1}{6}-\frac{1}{15}=\frac{1}{10}
$$

which means $c=10$.

Subtracting the second equation from the last, we get

$$
\frac{1}{b}=\frac{1}{6}-\frac{1}{8}=\frac{1}{24}
$$

which means $b=24$.

Finally, substituting the obtained values into any of the initial equations, we get $a=40$.

Therefore, for the first tractor to plow the field alone, it needs 40 days, the second tractor needs 24 days, and the third tractor needs 10 days.