6.3. Grandma has two balls of wool: a large one and a small one. From the large one, she can knit either a sweater and three socks, or five identical caps. $A$ from the small one - either half a sweater, or two caps. (In both cases, all the wool will be used.) What is the maximum number of socks that Grandma can knit using both balls? Justify your answer.
# Solution:

Method 1. Half a sweater requires as much wool as 2 hats, so a sweater requires as much wool as 4 hats. Then 4 hats and 3 socks require as much wool as 5 hats. Therefore, one hat is equivalent to three socks. In total, 5 + 2 = 7 hats or 21 socks can be knitted.

Method 2. Let the amount of wool required for a sock, a hat, and a sweater be \( x, y, \) and \( z \) respectively (it doesn't matter in what units the unknowns are taken). Then the grandmother has \( 3x + z = 5y \) in the first ball and \( 0.5z = 2y \) in the second ball. We need to express the sum \( 5y + 2y \) in terms of \( x \). From the second equation, express \( z \) and substitute the result into the first: \( 3x + 4y = 5y \), from which \( y = 3x \). Then \( 5y + 2y = 7y = 21x \), that is, 21 socks can be knitted from all the wool.

Answer: 21 socks.

| in progress | points |
| :--- | :--- |
| Correct and fully justified answer | 7 points |
| The solution process is completely correct, but there are computational errors, possibly leading to an incorrect answer | 6 points |
| The problem condition is correctly written as a system of equations, but the further solution is incorrect or missing | 3 points |
| There are correct logically justified conclusions about the amounts of wool required for socks, sweaters, and hats, but they did not lead to an answer | 2 points |
| Correct answer without justification or with incorrect justification | 1 point |
| Incorrect answer without justification or with incorrect justification | 0 points |