3. On the table, there are candies of three types: caramels, toffees, and lollipops. It is known that there are 8 fewer caramels than all the other candies, and there are 14 fewer toffees than all the other candies. How many lollipops are on the table? Be sure to explain your answer.
# Solution.

Method 1. Since there are 8 fewer caramels than other candies, there are 4 fewer caramels than half of the candies. Since there are 14 fewer toffees than all other candies, there are 7 fewer toffees than half of the candies. Thus, if we remove all caramels and toffees, 4 + 7 = 11 candies will remain. Since the remaining candies are exactly the lollipops, there are 11 lollipops.

![](https://cdn.mathpix.com/cropped/2024_05_06_6efaa98e7c8f41148dbdg-2.jpg?height=425&width=1268&top_left_y=593&top_left_x=400)

## Method 2.

Let $л, \kappa, u$ be the number of lollipops, caramels, and toffees, respectively.

According to the problem,

$\pi + \kappa = 8 + u$,

$\pi + u = 14 + \kappa$.

Adding these equations, we get: $2 \pi + \kappa + u = 22 + \kappa + u$

Therefore, $2 \pi = 22$, from which $\pi = 11$.

## Grading Criteria.

Complete correct solution - 7 points.

One arithmetic error in solving the system, but the solution method is correct - 4-5 points.

Correct answer and verification that it fits - 3 points.

The problem is solved with a specific example (e.g., "let there be 30 candies in total, then...") - 2 points.

Correct answer without justification - 2 points.

Other cases - 0 points.