Problem 1. Vlad, Luca, and Adina bought stationery from a bookstore for a total value of 118 lei. Vlad bought 5 pens, 4 notebooks, and 3 boxes of colored pencils, Luca bought 7 pens, 3 notebooks, and 4 boxes of colored pencils, and Adina bought 8 pens, 7 notebooks, and 7 boxes of colored pencils.

Knowing that Luca paid 5 lei more than Vlad, and Adina paid 4 lei less than Vlad and Luca together, find out how much a pen costs, how much a notebook costs, and how much a box of colored pencils costs.
Solution. We use the graphical method to find the amounts paid by each child. In the representation below, $p$ is the amount paid by Vlad:

![](https://cdn.mathpix.com/cropped/2024_06_07_fc27702d722f631f1f86g-1.jpg?height=258&width=1028&top_left_y=814&top_left_x=516)

$p + p + 5 + 2p + 1 = 118$, from which $p = 28$, so Vlad paid 28 lei, Luca paid 33 lei, and Adina 57 lei $2p$.

We use the comparison method to find the price of each type of stationery:

5 pencils 4 notebooks $\qquad$ 3 boxes of crayons $\qquad$ 28 lei

7 pencils

3 notebooks $\qquad$
4 boxes of crayons $\qquad$
8 pencils

7 notebooks $\qquad$
7 boxes of crayons $\qquad$
Adding the first two relations, we get

$$
12 \text { pencils }
$$

$\qquad$ 7 notebooks $\qquad$ 7 boxes of crayons $\qquad$ 61 lei

Comparing with the third relation and performing the difference, we obtain that 4 pencils cost 4 lei, so one pencil costs 1 leu.

Substituting in the first two relations, we get:
4 notebooks $\qquad$ 3 boxes of crayons $\qquad$ 23 lei
3 notebooks $\qquad$ 4 boxes of crayons 26 lei

To bring to the same term of comparison, we will equalize the number of notebooks, multiplying the first relation by 3 and the second by 4:

12 notebooks ............. 9 boxes of crayons ............... 69 lei 10 lei
12 notebooks ........... 16 boxes of crayons ............ 104 lei

We obtain that 7 boxes of crayons cost 35 lei, so a box of crayons costs 5 lei .........1p Substituting in one of the relations, we get that a notebook costs 2 lei $1 p$