B4. Janko wishes to prepare paint for painting a room: he would like to mix blue and white paint in a ratio of $3: 7$.

A How many kilograms of white paint should he mix with $7.35 \mathrm{~kg}$ of blue paint to maintain the above ratio? Round the result to two decimal places.

B How many kilograms of blue and how many kilograms of white paint should he mix to get $21.6 \mathrm{~kg}$ of the mixture in the desired ratio? Round the result to two decimal places.

C When Janko prepared 20 kg of a mixture of blue and white paint in a ratio of 3 : 7, he added $1 \mathrm{~kg}$ of white paint. In what ratio were the blue and white paint in this mixture?

## Solutions to the problems and scoring guide

## A contestant who arrives at the solution by any correct method (even if the scoring guide does not provide for it) receives all possible points.

A correct method is considered any procedure that

- sensibly takes into account the wording of the problem,
- leads to the solution of the problem,
- is mathematically correct and complete.

A contestant who has only partially solved the problem, from otherwise correct procedures for solving it but the path to the final solution of the problem is not visible, cannot receive more than half of the possible points.

## PART I

The correct answers are recorded in the table. A correct answer by the contestant is scored with 2 points, an incorrect one with -1 point, and a blank field in the table with 0 points.

| Problem | A1 | A2 | A3 | A4 | A5 | A6 |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: |
| Answer | $\mathrm{B}$ | $\mathrm{B}$ | $\mathrm{D}$ | $\mathrm{C}$ | $\mathrm{A}$ | $\mathrm{B}$ |
B4. Ratio blue : white $=3: 7$

A We write the equation $7.35: x=3: 7$ and solve it. If we have $7.35 \mathrm{~kg}$ of blue paint, we need $x=17.15 \mathrm{~kg}$ of white paint.

B We write the equation $3 x+7 x=21.6$ and solve it. The solution is $x=2.16$. For $21.6 \mathrm{~kg}$ of the mixture, we need $3 x=6.48 \mathrm{~kg}$ of blue paint and $7 x=15.12 \mathrm{~kg}$ of white paint.

C We write the equation $3 x+7 x=20$ and solve it. The solution is $x=2$. In $20 \mathrm{~kg}$ of the mixture, there are $3 x=6 \mathrm{~kg}$ of blue paint and $7 x=14 \mathrm{~kg}$ of white paint. If we add $1 \mathrm{~kg}$ of white paint to the mixture, the new ratio of colors blue : white $=6: 15=2: 5$.

## Scoring: Total: 7 points

A Writing the equation: $7.35: x=3: 7$ Answer, e.g.: If we have $7.35 \mathrm{~kg}$ of blue paint, we need $17.15 \mathrm{~kg}$ of white paint. $1 \mathrm{p}$

![](https://cdn.mathpix.com/cropped/2024_06_07_ebb644290349f46f6f8bg-8.jpg?height=63&width=1551&top_left_y=1436&top_left_x=298)
Answer, e.g.: For $21.6 \mathrm{~kg}$ of the mixture, we need $6.48 \mathrm{~kg}$ of blue paint and $15.12 \mathrm{~kg}$ of white paint. . ..............................................................................................

C Calculation and conclusion: In $20 \mathrm{~kg}$ of the mixture, there are $6 \mathrm{~kg}$ of blue paint and $14 \mathrm{~kg}$ of white paint.

Answer, e.g.: If we add $1 \mathrm{~kg}$ of white paint to the mixture, the new ratio of colors blue : white $=6: 15$ (or blue : white $=2: 5$ ).

$1 \mathrm{p}$

(For each incorrectly rounded result, we deduct 1 point.)