B4. On a winter evening, all the family members of Grandpa Albert weighed themselves. Since they all love mathematics, they described what the scale showed in a more witty way.

Grandpa Albert: "I am $5 \cdot 10^{6}$ mg lighter than Dad."

Dad Miha: "I weigh as much as Mom and Žan together."

Mom Jana: "I weigh twice as much as Tomaž."

Metka: "I weigh twice as much as Žan."

Tomaž: "I weigh 0.004 t less than Metka."

Žan: “I weigh $10 \mathrm{~kg} 800$ dag $2000 \mathrm{~g}$."

A How much does each family member weigh?

B How much do they weigh together?

C Arrange them from the lightest to the heaviest. Write their names.
for vocational school students

## Solutions to the tasks and scoring

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

For a correct method, each procedure is considered that

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


## I. PART
B4. Žan weighs $20 \mathrm{~kg}$, Metka $40 \mathrm{~kg}$, Tomaž $36 \mathrm{~kg}$, mom $72 \mathrm{~kg}$, dad $92 \mathrm{~kg}$, grandpa $87 \mathrm{~kg}$. B All together they weigh $347 \mathrm{~kg}$.

C From the lightest to the heaviest, they are: Žan, Tomaž, Metka, Jana, Albert, Miha.

## SOLUTIONS AND SCORING GUIDES

## I. PART

$V$ The correct answers to the multiple-choice questions are listed in the table. Each correct answer is worth 2 points, an incorrect answer is worth -1 point, and an unanswered question is worth 0 points.

| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| $\mathrm{C}$ | $\mathrm{E}$ | $\mathrm{E}$ | $\mathrm{C}$ | $\mathrm{C}$ | $\mathrm{C}$ | $\mathrm{D}$ | $\mathrm{A}$ | $\mathrm{E}$ | $\mathrm{C}$ |

## II. PART

## 1. problem Total: 5 points

Calculated devil's number: $120: 3=40$

Calculated wizard's number: $2 \cdot 133=266$

$1 \mathrm{t}$
Calculated magic number: $10 \cdot(2 \cdot 40-266)=-1860$ ..... $2 \mathrm{t}$
Answer: The magic number is -1860. ..... $1 \mathrm{t}$

2. problem Total: 5 points

A Calculated side length of the cube $a=28.3 \mathrm{~cm}$

$1 \mathrm{t}$
B Calculated volume of the cube $V_{t}=320356 \mathrm{~cm}^{3}=0.320 \mathrm{~m}^{3}$ ..... $2 \mathrm{t}$
C Calculated volume of the sphere $V_{d}=502400 \mathrm{~cm}^{3}=0.502 \mathrm{~m}^{3}$ ..... $2 \mathrm{t}$

Note: In cases $\mathbf{B}$ and $\mathbf{C}$, the contestant gets 1 point for the correct solving procedure, 1 point for the correctly rounded result in the appropriate units.

3. problem Total: 5 points

Appropriate choice of the unknown, e.g., the number of seats on the balconies $b$, the number of seats in the stalls $b-290$.

![](https://cdn.mathpix.com/cropped/2024_06_07_fb8dcc2580df7ec25528g-7.jpg?height=60&width=1630&top_left_y=1986&top_left_x=276)

![](https://cdn.mathpix.com/cropped/2024_06_07_fb8dcc2580df7ec25528g-7.jpg?height=51&width=1628&top_left_y=2042&top_left_x=277)

Answer: There are 440 seats on the balconies and 150 seats in the stalls. .............................. 2 t

4. problem Total: 5 points

A Žan weighs $20 \mathrm{~kg}$. Metka weighs $40 \mathrm{~kg}$. Tomaž weighs $36 \mathrm{~kg}$. Mom Jana weighs $72 \mathrm{~kg}$. Dad Miha weighs $92 \mathrm{~kg}$. Grandpa Albert weighs $87 \mathrm{~kg}$. .................................. 3 t Note: The contestant gets 1 point for each correctly determined mass of two family members.

B All together they weigh 347 kg. ............................................................................. 1 t

C From the lightest to the heaviest, they are: Žan, Tomaž, Metka, Jana, Albert, Miha. . 1 t