5. Ema, Filip, and Goga divided a certain amount of money. Ema received $56 €$ and a quarter of the remaining money. After that, Filip received $78 €$ and a quarter of the new remainder. Finally, Goga received $90 €$ and a quarter of the new remainder. How should Ema, Filip, and Goga divide the remaining $108 €$ so that each of them receives an equal amount of money?

Each task is worth 10 points.

The use of a pocket calculator or any manuals is not allowed.

Ministry of Science and Education of the Republic of Croatia

Agency for Education and Upbringing

Croatian Mathematical Society

## NATIONAL COMPETITION IN MATHEMATICS  Poreč, April 24 - 26, 2023.  7th grade - elementary school

Each task is worth 10 points. In addition to the final result, the procedure is also graded. To earn all points, it is necessary to find all solutions and confirm that there are no others, write down the procedure, and justify your conclusions.
First solution.

Let $\mathrm{s} x$ be the total amount of money, $o_{1}$ the first remainder, $o_{2}$ the second remainder, and $o_{3}$ the third remainder.

![](https://cdn.mathpix.com/cropped/2024_05_30_ad9a4c9088848eb03059g-13.jpg?height=379&width=831&top_left_y=610&top_left_x=241)

After Goga received her share, $\frac{3}{4}$ of the third remainder remained, which is $108 €$.

This means the third remainder was $144 €$.

Goga received $90 €$ and $\frac{1}{4}$ of $144 €$, which is $90+36=126 €$.

$\frac{3}{4}$ of the second remainder is $90+144=234 €$.

After Filip received his share, $\frac{3}{4}$ of the second remainder remained, which is $234 €$.

This means the second remainder was $312 €$.

Filip received $78 €$ and $\frac{1}{4}$ of $312 €$, which is $78+78=156 €$.

$\frac{3}{4}$ of the first remainder is $78+312=390 €$.

After Ema received her share, $\frac{3}{4}$ of the first remainder remained, which is $390 €$.

This means the first remainder was $520 €$.

Ema received $56 €$ and $\frac{1}{4}$ of $520 €$, which is $56+130=186 €$.

The initial amount was $56+520=576 €$.

Goga received $126 €$, Filip $156 €$, Ema $186 €$, and $108 €$ remained.

Goga is short of $186-126=60 €$ to have as much as Ema.

Filip is short of $186-156=30 €$ to have as much as Ema.

From the remaining $108-90=18 €$, each of them needs to receive $18: 3=6 €$.

This means that from the remaining $108 €$, Goga will receive $66 €$, Filip $36 €$, and Ema $6 €$.