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Macedonian Mathematical Competitions

Algebra Difficulty 5.2 AIME, harder Prove it North Macedonia

Stefan, Filip and Nikola went on an excursion. They took with them $222$ denars. Stefan spent $\frac{1}{3}$ of his money, Filip spent $\frac{1}{5}$ of his and Nikola spent $\frac{7}{15}$ of his. On the end of the excursion they had equal amount of money. How much money did each of them take for the excursion?

Solution

Let Stefan took with him $x$ denars, Filip $y$ and Nikola $z$ . We have that $x + y + z = 222$ .

On the end of the excursion Stefan has left $\frac{2}{3}x$ , Filip $\frac{4}{5}y$ and Nikola $\frac{8}{15}z$ denars.

From the condition in the problem we have $\frac{2}{3}x = \frac{4}{5}y = \frac{8}{15}z$ , so $5x = 6y = 4z$ .

So $y = \frac{5}{6}x$ and $z = \frac{5}{4}x$ .

Substitution in $x + y + z = 222$ gives:

$x + \frac{5}{6}x + \frac{5}{4}x = 222$

After solving this equation we obtain $x = 72$ , $y = \frac{5}{6} \cdot 72 = 60$ , $z = \frac{5}{4} \cdot 72 = 90$ .

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