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Number Theory Level 2

During a party, someone starts pouring wine from 11 15 \frac{11}{15} -liter bottles into 2 17 \frac{2}{17} -liter glasses. After some time, all bottles have been emptied and all glasses have been filled.

What is the minimum possible number of filled glasses and emptied bottles, added together?


The answer is 217.

Yvonne Killian by Yvonne Killian , 19, Netherlands

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2 solutions

Raymond Chan
Aug 5, 2018

suppose there are b b bottles of wine and g g glasses. Therefore 11 b 15 = 2 g 17 \frac{11b}{15}=\frac{2g}{17} 187 b − 30 g 255 = 0 \frac{187b-30g}{255}=0 187 b − 30 g = 0 187b-30g=0 b g = 30 187 \frac{b}{g}=\frac{30}{187}

So min ⁡ b = 30 \min{b}=30 and min ⁡ g = 187 \min{g}=187 and min ⁡ b + g = 30 + 187 = 217 \min{b+g}=30+187=\boxed{217}

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Yvonne Killian
Aug 5, 2018

11/15 : 2/17 = 11/15 x 17/2 = 187/30, so one bottle fills exactly 187/30 glasses, so 30 bottles fill exactly 187 glasses. 187 + 30 = 217.

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