- Calculate the ratio of blue to purple marbles for each player: Rosa (\(\frac{10}{12} = \frac{5}{6}\)), Chi (\(\frac{5}{6}\)), Alberto (\(\frac{5}{12}\)), and Pedro (\(\frac{10}{6} = \frac{5}{3}\)).
- Convert each ratio to a common denominator of \(12\): Rosa (\(\frac{10}{12}\)), Chi (\(\frac{10}{12}\)), Alberto (\(\frac{5}{12}\)), and Pedro (\(\frac{20}{12}\)).
- Compare the ratios: \(\frac{5}{12} < \frac{10}{12} < \frac{20}{12}\).
- Conclude that Alberto has the lowest ratio and goes first. The final answer is \(\boxed{\text{Alberto}}\).
Explanation
- Analyzing the Marble Ratios
To determine which player goes first, we need to calculate the ratio of blue marbles to purple marbles for each of the four friends. The player with the lowest ratio will be the one to start the game.
The given data is:
- Rosa: \(10\) blue, \(12\) purple. The ratio is \(R_{\text{Rosa}} = \frac{10}{12}\).
- Chi: \(5\) blue, \(6\) purple. The ratio is \(R_{\text{Chi}} = \frac{5}{6}\).
- Alberto: \(5\) blue, \(12\) purple. The ratio is \(R_{\text{Alberto}} = \frac{5}{12}\).
- Pedro: \(10\) blue, \(6\) purple. The ratio is \(R_{\text{Pedro}} = \frac{10}{6}\).
- Comparing the Ratios
Now, we simplify the fractions or express them with a common denominator to compare them:
- \(R_{\text{Rosa}} = \frac{10}{12} = \frac{5}{6} \approx 0.833\)
- \(R_{\text{Chi}} = \frac{5}{6} \approx 0.833\)
- \(R_{\text{Alberto}} = \frac{5}{12} \approx 0.417\)
- \(R_{\text{Pedro}} = \frac{10}{6} = \frac{5}{3} \approx 1.667\)
To compare these, we can use a common denominator of \(12\):
- \(R_{\text{Rosa}} = \frac{10}{12}\)
- \(R_{\text{Chi}} = \frac{10}{12}\)
- \(R_{\text{Alberto}} = \frac{5}{12}\)
- \(R_{\text{Pedro}} = \frac{20}{12}\)
Comparing these values, we see that \(\frac{5}{12} < \frac{10}{12} < \frac{20}{12}\). Therefore, Alberto has the smallest ratio.
- Conclusion
Comparing the ratios \(\frac{5}{12}\), \(\frac{10}{12}\), and \(\frac{20}{12}\), it is clear that Alberto's ratio of \(\frac{5}{12}\) is the lowest. Consequently, Alberto gets to go first in the game. The answer is \(\boxed{\text{Alberto}}\).
Examples
In real-life scenarios, ratios are frequently used to compare proportions of different items. For example, when cooking, you might compare the ratio of sugar to flour in different recipes to see which one is less sweet. Similarly, investors use ratios to compare the financial health of different companies by looking at the ratio of their debts to assets. Calculating ratios helps in making informed decisions by standardizing the comparisons.