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Difficulty: MediumFractions, Decimals, and Percents Arithmetic

A commercial coffee roastery prepares a batch of specialty coffee beans containing Arabica, Robusta, and Liberica beans. Initially, Arabica beans make up 0.500.50 of the total weight of the batch, Robusta beans make up 310\frac{3}{10} of the total weight, and Liberica beans make up the remaining weight. After a specialized roasting process, the weight of the Arabica beans decreases by 10%10\%, the weight of the Robusta beans decreases by 50%50\%, and the weight of the Liberica beans remains unchanged. What percent of the final total weight of the coffee batch is made up of Liberica beans?

Answer: 25 %

Answer

25%
To find the percentage of Liberica beans in the final mixture, first determine the initial weight breakdown assuming a total weight of 100 units: Arabica is 50 units, Robusta is 30 units, and Liberica is 20 units. After roasting, Arabica weight decreases by 10% to 45 units, Robusta weight decreases by 50% to 15 units, and Liberica weight stays at 20 units. The new total weight is 45 + 15 + 20 = 80 units. The proportion of Liberica in the final batch is 20 out of 80 units, which equals 1/4 or 25%.

Step-by-Step Solution

1
Find the initial fractional and percentage composition of the coffee blend.
Arabica accounts for 50% (0.50), Robusta accounts for 30% (3/10), and Liberica accounts for 20% (1 - 0.50 - 0.30 = 0.20).
The sum of all components in the initial blend must equal 1 (or 100%).
2
Assume an initial reference weight of 100 units to represent the batch.
Initial Arabica weight = 50 units, initial Robusta weight = 30 units, and initial Liberica weight = 20 units.
Using a convenient base value like 100 simplifies multi-step percentage change calculations without loss of generality.
3
Calculate the post-roasting weight for each component.
New Arabica weight = 50 * (1 - 0.10) = 45 units. New Robusta weight = 30 * (1 - 0.50) = 15 units. New Liberica weight = 20 units.
Apply the respective percentage decreases to each individual component weight.
4
Determine the new total weight of the batch.
Total final weight = 45 + 15 + 20 = 80 units.
The final total weight is the sum of the remaining weights of all three bean types.
5
Compute the final percentage of Liberica beans.
(20 / 80) * 100% = 25%.
Divide the final weight of Liberica beans by the final total weight of the batch and convert to a percentage.

Key Concept

Combining initial fractions and percentages to compute multi-step composition changes
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Topics

Fractions, Decimals, and Percents Arithmetic

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