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Zorluk: OrtaWeighted Averages in Applied Contexts

A specialty coffee roaster creates a signature espresso blend by combining Arabica coffee beans costing 18.00perkilogramwithRobustacoffeebeanscosting18.00 per kilogram with Robusta coffee beans costing 12.00 per kilogram. The final blend weighs 50 kilograms and has an overall average cost of $15.60 per kilogram. How many kilograms of Arabica beans are included in the blend?

Cevap: 30 kg

Cevap

The blend contains 30 kilograms of Arabica coffee beans.
To determine the required quantity of Arabica beans, establish the total cost equation: 18A+12(50−A)=50(15.60)18A + 12(50 - A) = 50(15.60). Expanding the terms gives 18A+600−12A=78018A + 600 - 12A = 780, which simplifies to 6A=1806A = 180, yielding A=30A = 30 kilograms.

Adım Adım Çözüm

1
Define variables and relate component weights
Let AA represent the mass of Arabica beans in kilograms. The mass of Robusta beans is (50−A)(50 - A) kilograms.
The sum of the individual component weights must equal the total blend weight of 50 kilograms.
2
Formulate the weighted average total cost equation
18A+12(50−A)=15.60×50=78018A + 12(50 - A) = 15.60 \times 50 = 780
The combined monetary cost of both bean types equals the total value of the 50 kg blend.
3
Solve for the unknown variable AA
6A+600=780  ⟹  6A=180  ⟹  A=306A + 600 = 780 \implies 6A = 180 \implies A = 30
Simplifying the algebraic linear equation determines the exact weight of Arabica beans.

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Weighted Averages in Applied Contexts
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Weighted Averages in Applied Contexts

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