Examkin

Question

Difficulty: MediumAlgebraic Word Problems and Modeling

A commercial coffee roaster creates a custom blend by mixing two existing bean blends: Blend X and Blend Y. Blend X consists of 60% Arabica beans and 40% Robusta beans by weight, whereas Blend Y consists of 30% Arabica beans and 70% Robusta beans by weight. The roaster mixes a quantity of Blend X with a quantity of Blend Y to produce a total of 50 pounds of a new mixture that is 42% Arabica beans by weight. How many pounds of Blend X are in the final mixture?

Answer: 20 pounds

Answer

20 pounds
Let xx represent the number of pounds of Blend X. The remaining weight of the mixture, (50−x)(50 - x) pounds, comes from Blend Y. Setting up the equation for the total weight of Arabica beans gives 0.60x+0.30(50−x)=0.42(50)0.60x + 0.30(50 - x) = 0.42(50). Simplifying this expression yields 0.60x+15−0.30x=210.60x + 15 - 0.30x = 21, which reduces to 0.30x=60.30x = 6. Dividing by 0.300.30 gives x=20x = 20. Therefore, 20 pounds of Blend X were used.

Step-by-Step Solution

1
Define variables for component weights
Let xx be the pounds of Blend X. The weight of Blend Y used is 50−x50 - x pounds.
The total combined weight of the mixture is given as 50 pounds.
2
Formulate an equation for the total weight of Arabica beans
0.60x+0.30(50−x)=0.42(50)0.60x + 0.30(50 - x) = 0.42(50), which simplifies to 0.60x+15−0.30x=210.60x + 15 - 0.30x = 21.
The sum of Arabica beans contributed by each blend must equal the total weight of Arabica beans in the combined mixture.
3
Solve the linear equation for xx
0.30x+15=21  ⟹  0.30x=6  ⟹  x=200.30x + 15 = 21 \implies 0.30x = 6 \implies x = 20.
Subtract 15 from both sides to isolate the variable term, then divide by 0.30.

Key Concept

Linear Modeling and Mixture Problems
Estimated Time:1m 30s
Rate this question

Topics

Algebraic Word Problems and Modeling

More questions from this topic