A commercial coffee roasting facility operates three roasters: X, Y, and Z. - Roaster X pr… — GRE General Test

A commercial coffee roasting facility operates three roasters: $X$ , $Y$ , and $Z$ .

- Roaster $X$ processes coffee beans at a constant rate of $60\text{ kg/hr}$ , producing a blend with an Arabica-to-Robusta ratio of $3:2$ by weight.
- Roaster $Y$ processes coffee beans at a constant rate of $90\text{ kg/hr}$ , producing a blend with an Arabica-to-Robusta ratio of $2:1$ by weight.
- Roaster $Z$ processes coffee beans at a constant rate of $150\text{ kg/hr}$ , producing a blend with an Arabica-to-Robusta ratio of $1:4$ by weight.

All three roasters operate simultaneously for $4$ hours to complete a production order. Which of the following statements regarding the production output over this $4$ -hour period must be true? Select all such statements.

The total weight of Arabica beans processed across all three roasters is $504\text{ kg}$ .Answer
Arabica beans account for exactly $42\%$ of the total weight of coffee beans processed.Answer
The overall ratio of total Arabica weight to total Robusta weight produced is $21:29$ .Answer
D
The combined hourly production rate of Robusta beans across the three roasters is $150\text{ kg/hr}$ .
E
Roaster $Z$ produces a greater total weight of Arabica beans than Roaster $X$ .

Answer

The statements confirming that total Arabica weight is

504\text{ kg}

, Arabica accounts for

42\%

of total weight, and the overall ratio of Arabica to Robusta is

21:29

are all correct.

The total production from Roaster

X

240\text{ kg}

(

144\text{ kg}

Arabica,

96\text{ kg}

Robusta), from Roaster

Y

360\text{ kg}

(

240\text{ kg}

Arabica,

120\text{ kg}

Robusta), and from Roaster

Z

600\text{ kg}

(

120\text{ kg}

Arabica,

480\text{ kg}

Robusta). The total Arabica weight is

144 + 240 + 120 = 504\text{ kg}

, which constitutes

\frac{504}{1,200} = 42\%

of the

1,200\text{ kg}

total output. The remaining

696\text{ kg}

is Robusta, giving an Arabica to Robusta ratio of

504:696 = 21:29

Step-by-Step Solution

Calculate total output and Arabica quantity for each roaster over 4 hours

Roaster

X

240\text{ kg}

total,

\frac{3}{5} \times 240 = 144\text{ kg}

Arabica. Roaster

Y

360\text{ kg}

total,

\frac{2}{3} \times 360 = 240\text{ kg}

Arabica. Roaster

Z

600\text{ kg}

total,

\frac{1}{5} \times 600 = 120\text{ kg}

Arabica.

Converting hourly rates to total batch weights over

4

hours and applying part-to-whole fractions determined by each ratio.

Aggregate total Arabica and Robusta weights across all roasters

Total Arabica =

144 + 240 + 120 = 504\text{ kg}

. Total coffee =

240 + 360 + 600 = 1,200\text{ kg}

. Total Robusta =

1,200 - 504 = 696\text{ kg}

Adding individual component weights to get total aggregate weights.

Evaluate percentage and ratio statements

Arabica percentage =

\frac{504}{1,200} \times 100\% = 42\%

. Arabica to Robusta ratio =

\frac{504}{696} = \frac{21}{29}

Verifying overall proportions and simplifying the fraction by dividing by

24

Key Concept

Multi-stage weighted component ratios and rate conversion

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