A coffee roaster blends Arabica, Robusta, and Liberica beans in the ratio 4 : 3 : 2 by wei… — GRE General Test

A coffee roaster blends Arabica, Robusta, and Liberica beans in the ratio $4 : 3 : 2$ by weight to prepare a master batch. To adjust the flavor profile, the roaster adds $15\text{ kg}$ of Robusta beans and $15\text{ kg}$ of Liberica beans to the batch without changing the amount of Arabica beans. In the modified batch, Arabica beans constitute exactly $\frac{1}{3}$ of the total weight. Which of the following statements must be true? Select all such statements.

The initial weight of Arabica beans in the master batch was $40\text{ kg}$ .Answer
In the modified batch, the ratio of Robusta beans to Liberica beans is $9 : 7$ .Answer
In the initial master batch, Robusta beans constituted $\frac{1}{3}$ of the total weight.Answer
D
The total weight of the batch increased by $25\%$ after the additions.
E
In the modified batch, Liberica beans account for $35\%$ of the total weight.

Answer

The correct statements are those asserting that the initial weight of Arabica beans was 40 kg, that the modified ratio of Robusta to Liberica beans is 9 : 7, and that Robusta beans constituted 1/3 of the total weight in the initial batch.

Solving the proportion equation reveals that the scaling factor for the initial ratio is 10, giving an initial batch weight of 90 kg with 40 kg Arabica, 30 kg Robusta, and 20 kg Liberica. Consequently, Arabica originally weighed 40 kg, Robusta originally made up 30/90 = 1/3 of the batch, and the new Robusta-to-Liberica ratio is 45 : 35 = 9 : 7. All three of these statements are mathematically true.

Step-by-Step Solution

Represent initial component weights and total weight using a multiplier variable.

For ratio

4 : 3 : 2

, Arabica =

4x

, Robusta =

3x

, Liberica =

2x

, and initial total weight

W = 9x

Ratios define proportional components in terms of a common scalar variable.

Set up the equation for the modified batch based on the new total and Arabica ratio.

Added weight =

15 + 15 = 30\text{ kg}

. New total weight =

9x + 30

. Arabica weight remains

4x

. Since Arabica is

\frac{1}{3}

of the new total:

\frac{4x}{9x + 30} = \frac{1}{3} \implies 12x = 9x + 30 \implies 3x = 30 \implies x = 10

Equating the unchanged part to the given fraction of the new total allows solving for the scale factor.

Calculate all initial and modified quantities.

Initial total

W = 90\text{ kg}

. Initial Arabica =

40\text{ kg}

, Robusta =

30\text{ kg}

, Liberica =

20\text{ kg}

. Modified total =

120\text{ kg}

. Modified Robusta =

45\text{ kg}

, Modified Liberica =

35\text{ kg}

Determining exact values enables verification of each statement.

Evaluate each given statement.

Initial Arabica =

40\text{ kg}

(True). Modified Robusta : Liberica =

45 : 35 = 9 : 7

(True). Initial Robusta fraction =

\frac{30}{90} = \frac{1}{3}

(True). Percent weight increase =

\frac{30}{90} = 33.33\%

(False). Modified Liberica percentage =

\frac{35}{120} = 29.17\%

(False).

Direct comparison of calculated quantities with statement claims identifies all correct choices.

Key Concept

Ratio scale factors and part-to-whole proportions under quantity modifications

Estimated Time:2m 0s

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