A commercial roastery blends two batches of coffee beans, Batch A and Batch R. Batch A con… — GMAT

A commercial roastery blends two batches of coffee beans, Batch A and Batch R. Batch A contains whole beans and cracked beans in a mass ratio of $7:3$ . Batch R contains whole beans and cracked beans in a mass ratio of $3:2$ . A roaster combines $x$ kilograms of Batch A with $y$ kilograms of Batch R to create an unroasted blend in which the overall mass ratio of whole beans to cracked beans is $13:7$ .

During the roasting process, moisture evaporation causes whole beans to lose $10\%$ of their mass and cracked beans to lose $20\%$ of their mass. If the total mass of the roasted blend is $346$ kilograms, what was the initial mass $x$ of Batch A, in kilograms?

Answer: 200 kg

Answer

The initial mass of Batch A (

x

) is 200 kg.

The correct calculation shows that equal masses of Batch A and Batch R (

x = y

) satisfy the combined unroasted ratio of

13:7

. Applying the respective

10\%

and

20\%

mass losses for whole and cracked beans results in a total roasted mass factor of

1.73x

. Equating

1.73x = 346

gives

x = 200

kg.

Step-by-Step Solution

Set up expressions for initial whole and cracked bean masses

Batch A has

0.7x

kg whole and

0.3x

kg cracked beans; Batch R has

0.6y

kg whole and

0.4y

kg cracked beans.

Ratios

7:3

and

3:2

convert directly to component fractions

7/10 = 0.7

3/10 = 0.3

for A, and

3/5 = 0.6

2/5 = 0.4

for R.

Equate the total component ratio to

13:7

to find the relationship between

x

and

y

x = y

Setting

\frac{0.7x + 0.6y}{0.3x + 0.4y} = \frac{13}{7}

and cross-multiplying yields

4.9x + 4.2y = 3.9x + 5.2y

, simplifying to

x = y

Apply roasting mass loss percentages to compute total roasted mass in terms of

x

Total roasted mass =

1.73x

Whole beans retain

90\%

mass (

0.90 \times 1.3x = 1.17x

) and cracked beans retain

80\%

mass (

0.80 \times 0.7x = 0.56x

). Summing yields

1.17x + 0.56x = 1.73x

Solve for

x

given total roasted mass of

346

x = 200

Dividing

346

1.73

yields

200

kg.

Key Concept

Weighted average ratios combined with proportional percentage change

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