A artisanal coffee shop creates a custom blend using three grades of coffee beans: Grade P… — GMAT

A artisanal coffee shop creates a custom blend using three grades of coffee beans: Grade P costing $10 per pound, Grade Q costing$ 15 per pound, and Grade R costing $25 per pound. The weight ratio of Grade P to Grade Q in the mixture is$ 3 : 2 $. If the weighted average cost of the total mixture is$ 18 per pound, what fraction of the total mixture, by weight, consists of Grade R?

A
$\frac{5}{12}$
B
$\frac{11}{25}$
$\frac{6}{13}$ Answer
D
$\frac{6}{7}$
E
$\frac{7}{13}$

Answer

The fraction of the total mixture, by weight, that consists of Grade R is

\frac{6}{13}

First, find the weighted average price of Grades P and Q together using their given weight ratio of

3 : 2

\frac{3(10) + 2(15)}{3 + 2} = \frac{60}{5} = \$12

per pound. Next, treat the entire mixture as a combination of two parts: the (P+Q) group at

12 per pound and Grade R at

25 per pound. Let

W_{PQ}

be the weight of the (P+Q) group and

W_R

be the weight of Grade R. Setting up the weighted average gives

12 W_{PQ} + 25 W_R = 18(W_{PQ} + W_R)

, which simplifies to

7 W_R = 6 W_{PQ}

, or

W_R = \frac{6}{7} W_{PQ}

. The fraction of the total mixture represented by Grade R is

\frac{W_R}{W_{PQ} + W_R} = \frac{6/7 W_{PQ}}{W_{PQ} + 6/7 W_{PQ}} = \frac{6}{13}

Step-by-Step Solution

Calculate the average cost per pound of the combined P and Q mixture.

Average cost of P and Q mixture =

\frac{3(\$10) + 2(\$15)}{3 + 2} = \frac{30 + 30}{5} = \$12

per pound.

Since Grade P and Grade Q are present in a weight ratio of

3 : 2

, their combined per-pound cost must be weighted by these relative amounts.

Set up a weighted average equation for the combined mixture of (P + Q) and Grade R.

Let

W_{PQ}

be the weight of the P and Q combined mixture, and

W_R

be the weight of Grade R. The total cost equation is

12 W_{PQ} + 25 W_R = 18 (W_{PQ} + W_R)

The entire mixture consists of the combined P+Q component costing

12/lb and Grade R costing

25/lb, yielding a final weighted average of $18/lb.

Solve for the ratio of

W_R

W_{PQ}

12 W_{PQ} + 25 W_R = 18 W_{PQ} + 18 W_R \implies 7 W_R = 6 W_{PQ} \implies \frac{W_R}{W_{PQ}} = \frac{6}{7}

Expanding and rearranging terms isolates the relationship between the weight of Grade R and the combined weight of Grades P and Q.

Convert the part-to-part ratio into the part-to-whole fraction for Grade R.

\text{Fraction of Grade R} = \frac{W_R}{W_{PQ} + W_R} = \frac{6}{7 + 6} = \frac{6}{13}.

The question asks for the fraction of the total mixture, which requires dividing the weight of Grade R by the total combined weight of all three grades.

Key Concept

Weighted Average of Sub-groups and Combined Sets

Estimated Time:2m 0s

Question

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