A specialty coffee shop blends three types of coffee beans: Colombian (8.00 dollars per po… — ACT

A specialty coffee shop blends three types of coffee beans: Colombian ( $8.00$ dollars per pound), Ethiopian ( $11.00$ dollars per pound), and Sumatran ( $14.00$ dollars per pound). The shop manager wants to create a $50$ -pound blend that costs exactly $11.60$ dollars per pound. Due to supply constraints, the weight of the Ethiopian beans in the blend must be exactly $2$ pounds more than $\frac{1}{5}$ of the combined weight of the Colombian and Sumatran beans. What is the value of $2s - c$ , where $s$ is the amount of Sumatran beans, in pounds, and $c$ is the amount of Colombian beans, in pounds, in the blend?

Answer: 35

Answer

The correct answer is

35

The correct answer of

35

is found by translating the given constraints into three linear equations, solving for

e = 10

using substitution of the sum

c+s = 50 - e

, and then solving the resulting two-variable system to find

s = 25

and

c = 15

. Evaluating

2s - c

yields

2(25) - 15 = 35

Step-by-Step Solution

Define variables and write the system of equations representing the constraints.

Let

c

e

, and

s

represent the weight in pounds of Colombian, Ethiopian, and Sumatran beans, respectively. The system is:
1)

c + e + s = 50

(total weight)
2)

8c + 11e + 14s = 50 \times 11.60 = 580

(total cost)
3)

e = \frac{1}{5}(c + s) + 2

(Ethiopian bean weight constraint)

This translates the word problem's conditions into algebraic expressions.

Solve for the variable

e

using substitution.

From equation (1), we have

c + s = 50 - e

. Substituting this expression into equation (3) yields:

e = \frac{1}{5}(50 - e) + 2

Multiply both sides by

5

5e = 50 - e + 10

6e = 60 \implies e = 10

Grouping

c + s

allows us to solve for

e

directly without dealing with three separate variable eliminations.

Substitute

e = 10

back into the first two equations to simplify the system to two variables.

Equation (1) becomes:

c + s = 40 \implies c = 40 - s

Equation (2) becomes:

8c + 11(10) + 14s = 580 \implies 8c + 14s = 470

This reduces the remaining problem to a standard system of two linear equations.

Solve for

c

and

s

by substitution.

Substitute

c = 40 - s

into the simplified cost equation:

8(40 - s) + 14s = 470

320 - 8s + 14s = 470

320 + 6s = 470 \implies 6s = 150 \implies s = 25

Then, find

c

c = 40 - 25 = 15

This gives the exact weight of both the Sumatran and Colombian beans.

Evaluate the expression

2s - c

requested by the prompt.

2s - c = 2(25) - 15 = 50 - 15 = 35

This satisfies the specific algebraic quantity requested in the question.

Key Concept

Translating word problems into systems of linear equations and solving them using algebraic substitution.

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