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?

Cevap: 35

Cevap

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

Adım Adım Çözüm

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.

Anahtar Kavram

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

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru