A specialty coffee shop sells two types of coffee beans: Type A for \8 per pound and Type … — GMAT

A specialty coffee shop sells two types of coffee beans: Type A for $\$8$ per pound and Type B for $\$12$ per pound. A customer buys a total of $x$ pounds of coffee, consisting of $a$ pounds of Type A and $b$ pounds of Type B, for a total cost of $C$ dollars. Which of the following equations correctly model the relationship between these quantities? Select all that apply.

$a = x - b$ Answer
$C = 12x - 4a$ Answer
$b = \frac{C - 8a}{12}$ Answer
D
$C = 20x$
E
$a = \frac{C - 12}{8x}$

Answer

The correct formulations are

a = x - b

C = 12x - 4a

, and

b = \frac{C - 8a}{12}

The system is defined by two fundamental linear relationships: weight total

a + b = x

and cost total

8a + 12b = C

. Isolating

a

from the weight total gives

a = x - b

. Substituting

b = x - a

into the cost equation yields

C = 8a + 12(x - a) = 12x - 4a

. Isolating

b

from the cost equation gives

12b = C - 8a \implies b = \frac{C - 8a}{12}

. All three expressions are mathematically equivalent and correct.

Step-by-Step Solution

Set up the basic linear system from the word problem.

Total weight equation:

a + b = x

; Total cost equation:

8a + 12b = C

Word problems with total quantity and total monetary value translate into two distinct linear equations.

Rearrange the total weight equation to express

a

in terms of

x

and

b

a = x - b

Subtracting

b

from both sides isolates

a

Substitute

b = x - a

into the total cost equation.

C = 8a + 12(x - a) = 8a + 12x - 12a = 12x - 4a

Expressing cost in terms of a single variable

a

and total weight

x

simplifies multi-variable models.

Isolate

b

in the total cost equation.

12b = C - 8a \implies b = \frac{C - 8a}{12}

Subtracting the cost contribution of Type A (

8a

) and dividing by the unit cost of Type B (

12

) isolates

b

Key Concept

Linear Equation Modeling in Word Problems

Question

Answer

Step-by-Step Solution

Key Concept

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