A biotechnology laboratory formulates three custom reagent mixtures—Solution X, Solution Y… — GMAT

A biotechnology laboratory formulates three custom reagent mixtures—Solution $X$ , Solution $Y$ , and Solution $Z$ —using three chemical compounds: Alpha, Beta, and Gamma.

- Solution $X$ contains $2\text{ mL}$ of Alpha, $3\text{ mL}$ of Beta, and $1\text{ mL}$ of Gamma, and costs $\$13.00$ .
- Solution $Y$ contains $1\text{ mL}$ of Alpha, $2\text{ mL}$ of Beta, and $4\text{ mL}$ of Gamma, and costs $\$11.00$ .
- Solution $Z$ contains $3\text{ mL}$ of Alpha, $1\text{ mL}$ of Beta, and $2\text{ mL}$ of Gamma, and costs $\$13.00$ .

What is the cost of a mixture containing $5\text{ mL}$ of Alpha, $4\text{ mL}$ of Beta, and $3\text{ mL}$ of Gamma?

A
$\$22.00$
B
$\$24.00$
C
$\$25.00$
$\$26.00$ Answer
E
$\$30.00$

Answer

\$26.00

The target quantity consists of

5\text{ mL}

of Alpha,

4\text{ mL}

of Beta, and

3\text{ mL}

of Gamma. By inspecting the given system, adding Solution

X

(

2\text{ mL}

Alpha,

3\text{ mL}

Beta,

1\text{ mL}

Gamma) and Solution

Z

(

3\text{ mL}

Alpha,

1\text{ mL}

Beta,

2\text{ mL}

Gamma) directly yields

5\text{ mL}

Alpha,

4\text{ mL}

Beta, and

3\text{ mL}

Gamma. Therefore, the required cost is simply the sum of the costs of Solution

X

and Solution

Z

\$13.00 + \$13.00 = \$26.00

Step-by-Step Solution

Set up the linear system of equations representing the cost of each solution.

Let

a

b

, and

g

be the cost per mL of Alpha, Beta, and Gamma, respectively.
Equation 1:

2a + 3b + g = 13

Equation 2:

a + 2b + 4g = 11

Equation 3:

3a + b + 2g = 13

Translate the given word problem into algebraic equations representing system relationships.

Identify the requested quantity and evaluate whether it can be formed as a linear combination of the given equations.

Target expression:

5a + 4b + 3g

Recognizing linear combinations avoids solving for individual variable values when not required.

Add Equation 1 and Equation 3.

(2a + 3b + g) + (3a + b + 2g) = 13 + 13 \implies 5a + 4b + 3g = 26

The sum of coefficients for Alpha (

2+3=5

), Beta (

3+1=4

), and Gamma (

1+2=3

) exactly matches the target mixture.

Key Concept

Solving systems of linear equations using linear combinations without full variable elimination

Estimated Time:2m 0s

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