A craft brewery produces two specialty beverages, Batch A and Batch B, using two primary i… — GMAT

A craft brewery produces two specialty beverages, Batch A and Batch B, using two primary ingredients: hops and malt. To produce 1 barrel of Batch A, the facility requires 4 kilograms of hops and 10 kilograms of malt. To produce 1 barrel of Batch B, the facility requires 6 kilograms of hops and 15 kilograms of malt. Let $a$ represent the number of barrels of Batch A produced and $b$ represent the number of barrels of Batch B produced in a week, where $a > 0$ and $b > 0$ . The total mass of hops used is $H$ kilograms, and the total mass of malt used is $M$ kilograms. Which of the following statements must be true for any valid production quantities of $a$ and $b$ ? Select all that apply.

$2M = 5H$ Answer
$M > H$ Answer
C
$3H - 2M = 0$
D
The ratio of total malt used to total hops used, $\frac{M}{H}$ , depends on the specific values of $a$ and $b$ .
E
If $H = 120$ , then the number of barrels of Batch A produced ( $a$ ) must be equal to 15.

Answer

The statements

2M = 5H

and

M > H

must both be true.

Expressing

H = 4a + 6b = 2(2a + 3b)

and

M = 10a + 15b = 5(2a + 3b)

reveals that

M = 2.5H

, or equivalently

2M = 5H

. Because

H

is strictly positive for positive values of

a

and

b

M = 2.5H

also guarantees that

M > H

Step-by-Step Solution

Set up algebraic equations for total hops (

H

) and total malt (

M

) in terms of

a

and

b

H = 4a + 6b

and

M = 10a + 15b

Each barrel of Batch A uses 4 kg hops and 10 kg malt, while each barrel of Batch B uses 6 kg hops and 15 kg malt.

Factor common numerical terms from both algebraic expressions.

H = 2(2a + 3b)

and

M = 5(2a + 3b)

Factoring isolates the common linear factor

(2a + 3b)

present in both quantities.

Calculate the ratio

\frac{M}{H}

and clear fractions to find the invariant linear equation.

\frac{M}{H} = \frac{5(2a + 3b)}{2(2a + 3b)} = \frac{5}{2} \implies 2M = 5H.

Since

a > 0

and

b > 0

(2a + 3b) \neq 0

, so the variable terms cancel completely.

Evaluate the inequality relationship between

M

and

H

M = 2.5H \implies M - H = 1.5H > 0 \implies M > H

Because

a, b > 0

H > 0

, making

2.5H

strictly greater than

H

Key Concept

Linear Equation Modeling and Proportional Invariants

Question

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