A freight logistics company charges fixed rates per unit for two types of cargo containers… — GMAT

A freight logistics company charges fixed rates per unit for two types of cargo containers: Type $X$ and Type $Y$ . Shipping $4$ Type $X$ containers and $3$ Type $Y$ containers costs a total of $\$290$ . Shipping $6$ Type $X$ containers and $5$ Type $Y$ containers costs a total of $\$460$ . Based on these rates, what is the total cost to ship $7$ Type $X$ containers and $6$ Type $Y$ containers?

A
$375
B
$495
$545Answer
D
$630
E
$750

Answer

$545

By solving the system

4x + 3y = 290

and

6x + 5y = 460

, subtracting the first equation from the second yields

2x + 2y = 170

, so

x + y = 85

. Adding

x + y = 85

6x + 5y = 460

gives

7x + 6y = 545

. Alternatively, solving individually yields

x = 35

and

y = 50

, giving

7(35) + 6(50) = 545

Step-by-Step Solution

Set up a system of linear equations representing the given shipping costs.

Let

x

be the cost per Type

X

container and

y

be the cost per Type

Y

container. Equation (1):

4x + 3y = 290

. Equation (2):

6x + 5y = 460

Translate the word problem statements into linear equations.

Subtract Equation (1) from Equation (2) to find a linear combination.

(6x + 5y) - (4x + 3y) = 460 - 290 \implies 2x + 2y = 170 \implies x + y = 85.

Determining the combined cost of one container of each type simplifies calculating larger combinations.

Add the expression for

x + y

to Equation (2) to obtain

7x + 6y

(6x + 5y) + (x + y) = 460 + 85 \implies 7x + 6y = 545.

Combining

6x + 5y

and

x + y

directly gives the cost for

7

Type

X

and

6

Type

Y

containers without requiring individual variable solving.

Key Concept

Linear combinations and systems of linear equations in two variables

Estimated Time:1m 30s

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