A logistics terminal at a seaport utilizes two types of automated cranes, Model X and Mode… — GMAT

A logistics terminal at a seaport utilizes two types of automated cranes, Model X and Model Y, to unload container ships. During the morning shift, a fleet of 4 Model X cranes and 5 Model Y cranes unloaded a total of 330 containers. During the afternoon shift, a fleet of 6 Model X cranes and 2 Model Y cranes unloaded the same total of 330 containers. Assuming all cranes of a given model operate at a constant unloading rate per shift, select the value in the table that represents the number of containers unloaded per shift by a single Model X crane and the value that represents the number of containers unloaded per shift by a single Model Y crane.

Model X Crane45
Model Y Crane30

Answer

Model X Crane unloads 45 containers per shift, and Model Y Crane unloads 30 containers per shift.

Setting up the system

4x + 5y = 330

and

6x + 2y = 330

yields

x = 45

for Model X and

y = 30

for Model Y. Substituting these values into both equations confirms accuracy:

4(45) + 5(30) = 180 + 150 = 330

and

6(45) + 2(30) = 270 + 60 = 330

Step-by-Step Solution

Define variables and construct the system of linear equations from the given problem statement.

Let

x

be the number of containers unloaded per shift by a Model X crane, and

y

be the number of containers unloaded per shift by a Model Y crane. Morning shift equation:

4x + 5y = 330

. Afternoon shift equation:

6x + 2y = 330

Translating word problem constraints into explicit algebraic equations is required to solve for both unknowns.

Express

y

in terms of

x

using the afternoon shift equation.

2y = 330 - 6x \implies y = 165 - 3x

Using substitution allows reducing the two-variable system to a single equation in one variable.

Substitute the expression for

y

into the morning shift equation and solve for

x

4x + 5(165 - 3x) = 330 \implies 4x + 825 - 15x = 330 \implies -11x = -495 \implies x = 45

Solving the single-variable equation determines the exact unloading capacity of a Model X crane.

Substitute

x = 45

back into the expression for

y

to find the capacity of a Model Y crane.

y = 165 - 3(45) = 165 - 135 = 30

Substituting the value of

x

gives the specific unloading rate for Model Y.

Key Concept

Simultaneous Linear Equations in Two Variables

Question

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