A semiconductor manufacturing facility operates two models of photolithography equipment: … — GMAT

A semiconductor manufacturing facility operates two models of photolithography equipment: Model D and Model E. Over a 10-hour operational cycle, each Model D scanner consumes $400\text{ kWh}$ of energy and processes $600$ silicon wafers, while each Model E scanner consumes $1,200\text{ kWh}$ of energy and processes $1,000$ silicon wafers. If the combined fleet consumed a total of $9,600\text{ kWh}$ of energy and processed $10,400$ wafers over the cycle, which of the following correctly identifies the number of Model D scanners and Model E scanners operated by the facility?

9 Model D scanners and 5 Model E scannersAnswer
B
5 Model D scanners and 9 Model E scanners
C
12 Model D scanners and 4 Model E scanners
D
6 Model D scanners and 6 Model E scanners
E
15 Model D scanners and 3 Model E scanners

Answer

9 Model D scanners and 5 Model E scanners

The correct answer states 9 Model D scanners and 5 Model E scanners. Setting up the system

d + 3e = 24

(energy) and

3d + 5e = 52

(production) and solving by substitution gives

e = 5

and

d = 9

, which uniquely satisfies both constraints.

Step-by-Step Solution

Define variables and establish equations based on given rates and totals.

Let

d

be the number of Model D scanners and

e

be the number of Model E scanners.
Energy equation:

400d + 1200e = 9600

Wafer equation:

600d + 1000e = 10400

Translate the physical constraints into a system of two linear equations.

Simplify both linear equations by dividing by their greatest common divisors.

Dividing the energy equation by

400

gives:

d + 3e = 24 \quad \text{--- (Equation 1)}

Dividing the wafer equation by

200

gives:

3d + 5e = 52 \quad \text{--- (Equation 2)}

Simplifying constants reduces calculation complexity and prevents arithmetic errors.

Solve the system using substitution or elimination.

From Equation 1, express

d

in terms of

e

d = 24 - 3e

Substitute this expression into Equation 2:

3(24 - 3e) + 5e = 52

72 - 9e + 5e = 52

72 - 4e = 52

4e = 20 \implies e = 5

Now calculate

d

d = 24 - 3(5) = 24 - 15 = 9

Isolating one variable yields exact values for both equipment models.

Verify values against both constraints.

Energy:

400(9) + 1200(5) = 3600 + 6000 = 9600\text{ kWh}

Wafers:

600(9) + 1000(5) = 5400 + 5000 = 10400\text{ wafers}

Ensures the solution satisfies both independent constraints simultaneously.

Key Concept

Simultaneous Linear Equations in Two Variables

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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