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Difficulty: MediumSimultaneous Equations and Systems

A luxury boutique hotel offers two types of guest accommodations: Deluxe Suites and Executive Suites. On a busy Saturday, housekeeping spends 45 minutes cleaning each Deluxe Suite and 75 minutes cleaning each Executive Suite, accumulating a total of 38 hours of cleaning time. On the same day, laundry services supply 4 sets of premium towels for each Deluxe Suite and 7 sets of premium towels for each Executive Suite, delivering a total of 208 towel sets. Which of the following correctly pairs the number of Deluxe Suites cleaned with the number of Executive Suites cleaned on that day?

  1. Deluxe Suites = 24; Executive Suites = 16Answer
  2. B
    Deluxe Suites = 16; Executive Suites = 24
  3. C
    Deluxe Suites = 28; Executive Suites = 12
  4. D
    Deluxe Suites = 20; Executive Suites = 20
  5. E
    Deluxe Suites = 32; Executive Suites = 8

Answer

Deluxe Suites = 24; Executive Suites = 16
Solving the simultaneous system of linear equations 3d+5e=1523d + 5e = 152 and 4d+7e=2084d + 7e = 208 yields d=24d = 24 Deluxe Suites and e=16e = 16 Executive Suites. The choice indicating 24 Deluxe Suites and 16 Executive Suites correctly matches these calculated values to their respective variables.

Step-by-Step Solution

1
Define variables and convert cleaning time to minutes.
Let dd be the number of Deluxe Suites and ee be the number of Executive Suites. Total cleaning time = 38 hours×60 minutes/hour=2280 minutes38 \text{ hours} \times 60 \text{ minutes/hour} = 2280 \text{ minutes}.
Units must be consistent across variables before setting up algebraic equations.
2
Formulate the linear equations system.
Equation (1) [Cleaning time]: 45d+75e=228045d + 75e = 2280
Equation (2) [Towel sets]: 4d+7e=2084d + 7e = 208
Translates the real-world operational constraints into simultaneous linear algebraic equations.
3
Simplify Equation (1) by dividing all terms by 15.
3d+5e=1523d + 5e = 152
Simplifying coefficients makes simultaneous elimination computationally easier.
4
Solve the system using elimination.
Multiply 3d+5e=1523d + 5e = 152 by 44: 12d+20e=60812d + 20e = 608.
Multiply 4d+7e=2084d + 7e = 208 by 33: 12d+21e=62412d + 21e = 624.
Subtracting the first equation from the second yields: (12d+21e)−(12d+20e)=624−608⇒e=16(12d + 21e) - (12d + 20e) = 624 - 608 \Rightarrow e = 16.
Eliminates dd to isolate variable ee.
5
Substitute e=16e = 16 back into the simplified Equation (1) to solve for dd.
3d+5(16)=152⇒3d+80=152⇒3d=72⇒d=243d + 5(16) = 152 \Rightarrow 3d + 80 = 152 \Rightarrow 3d = 72 \Rightarrow d = 24.
Determines the value of the second variable dd.

Key Concept

Setting up and solving a system of two linear equations in two variables
Estimated Time:2m 0s
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Topics

Simultaneous Equations and Systems

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