Friday, July 10, 2015

Linear Inequality problem

A motel charges a $100 flat fee per one night of stay plus 8.5% tax. Before tax, senior citizens receive a 20% discount, and children under 10 receive a 15% discount. If the motel wishes to collect at least $1,000 per day while having an average of x children under 10, y adults, and z senior citizens staying each night, which inequality shows the average daily total collected by the motel?

A. 100(0.085)[1.085x + y +  1.080z] ≥ 1000
B. 100(1.085)[0.85x + y +  0.80z] ≤ 1000
C. 100(1.085)[0.85x + y +  0.80z] ≥ 1000
D. 100(0.085)[0.85x + y +  0.80z] ≥ 1000

A senior citizen gets a 20% discount, so his rate is 100(1 – 0.20) = 100(0.80) per night of stay. After tax, a senior citizen will pay 100(0.80)(1 + 0.085) = 100(0.80)(1.085) dollars per night. Thus the motel will receive 100(0.80)(1.085)z from senior citizens if z senior citizens stay per night. An adult gets no discount, so they will pay 100(1 + 0.085) = 100(1.085) dollars per night. A total of 100(1.085)y is collected per night by the motel if y adults stay per night. A child receives a 15% discount, so his rate including tax is 100(1 – 0.15)(1 + 0.085) = 100(0.85)(1.085) dollars per night. The motel will collect 100(0.85)(1.085)x if x children stay per night. Thus the total collected by the motel for x children, y adults, and z senior citizens staying for the night is 100(0.85)(1.085)x + 100(1.085)y + 100(0.80)(1.085)z = 100(1.085)[0.85x + y +  0.80z] dollars. Since this sum must be at least 1000 dollars, it must be greater than or equal to 1000. Thus the correct answer is (C).


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