A social club reserves a private lounge for a holiday party. They receive a 25\% promotion… — Armed Services Vocational Aptitude Battery (ASVAB)

A social club reserves a private lounge for a holiday party. They receive a $25\%$ promotional discount on the standard food and beverage package price. A local hospitality tax of $6\%$ is applied to the discounted price, but the contract requires a mandatory $15\%$ gratuity calculated on the standard, pre-discount package price.

If the final bill paid by the club—consisting of the discounted package, the tax, and the gratuity—amounts to $\$378.00$ , what was the standard package price before any discounts or fees?

Answer:The standard package price was $\【400】.

Answer

The standard package price was $400.

The total bill is the sum of the discounted price, the tax on the discounted price, and the tip on the original price. By setting the original price as

P

, we can express the total cost algebraically:

0.75P

(discounted price)

+ 0.06(0.75P)

(tax)

+ 0.15P

(gratuity)

= 0.945P

. Setting

0.945P = 378

and solving for

P

yields exactly

400

Step-by-Step Solution

Let

P

represent the standard package price. Express the discounted price in terms of

P

Discounted Price =

P - 0.25P = 0.75P

The club receives a

25\%

discount off the standard price.

Express the tax amount in terms of

P

Tax =

0.06 \times 0.75P = 0.045P

The problem states that the

6\%

hospitality tax is applied to the discounted price.

Express the gratuity amount in terms of

P

Gratuity =

0.15 \times P = 0.15P

The contract explicitly requires the

15\%

tip to be calculated on the original, pre-discount package price.

Set up an equation for the final bill and solve for

P

0.75P + 0.045P + 0.15P = 378 \implies 0.945P = 378 \implies P = 378 \div 0.945 = 400

The sum of the discounted package, the tax, and the gratuity equals the total final bill of

\$378.00

Key Concept

Solving for an original price when tax and tip are calculated using different base amounts (discounted vs. pre-discount).

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