An event catering service charges a fixed setup fee plus a constant hourly rate for event … — GMAT

An event catering service charges a fixed setup fee plus a constant hourly rate for event staff. For a 4-hour event, the total charge is $680. For a 7-hour event with the same staff requirements, the total charge is$ 1,070. What is the fixed setup fee, in dollars, charged by the catering service?

Answer: 160 dollars

Answer

The fixed setup fee charged by the catering service is 160 dollars.

Modeling the total cost as

C = S + rt

, where

S

is the fixed setup fee and

r

is the hourly rate, yields the system of equations

S + 4r = 680

and

S + 7r = 1070

. Subtracting the first equation from the second yields

3r = 390

, so

r = 130

. Substituting

r = 130

back into the first equation yields

S + 520 = 680

, which gives

S = 160

Step-by-Step Solution

Define the algebraic model for total cost.

Let

S

be the fixed setup fee in dollars and

r

be the hourly rate in dollars per hour. The total cost

C

for

t

hours is modeled by

C = S + rt

The problem presents a linear relationship between time and total cost, consisting of a fixed component and a variable component.

Construct a system of linear equations.

S + 4r = 680

and

S + 7r = 1070

Substitute the two given combinations of time (

t = 4

and

t = 7

) and total cost (

C = 680

and

C = 1070

) into the algebraic model.

Solve for the variable hourly rate

r

(S + 7r) - (S + 4r) = 1070 - 680 \implies 3r = 390 \implies r = 130

Subtracting the first equation from the second eliminates the fixed setup fee variable

S

Calculate the fixed setup fee

S

S + 4(130) = 680 \implies S + 520 = 680 \implies S = 160

Substitute

r = 130

back into the first equation to find the value of

S

Key Concept

Linear Equation Modeling and Systems of Equations

Question

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