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Chapter 9: Problem 95

Solve each problem. The Fan Cost Index (FCI) is a measure of how much it will cost a fam- ily of four to attend a professional sports event. In \(2010,\) the FCI prices for Major League Baseball and the National Football League averaged \(\$ 307.76 .\) The FCI for baseball was \(\$ 225.56\) less than that for football. What were the FCIs for these sports? (Source: Team Marketing Report.)

Short Answer

Expert verified
The FCI for baseball is \( 194.98 \) and for football is \( 420.54 \).

Step by step solution

01

- Define Variables

Let the FCI for baseball be denoted as \( B \) and the FCI for football be denoted as \( F \).
02

- Create Equations

From the information given, you can create the following two equations: 1. The average FCI for both sports combined is given by \( \dfrac{B + F}{2} = 307.76 \). 2. The FCI for baseball is \( 225.56 \) less than that for football, which gives us the equation \( B = F - 225.56 \).
03

- Express \( F \) in Terms of \( B \)

Substitute the second equation \( B = F - 225.56 \) into the first equation. This results in \( \dfrac{B + (B + 225.56)}{2} = 307.76 \).
04

- Solve the Equation

Simplify the equation: \[ \dfrac{2B + 225.56}{2} = 307.76 \] Multiply both sides by 2: \[ 2B + 225.56 = 615.52 \] Subtract 225.56 from both sides: \[ 2B = 389.96 \] Divide by 2: \[ B = 194.98 \]
05

- Find \( F \)

Use the value of \( B \) to find \( F \). Recall \( B = F - 225.56 \): Substitute \( B = 194.98 \) into the equation: \[ 194.98 = F - 225.56 \] Add 225.56 to both sides: \[ F = 420.54 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fan Cost Index (FCI)
The Fan Cost Index (FCI) is a valuable measure used to estimate the average cost for a family of four to attend a professional sports event. This index includes various expenses, such as tickets, food, beverages, and merchandise. In our problem, we are given the average FCI for Major League Baseball and the National Football League. Additionally, we know the FCI for baseball is less than that for football.

The concept behind FCI helps us understand the logistics of sports event pricing from a family perspective. This is essential for fans who want to manage their expenses better and for marketers to gauge the affordability of their events.
Algebraic Manipulation
Algebraic manipulation is a fundamental skill in solving equations. It involves rearranging equations and expressions to find the values of unknown variables. In our exercise, we created two equations based on the problem statement and then manipulated them to find the FCIs for both sports.

First, we set up the equations based on the given information:
  • The average FCI is \(\frac{B + F}{2} = 307.76\).
  • The FCI for baseball is \(225.56\) less than football, so \(B = F - 225.56\).
By substituting one equation into another, we perform algebraic manipulation to solve for one variable first.
Solving Linear Equations
Solving linear equations is a critical method in algebra that allows us to find solutions for variables. When we substituted the equation \(B = F - 225.56\) into the average FCI equation, we got:
\[ \frac{B + (B + 225.56)}{2} = 307.76 \]
We then simplified this to:
\[ \frac{2B + 225.56}{2} = 307.76 \]

Further simplification and solving for \(B\) involved:
  • Multiplying both sides by 2
  • Subtracting 225.56
  • Dividing by 2
Finally, we found \( B = 194.98\).
Application of Systems of Equations
Systems of equations are sets of equations with multiple variables that can be solved simultaneously. In our problem, we used a system of equations to find both the FCIs for baseball and football:

1. The average FCI: \( \frac{B + F}{2} = 307.76 \)
2. The difference in FCI: \( B = F - 225.56 \)

By substituting and simplifying these equations step by step, we determined the values of both variables. After finding \(B\), we used it to calculate \(F\):
\[ 194.98 = F - 225.56 \]
Adding 225.56 to both sides, we obtained \( F = 420.54\).

This systematic approach demonstrates how systems of equations are applied to real-world problems, such as determining expenses in sports events.

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