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Understanding ticket pricing involves analyzing how changes in price affect demand. In the context of school dances, setting the right ticket price is crucial for maximizing revenue. Here, tickets start at $4, which is the base price.
Changes in pricing are expressed in increments of $0.10. For every $0.10 increase, fewer people are expected to attend. Specifically, for this exercise, 5 less people attend for every $0.10 increase in ticket price.
This reflects the law of demand in economics, where an increase in price typically leads to a decrease in quantity demanded. Finding the optimal ticket price is a balancing act between the price increase and attendance decrease.

• Start price: $4
• Increment: $0.10
• Attendance decrease per increment: 5 people

Understanding these components helps set a goal, like achieving a specific revenue mark, in this case, $1,248.

Algebraic expressions are tools to represent situations where elements change with variables. In this instance, algebra helps model the relationship between ticket price and attendance.
The exercise uses algebra to relate price to attendance in a formula:
- The price per ticket becomes \(4 + 0.10x\), representing the original $4 price plus an increase.- The projected attendance decreases by 5 for each price increment, leading to the expression \(300 - 5x\).The receipt total, which combines both aspects, is represented as the product \((4 + 0.10x)(300 - 5x)\).

• Expression for ticket price: \(4 + 0.10x\)
• Expression for attendance: \(300 - 5x\)
• Total receipt formula: \((4 + 0.10x)(300 - 5x)\)

Algebraic expressions simplify real-life problems by converting conditions into calculable formats.

Solving equations is a fundamental skill in algebra that involves finding values of unknowns that satisfy given relationships. In this problem, we are tasked with finding a ticket price that results in a total receipt of \(1,248.
We begin by equating the expression for total receipts to 1,248: \[(4 + 0.10x)(300 - 5x) = 1248\]
The equation is expanded and simplified into a quadratic form:\[-0.5x^2 + 10x + 1200 - 1248 = 0\]
Further simplification leads to a standard quadratic equation:\[x^2 - 20x + 96 = 0\]Quadratic equations can be solved using diverse methods like factoring, completing the square, or the quadratic formula. Here, we use the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]For this problem:

• \(a = 1\), \(b = -20\), \(c = 96\)
• Discriminant \(b^2 - 4ac = 16\)
• Solutions are \(x = 12\) or \(x = 8\)

Substituting these \(x\) values back verifies the correct ticket prices of \)5.20 and $4.80, both yielding the desired receipts.