S95-24

Official Solution:

An amusement park currently charges the same price for each ticket of admission. If the current price of admission were to be increased by \($3\), 12 fewer tickets could be bought for \($160\), excluding sales tax. What is the current price of each ticket?

A. \($3\)
B. \($5\)
C. \($8\)
D. \($20\)
E. \($32\)


The question asks us to determine the current price of each ticket of admission.

If we let \(p\) equal the current price per ticket, and \(n\) equal the number of tickets that can be bought for \($160\), we can set up some equations. First, we know that \(pn = 160\).

Second, we are told that if \(p\) is increased by 3, then 12 fewer tickets can be bought with \($160\). This equation also expresses how many tickets can be bought for \($160\), giving us: \((p + 3)(n - 12) = 160\).

Solve the first equation for \(n\), giving: \(n = \frac{160}{p}\).

Substitute this value for \(n\) into the second equation, solving for \(p\):

\((p + 3)(\frac{160}{p} - 12) = 160\)

Multiply both sides by \(p\) to get rid of the fraction: \((p + 3)(\frac{160}{p} - 12)p = (p + 3)(160 - 12p) = 160p\).

Multiply through to get rid of the parentheses: \(160p - 12p^2 + 480 - 36p = 160p\).

Combine like terms: \(124p - 12p^2 + 480 = 160p\). Set the equation equal to 0: \(0 = 12p^2 + 36p - 480\).Divide both sides by 12: \(0 = p^2 + 3p - 40\).

Factor: \(0 = (p - 5)(p + 8)\).

Therefore, \(p = 5\) or \(p = -8\). Since the the price cannot be negative, \(p = 5\).


Answer: B