Three museums charge an entrance fee based on the number of visitors in the group. The table lists the fees charged by the museums. At which museum is the entrance fee proportional to the number of visitors?
\begin{tabular}{|r|r|r|r|r|r|}
\hline \multicolumn{2}{|c|}{ Museum A } & \multicolumn{2}{c|}{ Museum B } & \multicolumn{2}{c|}{ Museum C } \\
\hline Visitors & Fee ($) & Visitors & Fee ($) & Visitors & Fee ($) \\
\hline 2 & 4 & 1 & 2 & 3 & 4 \\
\hline 3 & 5 & 4 & 8 & 12 & 16 \\
\hline 4 & 6 & 6 & 11 & 18 & 24 \\
\hline
\end{tabular}
A. museum $A$
B. museum B
C. museum C
D. museum $A$ and museum $B$
Answer (1)
Calculate the ratio of fee to visitors for Museum A: 2 4 = 2 , 3 5 ≈ 1.67 , 4 6 = 1.5 . The ratios are not equal.
Calculate the ratio of fee to visitors for Museum B: 1 2 = 2 , 4 8 = 2 , 6 11 ≈ 1.83 . The ratios are not equal.
Calculate the ratio of fee to visitors for Museum C: 3 4 ≈ 1.33 , 12 16 ≈ 1.33 , 18 24 ≈ 1.33 . The ratios are equal.
Museum C has an entrance fee proportional to the number of visitors. $\boxed{C}
Explanation
Analyzing the Problem Let's analyze the entrance fees for each museum to determine if they are proportional to the number of visitors. A proportional relationship means that the ratio of the fee to the number of visitors is constant.
Museum A Analysis For Museum A, we calculate the ratios of fee to visitors for each given data point:
For 2 visitors, the ratio is 2 4 = 2 .
For 3 visitors, the ratio is 3 5 ≈ 1.67 .
For 4 visitors, the ratio is 4 6 = 1.5 .
Since these ratios are not equal, the entrance fee for Museum A is not proportional to the number of visitors.
Museum B Analysis For Museum B, we calculate the ratios of fee to visitors for each given data point:
For 1 visitor, the ratio is 1 2 = 2 .
For 4 visitors, the ratio is 4 8 = 2 .
For 6 visitors, the ratio is 6 11 ≈ 1.83 .
Since these ratios are not equal, the entrance fee for Museum B is not proportional to the number of visitors.
Museum C Analysis For Museum C, we calculate the ratios of fee to visitors for each given data point:
For 3 visitors, the ratio is 3 4 ≈ 1.33 .
For 12 visitors, the ratio is 12 16 ≈ 1.33 .
For 18 visitors, the ratio is 18 24 ≈ 1.33 .
Since these ratios are equal, the entrance fee for Museum C is proportional to the number of visitors.
Conclusion Based on our analysis, only Museum C has an entrance fee that is proportional to the number of visitors.
Examples
Understanding proportional relationships is useful in everyday scenarios, such as calculating the cost of buying multiple items at a store. If the price of one apple is $0.50, then the cost of buying 5 apples would be $0.50 \times 5 = $2.50, demonstrating a proportional relationship between the number of apples and the total cost. Similarly, understanding proportions helps in scaling recipes, converting currencies, and determining distances on maps.
