Question 12·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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At the start of a community festival, $40\%$ of the visitors had purchased their tickets online. During the festival, 18 visitors who had purchased tickets online left, and 12 visitors who had purchased tickets at the gate entered. At that point, $25\%$ of the visitors at the festival had purchased their tickets online.

By what percent did the number of visitors who had purchased tickets online decrease from the start of the festival?

Changing attendance affects both the online group and the total crowd, so this is a percent-change question with a moving whole. Let the starting total be unknown, update both counts, and use a Desmos regression to match the later online share. For a percent decrease, compare the number lost with the original count of that group.

Hints

Hint 1
A part is a group within a larger total. Let $x$ be everyone at the start. How can you use the starting percentage to write the number who bought online?
Hint 2
Someone leaving reduces the total as well as their own group. Someone entering through the gate increases only the total, not the online group. Write each later count separately.
Hint 3
The later percent is a share of everyone there later. Set the later online count equal to that percent of the later total, then compare the number lost with the online count at the start.

Step-by-step

Track both counts, then use Desmos
Step 1
Write the starting online count
Let $x$ be all visitors at the start. A part is a group within that total, so the number who bought online is $0.40x$ .
Step 2
Update the online count
The $18$ visitors who left bought online, so subtract them from the online group: $0.40x-18$ . This counts the online visitors still there.
Step 3
Update the total crowd
Both departures and arrivals change the total. Subtract the $18$ who left and add the $12$ who entered, even though those arrivals bought at the gate: $x-18+12=x-6$ . So the later total is $x-6$ .
Step 4
Match the later online share
The percent $25\%$ describes the online visitors as a share of everyone there at that point, not of the starting crowd. Set the updated online count equal to $0.25$ of the updated total: $0.40x-18=0.25(x-6)$ .
Step 5
Find the starting total
Type $0.40x_1-18\sim0.25(x_1-6)$ in Desmos. Replace $x$ with $x_1$ so Desmos finds its value rather than graphing with it; $\sim$ tells Desmos to match the two sides. Under PARAMETERS, it shows $x_1=110$ , so 110 visitors were there at the start.
Step 6
Find the original online count
Type $40\%\operatorname{of}110$ on the next line. Desmos returns $44$ . The $110$ is the starting total, so 44 online visitors is the starting count to use for the requested decrease.
Step 7
Calculate the percent decrease
The decrease is $18$ online visitors. A percent decrease divides the loss by the original online count, so type $\frac{18}{44}\cdot100$ . Desmos shows $40.909\ldots$ ; its fraction button gives $\frac{450}{11}$ . The number of visitors who bought online decreased by $\frac{450}{11}\%$ . Choice C.

Question 12·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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