Super-Hard SAT Data Analysis Question 5 | Free SAT Prep

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Question 5·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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A theater sells only standard tickets and student tickets. After a price adjustment, the number of tickets sold in each category remains the same as before the adjustment. The price of a standard ticket increased by $15\%$ , and the price of a student ticket decreased by $10\%$ . As a result, the theater's total ticket revenue increased by $4\%$ .

To the nearest $0.01\%$ , which choice is the percentage of the theater's total ticket revenue after the price adjustment that came from standard tickets?

For percentage-change questions involving parts of a total, assign a variable to one category's original share and express the other category as the remaining share. Use the weighted changes to match the stated overall percent change. If the question asks about the new distribution, divide the changed amount for the category by the changed total rather than stopping at the original share.

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Define a fraction

Let $x$ represent the fraction of the original ticket revenue from standard tickets. The fraction from student tickets is then $1-x$ .

Translate each price change

A $15\%$ increase means multiply the standard-ticket revenue by $1.15$ . A $10\%$ decrease means multiply the student-ticket revenue by $0.90$ .

Use the correct total

After finding the original revenue split, remember that the question asks for a share of the revenue after the adjustment, not before it.

Desmos Guide

Use algebra first

Algebra is generally faster here because the relationship is a single linear equation. Desmos can verify the original standard-ticket revenue fraction.

Graph the revenue expressions

Enter $y=1.15x+0.90(1-x)$ and $y=1.04$ . Click their intersection to find the value of $x$ .

Calculate the new share

In a new expression line, enter $(1.15x)/1.04$ , using the intersection value for $x$ . Convert the resulting decimal to a percent and round to the nearest hundredth of a percent.

Step-by-step Explanation

Represent the original revenue mix

Let $x$ be the fraction of the original total revenue that came from standard tickets. Then $1-x$ is the fraction that came from student tickets.

Use the overall revenue increase

Measured as a fraction of the original total revenue, the new total revenue is $1.04$ . Standard-ticket revenue is multiplied by $1.15$ , and student-ticket revenue is multiplied by $0.90$ :

1.15x+0.90(1-x)=1.04

Solving gives $0.25x=0.14$ , so $x=0.56$ .

Find the share of the new total

After the adjustment, standard-ticket revenue is $1.15(0.56)=0.644$ of the original total revenue. The new total revenue is $1.04$ of the original total, so the new standard-ticket share is

\frac{0.644}{1.04}=0.619230\ldots

This is $61.92\%$ to the nearest $0.01\%$ .

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