Super-Hard SAT Math Question 94 | Free SAT Prep | Aniko

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Question 94·200 Super-Hard SAT Math Questions·Problem Solving and Data Analysis

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A theater sells adult tickets, student tickets, and child tickets. From week 1 to week 2, the number of adult tickets sold increased by $25\%$ , the number of student tickets sold decreased by $10\%$ , and the number of child tickets sold stayed the same.

In week 1, child tickets made up $20\%$ of all tickets sold. If the total number of tickets sold increased by $8\%$ from week 1 to week 2, what percentage of all tickets sold in week 1 were adult tickets? Anikο AI ⁠Tutοr

When one category’s share of the original total is given (here, child tickets are $20\%$ ), rewrite everything in terms of the original total $T$ . Then use multipliers for each category’s change to build a week 2 total equation. If two variables remain, eliminate one by combining the total equation with the known sum (like $A+S=0.80T$ ), and finish by converting $\frac{A}{T}$ to a percent. Pο‍wеred b‍у Anikο

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Use the child-ticket information

Let $T$ be the week 1 total. Since child tickets are $20\%$ of week 1, write $C$ in terms of $T$ , and then express $A+S$ in terms of $T$ .

Write week 2 totals using multipliers

Adult becomes $1.25A$ , student becomes $0.90S$ , and child stays $C$ . The overall week 2 total is $1.08T$ . A‍nік⁠o‌.‌ai -‍ ЅAТ ⁠Рre‌р

Eliminate a variable

You can get one equation from the week 2 total and another from $A+S=0.80T$ . Try scaling $A+S=0.80T$ by $0.90$ so a subtraction cancels $0.90S$ .

Convert to a percent

Once you have $\frac{A}{T}$ , multiply by $100$ to get the week 1 adult percentage.

Desmos Guide

Let the week 1 total be 1

In Desmos, assume the week 1 total is $1$ . Then child tickets are $0.2$ of the total.

Use a variable for the adult fraction of the total

Let $x$ be the adult fraction of the week 1 total. Then the student fraction is $0.8-x$ (since adult + student is $0.8$ of the total). А‌niко ‌- ‌Free ‍ЅАT Р‍reр

Enter the week 2 total equation and solve

Week 2 total is

1.25x+0.90(0.8-x)+0.2.

Set it equal to $1.08$ and solve:

1.25x+0.90(0.8-x)+0.2=1.08.

Convert the solution for $x$ to a percent by multiplying by $100$ and match to the choices (do not state the value until after solving).

Step-by-step Explanation

Define variables and use the 20% condition

Let $T$ be the total number of tickets sold in week 1.

Let $A$ , $S$ , and $C$ be the numbers of adult, student, and child tickets sold in week 1, so

A+S+C=T.

Child tickets are $20\%$ of week 1, so Аnіkо ⁠Q‍ueѕ‍tіоn Bank

C=0.20T,

which means

A+S=T-C=0.80T.

Translate week 2 in terms of week 1

Week 2 sales are:

Adult: $1.25A$
Student: $0.90S$
Child: $C$ (unchanged)

The total increased by $8\%$ , so week 2 total is

1.08T.

Write and simplify the total equation

Add the week 2 categories and set equal to $1.08T$ :

1.25A+0.90S+C=1.08T.

Substitute $C=0.20T$ :

1.25A+0.90S+0.20T=1.08T

1.25A+0.90S=0.88T.

Solve for the adult fraction of week 1 (final)

From step 1, $A+S=0.80T$ . Multiply this by $0.90$ :

0.90A+0.90S=0.72T.

Subtract from $1.25A+0.90S=0.88T$ :

\left(1.25-0.90\right)A=0.88T-0.72T

0.35A=0.16T

\frac{A}{T}=\frac{0.16}{0.35}=\frac{16}{35}.

So the percent of week 1 tickets that were adult is

\frac{16}{35}\cdot 100=\frac{1600}{35}=45\frac{5}{7}.

Therefore, the correct answer is $45\frac{5}{7}\%$ .

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Step-by-step Explanation

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