Super-Hard SAT Data Analysis Question 44 | Free SAT Prep

00:00

Try Full-Screen!

Question 44·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

0 / 50 Questions completed

A community center has 300 members. Of these members, 174 take a fitness class, 156 take an art class, and 78 take both a fitness class and an art class. Any member who does not take either class takes neither class.

The table shows the fraction of members in each class-participation group who use the center's pool.

Class-participation group	Fraction who use the pool
Fitness class only	$\frac{1}{4}$
Art class only	$\frac{1}{3}$
Both classes	$\frac{1}{2}$
Neither class	$\frac{3}{4}$

A member who does not use the pool is selected at random. What is the probability that the selected member takes an art class, whether or not the member also takes a fitness class?

For conditional-probability questions with overlapping groups, first rewrite the information as mutually exclusive categories, such as fitness only, art only, both, and neither. Then apply the given fractions within each category. Finally, let the condition determine the denominator: here, all members who do not use the pool. Include every category that satisfies the event in the numerator.

Hints

Click me!

Create non-overlapping groups

The fitness and art totals overlap. Subtract the number who take both classes from each total to find the fitness-only and art-only groups.

Use the complement

For each group, the fraction who do not use the pool is $1$ minus the fraction who use the pool.

Choose the correct numerator and denominator

Because the selected member does not use the pool, count all non-pool users in the denominator. In the numerator, include non-pool users who take art only and those who take both classes.

Desmos Guide

Use hand calculations first

Hand calculation is usually faster because the main task is sorting the members into groups. Find the four group sizes as $96$ , $78$ , $78$ , and $48$ .

Calculate the non-pool-user counts

For a quick verification, enter the following calculation into Desmos:

$96(1-1/4)+78(1-1/3)+78(1-1/2)+48(1-3/4)$

This calculates the total number of members who do not use the pool.

Calculate the art-class count within that group

Enter $78(1-1/3)+78(1-1/2)$ to count the non-pool users who take an art class. Divide this result by the total from the previous calculation.

Step-by-step Explanation

Find the size of each class group

The given class totals include the 78 members who take both classes.

\begin{aligned} \text{Fitness only} &= 174-78=96 \\ \text{Art only} &= 156-78=78 \\ \text{Both classes} &= 78 \\ \text{Neither class} &= 300-(96+78+78)=48 \end{aligned}

Count members who do not use the pool

Use the complement of each pool-use fraction.

Class-participation group	Members	Members who do not use the pool
Fitness class only	$96$	$96\left(1-\frac{1}{4}\right)=72$
Art class only	$78$	$78\left(1-\frac{1}{3}\right)=52$
Both classes	$78$	$78\left(1-\frac{1}{2}\right)=39$
Neither class	$48$	$48\left(1-\frac{3}{4}\right)=12$

Thus, the total number of members who do not use the pool is $72+52+39+12=175$ .

Apply the condition

Among members who do not use the pool, those who take an art class are the art-only members and the members who take both classes. Their count is $52+39=91$ .

Therefore, the conditional probability is

\frac{91}{175}=\frac{13}{25}

Study Hints

Hints

Click me!

Create non-overlapping groups

The fitness and art totals overlap. Subtract the number who take both classes from each total to find the fitness-only and art-only groups.

Use the complement

For each group, the fraction who do not use the pool is $1$ minus the fraction who use the pool.

Choose the correct numerator and denominator

Because the selected member does not use the pool, count all non-pool users in the denominator. In the numerator, include non-pool users who take art only and those who take both classes.

Desmos Guide

Use hand calculations first

Hand calculation is usually faster because the main task is sorting the members into groups. Find the four group sizes as $96$ , $78$ , $78$ , and $48$ .

Calculate the non-pool-user counts

For a quick verification, enter the following calculation into Desmos:

$96(1-1/4)+78(1-1/3)+78(1-1/2)+48(1-3/4)$

This calculates the total number of members who do not use the pool.

Calculate the art-class count within that group

Enter $78(1-1/3)+78(1-1/2)$ to count the non-pool users who take an art class. Divide this result by the total from the previous calculation.

Step-by-step Explanation

Find the size of each class group

The given class totals include the 78 members who take both classes.

\begin{aligned} \text{Fitness only} &= 174-78=96 \\ \text{Art only} &= 156-78=78 \\ \text{Both classes} &= 78 \\ \text{Neither class} &= 300-(96+78+78)=48 \end{aligned}

Count members who do not use the pool

Use the complement of each pool-use fraction.

Class-participation group	Members	Members who do not use the pool
Fitness class only	$96$	$96\left(1-\frac{1}{4}\right)=72$
Art class only	$78$	$78\left(1-\frac{1}{3}\right)=52$
Both classes	$78$	$78\left(1-\frac{1}{2}\right)=39$
Neither class	$48$	$48\left(1-\frac{3}{4}\right)=12$

Thus, the total number of members who do not use the pool is $72+52+39+12=175$ .

Apply the condition

Among members who do not use the pool, those who take an art class are the art-only members and the members who take both classes. Their count is $52+39=91$ .

Therefore, the conditional probability is

\frac{91}{175}=\frac{13}{25}

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation