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

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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?

A conditional probability question narrows who could be picked, so the selection condition sets the denominator. When two classes overlap, split members into only, both, and neither before using the table. In Desmos, turn each pool-use fraction into a nonuser count, then divide the art nonusers by all nonusers. Dividing by the original membership total ignores the selection condition.

Hints

Hint 1
The members taking both classes are included in each class total. Subtract that overlap from each total to find the fitness-only and art-only groups. How can you then find the group taking neither class?
Hint 2
The table gives the fraction who use the pool, but the selected member does not. A complement is the part left over: multiply each group's size by $1$ minus its pool-use fraction.
Hint 3
The pick is from nonusers, so the denominator counts every member who does not use the pool. For the numerator, include nonusers who take art only and nonusers who take both classes.

Step-by-step

Count nonusers in each group
Step 1
Separate the class-only groups
The $78$ members taking both classes are already included in each class total. Subtract that overlap from each total. Type $f=174-78$ and $a=156-78$ in Desmos. It shows $f=96$ fitness-only members and $a=78$ art-only members.
Step 2
Find the neither-class group
Fitness only, art only, and both classes are separate groups, so their counts can be subtracted from all $300$ members. Type $n=300-(f+a+78)$ . Desmos shows $n=48$ members who take neither class.
Step 3
Count pool nonusers in every group
Each table fraction describes pool users. Its complement, $1$ minus that fraction, describes nonusers in the same group. Type $u=f(1-\frac14)$ , $v=a(1-\frac13)$ , $w=78(1-\frac12)$ , and $z=n(1-\frac34)$ . Desmos shows $u=72$ fitness-only nonusers, $v=52$ art-only nonusers, $w=39$ both-class nonusers, and $z=12$ neither-class nonusers.
Step 4
Find everyone who could be selected
The member is selected from those who do not use the pool, so add all four nonuser counts. Type $t=u+v+w+z$ ; Desmos shows $t=175$ . Only the nonusers belong in the probability's denominator. The full $300$ members include people who cannot be selected.
Step 5
Count the nonusers who take art
Whether or not a member also takes fitness means both art-only and both-class members count. Type $h=v+w$ ; Desmos shows $h=91$ nonusers who take art. Counting only one of those groups would leave out members who qualify.
Step 6
Divide the art count by the selected group
Divide the art nonusers by all nonusers. Type $h/t$ ; Desmos shows $0.52$ , and its fraction button gives $\frac{13}{25}$ . So the probability that the selected nonuser takes art is $\frac{13}{25}$ . Choice C.

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

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