Question 27·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

There are nine values, including the four letters. The mean is the total divided by the count, so type $9(34)$ in Desmos. It shows $306$, the total of the data set.

The five known values are $18,24,35,43,49$. Type $9(34)-(18+24+35+43+49)$ in Desmos. It shows $137$, so $a+b+c+d=137$. Don’t treat the letters as zeros when finding this remaining total.

Let $S=a+d$ be the endpoint sum. Since the four letters add to $137$, the other pair has sum $b+c=137-S$. A pair’s mean is its sum divided by $2$. The $a,d$ mean is $4$ greater, so the $4$ belongs to the pair means, not directly to their sums:

Use a regression to ask Desmos for the number that makes both sides match. Replace $S$ with $x_1$ and $=$ with $\sim$; type $\frac{x_1}{2}\sim\frac{137-x_1}{2}+4$. Under PARAMETERS, Desmos shows $x_1=72.5$. That’s $a+d$, not $d$.

The list is ascending, so $a$ is the minimum and $d$ is the maximum. The range is maximum minus minimum, giving

Add $d-a=64$ to $a+d=S$ so $a$ cancels:

This is elimination: adding equations to remove a value you don’t need to find.

Divide by $2$:

Desmos already has $x_1=S$, so type $\frac{x_1+64}{2}$. It shows $68.25$; the fraction button gives $\frac{273}{4}$. That is the greatest data value, $d$. Choice B.