Super-Hard SAT Math Question 13 | Free SAT Prep | Aniko

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Question 13·200 Super-Hard SAT Math Questions·Problem Solving and Data Analysis

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A bag contains only red, blue, green, and black marbles.

There are 18 black marbles.
There are 12 green marbles.

Given that a randomly selected marble is not black, the probability of selecting a red marble is $\frac{2}{7}$ .

Given that a randomly selected marble is not green, the probability of selecting a blue marble is $\frac{3}{8}$ .

Which choice is the number of blue marbles in the bag?

When you see conditional probability with counts, immediately rewrite it as “part over whole within the condition.” Treat each condition (“not black,” “not green”) as defining a different denominator. After substituting any given fixed counts, cross-multiply to create linear equations, then solve the system efficiently (often by expressing one variable in terms of the other and substituting).

Hints

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Translate “given that” into a fraction

Use $P(\text{event} \mid \text{condition})=\dfrac{\text{number in both}}{\text{number in the condition}}$ . For example, “not black” means you count only the non-black colors in the denominator.

Set up two equations in two unknowns

Let $R$ be red and $B$ be blue. You already know green and black, so each conditional probability becomes an equation involving only $R$ and $B$ .

Clear the fractions carefully

Cross-multiply each equation to turn it into a linear equation. Then solve the resulting system.

Desmos Guide

Enter the two line equations in slope-intercept form

From the first condition you can rewrite $5R=2B+24$ as $B=\frac{5R-24}{2}$ .

In Desmos, graph:

$y=(5x-24)/2$

Graph the second line

Rewrite $5B=3R+54$ as $B=\frac{3R+54}{5}$ .

In Desmos, graph:

$y=(3x+54)/5$

Find the intersection

Click the intersection point of the two lines. The $x$ -coordinate is $R$ and the $y$ -coordinate is $B$ . Use the $y$ -coordinate to select the correct answer choice.

Step-by-step Explanation

Define variables and write the conditional-probability equations

Let $R$ be the number of red marbles and $B$ be the number of blue marbles.

Not black means the marble is red, blue, or green, so the total is $R+B+12$ .

\frac{R}{R+B+12}=\frac{2}{7}

Not green means the marble is red, blue, or black, so the total is $R+B+18$ .

\frac{B}{R+B+18}=\frac{3}{8}

Clear fractions to make a system of linear equations

From

\frac{R}{R+B+12}=\frac{2}{7}

cross-multiply:

7R=2(R+B+12) \Rightarrow 5R=2B+24

From

\frac{B}{R+B+18}=\frac{3}{8}

cross-multiply:

8B=3(R+B+18) \Rightarrow 5B=3R+54

Solve the system

From $5B=3R+54$ , write $B$ in terms of $R$ :

B=\frac{3R+54}{5}

Substitute into $5R=2B+24$ :

\begin{aligned} 5R &= 2\left(\frac{3R+54}{5}\right)+24 \\ 25R &= 6R+108+120 \\ 19R &= 228 \\ R &= 12 \end{aligned}

Find the number of blue marbles

Use $5B=3R+54$ with $R=12$ :

5B=3(12)+54=90 \Rightarrow B=18

Therefore, the number of blue marbles is 18.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation