A jar contains x yellow marbles, x-2 white marbles, and x+2 black marbles. The probability of randomly picking 2 yellow marbles is $\frac{1}{12}$.
What is the total number of marbles in the jar?

The correct answer is 9.

Step 1: Express the total number of marbles in terms of $x$.
We are given the number of marbles of each color in the jar:
- Yellow marbles = $x$
- White marbles = $x-2$
- Black marbles = $x+2$

The total number of marbles in the jar is the sum of all the marbles:

$\text{Total marbles}=x+(x-2)+(x+2)=3x$

Step 2: Write the probability equation for randomly picking 2 yellow marbles.
When drawing two marbles sequentially without replacement, the probability of selecting two yellow marbles is the probability of choosing a yellow marble first, multiplied by the probability of choosing another yellow marble second:

$\text{P(2 yellow marbles)}=\frac{\text{Yellow marbles}}{\text{Total marbles}}\times \frac{\text{Remaining yellow marbles}}{\text{Remaining total marbles}}$

Substituting the expressions in terms of $x$:

$\text{P(2 yellow marbles)}=\frac{x}{3x}\times \frac{x-1}{3x-1}$

Since $\frac{x}{3x}=\frac{1}{3}$, this simplifies to:

$\text{P(2 yellow marbles)}=\frac{1}{3}\times \frac{x-1}{3x-1}=\frac{x-1}{3(3x-1)}$

Step 3: Solve for $x$.
We are given that the probability of picking 2 yellow marbles is $\frac{1}{12}$. Setting up the equality:

$\frac{x-1}{3(3x-1)}=\frac{1}{12}$

Multiply both sides by 3:

$\frac{x-1}{3x-1}=\frac{3}{12}=\frac{1}{4}$

Cross-multiply to solve for $x$:

$4(x-1)=1(3x-1)$

$4x-4=3x-1$

$4x-3x=4-1$

$x=3$

Step 4: Calculate the total number of marbles.
The total number of marbles is given by $3x$.
Substituting $x=3$:

$\text{Total marbles}=3\times 3=9$

Therefore, the total number of marbles in the jar is 9.